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say, it will not pay out more than $463.62 in cash), however, then you have no incentive to withdraw in t + 1, even if you think that some others might choose to do so. This is because if you know that the bank is not going to honor withdrawal demands above a certain threshold, the bank will be able to pay you your promised return in t + 2. A credible announcement of the suspension of convertibility ought to prevent a bank run from starting and ought to stop it in its tracks once started. Another thing that the clearinghouses did was to band together and collude to not publish bank-speciﬁc information about assets and liabilities. You might choose to run on your bank if you think that the bank has made bad investments and that its assets are worth less than previously thought. Withdrawing from the bank could cause otherwise healthy banks to have problems – if your bank tries to sell assets to come up with cash, then this will depress the prices of assets, which will make other banks look weak. By not publishing bank-speciﬁc balance sheet information, the clearinghouses hoped to stop this process in its tracks. The idea is that the consortium of banks would stand together as one – because banks are interconnected through the market prices of the assets they hold, reducing information about the health of particular banks reduced the incentive for depositors to run on these banks, which limited the amount of assets that needed to be sold to come up with cash. The Federal Reserve implemented a similar procedure during the recent ﬁnancial crisis with the Term Auction Facility (TAF), as we will discuss in Chapter 37. While suspension of convertibility was used as the primary defense mechanism against bank runs prior to the founding of the Fed, it did not in practice prevent runs (although many scholars believe suspension of convertibility lessened the severity of banking panics). Further, suspension of convertibility was not without its own problems. If a bank suspends convertibility, then there may well be depositors who need to access their funds who cannot. This is undesirable. For these reasons, in the wake of the famed Panic of 1907, the Federal Reserve was founded in the United States in 1913. Among other day-to-day operational 770 details, the Fed’s mandate was to stand as a lender of last resort to banks facing a liquidity crisis. The discount window was introduced as a facility wherein commercial banks could borrow from the Fed in the event of a liquidity crisis. How
this was intended to work is best seen through an example. Suppose that a bank begins with a balance sheet as follows: Table 33.3: T-Account for Hypothetical Bank Assets Loans: $100 Securities: $10 Cash Reserves: $10 Equity $20 Liabilities plus Equity Deposits: $100 Borrowings: $0 Suppose that the bank faces a withdrawal of $40. It only has $10 in cash reserves and can only raise $10 through the selling of securities. Raising more liquidity through the sales of loans might require selling at a loss. Instead of selling loans, a bank could instead borrow from the Fed directly through its discount window. The bank could draw down its cash reserves and sell its securities to come up with $20 in cash, and could borrow the other $20 from the Fed. The new balance sheet would look like: Table 33.4: T-Account for Hypothetical Bank Assets Loans: $100 Securities: $0 Cash Reserves: $0 Equity $20 Liabilities plus Equity Deposits: $60 Borrowings: $20 In this way, through the discount window the bank can attract more liabilities (borrowings) to make up for an unexpected withdrawal, which can allow it to meet that withdrawal without being forced to sell assets. This is not necessarily a freebie for the bank – it has to pay interest on the loan from the Fed (at the so-called discount rate), which means that in a dynamic sense there is a cost to lacking liquidity and having to go to the Fed for help. But in a liquidity-driven panic going to the Fed for funds at a small interest rate is likely a better deal than selling illiquid assets at a loss. The hope was that the existence of the Fed and its discount window would eliminate bank runs. If banks were run on, they could go to the Fed for liquidity, which would stop the run in its place. Furthermore, private sector expectations of banks making use of the discount window might stop runs before they start. In the context of the model we have developed and studied in this chapter, if you do not need your funds in t + 1, the only reason you would withdraw is if you believe that you will suﬀer a loss by waiting until t + 2. But if you believe 771 that a bank will make use of the central bank’s discount window, you should not have any reason to fear a loss, and should not withdraw earlier than needed.
The lender of last resort function and the discount window did not work as planned during the Great Depression, which featured waves of bank runs. In part this was because the Fed itself did not fully understand and appreciate its role as lender of last resort, as argued for example in Friedman and Schwartz (1971). In part this was because of a stigma associated with going to the discount window for a loan. Banks feared that if they went to the discount window, they would be perceived as ﬁnancially unsound, which would increase the pressures on the bank. The most recent policy to deal with bank runs was the institution of deposit insurance. This happened with the creation of the Federal Deposit Insurance Corporation (FDIC) in 1933. The basic idea of deposit insurance is as follows. Banks can choose to purchase deposit insurance for a small fee. In exchange for this small fee, deposits at a bank are insured by the FDIC up to a given amount. Initially deposits were insured up to $2500; today the limit is $250,000. By “insured” we mean that in the event that the bank becomes insolvent and cannot meet withdrawal demands, if it is FDIC insured then the FDIC will step in and provide those funds. The idea behind the FDIC is similar to the lender of last resort function of a central bank. If the public believes that deposits will be honored (either through lending from a central bank or from deposit insurance), then depositors ought to have no incentive to run on their institution. In practice, it was not until the institution of deposit insurance in 1933 that the bank runs and panics that plagued the American economy for much of its history ceased to occur with regularity. Indeed, there were really no banking panics in the United States from the institution of deposit insurance until the ﬁnancial crisis and Great Recession. As we will discuss more in Chapter 37, the Great Recession featured a banking panic, but it was diﬀerent than past panics. Individuals did not run on deposits, and commercial banks did not fail. Rather, institutions “ran” on other institutions in the sense that short term credit was not rolled over, forcing massive liquidations of assets and a general ﬁnancial panic. Because this “run” occurred outside of the conventional banking system, the tools that had been developed to deal with runs were not immediately applicable. 33.5 Summary • A primary objective of banks is liquidity transformation whereby the pooling of deposits from many di
ﬀerent individuals can ﬁnance illiquid projects while at the same time give the depositors access to their deposits on demand. • While the process of liquidity transformation is typically beneﬁcial for the economy, 772 banks can be susceptible to runs. A run occurs if enough depositors think that the bank will not be able to meet their demand for future withdrawals. This causes all the depositors to withdraw in the present. Since a large portion of the bank’s assets are tied up in illiquid projects, it may not be able to satisfy all the withdrawal demand. This further increases demand for withdrawals in the present creating a self-fulﬁlling prophecy. • Banks have historically implemented a number of policies to reduce the probability of a bank run including suspending convertibility and withholding information. Suspending convertibility involved many banks joining together in refusing to meet withdrawal demand. Banks would also join together and withhold information about their balance sheet. The idea was that by withholding information this would stop depositors from running on any one bank. Otherwise healthy banks had an incentive to do this because a ﬁre sale from an unhealthy bank would depress the value of assets. • Government policy has also played a role in mitigating bank runs. By standing as the lender of last resort, the Federal Reserve gives banks a place to borrow without having to quickly sell its illiquid assets. The FDIC gives banks an opportunity to insure individual deposits up to $250,000. Depositors are less likely to run on the bank if their deposits are insured. Questions for Review 1. Suppose there exists an investment project that has a positive expected return. Would a household always ﬁnd it optimal to invest in such a project? 2. How might deposit insurance create moral hazard? 3. We usually think that having complete information about a product or service is superior to having incomplete information. Describe the extent to which this is true in the ﬁnancial intermediation industry. 4. Are bank runs irrational? 773 Chapter 34 Bond Pricing and the Risk and Term Structures of Interest Rates A bond is a type of ﬁnancial security which entitles the holder to periodic cash ﬂows until maturity. With one exception discussed below, bonds have a ﬁnite life span after which they are retired – this ﬁnite life span is known as the maturity date. Bonds are often called ﬁxed income securities because, outside of default, the cash �
��ows accruing to the owner of the bond are known at the time of purchase. Default is a situation in which the bond issuer (i.e. the borrower) fails to make good on promised repayment, either in whole or in part. While bonds promise ﬁxed income streams to the holder, they are not without risk. Risk arises for two reasons – ﬁrst, because of the possibility of default, and second, because of unexpected price ﬂuctuations (what is sometimes called interest rate risk). If a bondholder needs to sell his or her bond before its maturity date, he or she may do so at a price that entails a lower or higher return than what the bondholder expected. Governments and corporations are the chief issuers of bonds. They issue bonds to raise funds to ﬁnance either current expenditures or long term investments.1 There is an active secondary market for bonds – one can buy an already issued bond from another individual or institution, which then entitles the new owner to any future cash ﬂows coming from the bond. The other main type of ﬁnancial security is a stock or equity, which is the subject of Chapter 35. Both stocks and bonds are ways for corporations to raise funds to ﬁnance operations. Whereas bonds typically have ﬁnite life spans, equities are inﬁnitely-lived. Unlike bonds, equities do not promise ﬁxed cash ﬂows. Dividends are the periodic cash ﬂows returned to equity holders, but dividends are not known in advance and depend upon how proﬁts evolve. Bond and stockholders have diﬀerent seniority in the event of a corporation’s failure. 1A bond is a kind of debt instrument that is in many ways similar to a simple bank loan, and in fact the terms loan and bond are often used interchangeably (including in this very book). There are some subtle diﬀerences between bank loans and bonds in practice, however. First, because of asymmetric information, only very well-established ﬁrms can raise funds through issuing bonds, whereas smaller and less-established entities typically must rely on bank loans. Second, bank loans often include far more restrictive covenants and restrictions on the borrower uses of funds than do bonds. Third, while it is possible to trade bank loans in secondary markets, the markets for such loans are less liquid
than the market for government and corporate bonds. 774 Stockholders are residual claimants who are entitled to their share of a company’s assets only after all creditors, including bond holders, have been paid. Both because of unpredictability in dividends as well as less seniority in the event of a ﬁrm failure, stocks are considered riskier than bonds. Bonds diﬀer along several dimensions. First, they diﬀer according to speciﬁcs of cash ﬂow repayments. For example, some bonds make periodic monthly or quarterly cash payments to bond holders, whereas other bonds only oﬀer one cash payment upon maturity to the bondholder. Second, bonds diﬀer according to default risk – the probabilities that bond issuers will fail to make good on promised repayment naturally diﬀer according to the ﬁnancial health of the issuer. Government bonds in well-established countries are considered to be (essentially) default risk-free – that is, there is virtually no chance that the government will fail to make good on promised repayments. Third, bonds diﬀer according to their life span, or time to maturity. For example, the US government issues both very short maturity debt (known as Treasury bills), medium term securities known as Treasury notes, and long maturity securities known as Treasury bonds. In spite of these diﬀerent terms, we will refer to all three types of securities as “bonds.” In this chapter we will ﬁrst discuss the main type of cash ﬂow repayment plans oﬀered by bonds. Then we will discuss how to infer the interest rate on a bond in terms of the market price of the bond and the promised cash ﬂows. This interest rate will be referred to as the yield to maturity (or just the “yield” for short). Yields can diﬀer on bonds with similar cash ﬂow repayment plans if these bonds have diﬀerent default risk or time to maturities. How yields diﬀer according to default risk is called the “risk structure” of interest rates and how yields diﬀer according to time to maturity is called the “term structure” of interest rates. After discussing some speciﬁcs of bond cash ﬂows and the concept of yield to maturity,
we use a micro-founded general equilibrium approach to price bonds with diﬀerent characteristics so as to discuss the risk and term structure of interest rates. 34.1 Bond Cash Flow Repayment Plans The simplest type of bond is a discount bond. A discount bond is issued at date t with a face value, F V, and a maturity date, t + m. The holder of a discount bond receives one cash ﬂow payment equal to the face value on the maturity date; there are no other promised cash ﬂows between the date of issuance and the maturity date. Discount bonds typically sell for a discount relative to face value (face value is also sometimes called par value). For example, you might pay $90 for a $100 face value discount bond with time to maturity of 1 year. The diﬀerence between the buy price, in this case $90, and the face value amounts to 775 interest earned on holding the bond. Examples of discount bonds in the real world include US Treasury Bills and commercial paper (a type of short term debt instrument issued by large corporations). The other main type of bond is a coupon bond. Like a discount bond, a coupon bond is issued at date t with a face value and a maturity date. Diﬀerently than a discount bond, a coupon bond makes regular (monthly or quarterly) cash payments to the holder of the bond at intervals between the issue date and the maturity date. Like a discount bond, the face value is returned to the bond holder at the maturity date. Let CO be the coupon payment, and co = CO F V is deﬁned as the coupon rate (deﬁned as the (annual) coupon payment divided by the face value). The coupon rate is often referred to as an interest rate, though as we shall will see this is not quite right. The implicit interest on a coupon bond depends both on the coupon rate as well as how much of a discount (or not) the bond trades at relative to face or par value. Examples of coupon bonds include Treasury bonds and corporate bonds. A third type of bond is what is called a perpetuity (sometimes also a consol as this the name of these bonds issued in Great Britain). Perpetuities have no maturity date and hence no face value. They simply promise the holder of the bond a ﬁxed coupon payment at regular intervals. Hence, the holder of a perpetuity is entitled to known and regular
coupon payments for long as that holder maintains possession of the bond. 34.1.1 Yield to Maturity The interest rate on a bond is rarely if ever explicit (although it is fairly common to confuse coupon rates on coupon paying bonds with the interest rate on the bond). Rather, the interest rate on a bond is implicitly deﬁned given a market price of the bond and cash ﬂow details about the bond. The most common measure of the interest rate on a bond is the yield to maturity (or YTM or just yield for short). The YTM is deﬁned as the (constant) discount rate which equates the price of the bond with the expected present discounted value of cash ﬂows coming from ownership of the bond. The YTM represents the expected rate of return a holder of a bond would earn by holding that bond until maturity. We will refer to the YTM both as the interest rate on a bond as well as the “yield.” It is worth noting that the realized return on holding a bond need not equal the yield on the bond – if the bond price goes up or down in the future and the holder must sell the bond, then the realized return may diﬀer from the yield. Given a price of the bond (which we will discuss below) and cash ﬂow details, we can determine the yield to maturity by equating the bond price with the present discounted value 776 of expected future cash ﬂows. This will be easiest to see with examples based on the diﬀerent kinds of cash payout streams discussed above. For simplicity, we assume in this section that there is no default risk, and hence no uncertainty about the cash ﬂows accruing to the holder of a bond over that bond’s lifespan. First, consider a one period discount bond. The bond is sold in period t at price P B t.2 It promises the face value, F V, to the holder the next period. The yield to maturity, or what we will label as rt,3 is implicitly deﬁned by: 1 1 + rt In other words, the yield to maturity is simply the discount rate which equates the price of the bond to the future cash ﬂows from the bond. For a one period discount bond, it is simply: (34.1) P B t F V = 1 + rt = F V P B
t (34.2) Provided the bond trades at less than face value, i.e. P B t < F V, the yield to maturity is positive. This is why most discount bonds do indeed trade at a discount relative to face – the holder of the bond requires a positive expected return (i.e. the yield) to hold the bond. The bigger is the gap between the price and the face value, the bigger is the implied yield. Next, consider a discount bond with a greater than one period time to maturity. Denote this maturity by m. The yield to maturity on this bond satisﬁes: 1 (1 + rt)m Since the face value on the bond is not received for m periods, it gets discounted by (1 + rt)m. This implicitly accounts for compounding. Given a price of the bond, the yield to maturity can be solved for as: (34.3) P B t F V = 1 + rt = ( F V P B t Note that (34.2) is just a special case of (34.4) when m = 1. Also note that the price of the bond and its implied yield are negatively related. This is an important point in bond (34.4) ) 1 m 2Here and throughout the remainder of the chapter we use P B t to refer to the price of a bond (hence the B superscript). This superscript is included so as to note confuse the bond price with the money price of goods, Pt in earlier notation. 3We are using the same notation as the real interest rate from other parts of the book, though of course in practice what one directly can observe in the data are nominal yields. For what follows in this section, we abstract from risk arising from unexpected inﬂation movements and hence simply treat everything as real. 777 pricing – prices and yields move opposite one another. Consider next a coupon bond. A coupon bond makes coupon payments of CO each period until maturity when it returns the face value to the holder. Taking the price as given, the yield to maturity on the coupon bond satisﬁes the following expression: P B t = CO 1 + rt + CO (1 + rt)2 + CO (1 + rt)3 +... CO (1 + rt)m + F V (1 + rt)m This can be written more compactly using summation operators as: P B t
= CO 1 + rt m−1 ∑ j=0 1 (1 + rt)j + F V (1 + rt)m The summation in (34.6) can be written: S = m−1 ∑ j=0 1 (1 + rt)j = 1 + 1 1 + rt + ( 1 1 + rt 2 ) +... ( 1 1 + rt ) m−1 So as to economize on notation, deﬁne γ = 1 1+rt. (34.7) can be written: Then: Which means: S = 1 + γ + γ2 +... γm−1 γS = γ + γ2 +... γm S − γS = 1 − γm Now solve for S in terms of γ: S = 1 − γm 1 − γ = 1 + rt rt (1 − ( 1 1 + rt m ) ) Which means (34.6) can be written: P B t = CO rt (1 − ( 1 1 + rt m ) ) + F V (1 + rt)m (34.12) can equivalently be written: P B t = CO F V rt (F Vt − F V (1 + rt)m ) + F V (1 + rt)m 778 (34.5) (34.6) (34.7) (34.8) (34.9) (34.10) (34.11) (34.12) (34.13) As noted above, let co = CO/F V be the coupon rate. Then we can write (34.13) as: P B t F V + = co rt (1 − co rt From (34.14), one observes that if P B t F V (1 + rt)m = F V, then it must be the case that co = rt – i.e. the yield and the coupon rate are equal. Conversely, if the price of the bond is above par > F V ), it must be that the yield is less than the coupon rate and vice-versa. This (i.e. P B t illustrates nicely that “the” interest rate on a bond is not necessarily the coupon rate on the bond. (34.14
) ) Finally, we examine the relationship between the price and implied yield on a perpetuity. Given a price and coupon payment, the yield on a perpetuity must satisfy: P B t = CO 1 + rt ∞ ∑ j=0 1 (1 + rt)j (34.15) The term inside the summation operator in (34.15) reduces to 1+rt rt m → ∞ in (34.12)). Then the yield on a perpetuity is given by: (to see this, simply let rt = CO P B t (34.16) It is relatively easy to work with discount bonds, and for the remainder of the chapter we will do so. It turns out that a coupon bond can simply be thought of as a portfolio of discount bonds with diﬀerent face values and maturities. For example, suppose that there is bond paying a regular coupon payment of CO, with face value of F V, and two periods to maturity. Its yield to maturity is implicitly deﬁned by: P B t = CO 1 + rt + CO (1 + rt)2 + F V (1 + rt)2 We could write (34.17) as the sum of the prices of two diﬀerent discount bonds: Pt = P B 1,t + P B 2,t (34.17) (34.18) In (34.18), P1,t is the price of a one period discount bond with face value of CO and P2,t is the price of a two period discount bond with face value of CO + F V. For a perpetuity, we can think about it as the sum of discount bonds with face values of the coupon payment going oﬀ into the inﬁnite future. For all three types of cash ﬂow repayment systems considered – (34.4), (34.14), and (34.16) we observe the general point that the price and yield on a bond are inversely related. Secondly, the yield is simply another way to express the price. The reason bond prices are 779 typically expressed in terms of yields and not prices is that it makes it easier to compare bonds with diﬀerent maturities or cash ﬂow characteristics. For example, a bond with a face value of $10,000 will almost certainly trade for a higher price than a bond with a face
value of $1,000. But this doesn’t necessarily mean that the $10,000 bond will provide a better expected return if held to maturity. Similarly, a discount bond with a longer time to maturity will generally trade at a lower price than a discount bond with a shorter maturity. Again, this does not necessarily mean that the longer maturity bond will oﬀer a higher expected return. Expressing bond prices in terms of yields puts bonds with diﬀerent face values and maturities on a “level playing ﬁeld” that makes it easier to compare them. But even if we compare bonds in terms of yields, which eliminates potential issues arising from diﬀerent face values or times to maturity, the yields on two diﬀerent bonds are unlikely to be the same at any point in time. The question is how and why yields might diﬀer. In the next section, we introduce a micro-founded approach to determining bond prices and thereby inferring yields. We do so in a dynamic consumption-saving environment where in equilibrium consumption must equal an exogenous endowment of income. We then extend the analysis to allow for uncertainty, which allows us to discuss both the risk and term structures of interest rates. 34.2 Bond Pricing with No Uncertainty: A General Equilibrium Approach We have to this point deﬁned bonds, discussed repayment schemes on diﬀerent kinds of bonds, and introduced the concept of the yield to maturity as a measure of the interest rate on a bond. In ﬁnding the yield, we took the price of the bond as given, and noted that bond prices and yields move opposite one another. But how does the price of the bond (and hence its yield) get determined? We will explore bond pricing within an optimizing, dynamic framework. We will focus on a representative agent model with as few periods as possible (for the most part this will be the two period framework used throughout the book, but when studying the term structure of interest rates, we will have to move beyond two periods). Since we want to focus on the determination of prices, which are an equilibrium construct, we need to work in the conﬁnes of a model with general equilibrium. We will do so using the simplest possible framework – the endowment economy framework studied in Chapter 11. We will focus on purely real frameworks in which we abstract from money and nominal prices. We will make three additional assumptions
meant to simplify the analysis. First, we will assume that all bonds are discount bonds. This obviously abstracts from the multitude of 780 repayment plans that bonds oﬀer, but it simpliﬁes the analysis and no important insights are lost because a coupon-paying bond can always be thought of as a portfolio of discount bonds. Second, we will assume, except where noted, that all bonds are in ﬁxed supply, and in particular for the most part in zero net supply. This follows the important work of Lucas (1978). In this environment, in equilibrium the representative household does not actually hold any bonds (i.e. as in our endowment economy model, in equilibrium it is not possible for the household to save or borrow). Bond prices (equivalently yields) adjust so that in equilibrium the household is content not borrowing or saving. Allowing for bonds to be in non-zero net supply would not fundamentally alter any of the conclusions to follow but would make the analysis more complicated. Third, for now we shall assume that there is no uncertainty over the future. Uncertainty could arise because of uncertainty in bond repayments (i.e. default) or in income/endowment uncertainty. We will address both types of uncertainty later. Suppose that a representative household lives for two periods, t and t + 1. It has the following lifetime utility: U = u(Ct) + βu(Ct+1) (34.19) The household begins life with no assets (and in this framework the only possible asset is a bond). In period t, the household earns some exogenous stream of income, Yt. With this income it can consume or save/borrow through a discount bond, Bt. One unit of the discount bond held from t to t + 1 yields one unit of income in t + 1. The discount bond trades for a price of P B in period t. The household takes this price as given. The household’s period t t budget constraint is: Ct + P B t Bt ≤ Yt (34.20) In period t + 1, the household earns an exogenous income stream and receives income from its holdings of bonds. In principle, it could accumulate additional bonds (or borrow through additional bonds), but we will go ahead and impose the terminal condition that the household cannot die with a non-zero stock of assets. Hence, its period t + 1 budget constraint is: Ct+1 ≤ Yt+
1 + Bt (34.21) The household’s objective is to maximize (34.19) subject to (34.20)-(34.21). We could form a uniﬁed intertemporal budget constraint as we did earlier in the course, and think about the household as choosing a consumption plan, Ct and Ct+1. Because it is more closely 781 aligned with the approach we will take later, we will instead think about the problem as one of simply choosing Bt in period t. Assuming that both constraints hold with equality, plugging in so as to eliminate Ct and Ct+1 yields an unconstrained problem: max Bt U = u(Yt − P B t Bt) + βu(Yt+1 + Bt) The ﬁrst order optimality condition is: ∂U ∂Bt = 0 ⇔ P B t u′(Ct) = βu′(Ct+1) (34.22) (34.23) In taking the derivative, we have taken the liberty of noting that the argument of the utility function is simply Ct or Ct+1. (34.23) is an optimality condition that must hold if the household is behaving optimally. The intuition for this condition is straightforward. Purchasing one unit of the bond in period t requires foregoing P B t units of income in that period. In terms of utility, this income is valued at u′(Ct). Hence, P B t u′(Ct) represents the marginal utility cost of holding one additional unit of the discount bound. What is the beneﬁt of holding an additional unit of the bond? The beneﬁt is increasing period t + 1 consumption by 1, which is valued in terms of lifetime utility at βu′(Ct+1). Hence, βu′(Ct+1) represents the marginal utility beneﬁt of purchasing this bond. At an optimum, the marginal utility beneﬁt must equal the marginal utility cost if the household is behaving optimally. (34.23) can equivalently be written: P B t = βu′(Ct+1) u′(Ct) (34.24) The right hand side of (34.24) is an important concept in macroeconomics and ﬁnance known as the stochastic discount factor. The stochastic discount factor is simply the inverse marginal rate of substitution
between current and future consumption. In any micro-founded model of asset pricing, the price of an asset ought to satisfy: Pa,t = E [βu′(Ct+1) u′(Ct) × Da,t+1] (34.25) In (34.25) we allow for uncertainty (either over future consumption or future cash ﬂows from the asset or both), and hence the E operator appears on the right hand side. The subscript a indexes an asset and Da,t+1 is the cash ﬂow generated by the asset in the subsequent period. (34.25) says that the price of an asset ought to equal the expected value of the product of the stochastic discount factor with the payout generated by the asset in the future. (34.24) is simply a special case of this, because (i) the payout of the risk-free bond is 1, so Da,t+1 = 1, and (ii) we have assumed no uncertainty of any sort, so we can drop the 782 expectations operator. As we shall see, the basic pricing formula for risky debt and for stocks (see Chapter 35) will take the same general form of (34.25). Let us return to our particular example of a risk-free one period discount bond with no income uncertainty. On its own, (34.24) does not determine Pt because it is written as a function of endogenous variables, Ct and Ct+1. To determine Pt, we need some notion of what it means for markets to clear. As noted above, we are going to consider endowment economies in which bonds are in zero net supply. This means that, in equilibrium, Bt = 0, which requires that Ct = Yt and Ct+1 = Yt+1, where Yt and Yt+1 are taken to be exogenous. This means that the equilibrium bond price is: P B t = βu′(Yt+1) u′(Yt) With log utility, for example, (34.26) would become: P B t = β Yt Yt+1 (34.26) (34.27) The equilibrium bond price is inversely related to expected growth of the endowment. The intuition for this is straightforward and related to intuition discussed in Chapter 11. When Yt increases relative to Yt+1, other things being equal the household would like to save so as to transfer resources from the
present (when resources are plentiful) to the future (when resources are comparatively scarce). This amounts to an increase in the demand for the bond. In equilibrium, it is not possible for the household to hold more of the bond, so the price of the bond must rise to keep the household content not holding any of the bond. The intuition for this can be seen in a simple demand-supply diagram below. An increase in the current endowment makes the household want to save more, and so there is an increase in demand for the bond. With a ﬁxed supply of the bond, this necessitates an increase in the price of the bond. 783 Figure 34.1: Increase in Current Enowment Conversely, suppose that the household anticipates an increase in Yt+1 relative to Yt. Other things being equal, the household would like to borrow (or reduce its saving) so as to smooth its consumption. In other words, there is reduced demand for the bond. But in equilibrium, the household cannot borrow, so the price of the bond must fall. This is shown in the simple demand-supply diagram below. Figure 34.2: Increase in Future Endowment Once we know the bond price, we can determine the implied yield to maturity. Recall 784 𝑃𝑃𝑡𝑡 𝐵𝐵𝑡𝑡 0 𝑃𝑃0,𝑡𝑡 𝐵𝐵𝑠𝑠 𝐵𝐵𝑑𝑑(𝑌𝑌0,𝑡𝑡) 𝐵𝐵𝑑𝑑(𝑌𝑌1,𝑡𝑡) 𝑃𝑃1,𝑡𝑡 𝑌𝑌1,𝑡𝑡>𝑌𝑌0,𝑡𝑡 𝑃𝑃𝑡𝑡 𝐵𝐵𝑡𝑡 0 𝑃𝑃0,𝑡𝑡 𝐵𝐵𝑠𝑠 𝐵𝐵𝑑𝑑(𝑌𝑌0,𝑡𝑡+1) 𝐵�
��𝑑𝑑(𝑌𝑌1,𝑡𝑡+1) 𝑃𝑃1,𝑡𝑡 𝑌𝑌1,𝑡𝑡+1>𝑌𝑌0,𝑡𝑡+1 that the yield to maturity is the discount rate which equates the bond price with the present discounted value of expected future cash ﬂows from the bond. There is no uncertainty here, and the bond yields 1 unit of income in period t + 1. Hence, the yield must satisfy: Plugging in the expression for the equilibrium bond price, (34.29), we get: P B t = 1 1 + rt With log utility, for example, (34.29) becomes: 1 + rt = u′(Yt) βu′(Yt+1) 1 + rt = Yt+1 βYt (34.28) (34.29) (34.30) One will note that this expression for the equilibrium bond yield is exactly the same expression we derived for the equilibrium interest rate for an endowment economy with log utility in Chapter 11. In equilibrium, the bond yield / interest rate is a measure of the expected plentifulness of the future relative to the present. In that chapter, we talked about a supply of saving being increasing in rt. In this chapter, we are thinking about a demand for bonds that is decreasing in the price of the bond, Pt. Even though we are switching from “supply” to “demand,” in either case we are conveying exactly the same information and intuition. Bond prices / yields adjust to prevent the household from smoothing its consumption relative to its income. 34.3 Default Risk and the Risk Structure of Interest Rates In the previous section we used an optimizing dynamic model to discuss how the price of a bond is determined in general equilibrium. We did so in an environment with no uncertainty. We now modify the setup so that the future is uncertain, which allows us to study the risk structure of interest rates. The risk structure of interest rates refers to how bond prices (equivalently interest rates) vary according to the risk that the future cash ﬂows from the bond will diﬀer from what has been promised. To discuss the risk structure, we need to (i) allow for uncertainty, potentially both in
the household’s endowment stream as well as in payouts from the bond, and (ii) allow for diﬀerent types of bonds with potentially diﬀerent risk. Suppose that there is a representative household who lives for two periods. The household receives an exogenous income stream in period t and a potentially uncertain exogenous income stream in t + 1. In period t, the household can save through diﬀerent discount bonds. One of 785 these is risk-free and we will denote it with an rf subscript. Let Brf,t denote the quantity of the risk-free bond the household purchases to take from t to t + 1 at price P B rf,t. The bond is risk-free in the sense that one unit purchased in period t yields one unit of income in period t + 1 with certainty. We can think of the risk-free bond as a government-issued bond – there is essentially no risk of the government defaulting on repayment. In addition, the household has access to a risky discount bond. The quantity the household takes from t to t + 1 is denoted Br,t and the household pays a price of P B r,t for this bond. This bond is risky in the sense that there is potentially a probability that the bond defaults, which means it generates no income in period t + 1.4 In period t, household consumption plus purchases of both the risk-free and risky bond cannot exceed an exogenous endowment of income: Ct + P B rf,tBrf,t + P B r,tBr,t ≤ Yt (34.31) To think about the budget constraint in t+1, we need to describe the nature of uncertainty over what happens in period t + 1. Suppose that income in period t + 1 can take on two diﬀerent values, high and low – Y h t+1. Furthermore, the risky bond can make two t+1 diﬀerent payouts – no default (payout of 1) or default (payout of 0). This means that there are four diﬀerent possible “states of the world” in t + 1. These states are summarized below: ≥ Y l State 1 ∶ Yt+1 = Y h State 2 ∶ Yt+1 = Y h State 3 ∶ Yt+1 = Y l State 4 ∶ Yt+1 = Y h t
+1, risky bond pays t+1, risky bond defaults t+1, risky bond pays t+1, risky bond defaults Let the probabilities of these states occurring be denoted p1, p2, p3, and p4 = 1 − p1 − p2 − p3 (i.e. the probabilities must sum to one since one of the four states must occur). A ﬂow budget constraint must hold in period t + 1 in all four states of the world. Letting Ct+1(j), j = 1,..., 4, denote consumption in each state of the world, and going ahead and imposing that the constraint must hold with equality, we must have: 4One could also entertain partial default, wherein the bond would generate somewhere between 0 and 1 units of income period t + 1. 786 Ct+1(1) = Y h t+1 Ct+1(2) = Y h t+1 Ct+1(3) = Y l t+1 Ct+1(4) = Y l t+1 + Brf,t + Br,t + Brf,t + Brf,t + Br,t + Brf,t (34.32) (34.33) (34.34) (34.35) Regardless of the realized state of the world, consumption must equal available resources since the household wishes to die with no assets. In (34.32)-(34.35), the risk-free bond always generates one unit of income, while the risky bond only generates income in states 1 and 3. The household’s expected lifetime utility is simply a discounted probability-weighted sum of ﬂow utilities across time and states of the world: U = u(Ct) + β [p1u(Ct+1(1)) + p2u(Ct+1(2)) + p3u(Ct+1(3)) + (1 − p1 − p2 − p3)u(Ct+1(4))] (34.36) (34.36) can equivalently be written in terms of the expected utility of future consumption, since E [u(Ct+1)] = p1u(Ct+1(1)) + p2u(Ct+1(2)) + p3u(Ct+1(3)) + (1 − p1 − p2)u(Ct+1(4)): U
= u(Ct) + β E [u(Ct+1)] (34.37) The household’s objective is pick a consumption plan to maximize (34.36) subject to (34.31)-(34.35). It will be easiest to re-cast the problem not as one of choosing a consumption plan but rather bond holdings in period t. Doing so yields the following unconstrained maximization problem: max Brf,t,Br,t U = u [Yt − P B rf,tBrf,t − P B r,tBr,t] + βp1u [Y h t+1 + Brf,t + Br,t] + βp2u [Y h t+1 + Brf,t] + βp3u [Y l t+1 + Brf,t + Br,t] + β(1 − p1 − p2 − p3)u [Y l t+1 + Brf,t] (34.38) The ﬁrst order conditions are: ∂U ∂Brf,t = 0 ⇔ P B rf,tu′(Ct) = β [p1u′(Ct+1(1)) + p2u′(Ct+1(2)) + p3u′(Ct+1(3)) + (1 − p1 − p2 − p3)u′(Ct+1(4))] (34.39) 787 ∂U ∂Br,t = 0 ⇔ P B r,tu′(Ct) = βp1u′(Ct+1(1)) + βp3u′(Ct+1(3)) (34.40) In writing these ﬁrst order conditions, we have taken the (simplifying) liberty of writing the arguments of the utility function simply as consumption values at diﬀerent dates and states. In an endowment economy equilibrium, no bonds are held and consumption simply equals income. This means that (34.39)-(34.40), combined with this market-clearing condition, determine the equilibrium prices of the two bonds: P B rf,t = β p1u′(Y h t+1 ) + p2u′(Y h t+1 ) + p3u′(Y l t+1 u′(Yt) ) + (1
− p1 − p2 − p3)u′(Y l t+1 ) (34.41) P B r,t = β p1u′(Y h t+1 ) + p3u′(Y l t+1 u′(Yt) ) (34.42) Note that both (34.41) and (34.42) can be re-written in terms of expectations operators as: P B rf,t = β E[u′(Yt+1)] u′(Yt) = E [ βu′(Yt+1) u′(Yt) ] P B r,t = β E[u′(Yt+1)Dt+1] u′(Yt) = E [βu′(Yt+1)Dt+1 u′(Yt) ] (34.43) (34.44) In (34.44), Dt+1 is the payout on the risky bond (either 1 or 0). In both (34.43) and (34.44), the second equal signs follows because β and u′(Yt) are known at the time the expectation is made, so the pricing conditions can be written either with these inside or outside of the expectations operator. With these terms written inside the expectations operator, one observes that (34.43) and (34.44) are just special cases of (34.25). The price of either bond is the expected value of the product of the stochastic discount factor (evaluated at equilibrium levels of consumption Ct = Yt and Ct+1 = Yt+1) with the payout on the bond (1 in the case of the risk-free bond and Dt+1 = 1 or 0 in the case of the risky debt). We will deﬁne the risk premium as the diﬀerence between the yield on the risky bond and the yield on the risk-free bond. The yield on each bond equates the price of the bond to the expected cash ﬂows. For the risk-free bond, the yield simply satisﬁes: 1 + rrf,t = 1 P B r,t = u′(Yt) β E[u′(Yt+1)] (34.45) For the risky bond, the yield is the ratio of the expected payout, E[Dt+1],
to the price: 788 1 + rr,t = E[Dt+1] P B r,t = E[Dt+1]u′(Yt) β E[u′(Yt+1)Dt+1] (34.46) The risk premium can be deﬁned as (approximately) the ratio of the two gross yields minus one: rr,t − rf,t ≈ 1 + rr,t 1 + rrf,t − 1 = E[Dt+1] E[u′(Yt+1)] E[u′(Yt+1)Dt+1] − 1 (34.47) One would be tempted to look at (34.47) and distribute the expectations operator through the denominator, in which case the risk premium would be zero. In general, one cannot do this. As we shall see again and again in this and the next chapter, for two arbitrary random variables X and Y, E[XY ] = E[X] E[Y ] + cov(X, Y ). This means we can write: rr,t − rf,t = E[u′(Yt+1)Dt+1] − cov(u′(Yt+1), Dt+1) E[u′(Yt+1)Dt+1] − 1 Which means: rr,t − rf,t = − cov(u′(Yt+1), Dt+1) E[u′(Yt+1)Dt+1] (34.48) (34.49) In other words, the existence and sign of the risk premium depend on how the payouts from the risky bond co-vary with the marginal utility of future income. If the risky bond’s payout covaries negatively with u′(Yt+1), then a household will require a higher yield to be indiﬀerent between holding that bond and the risk-free bond. The household wishes to hold assets to facilitate consumption smoothing. It therefore “likes” assets whose payouts covary positively with u′(Yt+1) and “dislikes” assets whose payouts covary negatively with u′(Yt+1). The reason why is straightforward – u′(Yt+1) measures (in equilibrium) how
much the household values extra income. The household “likes” assets that have a high payout precisely when extra income is most valuable (periods in which u′(Yt+1) is high) and vice-versa. Note that given the assumption of diminishing marginal utility, u′(Yt+1) will be high (low) when Yt+1 is low (high). Let us now turn to some speciﬁc numerical examples to see these points clearly. Let us assume that the utility function is the natural log, that β = 0.95, that Yt = 1, and that = 0.9. In the subsections which follow, we will consider diﬀerent values of Y h t+1 p1, p2, and p3 and how these inﬂuence the prices of each bond as well as the implied yields on each type of bond. = 1.1 and Y l t+1 789 34.3.1 No Income Risk First, consider a speciﬁcation of uncertainty in which there is no income risk – future income is either high with probability 1 (so p1 + p2 = 1 and p3 = 0), or low with probability 1 (so p1 = p2 = 0). First, suppose that income is high with probability 1. This means that the price of the risk-free bond, (34.41), is: P B rf,t = β Yt Y h t+1 = 0.95 × 1 1.1 = 0.8636 The price of the risky bond, (34.42), can be written: P B r,t = βp1 Yt Y h t+1 (34.50) (34.51) To determine the price of the risky bond, we need to specify a value of p1, which works out to the probability that the bond does not default. If p1 = 0, for example, the bond would have an equilibrium price of 0. This makes sense because the bond defaults in t + 1 with probability 1, and is therefore worthless. If p1 = 1, in contrast, the price of the risky bond would be identical to the price of the risk-free bond. For intermediate values of p1, the price of the risky bond will between 0 and the price of the risk-free bond. The relationship between p1 and the price of each type of bond is
shown below in Figure 34.3. Figure 34.3: Bond Prices and p1: Income Always High The yield to maturity on either bond is simply the ratio of the expected future cash ﬂow 790 00.10.20.30.40.50.60.70.80.910.0250.10.1750.250.3250.40.4750.550.6250.70.7750.850.9251p1Bond PricesRisk-FreeRisky generated from the bond to the current bond price (since we are dealing with one period discount bonds). The (gross) yield on the risk-free bond works out to: 1 + rrf,t = Y h t+1 βYt = 1.1579 (34.52) In (34.52), the expected future cash ﬂow from the bond is just 1, so the yield is simply the inverse of the bond price. The yield to maturity on the risky bond is conceptually the same. The expected cash ﬂow from the bond is p1 – with probability p1, the bond pays out one unit of income, and with probability p2 the bond pays nothing. Hence, the yield to maturity on the risky bond is: 1 + rr,t = p1 × Y h t+1 p1βYt = Y h t+1 βYt = 1.1579 (34.53) In other words, for this particular example, the yield on the risky bond is exactly the same as the yield on the risk-free bond, and there is consequently no risk premium. This is shown in Figure 34.4, which plots the yields on each kind of bond as well as the risk premium as a function of p1. There is no risk premium in spite of the fact that the bond itself is risky in the sense that there is a chance it might default (unless p1 = 0 or p1 = 1). The reason there is no risk premium is evident from (34.49). If there is no income risk, then there can be no covariance between the bond’s payout and u′(Yt+1), and hence no risk premium. Figure 34.4: Yields and Risk Premium: Income Always High For completeness, we also consider the case where there is no uncertainty over future 791 -0.0200.020.040.060.080.10.120.
140.160.180.0250.10.1750.250.3250.40.4750.550.6250.70.7750.850.9251p1Yields and Risk PremiumYield: Risk-FreeYield: RiskyRisk Premium income, but it is always low instead of always high. This means that p1 = p2 = 0. Figure 34.5 plots the prices of the risk-free and risky bonds as a function of p3 (the probability the risky bond does not default). Figure 34.5: Bond Prices and p3: Income Always Low Figure 34.5 looks similar to Figure 34.3 in that the price of the risk-free bond is independent of p3 and the price of the risky bond is increasing in p3. What is diﬀerent however is that the price of the risk-free bond is substantially higher here than in Figure 34.3. The reason is that if the household knows it is going to have low income in the future, it wants to save today to try to smooth its consumption. This means there is high demand for the risk-free bond, which puts upward pressure on its price. Corresponding to upward pressure on its price, in Figure 34.6 we plot yields on both kinds of bond as well as the risk premium. Here we see that both bonds oﬀer negative yields. There is nothing conceptually wrong with negative (real) yields – it simply means that there is suﬃciently strong demand for the bond that market-clearing requires a negative yield. There is again no risk premium because there is no covariance between the bond’s payout and future income. 792 00.20.40.60.811.20.0250.10.1750.250.3250.40.4750.550.6250.70.7750.850.9251p3Bond PricesRisk-FreeRisky Figure 34.6: Yields and Risk Premium: Income Always Low 34.3.2 No Default Risk Next, let us suppose that there is income risk but there is no risk of the risky bond defaulting. That is, the economy is always in states 1 or 3 – income could be high or low in t + 1, but the risky bond always pays oﬀ. This means that p1 + p3 = 1, with p2 = p4 = 0
. Figure 34.7 plots the price of each bond as a function of p1 (the probability of the good income state): 793 -0.06-0.05-0.04-0.03-0.02-0.0100.010.0250.10.1750.250.3250.40.4750.550.6250.70.7750.850.9251Yields and Risk PremiumYield: Risk-FreeYield: RiskyRisk Premium Figure 34.7: Bond Prices and p1: No Default Risk There are two important things to emerge from Figure 34.7. First, the prices of each type of bond are identical regardless of p1. The reason for this is that there is no real diﬀerence between the bonds if there is no default risk – they both pay one in t + 1. Second, the price of either type of bond is decreasing in p1. When p1 is low, income in the future is expected to be low. The household would like to save to smooth consumption, so there is high demand for bonds and consequently a comparatively high price. In contrast, when p1 is large, in expectation future income is high, so the household doesn’t have much incentive to save. Consequently, there is not much demand for the bond and consequently the price of the bond is comparatively low. 794 00.20.40.60.811.20.0250.10.1750.250.3250.40.4750.550.6250.70.7750.850.9251p1Bond PricesRisk-FreeRisky Figure 34.8: Yields: No Default Risk Figure 34.8 plots yields on each type of bond as a function of p1. Since the prices of each bond are always the same, the yields are always equal and hence there is no risk premium. The yields move in the opposite direction of the bond price. When p1 is low, yields are low (even negative), because there is a strong incentive to save to prevent future consumption from being low and yields must be low (or even negative) to discourage this saving. The reverse is true when p1 is large. There is no risk premium because there is no risk of default on the risky bond. 34.3.3 Income Risk and Default Risk Above we considered two separate descriptions of uncertainty – one in which future income is certain and the
risky bond is in fact risky, and another in which future income is unknown but there is no probability of default. In neither case do the yields on the risk-free and risky bond diﬀer (even though their prices potentially do). To get a risk premium, we must have both income and the risky bond’s payout be risky in the sense of being uncertain. But even this is not suﬃcient to generate a risk premium, as we shall see. To generate a positive risk premium, it must the case that default is more likely in states in which income is low (and vice-versa). Table 34.1 considers diﬀerent values of p1, p2, p3, and p4. The examples are all constructed in which the expected value of future income is always one and the expected value of the risky bond’s payout is 0.5. We also show the expected risky bond payout conditional on ] for the expectation conditional on high income income being high or low, i.e. E[Dt+1 ∣ Y h t+1 795 -0.1-0.0500.050.10.150.20.0250.10.1750.250.3250.40.4750.550.6250.70.7750.850.9251Yields and Risk PremiumYield: Risk-FreeYield: RiskyRisk Premium in the future. If the expected risky bond payout is higher than its unconditional payout when income is high, then the bond payout is positively correlated with income (and negatively correlated with the marginal utility of future income) and vice-versa. The ﬁrst row considers the case where these probabilities are all 0.25, so that each state is equally likely. In this case, future income is uncertain and the bond is risky, yet there is no risk premium. The reason why is that there is no diﬀerence between the conditional expectations of the bond payout and the unconditional expectations. Put diﬀerently, there is no correlation between the risky bond payout and income – the risky bond is equally likely to default when income is high as when income is low. Table 34.1: The Nature of Uncertainty and Risk Premia Probabilities p1 = p2 = p3 = p4 = 0.25 p1 = 0.5, p2 = p3 = 0, p4 = 0
.5 p1 = p4 = 0, p2 = p3 = 0.5 p1 = 0.4, p2 = 0.1, p3 = 0.1, p4 = 0.4 E(D) E(Yt+1) E(Dt+1 ∣ Y h t+1 0.5 0.5 0.5 0.5 0.5 1 0 0.8 1 1 1 1 ) E(Dt+1 ∣ Y l t+1 ) 0.5 0 1 0.2 rr,t − rf,t 0.00 0.12 -0.09 0.07 The second row considers the case in which there is a 50 percent chance of high income and a 50 percent chance of low income, where the bond defaults with certainty when income is low and pays with certainty when income is high (i.e. p2 = p3 = 0). While somewhat extreme, this is a reasonable characterization of corporate debt – defaults are most likely when output is low and less likely when times are good. Here we observe a positive risk premium of 0.12 – the yield on the risky bond is substantially higher than the yield on the riskless bond. Intuitively, the reason why can be seen by looking at the conditional expectations. In the good income state, the payout on the bond is high, and in the bad income state, the payout on the bond is zero. A household does not like an asset with these characteristics – the household wants to smooth its consumption, so other things being equal it would prefer an asset whose payout is high when income is low (equivalently, high when extra income is most highly valued, i.e. when u′(Yt+1) is high). In this setup, the risky bond does not have these characteristics. For market-clearing, the household cannot hold any of either the risky or the risk-free bond, and hence must be indiﬀerent between them. For the household to be indiﬀerent between the two types of bonds, the yield on the risky bond must be higher than the yield on the risk-free bond. The third row considers the opposite case – income can be high or low, but the bond always defaults when income is high, and always pays face value when income is low. This results in a negative risk premium. The reason why the risk premium is negative is the mirror image of
why we get a positive risk premium when the bond defaults when income is low. 796 The risky bond helps the household smooth its consumption by giving it income precisely in the periods where additional income is most valuable (i.e. periods in which income is otherwise low). Hence, the household prefers the risky bond to the risk-less bond, and the risky bond accordingly must oﬀer a lower yield to make the household indiﬀerent between the two types of debt. The ﬁnal row considers a case similar to the second row, but less extreme. There is again a 50-50 chance of income being high or low. The risky bond is more likely to default in the low income state, but there is some probability of it not defaulting even if income in t + 1 is low. This results in a positive risk premium but not as large as in the case considered in the second row. The general pattern that emerges from Table 34.1 is that the risk premium depends ). When this diﬀerence is positive, the bond is positively on E(Dt+1 ∣ Y h t+1 most likely to default in precisely the periods in which income is most dear to the household (and hence a default is most costly). To compensate the houehold for this risk, the risky bond must oﬀer a comparatively high yield relative to the risk-free bond. ) − E(Dt+1 ∣ Y l t+1 Figure 34.9 below plots a time series of a popular measure of the aggregate risk premium. In particular, we show the diﬀerence between the average yield on Baa-rated corporate debt and the yield on a 10 year Treasury note. Figure 34.9: Yields: No Default Risk The risk premium is positive for the entire sample, typically hovering in the range of 1 to 2 percent (annualized). If anything, the risk premium seems to have risen over time. Another 797 19601970198019902000201001234567Moody's Seasoned Baa Corporate Bond Yield Relative to Yield on 10-Year Treasury Constant MaturityPercentShaded areas indicate U.S. recessionsSource: Federal Reserve Bank of St. Louismyf.red/g/klQF interesting pattern is that the risk premium is quite clearly countercyclical – i.e. it seems to rise during periods identiﬁed
as recession (the shaded gray bars) and falls during expansions. 34.4 Time to Maturity and the Term Structure of Interest Rates In the previous section we considered diﬀerent kinds of bonds with potentially diﬀerent probabilities of default. We showed that if a risky bond is more likely to default in a period in which output is low (so the marginal utility of consumption is high), then that bond must oﬀer a higher yield than a risk-free bond to make the household indiﬀerent between the two types of bonds. Aside from default risk, the other principal dimension along which bonds diﬀer is time to maturity. As noted above, the US government issues both short term (Treasury Bills), medium term (Treasury Notes), and long term (Treasury Bonds) debt securities. Similarly, corporations issue both short term (commercial paper) and longer term (corporate bond) securities. Holding the default risk ﬁxed (i.e. comparing only risk-free government securities with diﬀerent maturities, or risky corporate debt with diﬀerent maturities but the same default probabilities), how, if at all, do yields vary with time to maturity? If they do vary with time to maturity, why do they vary? We refer to the study of how yields vary with time to maturity (holding default risk ﬁxed) as the term structure of interest rates. What relevant macroeconomic information does the term structure convey? We study these questions in this section. Before considering theory, let us start with some facts. Figure 34.10 plots yields across time on Treasury debt with both a 10 year maturity (Treasury Note) and a three month maturity (Treasury Bill). There are several things worth noting. First, yields have been steadily falling for the last three decades. Second, short and long maturity yields tend to move together – when the long maturity yield declines, typically so too does the short maturity yield. Third, the yield on the long maturity debt is almost always higher than the yield on the short maturity debt. Since both short and long term Treasury securities presumably have the same (near-zero) default risk, this diﬀerence in yields must be due to something else. There are a couple of exceptions. In particular, short term yields tend to rise above long term yields immediately prior to recessions (denoted with gray shaded regions). 798 Figure 34.10
: Yields: 10 Yr Treasury Note and Three Month Treasury Bill A plot of yields (for securities with similar default probabilities) against time to maturity on the horizontal axis is known as a yield curve. It is most common to plot yield curves using US government debt, which, as noted above, comes in a variety of diﬀerent maturities. One can observe a yield curve at each particular date. Figure 34.11 below plots yield curves observed at diﬀerent points in the last decade. Consistent with the visual evidence in Figure 34.10, the typical yield curve is upward-sloping (i.e. long term yields are greater than short term yields), though this is not the case for the yield curve from 2007. 799 19902000201019851995200520150.02.55.07.510.012.515.03-Month Treasury Constant Maturity Rate10-Year Treasury Constant Maturity RatePercentShaded areas indicate U.S. recessionsSource: Board of Governors of the Federal Reserve System (US)myf.red/g/km1V Figure 34.11: Representative Yield Curves Yield curves typically “ﬂatten” immediately prior to recessions (i.e. long term yields fall in comparison to short term yields, perhaps so much so that the yield curve becomes downward-sloping or “inverted”). We can see this pattern for the last three documented recessions in the US in Figure 34.12 below. Figure 34.12: Yield Curves Prior to Recent Recessions 800 01234561 mo3 mo6 mo1 yr2 yr3 yr5 yr7 yr10 yr20 yr30 yrYield to Maturity Time to Maturity 20172014201120092008200701234567893 mo6 mo1 yr2 yr3 yr5 yr7 yr10 yr30yrYield to Maturity Time to Maturity 200720001990 34.4.1 No Uncertainty: The Expectations Hypothesis Let us now turn to theory to seek to understand the behavior of yields as a function of time to maturity. Let us again suppose that the economy is populated by a single representative household with an exogenous income stream. For now, let us assume that the household lives for three periods (instead of just two) and that there is no uncertainty over the future. In particular, there is no uncertainty over future realizations of income and there
is no uncertainty over future payouts on bonds (i.e. there is no default risk). The household begins in period t with no stock of wealth. It lives until period t + 2. Its lifetime utility is: U = u(Ct) + βu(Ct+1) + β2u(Ct+2) (34.54) In period t, the household earns exogenous income stream Yt. The household may save/borrow through two diﬀerent kinds of discount bonds. The ﬁrst, which we will denote Bt,t,t+1, is a one period maturity discount bond which sells at P B t,t,t+1. The notation here has the following interpretation – the ﬁrst subscript refers to the date at which a bond is purchased or a price is observed (this is the ﬁrst t subscript). The second subscript refers to the date of issue of the bond in question (in this case also t). The third subscript is the maturity date, in this case t + 1. If a household purchases one unit of this bond, it receives an income ﬂow of 1 in period t + 1 with certainty. The second bond to which the household has access will be denoted Bt,t,t+2. This is a two period maturity discount bond; it sells at P B t,t,t+2 in period t. If the household purchases one unit of this bond in period t and holds it until period t + 2, it receives an income ﬂow of 1 in t + 2. The household will also have the opportunity to buy or sell previously issued two period bonds in period t + 1. The household’s ﬂow budget constraint in period t is given in (34.55). The household may spend its income on consumption or on one or two period maturity bonds. Ct + P B t,t,t+1Bt,t,t+1 + P B t,t,t+2Bt,t,t+2 ≤ Yt (34.55) In period t + 1, the household faces the following budget constraint: Ct+1 + P B t+1,t+1,t+2Bt+1,t+1,t+2 + P B t+1,t,t+2 (Bt+1,t,t+2 − Bt,
t,t+2) ≤ Yt+1 + Bt,t,t+1 (34.56) What is going on in (34.56)? On the right hand side, the household receives an exogenous income ﬂow of Yt+1 and also receives one unit of income for each unit of the one period bond it purchased in period t (i.e. Bt,t,t+1). With this income, the household can either 801 consume or purchase/sell newly issued one period bonds (i.e. Bt+1,t+1,t+2 at price P B t+1,t+1,t+2) or it can purchase/sell two period bonds which were previously issued in period t. The term Bt+1,t,t+2 − Bt,t,t+2 denotes the change in the household’s holdings of two period bonds maturing in t + 2 – Bt,t,t+2 is the stock of such bonds the household brings from t to t + 1, while Bt+1,t,t+2 is the stock of such bonds the household takes from t + 1 to t + 2. If the household wishes to keep its stock of such bonds ﬁxed relative to what it purchased in t, it would simply set Bt+1,t,t+2 = Bt,t,t+2. P B t+1,t,t+2 is the market price in period t + 1 of bonds issued in period t which mature in t + 2. The household’s budget constraint in t + 2 (going ahead and imposing the terminal condition that it takes/leaves no bonds after this period) is: Ct+2 ≤ Yt+2 + Bt+1,t+1,t+2 + Bt+1,t,t+2 (34.57) Imposing the terminal conditions, the household simply consumes all of its available resources in the ﬁnal period of life. It has income from three sources – exogenous income ﬂow, Yt+2; maturing one period bonds brought from the previous period, Bt+1,t+1,t+2; and maturing two period bonds brought from the previous period, Bt+1,t,t+2. The household’s objective in period t is
to pick a consumption/saving plan which maximizes (34.54) subject to (34.55)-(34.57). It will be easiest to characterize optimal behavior by substituting out the consumption terms and instead writing the problem as choosing how many of each type of bond to purchase/sell. Doing so, the houehold’s problem can be written: max Bt,t,t+1,Bt,t,t+2,Bt+1,t+1,t+2,Bt+1,t,t+2 U = u [Yt − P B t,t,t+1Bt,t,t+1 − P B t,t,t+2Bt,t,t+2] + βu [Yt+1 + Bt,t,t+1 − P B t+1,t+1,t+2Bt+1,t+1,t+2 − P B t+1,t,t+2 (Bt+1,t,t+2 − Bt,t,t+2)]+β2u [Yt+2 + Bt+1,t+1,t+2 + Bt+1,t,t+2] The ﬁrst order conditions are: ∂U ∂Bt,t,t+1 = 0 ⇔ P B t,t,t+1u′(Ct) = βu′(Ct+1) ∂U ∂Bt,t,t+2 ∂U ∂Bt+1,t+1,t+2 ∂U ∂Bt+1,t,t+2 = 0 ⇔ P B t,t,t+2u′(Ct) = βP B t+1,t,t+2u′(Ct+1) = 0 ⇔ βP B t+1,t+1,t+2u′(Ct+1) = β2u′(Ct+2) = 0 ⇔ βP B t+1,t,t+2u′(Ct+1) = β2u′(Ct+2) 802 (34.58) (34.59) (34.60) (34.61) These ﬁrst order conditions have the usual marginal beneﬁt
= marginal cost interpretation. Start with (34.58). Suppose you purchase one unit of a one period bond. This reduces period t,t,t+1, which is valued in utility terms by u′(Ct). t consumption by the price of the bond, P B t,t,t+1u′(Ct), represents the marginal cost of buying a one period Hence, the left hand side, P B bond in period t. The marginal beneﬁt is extra income of one in period t + 1, which is valued in utility terms at βu′(Ct+1). Hence, βu′(Ct+1) is the marginal utility beneﬁt of purchasing a one period bond in period t. At any optimum, the marginal beneﬁt must equal the marginal cost. Consider next the ﬁrst order condition for one period bonds bought in period t + 1. Note that (34.60) can be re-written: t+1,t+1,t+2u′(Ct+1) = βu′(Ct+2) P B (34.62) (34.62) has exactly the same intuitive interpretation as (34.58), just led forward one period. Next, let us go to the ﬁrst order condition for the two period maturity bond. Note that (34.59) and (34.61) can be combined to yield: t,t,t+2u′(Ct) = β2u′(Ct+2) P B (34.63) (34.63) has a similar intuitive interpretation. If the household buys one unit of the two period bond in period t, it foregoes P B t,t,t+2 units of consumption in period t, which is valued in utility terms at u′(Ct). Hence, the left hand side of (34.63) represents the marginal utility cost of saving in the two period bond. The marginal beneﬁt of saving in the two period bond (and holding it until maturity) is one unit of additional consumption in period t + 2. This is valued in utility terms at β2u′(Ct+2). At any optimum, the marginal utility beneﬁt must equal the marginal utility cost. Note also that (34.60)-(34.61) together imply that P B t+1,t,t+2. In
other words, in period t + 1 the price of two period bonds issued in period t must equal the price of newly issued one period bonds. In other words, all that matters for a bond’s price is its remaining time to maturity, not its date of issue. t+1,t+1,t+2 = P B Now, note that we can combine (34.62)-(34.63) to get: βP B t+1,t+1,t+2u′(Ct+1) = P B t,t,t+2u′(Ct) But then we can use (34.58) to write (34.64) as: βP B t+1,t+1,t+2u′(Ct+1) = P B t,t,t+2 βu′(Ct+1) P B t,t,t+1 803 (34.64) (34.65) But then (34.65) implies: P B t,t,t+2 = P B t,t,t+1 × P B t+1,t+1,t+2 (34.66) (34.66) says that the price of a two period bond ought to equal the product of the prices of one period bonds issued today and in t + 1. At this point, it is useful to step back and relate bond prices back to yields. Recall that the yield to maturity is the discount rate which equates the price of the bond with the present discounted value of cash ﬂows if the bond is held to maturity. For the two period bond, the yield to maturity satisﬁes: P B t,t,t+2 = 1 (1 + r2,t)2 (34.67) (34.67) implicitly deﬁnes the yield on the two period bond because it generates a cash ﬂow of one unit two periods into the future. The yield on the one period bond in period t is implicitly deﬁned by: Similarly, the implied yield on the one period bond issued in period t + 1 is: P B t,t,t+1 = 1 1 + r1,t P B t+1,t+1,t+2 = 1 1 + r1,t+1 Combining (34.67)-(34.69) together with (
34.66) means that: (1 + r2,t)2 = (1 + r1,t)(1 + r1,t+1) (34.68) (34.69) (34.70) In other words, the gross compounded yield on the two period bond must equal the product of the sequence of gross yields on the one period bond. What is the intuition for this? Short maturity (in this case one period) and long maturity (in this case, two period) bonds are substitutes. If the household wishes to transfer income from period t to period t + 2, it can do so either by: (i) buying a two period bond and holding it until t + 2, or (ii) buying a one period bond in t and then taking the proceeds from this and purchasing another one period bond in t + 1 (what is sometimes called a “rollover”). For market-clearing, in equilibrium the household must be indiﬀerent between these two options of transferring resources from t to t + 2. Why is this? Suppose that (1 + r2,t)2 > (1 + r1,t)(1 + r2,t). The household could make an inﬁnite proﬁt by buying two period bonds and ﬁnancing this purchase by borrowing through one period bonds (i.e. demand negative quantities of these bonds). This would entail inﬁnite demand for two period bonds and negative inﬁnity demand for one period bonds. The 804 opposite situation would occur if (1 + r2,t)2 < (1 + r1,t)(1 + r2,t) – the household would borrow through two period bonds and save through one period bonds, in the process making a proﬁt. In equilibrium, since both bonds are in ﬁnite and ﬁxed supply, positive or negative inﬁnity demand is not possible. Hence, (1 + r2,t) = (1 + r1,t)(1 + r1,t+1) must hold in equilibrium. (34.70) can be written in approximate form by taking natural logs and using the approximation that the log of one plus a small number is approximately the small number. Doing so yields: rt,t+2 ≈ 1 2 [rt,t+1 + r
t+1,t+2] (34.71) In other words, the yield on the long bond ought to approximately equal the average of the yields on the sequence of short bonds over the maturity of the long bond. (34.71) can be extended for an arbitrary m period maturity bond: rt,t+m ≈ 1 m [rt,t+1 + rt+1,t+2 +... rt+m−1,t+m] (34.72) Expression (34.72) is a statement of the Expectations Hypothesis of the term structure. The expectations hypothesis says that the yield on a long maturity bond is approximately equal to the average of expected short maturity yields over the life of the long maturity bond. Put somewhat diﬀerently, according to (34.72) the behavior of long maturity yields ought to provide information on market expectations of future short maturity interest rates. The expectations hypothesis is empirically successful on many dimensions. First, because the long bond yield is simply an average of short bond yields, it can easily account for the fact that yields on bonds of diﬀerent maturities tend to move together. Second, changes in the “slope” of the yield curve (i.e. the diﬀerence between long maturity and short maturity yields at a particular point in time) will be predictive of future movements in income. Revert back to our three period example with a one and two period bond. Suppose that the household has log utility. In equilibrium, the bond yields will satisfy: 1 + rt,t+1 = 1 β (1 + rt,t+2)2 = 1 β2 1 + rt,t+1 = 1 β Yt+1 Yt Yt+2 Yt Yt+2 Yt+1 (34.73) (34.74) (34.75) Suppose that Yt = Yt+1, but Yt+2 < Yt+1 (i.e. a “recession” is coming in t + 2). This will not aﬀect the current one period bond yield, rt,t+1, but will push down rt,t+2. In essence, the yield 805 curve will ﬂatten or become inverted if the household anticipates a coming recession and lower short term yields. This is roughly consistent with the empirical
regularity documented in Figure 34.12, for example. A major failing of the expectations hypothesis of the term structure is that it is unable to account for why yield curves are almost always upward-sloping. In the data, short term yields over very long periods of time are either roughly constant or even trending down (see, e.g., Figure 34.10). Given this, if (34.72) holds, we would expect the typical yield curve to be ﬂat or even downard-sloping, but this is not what we see in the data. Evidently, investors demand a “term premium” in the form of a higher average yield for holding longer maturity debt. In the next subsection, we incorporate uncertainty over future income to motivate the existence of the term premium. 34.4.2 Uncertainty and the Term Premium Let us continue with the three period example from above in which the household can purchase either one or two period maturity bonds in period t. These bond are default risk-free. Diﬀerently from above, let us allow future realizations of the endowment to be uncertain. As we shall see, this uncertainty may be capable of generating a term premium. The household wishes to maximize expected utility, which in general form using expecta- tions operators is given below: U = u(Ct) + β E [u(Ct+1)] + β2 E [u(Ct+2)] (34.76) Because everything in the present is observed, the period t budget constraint is the same as in the case of no income uncertainty: Ct + P B t,t,t+1Bt,t,t+1 + P B t,t,t+2Bt,t,t+2 ≤ Yt (34.77) ≥ Y l Suppose that future income can take on two possible values: Y h t+j t+j for j = 1, 2. Assume that the probability of the high state is p, and the probability of the low state is 1 − p. Assume that the possible realizations of the endowment are the same in period t + 2 as in period t + 1, and that the probabilities of high or low realizations in t + 2 are independent of the realized values of income in period t + 1. We could instead assume that the income process is persistent in the precise sense that high income in period t + 1 portends high income (in expectation
) in period t + 2, but we do not lose much by making the simplifying assumption that the income draws in periods t + 1 and t + 2 are independent from one another. As when we considered default risk above, ﬂow budget constraints must hold in each possible state of the world. In period t + 1, there are two states of the world – either the 806 endowment is high (probability p), or it is low (probability 1 − p). The t + 1 ﬂow budget constraints in these states of the world are: C h t+1 C l t+1 + P B,h t+1,t+1,t+2Bh t+1,t+1,t+2 + P B,l t+1,t+1,t+2Bl t+1,t+1,t+2 + P B,h t+1,t,t+2 (Bh t+1,t,t+2 − Bt,t,t+2) ≤ Y h t+1 + Bt,t,t+1 + P B,l t+1,t,t+2 (Bl t+1,t,t+2 − Bt,t,t+2) ≤ Y l t+1 + Bt,t,t+1 (34.78) (34.79) In either state of the world, available resources are the exogenous income ﬂow plus one period bonds brought from the previous period. With these resources, the household can consume, accumulate more one period bonds, or change its stock of two period bonds. We index values of consumption, bond prices, and bond holdings in period t + 1 with an h or l superscript to refer to the realized state of nature. In period t + 2, while there are again just two states of the world in terms of income, there are an additional two states of the world depending on what happens in t + 1. The stock of one and two period bonds which payoﬀ in period t + 2 depend on the realized state of the world in t + 1. Let C h,l t+2, for example, denote consumption in period in period t + 2 when income is high in period t + 2 and when income was is low in period t + 1. The ﬁrst superscript references the t + 2 state
of the world, while the second references the t + 1 state of the world. The budget constraint for this state is summarized in (34.81). (34.80), (34.82), and (34.83) summarize the budget constraints in the other possible states – (h, h) (income is high in t + 2 and in t + 1); (l, l) (income is low in both t + 1 and t + 2); and (l, h) (income is low in t + 2 but high in t + 1). Going ahead and imposing the terminal conditions that the household will not choose to die with a positive stock of assets, and will not be allowed to die in debt, these constraints are: C h,h t+2 C h,l t+2 C l,l t+2 C l,h t+2 ≤ Y h t+2 ≤ Y h t+2 ≤ Y l t+2 ≤ Y l t+2 + Bh t+1,t+1,t+2 + Bh t+1,t,t+2 + Bl t+1,t+1,t+2 + Bl t+1,t,t+2 + Bl t+1,t+1,t+2 + Bl t+1,t,t+2 + Bh t+1,t+1,t+2 + Bh t+1,t,t+2 (34.80) (34.81) (34.82) (34.83) The probability of the (h, h) state is just p2 – the probability income is high in t + 1 times the probability income is high in t + 2. Similarly, the probability of the (h, l) state is p(1 − p) – the probability income is high in t + 2 times the probability it is low in t + 1. With this description of uncertainty, expected lifetime utility, (34.76), may be written: 807 U = u(Ct) + pβu(C h t+1 p2β2u(C h,h t+2 ) + (1 − p)βu(C l t+1 ) + p(1 − p)β2u(C h,l t+2 ) +... ) + (1 − p)2β2u(C l,l t+2 ) + (1 − p)p
β2u(C l,h t+2 ) (34.84) t+1, C h,h The household’s objective is to pick a state-contingent sequence of consumption (i.e. Ct, C h t+1, C l t+2, and so on) to maximize (34.84) subject to (34.77)-(34.83). As before, it is easier to think about the problem by substituting the consumption values out and instead thinking about an unconstrained problem of choosing a state-contingent sequence of bond holdings. The resulting unconstrained optimization problem is below: U = u [Yt − P B t,t,t+1Bt,t,t+1 − P B t,t,t+2Bt,t,t+2] + Bt,t,t+1,Bt,t,t+2,Bh t+1,t+1,t+2,Bl max pβu [Y h t+1 (1 − p)βu [Y l t+1 p2β2u [Y h + Bh t+2 (1 − p)2β2u [Y l + Bl t+2 t+1,t+1,t+2,Bh + Bt,t,t+1 − P B,h t+1,t,t+2,Bl t+1,t+1,t+2Bh t+1,t,t+2 + Bt,t,t+1 − P B,l t+1,t+1,t+2Bl t+1,t+1,t+2 t+1,t,t+2 t+1,t+1,t+2 t+1,t,t+2 + Bh + Bl − P B,h (Bh − Bt,t,t+2)] + t+1,t+1,t+2 t+1,t,t+2 − P B,l t+1,t+1,t+2 ] + p(1 − p)β2u [Y h t+2 ] + (1 − p)pβ2u [Y l t+2 t+1,t,t+2 (Bl t+1,t,t+2 t+1,t,
t+2 + Bl + Bh t+1,t+1,t+2 t+1,t+1,t+2 − Bt,t,t+2)] + ] + + Bl + Bh t+1,t,t+2 ] t+1,t,t+2 (34.85) The ﬁrst order conditions are: ∂U ∂Bt,t,t+1 = 0 ⇔ P B t,t,t+1u′(Ct) = β[pu′(C h t+1 ) + (1 − p)u′(C l t+1 )] (34.86) = 0 ⇔ P B t,t,t+2u′(Ct) = β[pP B,h t+1,t,t+2u′(C h t+1 ) + (1 − p)P B,l t+1,t,t+2u′(C l t+1 )] (34.87) ∂U ∂Bt,t,t+2 ∂U ∂Bh t+1,t+1,t+2 ∂U ∂Bl t+1,t+1,t+2 ∂U ∂Bh t+1,t,t+2 = 0 ⇔ pβP P,h t+1,t+1,t+2u′(C h t+1 ) = β2[p2u′(C h t+2 ) + (1 − p)pu′(C l t+2 )] (34.88) = 0 ⇔ (1−p)βP P,l t+1,t+1,t+2u′(C l t+1 ) = β2[p(1−p)u′(C h t+2 )+(1−p)2u′(C l t+2 )] (34.89) = 0 ⇔ pβP B,h t+1,t,t+2u′(C h t+1 ) = β2[p2u′(C h t+2 ) + (1 − p)pu′(C l t+2 )] (34.90) ∂U ∂Bl t+1,t,t+2 = 0 ⇔ (1 − p)βP
B,l t+1,t,t+2u′(C l t+1 ) = β2[p(1 − p)u′(C h t+2 + (1 − p)2u′(C l t+2 )] (34.91) These appear a bit nasty but have fairly intuitive interpretations. Note ﬁrst that in terms of the expectations operator (34.86) is simply: 808 t,t,t+1u′(Ct) = β E [u′(Ct+1)] P B (34.92) (34.92) is a familiar bond-pricing condition allowing for uncertainty over future income. In particular, the price of the one period bond in period t is simply the expected vale of the ] since stochastic discount factor (since we can re-write the condition as P B both Ct and β are known at the time the expectation is made). = E [ βu′ u′ (Ct+1) (Ct) t,t,t+1 Again in terms of expectations operators, note that (34.87) can be written: t,t,t+2u′(Ct) = β E[P B P B t+1,t,t+2u′(Ct+1)] (34.93) (34.93) can be re-written: Pt,t,t+2 = E [ βu′(Ct+1) ] u′(Ct) P B (34.94) is simply the standard asset pricing condition – the price of an asset is the expected value of the product of the stochastic discount factor with the payout of the bond. One way to think of the payout in period t + 1 of buying a two period bond in t is that the payout is simply the price of the two period bond in t + 1 (i.e. P B t+1,t,t+2), since the household can sell the bond and raise this amount of funds. (34.94) t+1,t,t+2 If one combines (34.88) with (34.90), and (34.89) with (34.91), one gets: P B,h P B,l t+1,t+1,t+2 t+1,t,t+2 t+1,t+1,t+2 t+1,t,
t+2 = P B,h = P B,l (34.95) (34.96) These are intuitive. They simply say that the price of newly issued one period bonds must equal the price of previously issued two period bonds in period t + 1 regardless of the state of the world. We also see this in the world without uncertainty. The only thing relevant for the price of a bond is its remaining time to maturity, not its date of issuance. (34.88)-(34.89) together imply that: E[P B t+1,t+1,t+2u′(Ct+1)] = β E[u′(Ct+2)] Plugging (34.97) into (34.93), we get: t,t,t+2u′(Ct) = β2 E[u′(Ct+2)] P B (34.98) can also be re-written: 809 (34.97) (34.98) P B t,t,t+2 = E [ β2u′(Ct+2) u′(Ct) ] (34.99) (34.99) has exactly the same interpretation as (34.93). The price of the asset must be the product of the stochastic discount factor and the payout from the bond. If the bond is held to maturity in t + 2, the payout is 1 with certainty in t + 2. The relevant stochastic discount factor is β2u′. Whether the bond is held to maturity (i.e. (34.99)), or sold after one u′ period (i.e. (34.93)), the basic asset pricing optimality condition must hold. (Ct+2) (Ct) As we did above in the case of no uncertainty, we wish to derive a relationship between the prices of bonds with diﬀerent maturities. Take (34.93), noting that P B t+1,t+1,t+2 regardless of the state of nature, and divide it by (34.92). After re-arranging terms a bit, one gets: t+1,t,t+2 = P B Multiply and divide (34.100) by E[P B P B t,t,t+2 = P B t,t,t+1 E[P B t+1,t+1,t+2u′
(Ct+1)] E[u′(Ct+1)] ]. One gets: t+1,t+1,t+2 P B t,t,t+2 = P B t,t,t+1 E[P B t+1,t+1,t+2 ] E[P B t+1,t+1,t+2u′(Ct+1)] ] E[u′(Ct+1)] t+1,t+1,t+2 E[P B (34.100) (34.101) One would be tempted to distribute the expectations operator through the term E[P B in (34.101). If one could do this, the fraction would cancel out, leaving just the term: t+1,t+1,t+2u′(Ct+1)] P B t,t,t+2 = P B t,t,t+1 E[P B t+1,t+1,t+2 ] (34.102) (34.102) would be the natural analog of (34.66) accounting for the fact that P B t+1,t+1,t+2 is not necessarily known in advance in period t. The expectations operator can only be distributed in this way if (i) there is no uncertainty over the future, as in the previous subsection, or (ii) the marginal utility of consumption is linear, so that u′′(⋅) = 0. If neither of these conditions are satisﬁed, it is not possible to distribute the expectations operator in this way. As noted above, the expected value of a product of random variables is the product of the expectations plus the covariance between the variables. Making use of this fact, we can write (34.101) as: P B t,t,t+2 = P B t,t,t+1 E[P B t+1,t+1,t+2 ] ( E[P B t+1,t+1,t+2 (34.103) simpliﬁes to: ] E[u′(Ct+1)] + cov(P B E[P B ] E[u′(Ct+1)] t+1,t+1,t+2 t+1,t+1,t+2, u′(Ct+1)) ) (34.
103) 810 P B t,t,t+2 = P B t,t,t+1 E[P B t+1,t+1,t+2 ] (1 + cov(P B E[P B t+1,t+1,t+2, u′(Ct+1)) ] E[u′(Ct+1)] t+1,t+1,t+2 ) (34.104) Since consumption equals income in equilibrium, the equilibrium two period bond price satisﬁes: P B t,t,t+2 = P B t,t,t+1 ] E[P B t+1,t+1,t+2 ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Expectations Hypothesis (1 + cov(P B E[P B t+1,t+1,t+2, u′(Yt+1)) ] E[u′(Yt+1)] ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„
„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Term Premium t+1,t+1,t+2 ) (34.105) t,t,t+1 E[P B t+1,t+1,t+2 The price of the two period bond in equilibrium is the product of two terms. The ﬁrst term we will call the expectations hypothesis term – this is simply the product of the current ]. The second term is what we and expected future short bond price, P B t+1,t+1,t+2,u′ will call the term premium term, and is given by (1 + cov(P B ). If there is no t+1,t+1,t+2] E[u′ uncertainty, if the marginal utility of consumption is constant, or if the covariance between the future one period bond price and the future marginal utility of
consumption term is zero, then this term is one, and the expectations hypothesis as laid out in the previous subsection under certainty would hold. Otherwise, the simple expectations hypothesis does not hold exactly, though the logic of the expectations hypothesis is still at play. In particular, changes in expected future short maturity bond prices ought to be reﬂected in current long maturity bond prices holding the term premium term ﬁxed. (Yt+1)) (Yt+1)] E[P B We would in general expect the covariance term in (34.105) to be negative. Why is this? Note that (34.88)-(34.89) together imply that, regardless of the state of nature in t + 1, we ). In other words, the following must hold: must have P h ) = P l t+1,t+1,t+2u′(C h t+1 t+1,t+1,t+2u′(C l t+1 Pt+1,t+1,t+2u′(Ct+1) = β Et+1[u′(Ct+2)] (34.106) In (34.106), Et+1[⋅] is the expectations operator conditional on the realization of the state in t + 1 (whereas in the notation we have been using E[⋅] is the expectation operator conditional on information observed in t). In other words, (34.106) must hold in all states of the world. In equilibrium, the price of the one period bond in t + 1, regardless of the state of nature in that period, will therefore be: P B t+1,t+1,t+2 = β Et+1[u′(Yt+2)] u′(Yt+1) (34.107) When Yt+1 is high, for example, u′(Yt+1) will be low, and therefore P B t+1,t+1,t+2 will be high. The reverse will be true when Yt+1 is in the low state. The intuition for this relates to the household’s desire to smooth consumption. If it receives a high endowment in t + 1, it 811 will want to increase its saving, which requires holding more of the one period bond. The increased demand for the bond pushes its equilibrium price up (and yield down
), because in equilibrium there can be no saving in an endowment economy. Hence, we would expect t+1,t+1,t+2 to be high when u′(Yt+1) is low. This means that the covariance term is negative, P B and hence the two period bond ought to trade at a discount relative to the product of the expected sequence of one period bond prices (i.e. (1 + cov(P B ) ought to be less than one). t+1,t+1,t+2,u′ t+1,t+1,t+2] E[u′ (Yt+1)) (Yt+1)] E[P B We therefore see that there is risk associated with the long bond even though we have assumed away default risk. This form of risk is sometimes called interest rate risk. There is an intuitive way to think about this form of risk. It is easiest to do so if we think about a situation in which the household buys a two period bond in period t and has to sell it in t + 1. In other words, focus on (34.93). Since the price of the long bond in period t + 1 must equal the price of the one period bond in t + 1, if P B t+1,t+1,t+2 is lower than anticipated (i.e. the yield is higher), the household gets a lower payout on the long bond than it anticipated. P B t+1,t+1,t+2 is likely to be lower than expected when Yt+1 is lower than expected (low Yt+1 makes the household want to borrow in t + 1 and reduces demand for the bond). Low Yt+1 means that the marginal utility of consumption is comparatively high. In other words, the long bond has a low return in a period in which the household most values a high return. If P B t+1,t+1,t+2 is higher than expected (short term yields are lower than expected), in contrast, the household gets a bigger payout on the long bond than it anticipated because P B t+1,t,t+2 will be higher than anticipated. But P B t+1,t+1,t+2 is likely to be high when Yt+1 is high, which is a period in which the household places a comparatively small weight on an extra payout (i.e. u′
(Yt+1) is small). For this reason, the household requires a premium to hold the long bond in the form of a lower price than would be predicted by the simple expectations hypothesis. (34.105) is written in terms of bond prices, whereas for the usual reasons we would prefer to work with yields. Take the inverse of (34.105) to get: 1 P B = 1 P B t,t,t+2 t,t,t+1 Or, in terms of yields: 1 E[P B t+1,t+1,t+2 (1 + ] cov(P B E[P B t+1,t+1,t+2, u′(Yt+1)) ] E[u′(Yt+1)] t+1,t+1,t+2 −1 ) (34.108) (1 + rt,t+2)2 = (1 + rt,t+1) 1 E[P B t+1,t+1,t+2 (1 + ] cov(P B E[P B t+1,t+1,t+2, u′(Yt+1)) ] E[u′(Yt+1)] t+1,t+1,t+2 −1 ) (34.109) A complication arises in (34.109). This is that the expected one period yield is E [ 1 P B t+1,t+1,t+2 ], 812 1 E[P B t+1,t+1,t+2]. This would be the case if there were no uncertainty (or if which is in general not the second derivative of the utility function were zero), but neither of these conditions will in general hold. We will ignore this complication and treat as the expected yield on the one period bond. More precisely, we can think of this is as expectation of the risk-neutral one period yield (i.e. the yield that would be expected in the absence of the risk aversion (positive second derivative of utility function) and/or the absence of uncertainty). Doing so, we can write (34.109) as: t+1,t+1,t+2] E[P B 1 (1 + rt,t+2)2 = (1 + rt,t+1) E[1 + rt+1,t+2
] (1 + cov(P B E[P B t+1,t+1,t+2, u′(Yt+1)) ] E[u′(Yt+1)] t+1,t+1,t+2 −1 ) (34.110) If we take logs of (34.110), making use of the approximation that the log of one plus a small number is approximately the small number, we get:5 rt,t+2 ≈ 1 [rt,t+1 + E[rt+1,t+2]] 2 ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Expectations Hypothesis + 1 tpt 2 – Term Premium (34.111) (Yt+1)) (Yt+1)] t+1,t+1,t+2,u′ t+1,t+1,t+2] E[u′ (34.111) is the same as (34.71), with an additional term we call the term premium. In (34.111), the term tpt = − ln (1 + cov(P B ). If the covariance is negative, as we E[P B expect, the term inside the parentheses will be less than 1, so the natural log of this will be negative. Since a
− multiplies this term, we would therefore expect the term premium to be positive. This is the natural analog to the idea that the long bond should sell at a discount relative to the expected sequence of short bond prices; if this is true, the long bond must oﬀer a higher yield. This higher yield is a compensation for the additional risk that long bonds carry and is what we call the term premium. t+1 and Y h t+2 It may be beneﬁcial to use some numbers to generate a numeric example. Suppose that Yt = 1 and β = 0.95, with the utility function the natural log. Future income can take on a ≥ Y l high or a low value. Suppose Y h t+2. Suppose that the high state occurs t+1 = 1.1 and with probability p and the low state with probability 1 − p. Suppose that Y h t+1 = 0.90, with p = 0.5. This means that the expected value of future income that Y l equals the current value (i.e. 1). We can solve for bond prices and yields. The price of the = 0.96, with a yield of about 0.042. The price of the one period bond in period t is P B = 0.91. The implied yield to maturity on the two year bond is two period bond is P B 0.047. In other words, the yield curve slopes up – the long bond yield exceeds the short bond = Y h t+2 = Y l ≥ Y l t,t,t+2 t,t,t+1 t+2 t+1 5In addition, we are making another approximation, because in general ln(E[X]) ≠ E[ln(X)] for some random variable X. 813 yield – even though in expectation the future endowment looks just like the present. The expected future one period bond price is the same as the current bond price. Hence, the risk-free expected one period yield is the same as the current one period yield. This means that our model attributes the positive slope of the yield curve to a positive term premium of about 0.005 (50 basis points). Incorporating uncertainty and allowing for a term premium addresses an important failure of the expectations hypothesis of the term structure, which is that the expectations hypothesis cannot account for why the typical yield curve observed in the data is upward-sloping. This augmented model taking into account uncertainty
can generate an upward-sloping yield curve even if expected short term yields are not rising. Yet the augmented model with uncertainty retains some of the intuition of the expectations hypothesis for why the shape of the yield curve might change from time to time. To see this, continue with the numerical example = 0.85. In other words, outlined in the paragraph above, but assume that Y h t+2 the expected value of output in t + 2 is lower than in t + 1 or t – i.e. a “recession” is coming. What eﬀect does this have on the prices of short and long maturity bonds and the associated yields? The price of the one period bond in period t, and its associated yield, are the same as in the paragraph above. But the price of the two period bond rises, and hence its yield falls. = 0.96, which implies a yield of ( 1 2 = 0.02. Hence, the long In particular, we have P B 0.96 bond now has a lower yield than the short bond, i.e. the yield curve slopes down instead of up. This is consistent with the empirical facts documented above that ﬂat or inverted yield curves often precede recessions. The reason why the yield curve becomes inverted here is because E[P B ] rises (so the expected future one period yield falls). The implied term premium is still roughly 50 basis points, as in the example above. = 1.05 and Y l t+1,t+1,t+2 t,t,t+2 t+2 ) 1 It is reasonably straightforward, if not a bit laborious, to extend beyond three periods. A general pattern emerges which is already evident in the three period model. Suppose that time extends for four periods – t, t + 1, t + 2, and t + 3. In period t, the household can purchase newly issued one period bonds, two period bonds, or three period bonds. Its budget constraint in period t is: Ct + P B t,t,t+1Bt,t,t+1 + P B t,t,t+2Bt,t,t+2 + P B t,t,t+3Bt,t,t+3 ≤ Yt (34.112) Without explicitly laying out the nature of uncertainty or fully specifying the future budget constraints which must hold in each state of the world, we will skip straight ahead
to the optimality conditions which must hold in period t. These are analogous to (34.92) and (34.93) for the three period case, although there is an additional condition for the three period bond: 814 t,t,t+1u′(Ct) = β E [u′(Ct+1)] P B t,t,t+2u′(Ct) = β E [P B P B t,t,t+3u′(Ct) = β E [P B P B t+1,t,t+2u′(Ct+1)] t+1,t,t+3u′(Ct+1)] (34.113) (34.114) (34.115) In equilibrium, consumption will equal income regardless of date or state of nature. Hence we can impose this market-clearing condition to solve for equilibrium bond prices. (34.114) can be combined with (34.113) to yield: P B t,t,t+2 = P B t,t,t+1 E [P B t+1,t,t+2u′(Yt+1)] E [u′(Yt+1)] (34.116) Regardless of the state of nature in t+1, it again must be the case that P B t+1,t+1,t+2 (i.e. all that matters for the price of a bond is its remaining time to maturity, not its date of issuance). Making use of this fact, (34.116) can be written in exactly the same way as we did above in the three period case, giving: = P B t+1,t,t+2 P B t,t,t+2 = P B t,t,t+1 E[P B t+1,t+1,t+2 ] E [P B t+1,t+1,t+2u′(Yt+1)] ] E [u′(Yt+1)] t+1,t+1,t+2 E[P B (34.117) Making use of the relationship between covariance and expectations, (34.117) can be written: P B t,t,t+2 = P B t,t,t+1 E[P B t+1,t+1,t+2 ] (1
+ cov(P B E[P B t+1,t+1,t+2, u′(Yt+1)) ] E [u′(Yt+1)] t+1,t+1,t+2 ) (34.118) (34.118) expresses the price of the two period bond as a function of the product of the current and expected one period bond prices multiplied by a term related to covariance. It is exactly the same expression that we derived above, (34.105). Now let us turn to the three period bond. Take (34.115) and combine it with (34.114) to get: P B t,t,t+2 t,t,t+3 = P B E [P B E [P B In looking at (34.120), note that P B t+1,t,t+3u′(Yt+1)] t+1,t,t+2u′(Yt+1)] = P B t+1,t+1,t+3 (i.e. the price of a previously issued three period bond with two periods until maturity will equal the price of a newly issued two period bond) and P B t+1,t+1,t+2 (i.e. the price of a previously issued two period bond with one period to maturity will equal the price of a newly issued one period bond). We can then write: (34.119) = P B t+1,t,t+2 t+1,t,t+3 815 P B t,t,t+3 = P B t,t,t+2 E [P B E [P B t+1,t+1,t+3u′(Yt+1)] t+1,t+1,t+2u′(Yt+1)] (34.120) (34.113) and (34.114) will hold in expectation for period t + 1, eﬀectively determining the prices of the one and two period bonds in t + 1. In particular: E[P B t+1,t+1,t+2u′(Yt+1)] = β E[u′(Yt+2)] E[P B t+1,t+1,t+3u′(Yt+1)] = β E[
P B t+2,t+2,t+3u′(Yt+2)] Combine (34.121)-(34.122) with (34.120) to get: P B t,t,t+3 = P B t,t,t+2 E[P B t+2,t+2,t+3u′(Yt+2)] E[u′(Yt+2)] Multiplying and dividing the right hand side (34.123) by E[P B t+2,t+2,t+3 ], we get: P B t,t,t+3 = P B t,t,t+2 E[P B t+2,t+2,t+3 ] E[P B t+2,t+2,t+3u′(Yt+2)] ] E[u′(Yt+2)] t+2,t+2,t+3 E[P B (34.121) (34.122) (34.123) (34.124) Via logic used above, the fraction on the right hand side of (34.124) may be written in terms of covariance. Therefore: P B t,t,t+3 = P B t,t,t+2 E[P B t+2,t+2,t+3 ] (1 + cov(P B E[P B t+2,t+2,t+3, u′(Yt+2)) ] E[u′(Yt+2)] t+2,t+2,t+3 ) (34.125) Now, we can use (34.118) to substitute Pt,t,t+2 out of (34.125). Re-arranging terms a bit, we get: P B t,t,t+3 = P B t,t,t+1 E[P B ] t+2,t+2,t+3 ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„
„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Expectations Hypothesis ] E[P B t+1,t+1,t+2 (1 + cov(P B E[P B t+2,t+2,t+3, u′(Yt+2)) ] E[u′(Yt+2)] ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„
„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„�
�„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Term Premium t+1,t+1,t+2, u′(Yt+1)) ] E[u′(Yt+1)] cov(P B E[P B ) (1 + t+1,t+1,t+2 t+2,t+2,t+3 ) (34.126) Note that (34.126) is very similar to (34.105) but allows for more periods. In particular, the price of the three period bond is equal to the product of the current and expected sequence of one period bond prices over the life of the three period bond (the expectations hypothesis component) times two terms related to the covariance between the price of a one period bond and the marginal utility of consumption. Although one of these terms is dated t + 1 and the other t + 2, each is simply related to the covariance between the one period bond price in a period with the marginal utility of consumption in that same period. As such, these
terms 816 should be the same. In particular, deﬁne: T Pt = (1 + cov(P B E[P B t+1,t+1,t+2, u′(Yt+1)) ] E[u′(Yt+1)] t+1,t+1,t+2 ) (34.127) We can therefore write (34.126) as: P B t,t,t+3 = P B t,t,t+1 E[P B t+1,t+1,t+2 ] E[P B t+2,t+2,t+3 ]T P 2 t (34.128) Writing in terms of yields, and making the same approximation as above in the three period case that the inverse of the expected bond price can be treated as the expected yield, we can write (34.128) as: (1 + rt,t+3)3 = (1 + rt,t+1) E[1 + rt+1,t+2] E[1 + rt+2,t+3]T P −2 t (34.129) Taking logs and making use of the approximation that the log of one plus a small number is the small number, (34.129) may be written: tpt rt,t+3 ≈ 1 3 [rt,t+1 + E[rt+1,t+2] + E[rt+2,t+3]] + 2 3 This is very similar to (34.111) but allows for one more period. The three period yield is approximately the average of expected one period yields over the life of the three period bond (the expectations hypothesis component) plus another component. In (34.130), we have deﬁned tpt = − ln (T Pt). This is exactly the same term as in (34.111) for the two period yield, but is weighted by 2 2. This is the term premium term, but because it is weighted by 2 2, we would expect the term premium on the three period bond to be bigger than on 3 the two period bond. 3 instead of 1 (34.130) > 1 We will not do so explicitly, but if one extends to an arbitrary number of periods until maturity, m > 1, one arrives at the following expression: rt,t+m ≈
1 [rt,t+1 + ⋅ ⋅ ⋅ + E[rt+m−1,t+m]] m ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Expectations Hypothesis + m − 1 m tpt ·„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„¶ Term Premium (34.131) In other words, the yield on a m period bond is approximately the average of current and expected one period
bonds over the life of the m period bond (the expectations hypothesis component), plus a term premium component that is increasing in m. One can see that (34.111) and (34.130) are special cases of (34.131) when m = 2 or m = 3. As m gets big, the term m−1 m should settle down and approach one. Hence, the term premium should be positive, increasing in time to maturity (i.e. m−1 m is increasing in m), but increasing at a decreasing 817 rate (i.e. m−1 m settles down to one as m gets big). Put slightly diﬀerently, the diﬀerence in term premia for a two versus three period bond ought to be bigger than the diﬀerence in term premia between twenty and twenty-one period bond. In Figure 34.13 below, we postulate an expected sequence of one period interest rates over a 30 period horizon. We assume that the value of the tp term is ﬁxed at 0.8. We then use the postulated sequence of short term interest rates, the assumed value of the tp term, and (34.131) to measure yields on bonds ranging in maturity from one period to m = 30. In the upper row, we consider a situation in which the current and expected short term yields are constant at 4.5 percent. The yield curve is nevertheless upward-sloping, because, as show in the upper right panel, the term premium is positive for maturities greater than one period and increasing in the maturity. The shape of the implied yield curve in the upper left quadrant is roughly consistent with the typical yield curve observed in the data. Figure 34.13: The Yield Curve and the Term Premium In the lower row we postulate a diﬀerent path for the short term yield. In particular, we assume that it is constant at 4.5 for four periods. Then it drops to 2.5 in the ﬁfth period. 818 01020304.44.64.855.25.4010203000.20.40.60.801020302.533.544.555.5010203000.20.40.60.8 Thereafter it smoothly approaches the value of 4.5 We observe that this expectation of falling short term yields does in fact result in the yield curve inverting – though upward-sl
oping for a few periods, with yields on bonds with maturities greater than 5 periods less than the current one period yield. This inversion is roughly consistent with what is observed in the data immediately prior to most recessions. 34.5 Conventional versus Unconventional Monetary Policy Although this chapter is about bond pricing, we conclude it with a brief discussion of monetary policy. As discussed in Chapter 32, conventional monetary policy works through bond markets. Central banks buy or sell short term government debt. This buying or selling impacts the prices (yields) of short term government debt and ultimately spills over to the prices (yields) on debt instruments relevant for economic activity. We have studied a micro-founded general equilibrium model with both short and long term riskless debt as well as a model with both riskless and risky debt. We have not yet studied these together. The next subsection does so, though much of the analysis is repetitive with what has already been presented in this chapter and may be skipped. We brieﬂy summarize the key points here. Most private investment is ﬁnanced with long term debt (think about a 30 year mortgage for a household purchasing a home, or a 10 year corporate bond for a ﬁrm looking to ﬁnance a new factory). The reason this debt is mostly long term is because the underlying projects take a long time to generate signiﬁcant cash ﬂows. But if interest rates on longer term risky debt are what is relevant for economic activity, how does monetary policy, which typically inﬂuences short term, riskless interest rates, impact the economy? From a saver’s perspective, the key point is that bonds of diﬀerent types (either diﬀering in maturity or default risk) are substitutes. Purchasing bonds is simply a means by which to transfer resources intertemporally and hence to smooth consumption. In fact, once one adjusts for risk (potentially both default risk and maturity/duration risk), diﬀerent types of bonds are perfect substitutes. This means that the prices/yields on diﬀerent types of bonds are intimately related to one another. Conventional monetary policy works through adjusting short term, riskless interest rates. Because of the substitutability among diﬀerent types of debt, this in turn ﬁlters through to the longer maturity and risky rates relevant for important economic decisions
. Facing the ZLB post-2008, conventional policies were unavailable to central banks around the world since short term, riskless (nominal) interest rates were bound from below by zero. These central banks therefore resorted to unconventional policy actions which sought to impact the economically-relevant interest rates through means other than adjusting short term riskless 819 rates. The next subsection provides a formal model with three periods. It allows for both short and long term riskless government debt as well as long maturity debt which has default risk. We can think about the latter as corporate debt, a mortgage rate, or a corporate borrowing rate. It is the economically relevant interest rate for investment decisions. We then formally derive conditions that relate the prices (yields) on diﬀerent types of debt together. We conclude the section with a graphical discussion of conventional versus unconventional policy measures designed to impact economically-relevant interest rates. We should note here that we are presenting all of this material in a purely real model. Monetary policy requires some kind of nominal friction (either price or wage stickiness) to have real eﬀects. Furthermore, for monetary policy to have real eﬀects, output must be endogenous, whereas we are working in the conﬁnes of an endowment economy model in which output is exogenous. At the expense of signiﬁcantly complicating the analysis, we could modify our framework to account for these issues without fundamentally altering any of the conclusions. In the interest of transparency and brevity, we will not do so, but we do wish to point this issue out before proceeding further. 34.5.1 A Model with Short and Long Term Riskless Debt and Long Term Risky Debt Suppose that there is a representative household who lives for three periods – t, t + 1, and t + 2. As above, assume for simplicity that the household earns an exogenous income stream. It is potentially unknown in t + 1 and t + 2 from the perspective of period t. With its exogenous resource ﬂow in t, the household can consume, save/borrow through one period government (riskless) debt, two period government (riskless) debt, or two period private (risky) debt. The ﬂow budget constraint facing the household is: (34.132) Ct + P B t,t,t+1Bt,t,t+1 + P B t,t,t+2BR
t,t,t+2 ≤ Yt t,t,t+2Bt,t,t+2 + P RB Bt,t,t+1 denotes the stock of one period riskless bonds the household takes from t to t + 1. These bonds pay out one in t + 1 with certainty. Bt,t,t+2 is the stock of two period bonds the household purchases; if held to maturity, they pay out one with certainty in t + 2. BRt,t,t+2 is the stock of private, risky bonds that the household purchases in t. If held until t + 2, these bonds either payout one or zero (i.e. the bond issuer defaults). We discuss the nature of default risk below. The prices of the three bonds are P B Suppose that income in period t + 1 can be Y h t+1 with probability 1 − p. Use an h or l subscript to denote high or low realizations in t + 1. A ﬂow t,t,t+2, and P RB t,t,t+2. t+1 with probability p and Y l t+1 t,t,t+1, P B ≤ Y h 820 budget constraint must hold in t + 1 regardless of the realization of uncertainty. The budget constraints are: C h t+1 C l t+1 + P B,h t+1,t+1,t+2Bh t+1,t+1,t+2 + P B,h t+1,t,t+2 (Bh t+1,t,t+2 − Bt,t,t+2) + P RB,h t+1,t,t+2 (BRh t+1,t,t+2 − BRt,t,t+2) ≤ Y h t+1 + Bt,t,t+1 (34.133) + P B,l t+1,t+1,t+2Bl t+1,t+1,t+2 + P B,l (Bl t+1,t,t+2 P RB,l t+1,t,t+2 (BRl t+1,t,t+2 − Bt,t,t+2) + t+1,t,t+2 − BRt,t,t+2) ≤ Y l t+
1 + Bt,t,t+1 (34.134) In t + 1, regardless of the state of nature, the household can consume, accumulate newly issued one period government bonds, or buy/sell previously issued two period government or private bonds. In period t + 2, assume once again that income can be high or low with the same probabilities p and 1 − p, respectively. The realization of income in t + 2 is independent of the realization of income in t + 1. To make things as clean as possible, assume that the risky bond defaults with 100 percent probability if income is low in t + 2. Otherwise it generates one unit of income for the houehold. Because there are two possible states in t + 2 and two in t + 1, and what happens in t + 1 is potentially relevant for t + 2, there are eﬀectively four possible states in t + 2. Denote these with (h, h) for example, where the ﬁrst h denotes the t + 2 income state and the second entry corresponds to the state of nature in t + 1. In period t + 2, the household receives an exogenous income ﬂow plus payouts from bonds held between t + 1 and t + 2. The four budget constraints that must hold are: C h,h t+2 C h,l t+2 ≤ Y h t+2 ≤ Y h t+2 + Bh t+1,t+1,t+2 + Bh t+1,t,t+2 + BRh t+1,t,t+2 + Bl t+1,t+1,t+2 + Bl t+1,t,t+2 + BRl t+1,t,t+2 C l,l t+2 C l,h t+2 ≤ Y l t+2 ≤ Y l t+2 + Bl t+1,t+1,t+2 + Bl t+1,t,t+2 + Bh t+1,t+1,t+2 + Bh t+1,t,t+2 (34.135) (34.136) (34.137) (34.138) In (34.135)-(34.138), government bonds payout one with certainty regardless of whether income is high or low. The risky bond only pays if the household’s endowment of income is high; otherwise it
defauls and generates no income for the household. Hence, BRt+1,t,t+2 does not appear in (34.137)-(34.138), which correspond to the low income state in t + 2. Expected utility for the household is a discounted expected sum of ﬂow utilities across 821 time and states of nature: U = u(Ct) + β[pu(C h t+1 ) + (1 − p)u(C l t+1 )]+ β2[p2u(C h,h t+2 ) + p(1 − p)u(C h,l t+2 ) + (1 − p)2u(C l,l t+2 ) + (1 − p)pu(C l,h t+2 )] (34.139) The household’s objective is to maximize (34.139) subject to (34.132)-(34.138). It is once again easiest to transform this into an unconstrained problem of choosing bond holdings. Once one does so, the ﬁrst order optimality conditions may be written: Bt,t,t+1 ∶ P B t,t,t+1u′(Ct) = β[pu′(C h t+1 ) + (1 − p)u′(C l t+1 )] Bt,t,t+2 ∶ P B t,t,t+2u′(Ct) = β[pP B,h t+1,t,t+2u′(C h t+1 ) + (1 − p)P B,h t+1,t,t+2u′(C l t+1 )] (34.140) (34.141) BRt,t,t+2 ∶ P RB t,t,t+2u′(Ct) = β[pP RB,h t+1,t,t+2u′(C h t+1 ) + (1 − p)P RB,l t+1,t,t+2u′(C l t+1 )] (34.142) Bh t+1,t+1,t+2 ∶ βpP h t+1,t+1,t+2u′(C h t+1 ) = β2[p2u′(C
h,h t+2 ) + p(1 − p)u′(C h,l t+2 ) + (1 − p)2u′(C l,l t+2 ) + (1 − p)pu′(C l,h t+2 )] (34.143) Bl t+1,t+1,t+2 ∶ β(1 − p)P l ) = t+1,t+1,t+2u′(C l t+1 ) + p(1 − p)u′(C h,l t+2 β2[p2u′(C h,h t+2 ) + (1 − p)2u′(C l,l t+2 ) + (1 − p)pu′(C l,h t+2 )] (34.144) Bh t+1,t,t+2 ∶ βpP h t+1,t,t+2u′(C h t+1 ) = β2[p2u′(C h,h t+2 ) + p(1 − p)u′(C h,l t+2 ) + (1 − p)2u′(C l,l t+2 ) + (1 − p)pu′(C l,h t+2 )] (34.145) Bl t+1,t,t+2 ∶ β(1 − p)P l ) = t+1,t,t+2u′(C l t+1 ) + p(1 − p)u′(C h,l t+2 β2[p2u′(C h,h t+2 ) + (1 − p)2u′(C l,l t+2 ) + (1 − p)pu′(C l,h t+2 )] (34.146) BRh t+1,t,t+2 ∶ βpP Rh t+1,t,t+2u′(C h t+1 ) = β2[p2u′(C h,h t+2 ) + p(1 − p)u′(C h,l t+2 )] (34.147) BRl t+1,t,t+2 ∶ β(1 − p)P Rl t+1,t,t+2u
′(C l t+1 ) = β2[p2u′(C h,h t+2 ) + p(1 − p)u′(C h,l t+2 )] (34.148) 822 In terms of expectations operators, (34.140)-(34.142) are simply: t,t,t+1u′(Ct) = β E[u′(Ct+1)] P B t,t,t+2u′(Ct) = β E[P B P B t,t,t+2u′(Ct) = β E[P RB P RB t+1,t,t+2u′(Ct+1)] t+1,t,t+2u′(Ct+1)] (34.149) (34.150) (34.151) (34.149)-(34.151) have familiar intuitive interpretations based on work we have already done. In conjunction with the market-clearing condition that consumption equals income, these conditions determine the equilibrium bond prices. Via exactly the same arguments we made above, (34.149)-(34.150) can be combined and manipulated to relate the price of the two period riskless bond to the product of current and expected one period riskless bond prices with a term premium term: P B t,t,t+2 = P B t,t,t+1 E[Pt+1,t+1,t+2] (1 + cov(P B E[P B t+1,t+1,t+2, u′(Yt+1)) ] E[u′(Yt+1)] t+1,t+1,t+2 ) (34.152) To simplify notation, deﬁne the term in parentheses as T Pt and write (34.152) as: P B t,t,t+2 = P B t,t,t+1 E[Pt+1,t+1,t+2]T Pt (34.153) Next, turn to the pricing condition for the risky bond. In particular, combine (34.151) with (34.150) to get: P RB t,t,t+2 = P B t,t,t+2 E[P RB E[P B t+1,t,t+2u′(Yt+1
)] t+1,t,t+2u′(Yt+1)] (34.154) From (34.154), we can see that the price of risky long term debt is related to the price of long term riskless debt and another term related to future prices of risky and riskless government debt. To simplify this other term, note that if one adds (34.143)-(34.144) together, imposes the market-clearing condition that consumption equals income, and notes that P B t+1,t+1,t+2 (i.e. the price of the riskless debt depends only on remaining time to maturity, not date of issuance) one gets: = P B t+1,t,t+2 E[P B t+1,t,t+2u′(Yt+1)] = β E[u′(Yt+2)] (34.155) We can do something similar for the denominator in (34.154). Deﬁne Dt+2 as the payout t+2 and Dt+2 = 0 if t+2. With this new notation, one can add (34.147)-(34.148) together to get the analog on the risky bond in period t + 2. As discussed above, Dt+2 = 1 if Yt+2 = Y h Yt+2 = Y l of (34.155): 823 E[P RB t+1,t,t+2u′(Yt+1)] = β E[Dt+2u′(Yt+2)] (34.156) Now substitute (34.156) and (34.155) into (34.154) to get: P RB t,t,t+2 = P B t,t,t+2 E[Dt+2u′(Yt+2)] E[u′(Yt+2)] Multiply and divide the right hand side of (34.157) by E[Dt+2] to get: P RB t,t,t+2 = P B t,t,t+2 E[Dt+2] E[Dt+2u′(Yt+2)] E[Dt+2] E[u′(Yt+2)] (34.157) (34.158) One would be tempted to distribute the expectations
operator in the numerator and hence cancel out the fraction in (34.158), but as we have seen again and again, in general this is not possible. Rather, we can use the by-now-familiar relationship between covariance and expectations operators to write: P RB t,t,t+2 = P B t,t,t+2 E[Dt+2] (1 + cov(Dt+2, u′(Yt+2)) E[Dt+2] E[u′(Yt+2)] ) (34.159) (Yt+2)) (Yt+2)] In (34.159), the term (1 + cov(Dt+2,u′ E[Dt+2] E[u′ ) is a risk premium. Given our assumptions, this covariance term is negative. When income is high in t + 2, u′(Yt+2) is low and Dt+2 is higher than average. Conversely, when income is low in t + 2, u′(Yt+2) is higher than average and Dt+2 is low. This negative covariance means that the term in parentheses is less than one and that the risky bond will trade at a discount compared to the two period riskless bond (equivalently, it will demand a higher yield). So as to economize on notation, deﬁne RPt = (1 + cov(Dt+2,u′ E[Dt+2] E[u′ ) and write: (Yt+2)) (Yt+2)] P RB t,t,t+2 = P B t,t,t+2 E[Dt+2]RPt (34.160) (34.160) and (34.153) are the keys to understanding the transmission of monetary policy into the interest rates relevant for economic activity. The key point is that risky bond prices (yields) depend on longer maturity riskless bond prices (yields), which in turn depend on short term riskless bond prices (yields). Conventional monetary policy operates by increasing/decreasing the demand for short term riskless government bonds, and therefore impacting the price (yield) of such debt. Through (34.153) and (34.160), changes in the price (yield) of short run riskless debt ﬁlter
through to the price (yield) on longer maturity risky debt, which is what is relevant for the decision-making of households and ﬁrms. We can employ a simple graphical apparatus based on supply and demand curves to think about how the price (yield) of longer maturity risky debt is ultimately determined. Consider 824 Figure 34.14. Suppose that the supply of both short term and long term government debt are exogenously set by the (unmodeled) ﬁscal authority; hence the supply curves are vertical. There are downward-sloping demand curves for short term debt in both t and in expectation for t + 1 – i.e. the lower is the price (higher is the yield), the more the household would like to save, and hence the more bonds it wants to buy. The intersection of demand and supply determines P B ]. t,t,t+1 and E[P B t+1,t+1,t+2 Figure 34.14: The Markets for Short and Long Term Government Bonds The demand and supply curves for short term riskless debt (in both the present and future) are shown in the upper part of Figure 34.14. The demand for long term riskless debt, shown in the bottom part of the ﬁgure, is slightly diﬀerent. In particular, the demand for 825 𝑃𝑃𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵 𝔼𝔼�𝑃𝑃𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵� 𝑃𝑃𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵 𝐵𝐵𝑠𝑠 𝐵𝐵𝑠𝑠 𝐵𝐵𝑠𝑠 𝐵𝐵𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2 𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+1 𝐵𝐵�
��𝑡,𝑡𝑡,𝑡𝑡+2 𝐵𝐵𝑑𝑑 𝐵𝐵𝑑𝑑 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵 𝔼𝔼�𝑃𝑃0,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵� 𝐵𝐵𝑑𝑑 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵=𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵𝔼𝔼�𝑃𝑃0,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵�𝑇𝑇𝑃𝑃𝑡𝑡 long term debt is perfectly elastic at the price given in (34.153). If the price of long term debt were below this, there would be inﬁnite demand for long run bonds – the household would like to buy long term bonds and ﬁnance them by borrowing via short term bonds. The reverse would be true if the price of long term debt were above this. As a consequence, the demand curve for long term debt is perfectly horizontal at (34.153). This price depends upon the equilibrium prices of short term government debt (both in t as well as in expectation in t + 1), determined in the upper part of Figure 34.14, along with the term premium, which we shall take as given. Although we have not included production into our model, we can think about there being a representative ﬁrm that needs to ﬁnance its capital accumulation by issuing long term debt. The ﬁrm will want to do more investment, and hence issue more debt, the lower is the interest rate on that debt. Since the interest rate (yield) on
debt is inversely related to the price of debt, we can think about there being an upward-sloping supply curve of risky debt. The higher is the price of such debt, the cheaper it is for the ﬁrm to raise funds, and hence the more such debt it supplies. This is shown in Figure 34.15 below. Figure 34.15: The Market for Long Term Risky Bonds The demand curve for risky debt is determined by the household, who supplies savings by buying bonds. Like the demand curve for long term riskless debt, the demand curve for risky debt is perfectly elastic at the price given in (34.160). If the price of risky debt were greater than this, there would be inﬁnite demand for such debt – the household would want to borrow through riskless debt and use the proceeds from that to purchase the risky debt. 826 𝐵𝐵𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2 𝐵𝐵𝑠𝑠 𝐵𝐵𝑑𝑑 𝑃𝑃𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵𝔼𝔼[𝐷𝐷𝑡𝑡+2]𝐵𝐵𝑃𝑃𝑡𝑡 𝐵𝐵𝐵𝐵0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2 As such, the demand curve must be horizontal at (34.160). The position of this demand curve depends upon the price (yield) of long term riskless debt, the expected payout on the bond, and the risk premium. Because changes in short term riskless bond prices inﬂuence long term riskless bond prices, changes in short term yields on government debt should ﬁlter through to the yield (price) on longer term risky debt. 34.5.2 Conventional Monetary Policy We are now in a position to use
these graphs to think about the conventional channels through which central banks impact the interest rates relevant for households and ﬁrms. As discussed in Chapter 32, when a central bank wishes to increase the money supply it conducts an open market purchase. In particular, it buys government debt and ﬁnances this purchase with the creation of reserves. The reserves can then ﬁlter through to the money supply through the usual multiple deposit creation channel. Conventional open market operations deal in short term government debt (i.e. T-Bills in the US). When a central bank decides to conduct an open market purchase, it eﬀectively increases the demand for short term bonds. This is shown in Figure 34.16. The increase in the demand for short term government debt pushes up the price of such debt (i.e. lowers the yield). This then translates into more demand, and hence a higher price, of long term government debt. This is shown in the lower panel of Figure 34.16. 827 Figure 34.16: Conventional Monetary Policy: Open Market Purchase The higher price (lower yield) of long term government debt results in an increase in the demand for risky debt. Because P B t,t,t+2 is higher, the demand curve for risky bonds shifts up. This results in an increase in the price of risky debt and a decline in yield. The lower yield stimulates bond issuance and we move up the upward-sloping supply curve in Figure 34.17. As a consequence, there is more debt issuance and more investment. 828 𝑃𝑃𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵 𝔼𝔼�𝑃𝑃𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵� 𝑃𝑃𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵 𝐵𝐵𝑠𝑠 𝐵𝐵𝑠𝑠 𝐵𝐵𝑠𝑠 𝐵𝐵𝑡𝑡+1,𝑡𝑡+1,𝑡�
�+2 𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+1 𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2 𝐵𝐵𝑑𝑑 𝐵𝐵𝑑𝑑 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵 𝔼𝔼�𝑃𝑃0,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵� 𝐵𝐵𝑑𝑑 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵=𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵𝔼𝔼�𝑃𝑃0,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵�𝑇𝑇𝑃𝑃𝑡𝑡 𝐵𝐵𝑑𝑑′ 𝑃𝑃1,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵 𝑃𝑃1,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵=𝑃𝑃1,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵𝔼𝔼�𝑃𝑃0,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵�𝑇𝑇𝑃𝑃
𝑡𝑡 𝐵𝐵𝑑𝑑′ Figure 34.17: Conventional Monetary Policy: Impact on Market for Risky Long Term Debt Figures 34.16 and 34.17 describe the ordinary, or conventional, workings of monetary policy. Open market sales work in the opposite direction, resulting in lower prices of risky bonds and consequently higher yields and less investment. 34.5.3 Unconventional Policy The Federal Reserve in the US and other central banks around the world resorted to unconventional policies when the zero lower bound (ZLB) on nominal interest rates began to bind in late 2008. Although we have not explicitly incorporated or mentioned a ZLB constraint in the demand-supply graphs of this section, one could think about the demand for short term government debt becoming perfectly elastic (horizontal) at some upper bound, implying a lower bound on the yield on such debt. We show such a situation in Figure 34.18: 829 𝐵𝐵𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2 𝐵𝐵𝑠𝑠 𝐵𝐵𝑑𝑑 𝑃𝑃𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵𝔼𝔼[𝐷𝐷𝑡𝑡+2]𝐵𝐵𝑃𝑃𝑡𝑡 𝐵𝐵𝐵𝐵0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2 𝑃𝑃1,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵𝔼𝔼[𝐷𝐷𝑡𝑡+2]𝐵𝐵𝑃𝑃𝑡𝑡 𝐵𝐵𝑑𝑑′ 𝐵�
��𝐵𝐵1,𝑡𝑡,𝑡𝑡,𝑡𝑡+2 Figure 34.18: The Zero Lower Bound and the Market for Short Term Riskless Bonds If the demand curve intersects the supply curve in the ﬂat region of the demand curve, it is not possible to adjust the equilibrium yield on short term riskless debt through open market operations. Ultimately, however, short term riskless interest rates are not what is relevant for economic activity. Conventional monetary policy seeks to adjust such rates so as to inﬂuence longer term, risky interest rates. When the ZLB began to bind, the Federal Reserve and other central banks resorted to alternative policies designed to more directly inﬂuence the interest rates relevant for economic activity (whereas conventional monetary policy indirectly inﬂuences these rates because bonds of diﬀerent characteristics are substitutes). The two principal means of unconventional monetary policy were forward guidance and quantitative easing (also sometimes called large scale asset purchases, or LSAP). Forward guidance involves a central bank telegraphing its intentions for future short term interest rates. In the context of the model with which we have been working, it is easy to see why forward guidance might work. If long term risky yields depend on long term riskless yields, and long term riskless yields in turn depend on the current and expected sequence of short term yields, then promising lower future short term yields ought to result in lower long term yields in the present, which then ought to ﬁlter through to lower interest rates on risky debt. Of course, this will only work to the extent to which the public believes that the central bank will follow through on its promises, so credibility is important. Quantitative easing involves purchasing, in large amounts, non-traditional securities, such as longer term government debt or private-sector risky debt. The basic idea behind quantitative easing is straightforward – if one can increase the demand for these types of debt, one ought to be able to raise the 830 𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+1 𝑃𝑃𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵 𝐵𝐵𝑠𝑠 𝐵�
��𝑑𝑑 𝑃𝑃𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝑢𝑢𝑢𝑢,𝐵𝐵 price of such debt, and hence lower the yield. While such logic seems straightforward and unassailable, quantitative easing should not work in the type of model we have laid out. We can see this above. Because long term bonds (either risky or otherwise) are substitutable with short term riskless bonds, the demand for long term bonds is perfectly elastic, and it should not be possible to inﬂuence the price (yield) of such debt without aﬀecting either the sequence of short term yields or the risk or term premia. Figures 34.19-34.20 use our graphical demand-supply analysis to think about how forward guidance might work. Credibly promising to increase the demand for short term debt in the future (right panel of upper row of Figure 34.19) ought to increase the expected price of such debt (equivalently, lower the anticipated yield). This should result in an immediate increase in the price of long term debt (lower panel of Figure 34.19). The increased price of long term riskless debt ought to translate into a higher price of risky long term debt (Figure 34.20). The lower yield on such debt ought to simulate debt issuance and hence investment. 831 Figure 34.19: Unconventional Monetary Policy: Forward Guidance 832 𝑃𝑃𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵 𝔼𝔼�𝑃𝑃𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵� 𝑃𝑃𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵 𝐵𝐵𝑠𝑠 𝐵𝐵𝑠𝑠 𝐵𝐵𝑠𝑠 𝐵𝐵𝑡𝑡+1,𝑡𝑡+1,𝑡�
�+2 𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+1 𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2 𝐵𝐵𝑑𝑑 𝐵𝐵𝑑𝑑 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵 𝔼𝔼�𝑃𝑃0,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵� 𝐵𝐵𝑑𝑑 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵=𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵𝔼𝔼�𝑃𝑃0,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵�𝑇𝑇𝑃𝑃𝑡𝑡 𝐵𝐵𝑑𝑑′ 𝔼𝔼�𝑃𝑃1,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵� 𝑃𝑃1,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵=𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵𝔼𝔼�𝑃𝑃1,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵�
𝑇𝑇𝑃𝑃𝑡𝑡 𝐵𝐵𝑑𝑑′ Figure 34.20: Forward Guidance and the Market for Risky Long Term Debt In a sense, forward guidance in theory ought to work similarly to conventional policy. What is unconventional about it is that a central bank is hoping to alter expectations of future short term riskless yields rather than impacting current riskless yields. Quantitative easing is rather diﬀerent. While it involves trying to purchase bonds, it involves either purchasing long maturity riskless debt or privately-issued risky debt. As noted above, the idea is that by increasing the demand for such debt, the prices should rise and yields should fall. But this logic ignores the fact that long term bonds (either risky or riskless) are substitutes with short term debt. This substitutability pins down the prices of long term debt via (34.151) or (34.160). Without any change in the current or expected sequence of short term, riskless bond yields, the only way to inﬂuence the prices of long term debt (either risky or riskless) would be to impact the term or risk premia. These terms depend on the covariance of bond prices with output, and it is not clear how or why large scale asset purchases ought to be able to impact them. Because of this, former Fed chairman Ben Bernanke famously quipped “The problem with quantitative easing is that it works in practice but not in theory.” Quantitative easing in the US took two forms. In QE1 (Fall of 2008 throughout 2009) and QE3 (Fall of 2012 through 2014) the Fed purchased mortgage backed securities. In QE2 (November 2010), and at the tail end of QE1, the Fed also purchased longer maturity Treasury securities. In the context of the simple model we have laid out here, we can think about QE2 as trying to increase the demand for two period riskless debt, while QE1 and QE3 involved trying to increase the demand for private-issued risky debt. 833 𝐵𝐵𝐵𝐵,𝑡𝑡,𝑡𝑡+2 𝐵𝐵𝑠𝑠 𝐵𝐵𝑑𝑑 𝑃𝑃𝐵𝐵
𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵𝔼𝔼[𝐷𝐷𝑡𝑡+2]𝐵𝐵𝑃𝑃𝑡𝑡 𝐵𝐵𝐵𝐵0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2 𝑃𝑃1,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵𝔼𝔼[𝐷𝐷𝑡𝑡+2]𝐵𝐵𝑃𝑃𝑡𝑡 𝐵𝐵𝑑𝑑′ 𝐵𝐵𝐵𝐵1,𝑡𝑡,𝑡𝑡,𝑡𝑡+2 Figures 34.21-34.22 use our graphical demand-supply analysis to think about quantitative easing involving government securities. Because the demand curve for long term debt is perfectly elastic, absent a change in the term premium there is no change in the price (yield) of long term riskless debt, and hence no change in the price (yield) of long term risky debt (Figure 34.22). Figure 34.21: Unconventional Monetary Policy: Quantitative Easing, Government Securities 834 𝑃𝑃𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵 𝔼𝔼�𝑃𝑃𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵� 𝑃𝑃𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵 𝐵𝐵𝑠𝑠 𝐵𝐵
𝑠𝑠 𝐵𝐵𝑠𝑠 𝐵𝐵𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2 𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+1 𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2 𝐵𝐵𝑑𝑑 𝐵𝐵𝑑𝑑 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵 𝔼𝔼�𝑃𝑃0,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵� 𝐵𝐵𝑑𝑑=𝐵𝐵𝑑𝑑′ 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵=𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+1𝐵𝐵𝔼𝔼�𝑃𝑃0,𝑡𝑡+1,𝑡𝑡+1,𝑡𝑡+2𝐵𝐵�𝑇𝑇𝑃𝑃𝑡𝑡 Figure 34.22: Quantitative Easing (Government Securities) and Market for Risky Long Term Debt Figure 34.23 focuses on the market for privately-issued risky debt. Again, absent a change in the risk premium, it ought not to be possible to inﬂuence the price (yield) of risky debt by simply trying to buy more. In eﬀect, a central bank stimulating demand for such debt ought to cause a reduction in household demand for such debt, which in equilibrium results in the price (yield) being unchanged. 835 𝐵
𝐵𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2 𝐵𝐵𝑠𝑠 𝐵𝐵𝑑𝑑=𝐵𝐵𝑑𝑑′ 𝑃𝑃𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵𝔼𝔼[𝐷𝐷𝑡𝑡+2]𝐵𝐵𝑃𝑃𝑡𝑡 𝐵𝐵𝐵𝐵0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2 Figure 34.23: Quantitative Easing (Risky Long Term Debt) Under what theoretical conditions might quantitative easing work, as Ben Bernanke evidently believes it does in practice? What makes quantitative easing impotent in theory is that we have assumed that bonds of diﬀerent characteristics are perfect substitutes. Once one controls for risk (i.e. the term and risk premia), bond yields must be equalized across diﬀerent types of debt. A simple way to break this tight connection is to drop the assumption that bonds of diﬀerent characteristics are perfectly substitutable. Segmented markets theory instead assumes that bonds of diﬀerent characteristics are not substitutes at all. Some households prefer short term government debt but not the other two kinds of debt in our model; others are diﬀerent. If this is the case, the demand curves for all diﬀerent types of debt are downward-sloping and their prices (yields) are not intimately related. Increasing the demand for one kind of debt has no impact on the demands for other types of debt. The astute reader may note that the Federal Reserve was in essence hedging its bets in deploying both forward guidance and quantitative easing as unconventional policies. Under our standard theory laid out in this chapter, forward guidance (if credibly done) ought to be successful in inﬂu
encing economically relevant interest rates, but quantitative easing should not. Under segmented markets, quantitative easing could work in theory, but forward guidance should not be able to work. Why? If bonds of diﬀerent types are not substitutes, then the price (yield) of risky debt is unrelated to the price (yield) of riskless debt, and promising low interest rates on short term riskless debt into the far oﬀ future should not have any impact on yields on risky debt. 836 𝐵𝐵𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2 𝐵𝐵𝑠𝑠 𝐵𝐵𝑑𝑑=𝐵𝐵𝑑𝑑′ 𝑃𝑃𝐵𝐵𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵 𝑃𝑃0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2𝐵𝐵𝔼𝔼[𝐷𝐷𝑡𝑡+2]𝐵𝐵𝑃𝑃𝑡𝑡 𝐵𝐵𝐵𝐵0,𝑡𝑡,𝑡𝑡,𝑡𝑡+2 34.6 Summary • A bond is a type of security which entitles the holder to periodic cash ﬂows until maturity. Bonds diﬀer in their time to maturity, default risk, and cash ﬂow payments. • Interest rates on bonds are deﬁned implicitly. The most common measure of interest rates on binds is the yield to maturity (YTM). The YTM is the interest rate that equates the price of the bond to the expected present discounted value of cashﬂows coming from the bond. • There is an inverse relationship between a bond’s yield and its price. • We can formalize bond pricing within the context of a two-period general equilibrium model in an endowment economy. If bonds are in zero net supply, bond prices adjust so that households always consume their endowment in equilibrium.
• The existence and size of the risk premium depend on how payments from a risky bond covary with future income. If the bond payments are comparatively low when future resources are low, then the risky bond will need to pay a premium. • Empirical measures of the risk premium show that it is positive and, if anything, increasing over time. • Yields on long maturity debt are almost always higher than yields on short maturity debt. The exception is the time immediately prior to a recession where the yield curve ﬂattens or even inverts. • Decisions by households and corporations are often made on the basis of long term, risky yields. Conventional monetary policy involves buying and selling risk free, short run bonds. Through a term structure channel, the purchase of the short run riskless bonds can aﬀect the yields on longer maturity and riskier bonds. • Unconventional monetary policy can take the for of forward guidance, where the monetary policy maker signals its intentions over future short run interest rates, or quantitative easing, where the monetary policy maker purchases longer term and/or riskier bonds. Under our benchmark theory, only the former should inﬂuence yields on longer maturity debt. Key Terms 1. Bond 837 2. Maturity date 3. Default risk 4. Interest rate risk 5. Yield 6. Risk structure 7. Term structure 8. Discount bond 9. Coupon bond 10. Perpetuity bond 11. Expectations hypothesis 12. Interest rate risk 13. Forward guidance 14. Quantitative easing 15. Segmented markets theory Questions for Review 1. Why is it more common to refer to a bond’s yield rather than its price? 2. Does the coupon rate always equal a bond’s yield to maturity? Explain. 3. True or false: All risky bonds pay a premium relative to risk free bonds. Explain. 4. Discuss what the expectations hypothesis gets right and what it gets wrong. 5. Explain the transmission mechanism of conventional monetary policy. 6. Under what circumstances will forward guidance be eﬀective? Under what circumstances will quantitative easing be eﬀective? 838 Chapter 35 The Stock Market and Bubbles A stock is a type of ﬁnancial security that is sometimes called an equity. It is called an equity because an owner of a share of stock is an owner in the company issuing the stock. Shareholders are entitled to the current and future proﬁts generated by the company.
Like bonds, which are discussed in Chapter 34, a share of stock entitles the holder of the stock to periodic cash ﬂows. These cash ﬂows are called dividends and constitute distributed proﬁts to the owners of a company. As with most bonds, there is a highly liquid and active secondary market for shares of stock, so the holder of a share of stock can trade his/her shares. There are a couple of important diﬀerences between stocks and bonds, all of which generally make stocks riskier than bonds. First, in the event of a company’s failure, stockholders are junior claimants on the company’s assets – they only get their funds back after all debt holders have been paid. This exposes stockholders to more risk than bondholders in the event of a company’s failure. Secondly, whereas bonds oﬀer known cash ﬂows in the form of coupon payments and/or face value repayments (outside of default), the periodic cash ﬂows from stocks are unknown. Dividends can vary substantially, and are often quite procyclical – i.e. dividends are comparatively high when the economy is booming and low otherwise. This co-movement is undesirable from the perspective of a household wishing to smooth its consumption. Third, whereas bonds typically have ﬁnite maturities (and many types of bonds are very short maturity), stocks have no maturity. This makes them inherently riskier than bonds. For example, suppose that a household that is 55 years old wishes to save for retirement at 65. By matching maturity to the investment horizon (i.e. purchasing ten year bonds), a household can lock in an expected return by investing in bonds (assuming no default risk). This is not possible for stocks. For all these reasons, stocks are generally thought to be riskier than bonds. Because of this heightened risk, stocks trade at a higher average rate of return than do short term riskless government bonds. Figure 35.1 below plots the equity premium, which we deﬁne as the diﬀerence between the realized one year return on the S&P 500 less the realized one year return on three month Treasury Bills. On average, the equity premium so deﬁned is between 6 and 7 percent and is quite volatile. 839 Figure 35.1: The Equity Premium Figure 35.2 plots the time series of the Russell 3000, which is a total US
t. It is risk-free in the sense that the bond pays out its face value of 1 in period t + 1 as P B with certainty. The second issue is a share of stock in a ﬁrm, SHt. These shares trade at price Qt in period t. Shares of stock held in period t entitles the owner to a dividend payout of dt+1 per share in period t + 1. This dividend payout is not known with certainty in period t. The household’s period t ﬂow budget constraint is: Ct + P B t Bt + QtSHt ≤ Yt (35.1) ≥ Y l In period t + 1, there are two sources of uncertainty. First, the endowment of income in period t + 1 can either be high or low, with Y h t+1. Second, let us assume that the t+1 ≥ dl dividend payout on the stock can be high or low, with dh t+1. This means that there t+1 are four states of the world in t + 1 – think of these as (Y h ), t+1, dh t+1 t+1 t+1 ). The period t + 1 ﬂow budget constraint must hold in all four states oﬀ the and (Y l world. We will use double superscripts to denote the state of the world in t + 1 – (h, h) refers to both income and the dividend by high, whereas (h, l) refers to income being high and the dividend being low, and so on. These constraints are: t+1, dh t+1 ), (Y h t+1, dl t+1, dl ), (Y l C h,h t+1 C h,l t+1 + P B,h,h t+1 Bh,h t+1 t+1 Bh,l t+1 + P B,h,l + Qh,h t+1 + Qh,l t+1 (SH h,h t+1 (SH h,l t+1 − SHt+1) ≤ Y h t+1 − SHt+1) ≤ Y h t+1 + Bt + dh t+1SHt + Bt + dl t+1SHt (35.2) (35.3) 841 C l,l t+1 C l,h t+1 +
P B,l,l t+1 Bl,l t+1 t+1 Bl,h t+1 + Ql,l t+1 + Ql,h t+1 (SH l,l t+1 (SH l,h t+1 − SHt+1) ≤ Y l t+1 − SHt+1) ≤ Y l t+1 + Bt + dl t+1SHt + Bt + dh t+1SHt (35.4) (35.5) + P B,l,h Resources available to the household in period t + 1 include the exogenous income ﬂow, payouts on one period discount bonds brought from t to t + 1, and dividends received on shares of stock brought from t to t + 1. With these resources, the household can consume, purchase/sell new one period discount bonds, or change its stock of equity holdings. Since the household ceases to exist after t + 1, it will not choose to die with any positive stock of bonds or stock in any state of the world, and it may not die with negative stocks of these variables. Imposing these terminal conditions, the period t + 1 budget constraints may be written: C h,h t+1 C h,l t+1 C l,l t+1 C l,h t+1 ≤ Y h t+1 + Bt + (dh t+1 ≤ Y h t+1 + Bt + (dl t+1 ≤ Y l t+1 ≤ Y l t+1 + Bt + (dl t+1 + Bt + (dh t+1 + Qh,h t+1 + Qh,1 t+1 + Ql,l t+1 + Ql,h t+1 ) SHt ) SHt ) SHt ) SHt (35.6) (35.7) (35.8) (35.9) There are two potential sources of income from holding shares of stock, St. First, the stock makes a dividend payment in period t + 1, either dh t+1. Second, in principle one might be able to sell the stock for a price Qt+1 that depends on the realization of the state. We will say more about Qt+1 shortly. It is common to refer to these two components of the payout to a stock as the “dividend” component and the “capital gain�
� component, respectively. t+1 or dl Now let us address the nature of uncertainty facing the household. Let p1 be the probability of the ﬁrst state occurring in t + 1 (both income and the dividend are high), p2 be the probability of the second state (income is high, but the dividend is low), p3 be the probability of the third state (income is low and the dividend is low), and p4 = 1 − p1 − p2 − p3 be the probability of the ﬁnal state (income is low but the dividend is high). Period t + 1 consumption will depend on the realization of the state. We can write expected lifetime utility as: U = u(Ct) + p1βu(C h,h t+1 ) + p2βu(C h,l t+1 ) + p3βu(C l,l t+1 ) + (1 − p1 − p2 − p3)βu(C l,h t+1 ) (35.10) The household’s objective is to pick a consumption plan which maximizes (35.10) subject to (35.1) and (35.6)-(35.9). It is easiest to write the problem by eliminating the consumption 842 terms and instead think about the household as just picking Bt and SHt in period t. Doing so yields the following unconstrained problem: max Bt,SHt U = u[Yt−P B t Bt−QtSHt]+p1βu[Y h t+1 +Bt+(dh t+1 +Qh,h t+1 )SHt]+p2βu[Y h t+1 +Bt+(dl t+1 +Qh,l t+1 )SHt]+ p3βu[Y l t+1 + Bt + (dl t+1 + Ql,l t+1 )SHt] + (1 − p1 − p2 − p3)u[Y l t+1 + Bt + (dh t+1 + Ql,h t+1 )SHt] (35.11) The ﬁrst order conditions are: ∂U ∂Bt = 0 ⇔ P B t u′(Ct) = p1βu′(C h,h t+1 ) + p
2βu′(C h,l t+1 ) + p3βu′(C l,l t+1 ) + (1 − p1 − p2 − p3)βu′(C l,h t+1 (35.12) ) ∂U ∂SHt = 0 ⇔ Qtu′(Ct) = p1(dh t+1 + Qh,h t+1 )βu′(C h,h t+1 ) + p2(dl t+1 + Qh,l t+1 )βu′(C h,l t+1 )+ p3(dl t+1 + Ql,l t+1 )βu′(C l,l t+1 ) + (1 − p1 − p2 − p3)(dh t+1 + Ql,h t+1 )βu′(C l,h t+1 ) (35.13) (35.12)-(35.13) can be re-written in terms of expectations operators, since the right hand side of (35.12) is the expected value of the future marginal utility of consumption, while the right hand side of (35.13) is the expected value of the product of the future dividend with the future marginal utility of consumption. t u′(Ct) = β E [u′(Ct+1)] P B Qtu′(Ct) = β E [(dt+1 + Qt+1)u′(Ct+1)] (35.14) (35.15) Both (35.14) and (35.15) have intuitive marginal beneﬁt equals marginal cost interpretations. Purchasing an additional unit of the risk-free bond in t entails foregoing P B t units of t u′(Ct) is consumption in that period, which is valued in terms of utility at u′(Ct). Hence, P B the marginal utility cost of purchasing an additional unit of the bond. The marginal utility beneﬁt of purchasing an additional bond in t is an additional unit of income in t + 1, which is valued at β E [u′(Ct+1)] (i.e. the expected marginal utility of consumption). Hence, the right hand side of (35.14) is the marginal utility beneﬁt of purchasing an additional unit of the bond. At an optimum, the marginal utility
beneﬁt and cost must equal if the household is behaving optimally. The intuitive interpretation for why (35.15) must also hold is similar. Purchasing one unit of stock costs Qt units of consumption in period t, which is valued in terms of utility 843 at u′(Ct). Hence, Qtu′(Ct) is the marginal utility cost of purchasing stock. The beneﬁt of purchasing stock is the dividend to which the owner is entitled in t + 1 plus the share price of the stock in that period. This payout (dividend plus capital gain) is valued at the marginal utility of future consumption. In evaluating (35.15), it is important to note that E [(dt+1 + Qt+1)u′(Ct+1)] ≠ E [(dt+1 + Qt+1)] E [u′(Ct+1)] in general, as we shall see below. Both (35.14) and (35.15) can be written in such a way as to isolate the price of the asset on the left hand side. In so doing, we note that since u′(Ct) is known in period t, it can be moved “inside” the expectation operator on the right hand side or placed on the outside. The same thing holds true for β, which is a constant parameter. Doing so yields: P B t = E [β u′(Ct+1) u′(Ct) ] Qt = E [β u′(Ct+1) u′(Ct) (dt+1 + Qt+1)] (35.16) (35.17) In both (35.16) and (35.17), the price of the asset under consideration equals the expected value of the product of the cash ﬂows generated by the asset in question in t + 1 (1 for the risk-free bond, and (dt+1 + Qt+1) for the equity) with the term β u′, which is also known as the stochastic discount factor. For either the stock or the bond, the basic interpretation of the pricing condition is the same. (Ct+1) u′ (Ct) We can deﬁne the (gross) expected yield on each kind of asset (i.e. the expected rate of return) as the expected cash ﬂow generated by the asset in period t + 1 divided by the price paid for the asset
in period t. Let rt be the yield on the bond, and rs,t the yield on the stock. We get: 1 + rt = 1 P B t = 1 + rs,t = E(dt+1 + Qt+1) Qt = ] 1 E [β u′ (Ct+1) u′ (Ct) E(dt+1 + Qt+1) E [β u′ (Ct+1) u′ (Ct) (dt+1 + Qt+1)] (35.18) (35.19) The ratio of (gross) yields, which is approximately equal to one plus the diﬀerence between net yields, is: 1 + rs,t 1 + rt = E(dt+1 + Qt+1) E [β u′ E [(dt+1 + Qt+1)β u′ (Ct+1) u′ (Ct) (Ct+1) u′ (Ct) ] ] (35.20) One is tempted to look at (35.20) and conclude that the ratio of gross yields is 1, meaning rs,t = rt – i.e. the expected return on both the stock and the bond are the same. Indeed, 844 one might naturally expect an outcome such as this. From the household’s perspective, the bond and the stock are substitutes – they are both means by which to transfer resources intertemporally. If they are perfect substitutes, then in any equilibrium we would expect the expected returns to be equalized. But it turns out, that, in general, the stock and the bond are not perfect substitutes. While both securities are means by which to transfer resources intertemporally, they diﬀer in an important way. In particular, the bond generates one unit of income in period t + 1 with certainty. The share of stock, in contrast, generates an uncertain level of income in the future. To the extent to which the representative household dislikes such uncertainty, one might expect the household to demand compensation, in the form of a higher expected yield, to hold the risky asset. As it turns out, it is not uncertainty per se that might result in stocks oﬀering higher expected returns than risk-free bonds, but rather a particular form of uncertainty. In particular, the extent to which the expected yield for the stock diﬀers from the bond depends on how the payout from the stock co-varies
with u′(Ct+1). For the stock payout to co-vary with u′(Ct+1), the stock payout (as well as endowment income) must be uncertain. But this is only necessary, not suﬃcient, for stocks to have a diﬀerent yield than bonds. As was previously discussed in Chapter 34, for two random variables X and Y, E(XY ) ≠ E(X) E(Y ). In fact, one can show that E(XY ) = E(X) E(Y ) + cov(X, Y ). Only if X and Y are uncorrelated is the expectation of a product, i.e. E(XY ), equal to the product of the expected values, i.e. E(X) E(Y ). In this particular example, we can think about X as being the stochastic discount factor, β u′ (Ct+1), and Y as the cash ﬂow from holding the u′ (Ct) stock, dt+1 + Qt+1. Only when the cash ﬂow from holding the stock is uncorrelated with the stochastic discount factor will the risk-free bond and the stock have the same expected return. It is most reasonable to think that the cash ﬂow from the stock will be negatively correlated with the stochastic discount factor. In particular, it stands to reason that when dividends and stock prices are relatively high, so dt+1 + Qt+1 is relatively high, consumption will also be high, so that u′(Ct+1) will be low. If the cash ﬂow from the stock and the stochastic discount factor are negatively correlated, then E [(dt+1 + Qt+1)β u′ (Ct+1) ], u′ (Ct) and from (35.20) we should expected rs,t > rt. In other words, we should expect the stock to deliver a higher expected return than the bond. We will refer to this excess return of equity over a risk-free bond, rs,t − rt, as the equity premium. ] < E(dt+1 + Qt+1) E [β u′ (Ct+1) u′ (Ct) Having derived the optimality conditions and introduced some new terminology, we are now in a position to apply a market-clearing concept to solve for equilibrium prices and yields on
both the bond and the stock. As in Chapter 34, we are assuming that both the stock and the bond are in zero net supply (more generally assuming ﬁxed but non-zero supply 845 would yield identical results). This means that, in equilibrium, Bt = 0 and SHt = 0, so Ct = Yt. Furthermore, we can conclude that Qt+1 = 0 regardless of the realization of the state of nature in t + 1. Why is this? The stock is a claim on future cash ﬂows. But from the perspective of t + 1, there is no future, and so the asset ought to be worthless. This is a kind of terminal condition in its own right which is related to “bubbles,” and we shall return to it more below. Imposing these conditions, we arrive at expressions for equilibrium prices and yields of: Pt = E [β Qt = E [β 1 + rt = E [β ] u′(Yt+1) u′(Yt) u′(Yt+1) u′(Yt) dt+1] u′(Yt+1) u′(Yt) −1 ] 1 + rs,t = E[dt+1] [E [β −1 u′(Yt+1) u′(Yt) dt+1]] (35.21) (35.22) (35.23) (35.24) We are now in a position to give some values for future output, the future dividend, and the probabilities of future states, and then use those numbers to obtain numeric values for the prices of both assets as well as their relative yields (i.e. the equity premium). We do so in Table 35.1. Table 35.1: Uncertainty and the Equity Premium E(dt+1) E(dt+1 ∣ Y h t+1 ) E(dt+1 ∣ Y l t+1 ) Pt Qt rs,t − rt Probabilities = 1.5, dl t+1 = 0.5 dh t+1 p1 = p3 = 0.5, p2 = p4 = 0.5 p1 = p2 = p3 = p4 = 0.25 p1 = p3 = 0.4, p2 = p4 = 0.1 p1 = p3 = 0.2, p2 =
p4 = 0.3 = 1 = 3, dl dh t+1 t+1 p1 = p3 = 0.5, p2 = p4 = 0.5 p1 = p2 = p3 = p4 = 0.25 p1 = p3 = 0.4, p2 = p4 = 0.1 p1 = p3 = 0.2, p2 = p4 = 0..5 1 1.3 0.9 3 2 2.6 1.8 0.5 1 0.7 1.1 1 2 1.4 2.2 0.959 0.959 0.959 0.959 0.912 0.959 0.931 0.969 0.055 0 0.032 -0.010 0.959 0.959 0.959 0.959 1.823 0.959 1.862 1.938 0.055 0 0.032 -0.010 For all entries in the table, we suppose that Yt = 1, β = 0.95, Y h t+1 = 0.9. Probabilities are always such that the expected value of future income is one, E(Yt+1) = 1. We ﬁrst consider a case where the dividend payout is 1.5 in the high state and 0.5 in the low = 1.1, and Y l t+1 846 state; probabilities are restricted such that the expected dividend payout is always one. The ﬁrst row considers the case where there is a 50 percent chance income is high and a 50 percent it is low. The dividend payout is perfectly positively correlated with income – the dividend is high when income is high and low when income is low. In this case, the price of the bond is 0.959 and the price of the stock is 0.931. The equity premium (diﬀerence in yields on the two assets) works out to 0.055, or about 5.5 percent. In other words, the household demands a 5.5 percent premium in the expected return to be willing to be indiﬀerent between holding the stock and the bond. Why does the household demand this premium? The bond pays out 1 in the future regardless of the realization of income. The stock pays out comparatively high dividend when income is high and a comparatively low dividend when income is low.
This does not help the household smooth its consumption. In particular, the valuation of the dividend, β u′ (Yt+1), is low when income is high (because high income means u′ (Yt) low u′(⋅) and vice-versa). This means that you care more about the low dividend than the high dividend when pricing the stock. As a result, you demand a comparatively high expected return (or yield) to be indiﬀerent between holding the stock and the bond. In equilibrium, since both stock and bond are in ﬁnite supply, they must be priced such that the household is indiﬀerent between the two. The second row considers the case where all future states are equally likely. In this case, there is no equity premium. Note that there is no equity premium in spite of the fact that stock’s payout is risky compared to the bond. As in Chapter 34, it is not risk per se which the household seeks to avoid, but rather the household dislikes assets whose payouts covary positively with income (equivalently, negatively with the marginal utility of consumption). In other words, it dislikes assets (and therefore demands a high expected return) which hurt it from smoothing its consumption. In this case, the stock’s dividend is uncorrelated with the realization of income – the expected value of the dividend is 1 whether income is high or low. Since the payout on the bond is also uncorrelated with the realization of income, and since the bond and the stock oﬀer the same expected payout, they trade for the same price and there is no diﬀerence in yields. The third row considers an intermediate case; the dividend payout is positively correlated with income, but not as strongly as in the ﬁrst row. We can see this by noting that the conditional expectations of the dividend payout in the high and low state are closer to the unconditional expectation of the dividend compared to the ﬁrst row. As a result, the equity premium is positive, but not as large as in the ﬁrst case. The ﬁnal row of the upper part of Table 35.1 considers the case where the dividend payout covaries negatively with the realization of income. We can see this by nothing that the expectation of the dividend conditional on income being low is higher than the unconditional expectation. In this case, 847
the stock is priced higher than the bond and the equity premium is negative. Put somewhat diﬀerently – if the stock has a high dividend when income is low, this helps the household smooth consumption, as it gets an income kick exactly when it most values it (i.e. when u′(⋅) is comparatively high). This means that there is greater demand for the stock than the bond, and since they oﬀer the same expected return the yield on the stock must be lower than the bond. t+1 t+1 The second part of Table 35.1 is similar to the ﬁrst part, but supposes that the possible = 3 realizations of the dividend are double what they are in the ﬁrst part of the table; i.e. dh = 1. This means that the expected value of the future dividend is 2 instead of 1. and dl This naturally results in the stock price being higher compared to the earlier case – the stock is worth more, because it pays out more in expectation. However, the yields on the stock are identical to the ﬁrst part of the table, and hence the equity premia presented in the last column are also identical. As noted in Chapter 34, when comparing diﬀerent types of assets comparing them by price is not always particularly useful. A stock which pays a higher dividend in expectation will naturally have a higher price. Whether it is a better investment opportunity in the sense of oﬀering a higher expected return compared to a stock with a low dividend is not clear. To determine that, it is best to compare yields on diﬀerent kinds of stocks. 35.2 Comparing Diﬀerent Kinds of Stocks It is straightforward (though somewhat laborious) to extend our analysis to a world with multiple diﬀerent kinds of equities. Take the setup from the previous section with a risk-free bond, but allow for two diﬀerent stocks, SH1,t and SH2,t. These stocks trade at Q1,t and Q2,t in period t, and will pay dividends in period t + 1. The household’s period t budget constraint is: Ct + P B t Bt + Q1,tSH1,t + Q2,tSH2,t ≤ Yt (35.25) The speciﬁcation of constraints in period
t + 1 is somewhat more complicated, because there are additional sources of uncertainty. Suppose again that income could be high or In addition, suppose that the dividends on stocks 1 and 2 could also be high or low. low. This means that there are eight possible states of the world in period t + 11 – which we shall denote (Y h ), t+1, dl 1,t+1, dl 1,t+1, dh ). Denote the (Y l t+1, dl 1,t+1, dh 1,t+1, dh t+1, dh 1,t+1, dl t+1, dl t+1, dh ), (Y l t+1, dh ), (Y l 2,t+1 1,t+1, dh t+1, dl ), (Y l 2,t+1 1,t+1, dl 2,t+1 1,t+1, dl t+1, dh ), (Y h ), (Y h ), (Y h 2,t+1 2,t+1 2,t+1 2,t+1 2,t+1 1The number of possible states of 8 is 23 = 8. When there was only one kind of stock, as above, there were 22 = 4 possible states. 848 j=1 pj = 1. probabilities of each of these eight states materializing as pj for j = 1,... 8, with ∑8 This means that there will be eight diﬀerent budget constraints which must hold in period t + 1. There will be diﬀerent consumption values in diﬀerent states. Use a triple superscript to denote the consumption value in a particular realization of the state. Let the ﬁrst superscript denote whether income in t + 1 is high or low, the second superscript whether the dividend on stock 1 is high or low, and the third superscript whether the dividend on the second stock is high or low. Going ahead and imposing the terminal conditions that the household will not die with positive stocks of debt, these t + 1 budget constraints are given below: ≤ Y h t+1 + Bt + (dh 1,t+1 ≤ Y h t+1 + Bt + (dh 1,t+1 ≤ Y h t+1 + Bt + (dl 1,t+1 ≤ Y h t+

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