Princeton ECO 342 - Asset Pricing Term Structure

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Eco 342 Fall 2011 Chris SimsAsset Pricing and Term StructureOctober 4, 2011c2011 by Christopher A. Sims. This document is licensed under theCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported LicenseThis lecture is related to “Operation Twist”• The Fed is trying to lower long term rates by buying long debt, sellingshort debt.• The theory we are going to discuss suggests this will not work.1Asset pricing without uncertainty• Price of one dollar delivered one year from now is φ. φ is the one-yeardiscount factor.• In a competitive asset market, we can buy or sell any amount of futuredollars at this same unit price.• Treasury bills are priced this way. The “discount basis” one year interestrate is 1 − φ.• If we purchase one dollar’s worth of this security today, we get 1/φdollars in one year.2• The “coupon basis” interest rate is 1/φ −1. When φ is close to one, thecoupon basis and discount basis rates are nearly the same.• At high interest rates, the coupon basis rate is higher.3Multi-year rates• Notation: φstis the s-period discount factor that prevails at time t: I canbuy one dollar delivered at t + s by paying φstat t. (The superscript s isnot an exponent.)• Suppose one dollar delivered two years from now can be obtained bypaying now θ dollars.• The two-year rate, annualized, discount-basis, is 1 −√θ.• The idea is that if the one-year discount factor were the same value φthis year and next, we would need φ dollars at the beginning of next yearto produce one dollar two years from now, and to get φ dollars at the4beginning of next year we would need φ2dollars at the beginning of thisyear.• So with a constant one-year rate, θ = φ2, φ =√θ, and the annualizedtwo-year rate is 1 − φ = 1 −√θ.• Generally, if θ is the discount rate at which one dollar s periods fromnow is sold, the s-period rate is 1 −θ1/s. This works even for s < 1, e.g.s = 1/12 in the case of a monthly rate.5Mult-year rates and one-year rates• Suppose one-year discount factors φ1t+sthat will prevail in the future areknown at t.• Then, since one can deliver a dollar in two years by buying at t enoughone-year security so that we can at t + 1 buy another one-year securitythat delivers a dollar at t + 2, it must be thatφ2t= φ1tφ1t+1.6• More generally, when the future one-year discount factors are known,φst=s−1∏v=0φ1t+v.• If we take logs, log φst=∑s−1v=0log φ1t+v.7Multi-year rates as averages of future one-year rates• Recall that the discount-basis s-period net interest rate is 1 − (φst)1/s.When the net interest rate is only a few percent, i.e. (φst)1/sis close toone, the net rate rstis close to−log((φst)1/s)= −1slog φst= −1ss−1∑v=0log φ1t+v.=1ss−1∑v=0r1t+v8Pricing anything from current and future one-year rates• Suppose that at date t we know φst, the s-period discount factor (or1 − (φst)1/s, the s-period interest rate) for every s.• Then suppose we had to determine the price of a security that pays aknown dividend δt+sat each date t + s for s = 1, . . . , T .• This security is equivalent to a collection of s-period discount bonds, sowe can see its price has to beT∑s=1φstδt+s.9• Converting a stream of future payments to a current price in this way isknown as finding the present value of the payment stream.10The term structure• The term structure of interest rates is the set of s-period rates {rst}prevailing at a given date t.• It is often displayed in a plot of rstagainst s, called a yield curve.• In the absence of uncertainty about the future, we could deduce all thefuture one-year rates from knowledge of the term structure.• Even with uncertainty, we can interpret the term structure as suggestiveof beliefs about future short-term rates: when the plot is sharply rising,interest rates are expected to be rising, when the plot is falling, they areexpected to be falling.11• Uncertainty does make a difference. For example, because of uncertaintyusually short rates are lower than long rates. (We’ll see why later.)• When market beliefs that current short rates are unusually high arestrong enough, though, the yield curve slopes downward. This is calledan “inverted yield curve”.12Pricing long term bonds• A typical long term bond with maturity T and face value B pays acoupon r · B every year until T , when it returns the principal B.• If all future one-year rates are equal to r and known, then pricing thebond by discounting its coupons and principal payment makes it worthB at the date of issue.• However, interest rates change, and this makes the value of the bondchange.13Pricing consols• A consol is an infinite-maturity (T = ∞) bond. If its face value is B andits interest rate at issue was r, it just pays rB every year forever.• If the term structure is flat at t, with all rates equal to rt, withrt6= r possibly, then there is a constant one-period discount factorφ = 1/(1 + rt) and the consol is worth∞∑s=1φsrB =φrB1 − φ=rBrt.• The income stream from the consol is riskless, but the price will varysharply as interest rates vary, so for someone who might have to sell theconsol at a future date, it is a risky investment.14A general coupon bondSuppose the bond has the coupon rate r, maturity T , and a principalvalue B, and that the current interest rate is rtand expected to remain atrtin the future. Let φ = 1/(1 + rt) be the discount factor. Then the bondprice today is (again under our assumption of perfect foresight)Qt= φTB +T −1∑s=0rBφs= B(φT+ rφ(1 − φT)1 − φ)= B((1 + rt)−T(1 −rrt)+rrt).15This equation follows from the formula for the sum of a geometric series:T −1∑s=0ρs=1 − ρT1 − ρ,together with some algebra and use of the φ = 1 /(1 + rt) definition. Notethat with T = ∞, so long as rt> 0, this is just the formula for the price ofa consol that we derived above. Also, with rt= r, Qt= B.16Real vs. nominal returns, TIPS• US Treasury securities are for practical purposes risk free in dollar terms.• But for, say, someone saving for retirement 20 years from now, there isnonetheless a big risk, from inflation.• What such a person cares about is not the dollar value of the investmentin 20 years, but rather how much the investment will buy.• Unexpectedly high inflation over the next 20 years could make hisinvestment inadequate to support him in retirement.• Treasury Inflation-Protected


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