Lecture Notes on Rate of Return

Transcription

1 New York University Stern School of Business Professor Jennifer N. Carpenter Debt Instruments and Markets Lecture Notes on Rate of Return De nition Consider an investment over a holding period from time 0 to time T. Suppose the amount invested at time 0 is P and the payo at time T is F. F might not be known at time 0. In general F is a random variable whose value is realized on or before the investment horizon date T. To compare the performance of di erent investments, and adjust for scale, one might consider the gross or unannualized rate of return (ror) on the investment, F=P 1. To adjust for di erences in the length of the holding period, one might annualize the ror. We'll use semi-annual compounding to be consistent with US bond market interest rate quote conventions. The annualized ror with semi-annual compounding is R 2[( F P )1=2T 1] : (1) Rates of return on zero-coupon bonds What is the rate of return on an investment in a zero? Case 1) The zero maturity t matches the horizon date T. Then the ror is identically equal to the zero rate: P = d t, F = 1, and T = t implies R = 2[( 1 d t ) 1=2t 1] r t : (2) Case 2) The zero is longer than the investment horizon, i.e., t > T. Example 1) Suppose you buy a 1-year zero at 5.5% for a price of and sell it after six months at 8% for a price of Then the ror over the six months is R = 2[( 0: :94719 )1 1] = 3:03% : (3) Case 3) The zero is shorter than the horizon, i.e., t < T. Example 2) Suppose your investment horizon is one year, you buy a 0.5-year zero at 5% and roll it into another 0.5 year zero at 8%. Then the ror over the year is 1:04 R = 2[( 0:97561 )1=2 1] = 6:495% : (4) 1

2 Observation #1) Rates of returnandzerorates are di erent. They're di erent concepts. The zero rate is known at time zero; it is a transformation of price. The rate of return describes performance over a holding period and generally isn't known until the end. Only in the case that the zero maturity matches the horizon, or the zero is sold at the same rate at which it was purchased, or if the zero payo is reinvested at the same rate as at the purchase date, is the zero rate the same as the ror. Expected rates of return Consider a simple model of risk in the bond market. At time 0 we know current prices. We also know that at time 0.5 there will be either a bull market or a bear market and we know the possible future prices but at time 0 we don't know which will occur. Example 3) Current and future zero rates Time 0 Time 0.5 r u 0:5;1 = 5.50% r 0;0:5 = 5:00% r 0;1 = 5:50% r d 0:5;1 = 8.00% In other words, the current six month rate is 5%, the current 1-year rate is 5.5%, and the future 0.5-year rate will be either 5.5% or 8%. The notation r t1;t2 means the zero rate at time t 1 for the zero maturing at time t 2. Since we know the zero rates, we know the current and possible future prices, so we can compute the possible rates of return over a 0.5 year investment horizon: Current and future zero rates Rates of return in di erent markets Time 0 Time year rate of return 0.5-year zero 1-year zero 5.50% 5.50% 5.50% 8.00% 3.03% Is there any arbitrage opportunity in this market? No. Neither bond uniformly outperforms the other. The longer bond does better than the shorter bond in the low interest rate (bull market) scenario, and the shorter bond does better in the high interest rate (bear market) scenario. 2

3 Suppose that in addition we know the probabilities of the two future states of the world, say 50%, 50%. Then we could compute expected (average) rates of return: Rates of return in di erent markets Time 0 Time year rate of return 0.5-year zero 1-year zero 5.50% 5.50% 5.50% 8.00% 3.03% expected: 4.265% Observation #2) are not necessarily even equal to expected returns. Could this be an equilibrium? Yes. Since the two bonds have di erent risk pro les, they can have di erent expected returns in equilibrium. It might seem more plausible that if the expected returns are di erent, it is the riskier bond that has the higher expected return, to compensate people who dislike risk. The problem with this thinking is that what's more risky for one horizon is less risky for another. Consider the expected returns over one-year horizons: 5.50% 5.50% 5.25% 5.50% 5.50% 8.00% 3.03% 6.495% 5.50% expected: 4.265% 5.87% 5.50% Over the one-year holding period, it is the longer bond that is riskless. Observation #3) Which bond is riskier depends on the investment horizon. Observation #4) Regardless of the investment horizon, the longer bond always does better in the bull market and the shorter bond always does better in the bear market. Of course the bull-bear market classi cation of this model is a simpli cation of the real world. Once we start considering future prices for more than one long bond, say a 1-year and a 10-year, we can't assume the prices move in tandem. I.e., one could be moving up while the other moves down. In general, bond prices are highly correlated, and a lot can be understood by thinking of bond prices moving together. But the correlation isn't perfect and accounting for that can be important in some contexts. 3

4 By cheapening the 1-year zero, we canmakeits expectedreturn exceedthat ofthe 0.5-year zero. Say we raise the 5.50% to, say 6.00%: Example 4) 5.50% 6.50% 5.25% 6.00% 6.00% 8.00% 4.02% 6.495% 6.00% expected: 5.26% 5.87% 6.00% This could also be consistent with equilibrium. In general, we can't tell how the 1-year zero will be priced in equilibrium. No-arbitrage bounds We can put no-arbitrage bounds on the price, however. It can't be cheaper than 6.50%, or it will uniformly outperform the 0.5-year bond: Example 5) 5.50% 7.50% 5.25% 6.50% 6.50% 8.00% 5.01% 6.495% 6.50% expected: 6.26% 5.87% 6.50% Similarly, it can't be richer than 5.25%, or it will uniformly underperform the 0.5-year bond: Example 6) 5.50% 5.25% 5.25% 5.25% 8.00% 2.54% 6.495% 5.25% expected: 3.77% 5.87% 5.25% 4

5 Bond pricing and covariance Given future payo s, what determines bond prices? Or, in other words, what determines expected returns? Remember that the speci c risk of a given security is not what's most important, because people don't hold securities in isolation. People hold portfolios. People consider how adding a security will a ect their whole position, including things such as real estate and labor income as well as other nancial assets. In particular, covariance matters. If a security does well when the rest of the portfolio does well, and poorly when the rest of the portfolio does poorly, it adds risk to the total position. But if it does well when the rest of the portfolio does poorly, and poorly when the rest of the portfolio does well, it reduces risk. It hedges the other risks. It acts as insurance. To the extent that people don't like risk, hedging is desirable. In general, we can imagine that people form ideas about the distribution of the future price of a bond, its mean, variance, covariance with their other assets. Then they decide what expected return to require based on their individual risk preferences and a market clearing price is determined. Observation #5) Di erent bonds can have di erent expected returns over the same holding period. This can be sustainable in equilibrium because di erent bonds have di erent risk pro les. (Armchair economics: Longer bonds are often thought to have higher expected returns, though this is an over-simpli cation. A crude explanation of this is through a kind of CAPM-like model of the bond market: Long positions (bullish positions) in the bond market command a risk premium (because the average portfolio is long bonds because some one has to hold the national debt? so adding a bond adds variance to the average portfolio and must provide extra return). Longer bonds have a higher \bond market beta," so must o er a higher expected return. Implicit in this is a notion that the natural horizon is very short, say one day. This might make sense if we think of our horizon date as being at our next trading opportunity.) 5

6 Information in the yield curve? People often think the current yield curve, such as the set of current zero rates on zeroes of di erent maturities, gives a forecast of future rates. For example, if expected rates of return on all bonds are equal, then if the longer rates are higher then shorter rates, it must be that people expect rates to rise: Example 7) 5.50% 6.24% 5.25% 5.87% 5.87% 8.00% 3.76% 6.495% 5.87% 7.25% expected: 5.001% 5.872% 5.87% However, if di erent bonds have di erent expected returns, then the yield curve won't be at, even if rates are expected stay the same: Example 8) 4.00% 6.51% 4.50% 5.25% 5.25% 6.00% 4.50% 5.50% 5.25% expected: 5.50% 5.25% In Example 7, the yield curve is upward-sloping because rates are expected to rise. In Example 8, the yield curve is upward-sloping because the longer zero commands a higher expected return. In general, it is hard to disentangle these e ects. 6