# Rate of Return. Reading: Veronesi, Chapter 7. Investment over a Holding Period

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1 Rate of Return Reading: Veronesi, Chapter 7 Investment over a Holding Period Consider an investment in any asset over a holding period from time 0 to time T. Suppose the amount invested at time 0 is P and the payoff at time T is F. F might not be known at time 0. In general, the payoff F is not known until the investment horizon date T. Invest P Get Payoff F Time T Rate of Return 1

2 Rate of Return over a Holding Period To compare the performance of different investments, and adjust for scale, one might consider the gross or unannualized rate of return (ROR) on the investment: Unannualized ROR = F/P - 1. To adjust for differences in the length of the holding period as well, 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] so that F/P=(1+R/2) 2T Example of Holding Period Return Suppose you invest \$100 in an asset at time 0 and at time 5 it is worth \$150. Your un-annualized ROR is 150/100-1=50%. Class Problem: What is your annualized ROR with semiannual compounding? Rate of Return 2

3 Rate of Return on a Zero: Case 1) Maturity Equal to Investment Horizon If you buy a zero-coupon bond and hold it to maturity, the ROR on your investment is the zero rate at which you bought the bond: T = t, P = d t, F = 1 so R = 2[(1/d t ) (1/(2t)) -1]=r t Example: If you buy a 1-year zero at 5.25% and hold it to maturity your ROR over 1-year is 1 R = 2[( 1/( /2) 2 )1/ 2 1] = 5.25% Rate of Return on a Zero: Case 2) Maturity Longer than Investment Horizon If you buy a t-year zero-coupon bond and sell it at time T<t, the ROR on your investment depends on market conditions at the selling time T. Class Problem: Suppose you buy the 1-year zero at 5.25% for a price of 1/( /2) 2 = and sell it after 6 months at 6% for a price of 1/(1+0.06/2) = What is your ROR over 6 months? Class Problem: Suppose you buy a 1-year zero at 5.25% and sell it after 6 months at 4%. What is your ROR over 6 months? Rate of Return 3

4 Zero Rates and Rates of Return are Different Zero rate and rate of return are simply different concepts. The zero rate is only for a zero, and is known at the time the zero is purchased. Rate of return is a concept that applies to any kind of investment, bond, stock, currency, etc. In general it is not known at the start of the investment it is a random amount realized only at the end of the holding period. The only time the ROR is known at the start is when the investment is in the zero with maturity matching the horizon date. In that case, this known ROR is the zero rate. If the zero maturity differs from the horizon date, the ROR will generally differ from the rate at which the zero is purchased. It will be the same if the (longer) zero is sold at the same zero rate at which it was purchased or if the (shorter) zero payoff is reinvested at the same rate at which the zero was purchased. Zero Rates Are Not Even Expected Returns The rate of return on an investment that is realized over a holding period is not generally known in advance. However, one could imagine that the market might form an expectation or forecast of what the ROR will be. By definition, this expected rate of return is known in advance. Does the zero rate equal the expected rate of return from investing in a zero? In general the answer is no Rate of Return 4

5 Bond Market with Risk: Current and Future Possible Zero Rates Let t r u represent the zero rate at time t for maturity at time u. For example 0 r 0.5 means the 0.5-year rate at time 0 and 0.5 r 1 means the 0.5-year rate at time 0.5. Consider the following current and future possible zero rates: 0 r 0.5 = 5.00% r 1 u = 6.00% 0.5 r 1 d = 4.00% Bond Market with Risk: Future Possible 0.5-Year RORs Now consider investing in this market for 6 months. You could buy a 0.5-year zero and earn 5% risklessly. Or you could buy a 1-year zero and sell it at time 0.5. The ROR from this strategy is risky it would depend on market conditions at time 0.5:.5 ROR on ROR on Zero rate 0.5-yr zero 1-yr zero 0.5 r 1 u =6.00% 5.00%? 0.5 r 1 d =4.00% 5.00%? Rate of Return 5

6 Bond Market with Risk: Expected 0.5-Year ROR Now suppose each possible time 0.5 interest rate state occurs with probability 50%. Then we can compute the expected ROR on each strategy: Year ROR ROR on ROR on Zero rate 0.5-yr zero 1-yr zero 0.5 r 1 u =6.00% 5.00%? 0.5 r 1 d =4.00% 5.00%? Expected ROR: 5.00%? Different Bonds Can Have Different Expected Returns over the Same Holding Period This can be sustainable in equilibrium because different bonds have different risk profiles. Asset pricing: Given its future payoffs, what determines the price of an asset? Or, in other words, what determines its expected return? Recall from the CAPM for equities that the specific risk of an asset isn t what s most important, because people don t hold securities in isolation. People hold portfolios, including real estate and labor income. What matters is how the asset contributes to the total portfolio risk i.e, its covariance with other risks. Does it hedge other risks? Or add to them? Bond pricing: Roughly speaking, longer bonds are believed to have higher expected returns. Why? They do essentially have higher bond market beta because of higher duration, as we will see, but why is the bond market risk premium positive? Is the average investor long the bond market, or short inflation risk, so that adding a bond adds variance? No easy story. Rate of Return 6

7 Rate of Return on a Zero: Case 3) Maturity Shorter than Investment Horizon If you buy a t-year zero-coupon bond, hold it to maturity, and then reinvest the payoff to some later date T>t, your ROR depends on your reinvestment rate from time t to T. Class Problem: Suppose your investment horizon is T = 1 year and you buy the 0.5-year zero at 5% and then roll it into another 0.5-year zero at 6%. What is your ROR over the year? Class Problem: Suppose you buy a 0.5-year zero at 5% for a price of 1/(1+0.05/2) = and then at time t = 0.5 you roll it into another 0.5-year zero at 4%. What is your ROR over the year? Bond Market with Risk: Future Possible 1-Year RORs To invest for 1 year, you could buy a 1-year zero and earn 5.25% risklessly. Or you could buy a 0.5-year zero and reinvest its payoff at time 0.5 in another 0.5-year zero. This strategy is subject to reinvestment risk. Expected:.5 1-Year ROR ROR on ROR on Zero rate 0.5-yr zero 1-yr zero 0.5 r 1 u =6.00%? 5.25% 0.5 r 1 d =4.00%? 5.25% 5.00%? 5.25% Rate of Return 7

8 Example of Bond Market with Risk Premia: The longer bond has a higher expected return If the yield curve is upward-sloping, but rates are expected to stay the same, then longer bonds have higher expected return. Expected: yr horizon 1-yr horizon ROR on ROR on ROR on ROR on Zero rate 0.5-yr z. 1-yr z. 0.5-yr z. 1-yr z. 0.5 r 1 u =6.00% 5.00% 4.503% 5.499% 5.25% 0.5 r 1 d =4.00% 5.00% 6.508% 4.499% 5.25% 5.00% 5.00% 5.505% 4.999% 5.25% Risk-Neutral Bond Market Example: Different bonds have the same expected return If the yield curve is upward sloping, but expected returns across bonds are the same, it must be that rates are expected to rise. Expected: yr horizon 1-yr horizon ROR on ROR on ROR on ROR on Zero rate 0.5-yr z. 1-yr z. 0.5-yr z. 1-yr z. 0.5 r 1 u =6.50% 5.00% 4.008% 5.749% 5.25% 0.5 r 1 d =4.50% 5.00% 6.003% 4.750% 5.25% 5.50% 5.00% 5.005% 5.249% 5.25% Rate of Return 8

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