Rate of Return - New York University
Debt Instruments and Markets
Professor Carpenter
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 0
Time T
Rate of Return
1
Debt Instruments and Markets
Professor Carpenter
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
Debt Instruments and Markets
Professor Carpenter
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 = dt, F = 1 so
R = 2[(1/dt)(1/(2t))-1]=rt
Example: If you buy a 1-year zero at 5.25% and hold it to maturity your ROR over 1-year is
R
=
2[( 1/(1
+
1 0.0525
/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 ................
................
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