Rate of Return - New York University

[Pages:8]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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