Return Measures
Professor Lasse H. Pedersen
Prof. Lasse H. Pedersen
1
Outline
Quoted rate = APR
Compounding and EAR
Single period realized return:
– holding period return
Multiple-period realized return:
– Arithmetic average
– Geometric average
– IRR
Prof. Lasse H. Pedersen
2
Quoted Rates and EAR
Example:
– Interest rate quoted at 10% compounded semiannually
Effective Annual Rate EAR if interest is compounded
m times a year:
– EAR = (1+ quoted rate / m ) m - 1
Example: Which loan is cheapest:
– 15%, compounded daily
– 15.5%, compounded quarterly
– 16%, compounded annually
Prof. Lasse H. Pedersen
3
Continuous Compounding
Suppose the quoted rate is given.
Consider increasingly frequent compounding:
annually, quarterly, daily, every second,…
What happens to the EAR?
When compounding happens “all the time,” it
is called continuous compounding
EAR = exp(quoted rate) - 1.
Challenge: why? (Problem C1.)
Prof. Lasse H. Pedersen
4
APR
Lenders are required by law to report the
Annual Percentage Rate, APR.
APR = Quoted Rate
= interest per period
* number of periods per year
How do you make a loan seem cheaper?
Prof. Lasse H. Pedersen
5
Single-Period Realized Return
Holding period return:
HPR =
ending price + cash dividend – beginning price
beginning price
Annualized holding period return for a holding
period of t years:
annualized HPR = (1 + HPR )1/ t − 1
1/ t
 ending price + cash dividend 
 − 1
= 
beginning price


Prof. Lasse H. Pedersen
6
Multiple-Period Realized Return
Arithmetic Average:
1
(r1 + r2 + r3 + ... + rT )
T
– Not equivalent per-period return because it
neglects compounding
– Useful for forecasting the return next period
Prof. Lasse H. Pedersen
7
Multiple-Period Realized Return
Geometric Average
– Gives the equivalent per-period return
[(1 + r1 )(1 + r2 )(1 + r3 )...(1 + rT )]1/ T − 1
=
 accumulated valueT 


value0


1/ T
−1
Prof. Lasse H. Pedersen
8
Multiple-Period Realized Return
Internal rate of return, IRR
– Return if one can re-invest cash-flows at this
rate
– “Dollar-weighted average”
– The IRR in the rate that makes:
initial price = present value of future net profits
∞
P (0) = ∑
t =1
C (t )
(1 + IRR) t
Prof. Lasse H. Pedersen
9