FAU FIN 3403 Risk and Return (Students) - Risk and Return...

Risk and Return
Rates of Return
Holding Period Return
Assume you buy a stock in year t-1 and hold it until year t, and it pays a
dividend, d
t
.
Then the one-period holding return can be figured as follows:

R
t
P
t
– P
t-1
+ d
t
/ P
t-1
or = P
t
– P
t-1
/ P
t-1
if d
t
= 0.
This is for 1 year. What if we held a stock for more than a year? We could
compute this either using an arithmetic average return or a compound rate of
return. Naturally, in finance we prefer the latter, the former being calculated
by taking the simple average of the annual holding period returns.
eg. Price of stock at end of year, first paying no dividend, then paying a
dividend:
Price Return What if d≠0 Return w/dividend
P
t
= 20
 R
t
20 -15 / 15 = .3333 .75 .383
P
t-1
= 15
P
t-2
= 10


R
t 1
15 – 10 / 15 = .5 .5 .55
P
t-3
= 14


R
t 2
10 – 14 / 14 = -.286.25 -.268
IF there are no dividends, we could calculate this with a Financial
Calculator:
PV = -14 FV = 20
PMT = 0 I = ? = 12.62
N = 3
But this only works when there are no intermediate cash flows, i.e., when
there are no dividends. So, in general we need to use the GMR.

Geometric Mean Return
 
1 1
1
1







 


n
n
t
t t
R GMR
Where:

is the product operator similar to a summation operator. That is:
3 2 1
3
1
X X X
j
j
X
  


Then the GMR (for the without dividend example) is:
 
1 ) 714 )(. 5 . 1 )( 3333 . 1 (
3
1
 
GMR
t
= 1.126 – 1 = .126 or 12.6%
Notice that the .714 comes from 1+(-.286)=.714.
Note 1: It makes no sense to calculate a simple return over the whole period
unless you want to see how much you have made on your money over the
whole period.
% 9 . 42
14
14 20



R
t
Note 2: This is similar if we want to calculate the growth in, say, earnings or
dividends per share. That is, if we have data on earnings per share over 5