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5.1 Rates of Return
5-1
Measuring Ex-Post (Past) Returns
•An example: Suppose you buy one share of a stock today for
$45 and you hold it for one year and sell it for $52. You also
received $8 in dividends at the end of the year.
•(PB = $45, PS = $52
, CF = $8):
•HPR = (52 - 45 + 8) / 45 = 33.33%
5-2
Arithmetic Average
Finding the average HPR for a time series of returns:
• i. Without compounding (AAR or Arithmetic Average
Return):
n
HPR av g 

HPR T
n
T 1
• n = number of time periods
5-3
Arithmetic Average
An example: You have the following rates of return on a stock:
2000 -21.56%
2001 44.63%
2002 23.35%
2003 20.98%
2004
3.11%
2005 34.46%
2006 17.62%
n
HPR av g 

HPR av g 
(-.2156  .4463  .2335  .2098  .0311  .3446  .1762)
 17.51%
7
HPR T
n
T 1
AAR = 17.51%
5-4
Geometric Average
An example: You have the following rates of return on a stock:
2000 -21.56%
2001 44.63%
2002 23.35%
2003 20.98%
2004
3.11%
2005 34.46%
2006 17.62%
•With compounding (geometric average or GAR:
Geometric Average Return):
HPR av g
 n


(1  HPR T ) 
 T 1


1/ n
1
HPR avg  (0.7844 1.44631.2335 1.2098 1.03111.3446 1.1762)1/7 1  15.61%
GAR = 15.61%
5-5
Q: When should you use the GAR and when should you use
the AAR?
A1: When you are evaluating PAST RESULTS (ex-post):
 Use the AAR (average without compounding) if you ARE
NOT reinvesting any cash flows received before the end of
the period.
 Use the GAR (average with compounding) if you
ARE reinvesting any cash flows received before the end of
the period.
A2: When you are trying to estimate an expected return (exante return):

Use the AAR
5-6
Measuring Ex-Post (Past) Returns for a portfolio
•Finding the average HPR for a portfolio of assets for a
given time period:
J
HPR av g
VI 


HPR I  TV 

I1 

•where VI = amount invested in asset I,
•J = Total # of securities
•and TV = total amount invested;
•thus VI/TV = percentage of total investment invested in
asset I
5-7
•For example: Suppose you have $1000 invested in a stock
portfolio in September. You have $200 invested in Stock A, $300
in Stock B and $500 in Stock C. The HPR for the month of
September for Stock A was 2%, for Stock B the HPR was 4% and
for Stock C the HPR was - 5%.
•The average HPR for the month of September for this portfolio
is:
J
VI 

HPR av g 
HPR I  TV 

I1 
HPR avg  (.02 (200/1000))  (.04  (300/1000))  (-.05 (500/1000)) -0.9%

5-8
5.2 Risk and Risk
Premiums
5-9
Measuring Mean:
Scenario or Subjective Returns
a. Subjective or Scenario
Subjective expected returns
E(r) = S p(s) r(s)
s
E(r) = Expected Return
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states
5-10
Measuring Variance or
Dispersion of Returns
a. Subjective or Scenario
Variance
σ 2   p(s)  [rs  E(r)] 2
s
 = [2]1/2
E(r) = Expected Return
p(s) = probability of a state
rs = return in state “s”
5-11
Numerical Example: Subjective or
Scenario Distributions
State Prob. of State Return
1
.2
- .05
2
.5
.05
3
.3
.15
E(r) =
(.2)(-0.05) + (.5)(0.05) + (.3)(0.15) = 6%
σ 2   p(s)  [rs  E(r)] 2
s
2 = [(.2)(-0.05-0.06)2 + (.5)(0.05- 0.06)2 + (.3)(0.15-0.06)2]
2 = 0.0049%2
 = [ 0.0049]1/2 = .07 or 7%
5-12
Expost Expected Return & 
n HPR
T
r 
T 1 n
Expost Variance :  2
r  average HPR
n  # observatio ns
n

1

( ri  r ) 2
n  1 i 1
Expost Standard Deviation: σ  σ 2
Annualizing the statistics:
rannual  rperiod  # periods
 annual  period  # periods
5-13
Using Ex-Post Returns to estimate
Expected HPR
Estimating Expected HPR (E[r]) from ex-post data.
Use the arithmetic average of past returns as a
forecast of expected future returns and,
Perhaps apply some (usually ad-hoc) adjustment to
past returns
Problems?
• Which historical time period?
• Have to adjust for current economic
situation
5-14
Characteristics of Probability
Distributions
Arithmetic average & usually most likely _
1. Mean: __________________________________
2. Median:
Middle
observation
_________________
3. Variance or standard deviation:
Dispersion of returns about the mean
4. Skewness:_______________________________
Long tailed distribution, either side
5. Leptokurtosis: ______________________________
Too many observations in the tails

If a distribution is approximately normal, the distribution
1 and 3
is fully described by Characteristics
_____________________
5-15
Normal Distribution
Risk is the
possibility of getting
returns different
from expected.
Average = Median
 measures deviations
above the mean as well as
below the mean.
E[r] = 10%
 = 20%
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5.3 The Historical Record
5-17
Frequency distributions of annual HPRs,
1926-2008
5-18
Rates of return on stocks, bonds and
bills, 1926-2008
5-19
Annual Holding Period Returns Statistics 1926-2008
From Table 5.3
Series
Geom.
Arith.
Excess
Mean%
Mean%
Return%
Kurt.
Skew.
World Stk
9.20
11.00
7.25
1.03
-0.16
US Lg. Stk
9.34
11.43
7.68
-0.10
-0.26
11.43
17.26
13.51
1.60
0.81
World Bnd
5.56
5.92
2.17
1.10
0.77
LT Bond
5.31
5.60
1.85
0.80
0.51
Sm. Stk
• Geometric mean:
Best measure of
compound historical
return
• Deviations from
normality?
• Arithmetic Mean:
Expected return
5-20
Historical Real Returns & Sharpe
Ratios
Series
World Stk
US Lg. Stk
Sm. Stk
World Bnd
LT Bond
Real Returns%
6.00
6.13
8.17
Sharpe Ratio
0.37
0.37
0.36
2.46
2.22
0.24
0.24
• Real returns have been much higher for stocks than for bonds
• Sharpe ratios measure the excess return relative to standard
deviation.
• The higher the Sharpe ratio the better.
• Stocks have had much higher Sharpe ratios than bonds.
5-21
5.4 Inflation and Real Rates
of Return
5-22
Inflation, Taxes and Returns
The average inflation rate from 1966 to 2005 was _____.
4.29%
This relatively small inflation rate reduces the terminal
value of $1 invested in T-bills in 1966 from a nominal
value of ______
_____.
$10.08 in 2005 to a real value of $1.63
Taxes are paid on _______
nominal investment income. This
real investment income even further.
reduces _____
6% nominal, pre-tax rate of return and you
You earn a ____
15% tax bracket and face a _____
are in a ____
4.29%inflation rate.
What is your real after tax rate of return?
rreal  [6% x (1 - 0.15)] – 4.29%  0.81%; taxed on nominal
5-23
Real vs. Nominal Rates
Fisher effect: Approximation
real rate  nominal rate - inflation rate
rreal  rnom - i
rreal = real interest rate
Example rnom = 9%, i = 6%
rnom = nominal interest rate
rreal  3%
i = expected inflation rate
Fisher effect: Exact
rreal = [(1 + rnom) / (1 + i)] – 1
or
rreal = (rnom - i) / (1 + i)
rreal = (9% - 6%) / (1.06) = 2.83%
The exact real rate is less than the approximate
real rate.
5-24