Chapter 9
Measuring Asset Returns
Investments
© K. Cuthbertson and D. Nitzsche
Learning objectives
 Calculate asset returns – arithmetic mean, geometric




mean, continuously compounded returns
Sample statistics- mean, variance, standard
deviation, correlation, covariance
Random variable and probability distribution
Normal distribution
Central limit theorem
© K. Cuthbertson and D. Nitzsche
Measuring Asset Returns
 Nominal return, inflation and real return (Fisher
Effect)
 Holding Period Return (annualized return)
 Returns over several periods


Arithmetic average
Geometric average
 Compounding frequency
Table 1 : Compounding frequencies
Compounding frequency
Annually (q = 1)
Quarterly (q = 4)
Weekly (q = 52)
Daily (q = 365)
Continuously compounding
TV = $100e(0.1(1)) (n = 1)
Value of $ 100 at end of
year
(r = 10% p.a.)
110
© K. Cuthbertson and D. Nitzsche
110.38
110.51
110.5155
110.5171
Continuous Compounding
 Example $100e(0.1(1)) =110.5171
Continuously compounded 10% interest on 100
after a year will be 110.5171; (e is an irrational and
transcendental constant approximately equal to
2.718281828)
 The inverse problem $100e(x(1)) =122.14 we take the
difference of the natural logarithm
ln(122.14 ) - ln(100) = ln(122.14/ 100)=.20
© K. Cuthbertson and D. Nitzsche
Figure 1 : US real stock index, S&P500 (Jan 1915 – April 2004)
80
70
60
50
40
30
20
10
© K. Cuthbertson and D. Nitzsche
Jan-03
Jan-95
Jan-87
Jan-79
Jan-71
Jan-63
Jan-55
Jan-47
Jan-39
Jan-31
Jan-23
Jan-15
0
Figure 2 : US real return, S&P500 (Feb 1915 – April 2004)
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
Feb-15
Feb-27
Feb-39
Feb-51
Feb-63
© K. Cuthbertson and D. Nitzsche
Feb-75
Feb-87
Feb-99
Arithmetic Mean Return
8
 The arithmetic mean return is the arithmetic
average of several holding period returns measured
over the same holding period:
~
Ri
Arithmetic mean  
i 1 n
~
Ri  the rate of return in period i
n
Geometric Mean Return
9
 The geometric mean return is the nth root of the
product of n values:

~ 
Geometric mean   (1  Ri )
 i 1

n
1/ n
1
Arithmetic and
Geometric Mean Returns
10
Example
Assume the following sample of weekly stock
returns:
Week
Return
1
2
0.0084
–0.0045
3
0.0021
4
0.0000
Arithmetic and Geometric Mean Returns
(cont’d)
11
Example (cont’d)
What is the arithmetic mean return?
Solution:
~
Ri
Arithmetic mean  
i 1 n
0.0084  0.0045  0.0021  0.0000

4
 0.0015
n
Arithmetic and Geometric Mean Returns
(cont’d)
12
Example (cont’d)
What is the geometric mean return?
Solution:
1/ n


Geometric mean   (1  Ri ) 
 i 1

n
1
 1.0084  0.9955 1.00211.0000
1/ 4
 0.001489
1
Comparison of Arithmetic and
Geometric Mean Returns
13
 The geometric mean reduces the likelihood of
nonsense answers

Assume a $100 investment falls by 50 percent in period 1 and
rises by 50 percent in period 2

The investor has $75 at the end of period 2
Arithmetic mean = [(0.50) + (–0.50)]/2 = 0%
1/2
 Geometric mean = (0.50 × 1.50)
– 1 = –13.40%

Comparison of Arithmetic and
Geometric Mean Returns (Cont’d)
14
 The geometric mean must be used to determine the
rate of return that equates a present value with a
series of future values
 The greater the dispersion in a series of numbers, the
wider the gap between the arithmetic mean and
geometric mean
Standard Deviation and Variance
15
 Standard deviation and variance are the most
common measures of total risk
 They measure the dispersion of a set of observations
around the mean observation
Standard Deviation and Variance (cont’d)
16
 General equation for variance:
2
n
Variance     prob( xi )  xi  x 
2
i 1
 If all outcomes are equally likely:
n
2
1
    xi  x 
n i 1
2
Standard Deviation and Variance (cont’d)
17
 Equation for standard deviation:
Standard deviation     2 
2
n
 prob( x )  x  x 
i 1
i
i
Figure 3 : Histogram US real return (Feb 1915 – April 2004)
120
100
Frequency
80
60
40
20
0
-0.15
-0.11
-0.07
-0.03
0.01
© K. Cuthbertson and D. Nitzsche
0.05
0.09
0.13
Correlations and Covariance
19
 Correlation is the degree of association between
two variables
 Covariance is the product moment of two random
variables about their means
 Correlation and covariance are related and generally
measure the same phenomenon
Correlations and Covariance (cont’d)
20
COV ( A, B)   AB  E ( A  A)( B  B ) 
 AB 
COV ( A, B)
 A B
Example (cont’d)
21
The covariance and correlation for Stocks A and B
are:
 AB
1
  (0.5%  0.0%)  (2.5%  3.0%)  (2.5%  2.0%)  (0.5%  1.0%)
4
1
 (0.001225)
4
 0.000306
 AB 
COV ( A, B)
 A B
0.000306

 0.909
(0.018)(0.0187)
Correlations and Covariance (cont’d)
22
 Correlation ranges from –1.0 to +1.0.
 Two random variables that are perfectly positively correlated
have a correlation coefficient of +1.0

Two random variables that are perfectly negatively correlated
have a correlation coefficient of –1.0
Table 3 : Equity premium (% p.a.), 1900 - 2000
Over Bills
Over Bonds
Arith.
Geom.
Std.
error
Arith.
Geom.
UK
6.5
4.8
2.0
5.6
4.4
US
7.7
5.8
2.0
7.0
5.0
World (incl.
US)
6.2
4.9
1.6
5.6
4.6
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Figure 4 : Mean and std dev : annual averages (post 1947)
smallest “size sorted” decile
Average Return (percent)
20
= NYSE decile “size sorted” portfolios
16
Equally weighted, NYSE
12
S&P500
Value weighted,NYSE
8
Corporate Bonds
4
T-Bills
0
4
largest “size sorted”decile
Individual stocks in lowest size decile
Government Bonds
8
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12
16 20
24
28
32 40 45
Standard deviation of returns (percent)
50
Table 7 : UK stock market index and returns (2002-07)
Year
(June)
2002
FTSE100
Returns
2003
4031.17
-13.43%
2004
4464.07
10.74%
2005
5113.16
14.54%
2006
5833.42
14.86%
2007
6607.90
13.28%
4656.36
© K. Cuthbertson and D. Nitzsche
Figure 5 : Uniform distribution (discrete and continuous)
Continuous variable
Discrete variable
Probability
Probability
1/6
1/(b-a)
1
2
3 4
5
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6
a
b
Table 10 : Three scenarios for the economy
State k
Probability
of State k, pk
Return on
Stock A
Return on
Stock B
1. Good
0.3
17%
-3%
2. Normal
0.6
10%
8%
3. Bad
0.1
-7%
15%
© K. Cuthbertson and D. Nitzsche
Figure 6 : Normal distribution
Probability
One standard deviation
above the mean
5% of the area
5% of the area
-4
-3
-2
-1.65
© K. Cuthbertson and D. Nitzsche
-1
0

1
2
+1.65
3
4
Figure 7 : “Students’ t” and normal distribution
Students’ t-distribution
(fat tails)
Normal distribution
N(0,1)
2
© K. Cuthbertson and D. Nitzsche

0

2
Figure 8 : Lognormal distribution,  = 0.5,  = 0.75
0.45
0.4
Probability
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
Price level
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7
Figure 9 : Central limit theorem
© K. Cuthbertson and D. Nitzsche