© k. cuthbertson and d. nitzsche chapter 9 measuring asset returns investments
DESCRIPTION
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 frequencyTRANSCRIPT
© K. Cuthbertson and D. Nitzsche© K. Cuthbertson and D. Nitzsche
Chapter 9
Measuring Asset ReturnsInvestments
Learning objectives
© K. Cuthbertson and D. Nitzsche© K. Cuthbertson and D. Nitzsche
Calculate asset returns – arithmetic mean, geometric mean, continuously compounded returns
Sample statistics- mean, variance, standard deviation, correlation, covariance
Random variable and probability distributionNormal distributionCentral limit theorem
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
Compounding frequency Value of $ 100 at end of year
(r = 10% p.a.)Annually (q = 1) 110Quarterly (q = 4) 110.38Weekly (q = 52) 110.51Daily (q = 365) 110.5155Continuously compoundingTV = $100e(0.1(1)) (n = 1)
110.5171
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Table 1 : Compounding frequencies
Continuous Compounding
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Example $100e(0.1(1)) =110.5171Continuously compounded 10% interest on
100after 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
0
10
20
30
40
50
60
70
80
Jan-
15
Jan-
23
Jan-
31
Jan-
39
Jan-
47
Jan-
55
Jan-
63
Jan-
71
Jan-
79
Jan-
87
Jan-
95
Jan-
03
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Figure 1 : US real stock index, S&P500 (Jan 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 Feb-75 Feb-87 Feb-99
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Figure 2 : US real return, S&P500 (Feb 1915 – April 2004)
Arithmetic Mean Return88
The arithmetic mean return is the arithmetic average of several holding period returns measured over the same holding period:
iR
nR
i
n
i
i
periodin return of rate the~
~mean Arithmetic
1
99
Geometric Mean Return
The geometric mean return is the nth root of the product of n values:
1)~1(mean Geometric/1
1
nn
iiR
1010
Arithmetic and Geometric Mean Returns
Example
Assume the following sample of weekly stock returns:
Week Return
1 0.00842 –0.00453 0.00214 0.0000
1111
Arithmetic and Geometric Mean Returns (cont’d)
Example (cont’d)
What is the arithmetic mean return?
Solution:
0015.04
0000.00021.00045.00084.0
~mean Arithmetic
1
n
i
i
nR
1212
Arithmetic and Geometric Mean Returns (cont’d)
Example (cont’d)
What is the geometric mean return?
Solution:
1/
1
1/ 4
Geometric mean (1 ) 1
1.0084 0.9955 1.0021 1.0000 1
0.001489
nn
ii
R
1313
Comparison of Arithmetic andGeometric Mean Returns
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% Geometric mean = (0.50 × 1.50)1/2 – 1 = –13.40%
1414
Comparison of Arithmetic andGeometric Mean Returns
(Cont’d)
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
1515
Standard Deviation and Variance
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
1616
Standard Deviation and Variance (cont’d)
General equation for variance:
If all outcomes are equally likely:
2
2
1
Variance prob( )n
i ii
x x x
2
2
1
1 n
ii
x xn
1717
Standard Deviation and Variance (cont’d)
Equation for standard deviation:
2
2
1
Standard deviation prob( )n
i ii
x x x
0
20
40
60
80
100
120
-0.15 -0.11 -0.07 -0.03 0.01 0.05 0.09 0.13
Frequency
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Figure 3 : Histogram US real return (Feb 1915 – April 2004)
1919
Correlations and Covariance
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
2020
Correlations and Covariance (cont’d)
( , ) ( )( )ABCOV A B E A A B B
( , )AB
A B
COV A B
2121
Example (cont’d)
The covariance and correlation for Stocks A and B are:
1 (0.5% 0.0%) ( 2.5% 3.0%) (2.5% 2.0%) ( 0.5% 1.0%)41 (0.001225)40.000306
AB
( , ) 0.000306 0.909(0.018)(0.0187)AB
A B
COV A B
2222
Correlations and Covariance (cont’d)
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
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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Table 3 : Equity premium (% p.a.), 1900 - 2000
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Standard deviation of returns (percent)
Ave
rage
Ret
urn
(per
cent
)
0 4 8 12 16 20 24 28 32
4
8
12
16
Government Bonds
Corporate Bonds T-Bills
S&P500 Value weighted,NYSE
Equally weighted, NYSE
= NYSE decile “size sorted” portfolios
smallest “size sorted” decile
largest “size sorted”decile
40 45 50
20
Individual stocks in lowest size decile
Figure 4 : Mean and std dev : annual averages (post 1947)
Year (June)
FTSE100 Returns
2002 4656.362003 4031.17 -13.43%2004 4464.07 10.74%2005 5113.16 14.54%2006 5833.42 14.86%2007 6607.90 13.28%
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Table 7 : UK stock market index and returns (2002-07)
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1 2 3 4 5 6 a b
Discrete variable Continuous variable
Probability Probability
1/6 1/(b-a)
Figure 5 : Uniform distribution (discrete and continuous)
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%
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Table 10 : Three scenarios for the economy
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-4 -3 -2 -1 0 1 2 3 4
-1.65
Probability
5% of the area5% of the area
+1.65
One standard deviation above the mean
Figure 6 : Normal distribution
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Normal distribution N(0,1)
0
Students’ t-distribution (fat tails)
Figure 7 : “Students’ t” and normal distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 1 2 3 4 5 6 7
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Figure 8 : Lognormal distribution, = 0.5, = 0.75Pr
obab
ility
Price level
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Figure 9 : Central limit theorem