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Chapter 9

Measuring Asset ReturnsInvestments

Learning objectives

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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