© K. Cuthbertson and D. Nitzsche Chapter 9 Measuring Asset Returns Investments

Learning objectives © 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 distribution Normal distribution Central 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

Value of $ 100 at end of year (r = 10% p.a.) Annually (q = 1)110 Quarterly (q = 4) Weekly (q = 52) Daily (q = 365) Continuously compounding TV = $100e (0.1(1)) (n = 1) © K. Cuthbertson and D. Nitzsche Table 1 : Compounding frequencies

Continuous Compounding © K. Cuthbertson and D. Nitzsche Example $100e (0.1(1)) = Continuously compounded 10% interest on 100 after a year will be ; ( e is an irrational and transcendental constant approximately equal to )irrational transcendental The inverse problem $100e (x(1)) = we take the difference of the natural logarithm ln( ) - ln( 100 ) = ln( / 100)=.20

© K. Cuthbertson and D. Nitzsche Figure 1 : US real stock index, S&P500 (Jan 1915 – April 2004)

© K. Cuthbertson and D. Nitzsche Figure 2 : US real return, S&P500 (Feb 1915 – April 2004)

Arithmetic Mean Return 8 The arithmetic mean return is the arithmetic average of several holding period returns measured over the same holding period:

9 Geometric Mean Return The geometric mean return is the nth root of the product of n values:

10 Arithmetic and Geometric Mean Returns Example Assume the following sample of weekly stock returns: WeekReturn –

11 Arithmetic and Geometric Mean Returns (cont’d) Example (cont’d) What is the arithmetic mean return? Solution:

12 Arithmetic and Geometric Mean Returns (cont’d) Example (cont’d) What is the geometric mean return? Solution:

13 Comparison of Arithmetic and Geometric 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%

14 Comparison of Arithmetic and Geometric 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

15 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

16 Standard Deviation and Variance (cont’d) General equation for variance: If all outcomes are equally likely:

17 Standard Deviation and Variance (cont’d) Equation for standard deviation:

© K. Cuthbertson and D. Nitzsche Figure 3 : Histogram US real return (Feb 1915 – April 2004)

19 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

20 Correlations and Covariance (cont’d)

21 Example (cont’d) The covariance and correlation for Stocks A and B are:

22 Correlations and Covariance (cont’d) Correlation ranges from –1.0 to  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 BillsOver Bonds Arith.Geom.Std. error Arith.Geom. UK US World (incl. US) © K. Cuthbertson and D. Nitzsche Table 3 : Equity premium (% p.a.),

© K. Cuthbertson and D. Nitzsche Standard deviation of returns (percent) Average Return (percent) 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 Individual stocks in lowest size decile Figure 4 : Mean and std dev : annual averages (post 1947)

Year (June) FTSE100Returns % % % % % © K. Cuthbertson and D. Nitzsche Table 7 : UK stock market index and returns ( )

© K. Cuthbertson and D. Nitzsche a b Discrete variable Continuous variable Probability 1/61/(b-a) Figure 5 : Uniform distribution (discrete and continuous)

State kProbability of State k, p k Return on Stock A Return on Stock B 1. Good0.317%-3% 2. Normal0.610%8% 3. Bad0.1-7%15% © K. Cuthbertson and D. Nitzsche Table 10 : Three scenarios for the economy

© K. Cuthbertson and D. Nitzsche   Probability 5% of the area    One standard deviation above the mean Figure 6 : Normal distribution

© K. Cuthbertson and D. Nitzsche Normal distribution N(0,1)  0    Students’ t-distribution (fat tails) Figure 7 : “Students’ t” and normal distribution

© K. Cuthbertson and D. Nitzsche Figure 8 : Lognormal distribution,  = 0.5,  = 0.75 Probability Price level

© K. Cuthbertson and D. Nitzsche Figure 9 : Central limit theorem