Risk and Return Riccardo Colacito

Roadmap Rates of Return Summary Statistics of rates of return Holding Period Return Arithmetic and Geometric Averages Annual Percentage Rate and Effective Annual Rate Summary Statistics of rates of return Probability Distribution Expected Return Variance, Covariance and Standard Deviation Other properties Historical record of Bills, Bonds, and Stocks Risk premia from 1926-2003? Inflation and Real Rates of Return Foundations of Financial Markets

Holding Period Return Foundations of Financial Markets

Rates of Return: Single Period Example Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( 24 - 20 + 1 )/ ( 20) = 25% Foundations of Financial Markets

Roadmap Rates of Return Summary Statistics of rates of return Holding Period Return Arithmetic and Geometric Averages Annual Percentage Rate and Effective Annual Rate Summary Statistics of rates of return Probability Distribution Expected Return Variance, Covariance and Standard Deviation Other properties Historical record of Bills, Bonds, and Stocks Risk premia from 1926-2003? Inflation and Real Rates of Return Foundations of Financial Markets

Returns Using Arithmetic and Geometric Averaging Time 1 2 3 4 HPR .1 .25 -.20 Arithmetic ra = (r1 + r2 +... rn) / n ra = (.10 + .25 - .20 + .25) / 4 = .10 or 10% Geometric rg = [(1+r1) (1+r2) .... (1+rn)]1/n - 1 rg = [(1.1) (1.25) (.8) (1.25)]1/4 - 1 = (1.5150) 1/4 -1 = .0829 = 8.29% Foundations of Financial Markets

Roadmap Rates of Return Summary Statistics of rates of return Holding Period Return Arithmetic and Geometric Averages Annual Percentage Rate and Effective Annual Rate Summary Statistics of rates of return Probability Distribution Expected Return Variance and Standard Deviation Other properties Historical record of Bills, Bonds, and Stocks Risk premia from 1926-2003? Inflation and Real Rates of Return Foundations of Financial Markets

Quoting Conventions Annual Percentage Rate APR = (periods in year) X (rate for period) Effective Annual Rate EAR = ( 1+ rate for period)Periods per yr – 1 Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01)12 - 1 = 12.68% Foundations of Financial Markets

Roadmap Rates of Return Summary Statistics of rates of return Holding Period Return Arithmetic and Geometric Averages Annual Percentage Rate and Effective Annual Rate Summary Statistics of rates of return Probability Distribution Expected Return Variance, Covariance and Standard Deviation Other properties Historical record of Bills, Bonds, and Stocks Risk premia from 1926-2003? Inflation and Real Rates of Return Foundations of Financial Markets

Probability distribution Definition: list of possible outcomes with associated probabilities Example: State Outcome Prob 1 -2 .1 2 -1 .2 3 .4 4 5 Foundations of Financial Markets

Probability distribution: figure Foundations of Financial Markets

Normal distribution Foundations of Financial Markets

Notation Let p(i) denote the probability with which state i occurs Outcome Prob 1 -2 .1 2 -1 .2 3 .4 4 5 Let p(i) denote the probability with which state i occurs Then p(1)=0.1 p(2)=0.2 p(3)=0.4 p(4)=0.2 p(5)=0.1 Foundations of Financial Markets

Roadmap Rates of Return Summary Statistics of rates of return Holding Period Return Arithmetic and Geometric Averages Annual Percentage Rate and Effective Annual Rate Summary Statistics of rates of return Probability Distribution Expected Return Variance, Covariance and Standard Deviation Other properties Historical record of Bills, Bonds, and Stocks Risk premia from 1926-2003? Inflation and Real Rates of Return Foundations of Financial Markets

S Expected Return E ( r ) = p s Definition: p(s) = probability of a state r(s) = return if a state occurs 1 to s states E ( r ) = p s S Foundations of Financial Markets

E(r) = (.1)(-2) + (.2)(-1) + (.4)(0) + (.2)(1) + (.1)(2) = 0 Numerical Example State Prob Return 1 .1 -2 2 .2 -1 3 .4 4 5 E(r) = (.1)(-2) + (.2)(-1) + (.4)(0) + (.2)(1) + (.1)(2) = 0 Foundations of Financial Markets

Roadmap Rates of Return Summary Statistics of rates of return Holding Period Return Arithmetic and Geometric Averages Annual Percentage Rate and Effective Annual Rate Summary Statistics of rates of return Probability Distribution Expected Return Variance, Covariance and Standard Deviation Other properties Historical record of Bills, Bonds, and Stocks Risk premia from 1926-2003? Inflation and Real Rates of Return Foundations of Financial Markets

Why do we need the variance? Two variables with the same mean. What do we know about their dispersion? Foundations of Financial Markets

Measuring Variance or Dispersion of Returns Standard deviation = variance1/2 Variance = S s p ( ) [ r - E )] 2 Why do we take squared deviations? Foundations of Financial Markets

Numerical example State Prob Return 1 .1 -2 2 .2 -1 3 .4 4 5 4 5 Var = .1 (-2-0)2 + .2 (-1-0)2 + .4 (0-0)2 + .2 (1-0)2 + .1 (2-0)2 = 1.2 Std dev= (1.2)1/2 = 1.095 Foundations of Financial Markets

One important property of variance and standard deviation Let w be a constant Var(wxr) = w2 x Var(r) Similarly Std Dev(wxr) = w x Std Dev(r) Foundations of Financial Markets

Covariance: Preliminaries The extent at which two assets tend to move together Can be positive or negative Correlation Same idea of covariance, but bounded between -1 and 1 Foundations of Financial Markets

Covariance: definition Foundations of Financial Markets

Correlation: definition Foundations of Financial Markets

Correlation (cont’d) Foundations of Financial Markets

Other properties - Foundations of Financial Markets

Correlation=-1 r1 r2 probability 1 5 .2 2 4 3 Foundations of Financial Markets

Correlation=+1 r1 r2 probability 1 .2 2 3 4 5 Foundations of Financial Markets

Correlation=0 r1 r2 probability 2 .2 4 3 Foundations of Financial Markets

Roadmap Rates of Return Summary Statistics of rates of return Holding Period Return Arithmetic and Geometric Averages Annual Percentage Rate and Effective Annual Rate Summary Statistics of rates of return Probability Distribution Expected Return Variance, Covariance and Standard Deviation Other properties Historical record of Bills, Bonds, and Stocks Risk premia from 1926-2003? Inflation and Real Rates of Return Foundations of Financial Markets

Characteristics of Probability Distributions 1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics 1 and 2 Foundations of Financial Markets

Skewed Distribution: Large Negative Returns Possible Median Negative Positive r Foundations of Financial Markets

Skewed Distribution: Large Positive Returns Possible Median Negative r Positive Foundations of Financial Markets

Roadmap Rates of Return Summary Statistics of rates of return Holding Period Return Arithmetic and Geometric Averages Annual Percentage Rate and Effective Annual Rate Summary Statistics of rates of return Probability Distribution Expected Return Variance, Covariance and Standard Deviation Other properties Historical record of Bills, Bonds, and Stocks Risk premia from 1926-2003? Inflation and Real Rates of Return Foundations of Financial Markets

Risk premium An expected return in excess of that of a risk free rate Example The expected return on the S&P500 is 9% The return on a 1-month T-bill is 3% The risk premium is 6% (9%-3%) Foundations of Financial Markets

Annual Holding Period Returns From Table 5.3 of Text Geom. Arith. Stan. Series Mean% Mean% Dev.% World Stk 9.41 11.17 18.38 US Lg Stk 10.23 12.25 20.50 US Sm Stk 11.80 18.43 38.11 Wor Bonds 5.34 6.13 9.14 LT Treas 5.10 5.64 8.19 T-Bills 3.71 3.79 3.18 Inflation 2.98 3.12 4.35 Foundations of Financial Markets

Risk Premia Arith. Stan. Series Mean% Dev.% World Stk 7.37 18.69 US Lg Stk 8.46 20.80 US Sm Stk 14.64 38.72 Wor Bonds 2.34 8.98 LT Treas 1.85 8.00 Foundations of Financial Markets

Figure 5.1 Frequency Distributions of Holding Period Returns Foundations of Financial Markets

Figure 5.2 Rates of Return on Stocks, Bonds and Bills Foundations of Financial Markets

Roadmap Rates of Return Summary Statistics of rates of return Holding Period Return Arithmetic and Geometric Averages Annual Percentage Rate and Effective Annual Rate Summary Statistics of rates of return Probability Distribution Expected Return Variance, Covariance and Standard Deviation Other properties Historical record of Bills, Bonds, and Stocks Risk premia from 1926-2003? Inflation and Real Rates of Return Foundations of Financial Markets

Real vs. Nominal Rates Notation: Exact relationship R=nominal return i =inflation rate r =real return Exact relationship Approximate relationship Example R = 9%, i = 6%: what is r? Foundations of Financial Markets

Figure 5.4 Interest, Inflation and Real Rates of Return Foundations of Financial Markets