Finance 300 Financial Markets Lecture 3 Fall, 2001© Professor J. Petry

2 Chapter II-Portfolio Theory 1.Measuring Portfolio Risk & Return 2.Diversification 3.Capital Asset Pricing Model (CAPM) 4.Arbitrage Pricing Theory (APT)

3 –This measure of dispersion reflects the values of all the measurements. –The variance of a population of N measurements x 1, x 2,…,x N having a mean  is defined as –The variance of a sample of n measurements x 1, x 2, …,x n having a mean is defined as § The variance

4 Consider two small populations: Population A: 8, 9, 10, 11, 12 Population B: 4, 7, 10, 13, = = = = = = = = +6 Sum = 0 The mean of both populations is A B The sum of deviations is zero in both cases, therefore, another measure is needed.

= = = = = = = = +6 Sum = 0 A B The sum of deviations is zero in both cases, therefore, another measure is needed. The sum of squared deviations is used in calculating the variance.

6 Let us calculate the variance of the two populations Why is the variance defined as the average squared deviation? Why not use the sum of squared deviations as a measure of dispersion instead? After all, the sum of squared deviations increases in magnitude when the dispersion of a data set increases!!

7 Which data set has a larger dispersion? AB Data set B is more dispersed around the mean Sum A = (1-2) 2 +…+(1-2) 2 +(3-2) 2 + … +(3-2) 2 = 10 Sum B = (1-3) 2 + (5-3) 2 = 8 5 times However, when calculated on “per observation” basis (variance), the data set dispersions are properly ranked  A 2 = Sum A /N = 10/5 = 2  B 2 = Sum B /N = 8/2 = 4 !

8 –The standard deviation of a set of measurements is the square root of the variance of the measurements. – Example Rates of return over the past 10 years for two mutual funds are shown below. Which one have a higher level of risk? Fund A: 8.3, -6.2, 20.9, -2.7, 33.6, 42.9, 24.4, 5.2, 3.1, Fund B: 12.1, -2.8, 6.4, 12.2, 27.8, 25.3, 18.2, 10.7, -1.3, 11.4

9 –Solution Fund A should be considered riskier because its standard deviation is larger

10 Interpreting Standard Deviation The standard deviation can be used to –compare the variability of several distributions –make a statement about the general shape of a distribution. The empirical rule: If a sample of measurements has a mound-shaped distribution, the interval

11 And consequently with these intervals: For example, an actual data set with these characteristics: Mean = 10.26; Standard deviation = Interval Empirical Rule Actual percentage 5.97, %70% 1.68, %96.7% -2.61, %100% Interval Empirical Rule Actual percentage 5.97, %70% 1.68, %96.7% -2.61, %100%

12 Measures of Association Two numerical measures are presented, for the description of linear relationship between two variables depicted in the scatter diagram. –Covariance - is there any pattern to the way two variables move together? –Correlation coefficient - how strong is the linear relationship between two variables

13  x (  y ) is the population mean of the variable X (Y) N is the population size. n is the sample size. § The covariance

14 If the two variables move in two opposite directions, (one increases when the other one decreases), the covariance is a large negative number. If the two variables are unrelated, the covariance will be close to zero. If the two variables move the same direction, (both increase or both decrease), the covariance is a large positive number.

15 –This coefficient answers the question: How strong is the association between X and Y. § The coefficient of correlation

16 COV(X,Y)=0  or r = +1 0 Strong positive linear relationship No linear relationship Strong negative linear relationship or COV(X,Y)>0 COV(X,Y)<0

17 If the two variables are very strongly positively related, the coefficient value is close to +1 (strong positive linear relationship). If the two variables are very strongly negatively related, the coefficient value is close to -1 (strong negative linear relationship). No straight line relationship is indicated by a coefficient close to zero.

18 Benefits of Diversification An investor has two alternative investments Stocks return 17% w/ standard dev of 25% Bonds return 10% w/ standard dev of 12% She can choose any combination of these two investments in selecting her portfolio The return and risk characteristics of this portfolio are found by: 1.R p = w 1 R 1 + w 2 R 2   p = w 1 2    + w 2 2   2 +2w 1 w 2    1  2

19 Benefits of Diversification Using these two rules, find the risk and return for a portfolio that is 40% bonds and 60% stocks. Assume correlation coefficient of 1. Stocks return 17% w/ standard dev of 25% Bonds return 10% w/ standard dev of 12% R p = w 1 R 1 + w 2 R 2   (R p ) = w 1 2    + w 2 2   2 +2w 1 w 2    1  2

20 Benefits of Diversification

21 Benefits of Diversification Correlation Coefficient = 1

22 Benefits of Diversification Correlation Coefficient = 0

23 Benefits of Diversification Correlation Coefficient = -1

24 Benefits of Diversification The choice of portfolio will vary by individual, and would be enhanced by inclusion of a risk free alternative, but there are specific portfolios of particular interest. –Minimum variance portfolio is one such choice –w B(min) = (   s –  BS  B  S )/(  B 2 +  S 2 -2  BS  B  S ) –Extra credit for calculation of minimum variance portfolio weights (bonds vs. stocks) for all three pairs of correlation coefficients

25 Capital Asset Pricing Model (CAPM) First developed by Harry Markowitz in the 1950s Is now a fundamental underlying principle of financial analysis, and earned him a Nobel Prize Attempts to quantify the relationship between risk and return States that risk is either market risk (systematic risk, undiversifiable risk), or firm specific risk (diversifiable risk). Because individuals can diversify away firm specific risk, the only thing they should care about is market risk.

26 Capital Asset Pricing Model (CAPM) Finding “Beta” is easily done using simple linear regression. The risk of a stock is fully defined by its “Beta”, and the return to a stock should be directly proportional to its Beta.

27 Risk Prem Market Risk Prem or Index Risk Prem or Index Risk Prem i = the stock’s expected return if the market’s excess return is zero market’s excess return is zero ß i (r m - r f ) = the component of return due to movements in the market index movements in the market index (r m - r f ) = 0 e i = firm specific component, not due to market movements movements  Capital Asset Pricing Model (CAPM)