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# Risk And Return Of Security And Portfolio

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### Risk And Return Of Security And Portfolio

1. 1. Risk and Return
2. 2. Holding Period Return <ul><li>Three month ago, Peter Lynch purchased 100 </li></ul><ul><li>shares of Iomega Corp. at \$50 per share. Last </li></ul><ul><li>month, he received dividends of \$0.25 per </li></ul><ul><li>share from Iomega. These shares are worth </li></ul><ul><li>\$56 each today. </li></ul><ul><li>Compute Peter’s holding period return from his investment in Iomega common shares. </li></ul>
3. 3. Probability Concept <ul><li>Random variable </li></ul><ul><ul><li>Something whose value in the future is subject to uncertainty. </li></ul></ul><ul><li>Probability </li></ul><ul><ul><li>The relative likelihood of each possible outcome (or value) of a random variable </li></ul></ul><ul><ul><li>Probabilities of individual outcomes cannot be negative nor greater than 1.0 </li></ul></ul><ul><ul><li>Sum of the probabilities of all possible outcomes must equal 1.0 </li></ul></ul><ul><li>Moments </li></ul><ul><ul><li>Mean, Variance (or Standard deviation), covariance </li></ul></ul>
4. 4. Computing the Basic Statistics A security analyst has prepared the following probability distribution of the possible returns on the common stock shares of two companies: Compu-Graphics Inc. (CGI) and Data Switch Corp. (DSC).
5. 5. The Mean For CGI, the mean (or expected) return is: Similarly, the mean return for DSC is 24.00%
6. 6. The Variance and Standard Deviation The variance of CGI’s returns is: The Standard Deviation of CGI’s return is:
7. 7. The Covariance The covariance of the returns on CGI and DSC is:
8. 8. The Correlation Coefficient The correlation coefficient between CGI and DSC is:
9. 9. Summary of Results for CGI and DSC
10. 10. <ul><li>A portfolio is a combination of two or more securities. </li></ul><ul><li>Combining securities into a portfolio reduces risk. </li></ul><ul><li>An efficient portfolio is one that has the highest expected return for a given level of risk. </li></ul><ul><li>We will look at two-asset portfolios in fair detail. </li></ul>Portfolio Securities
11. 11. Portfolio Expected Return and Risk Expected Return Risk The Expected Returns of the Securities The Portfolio Weights The Risk of the Securities The Portfolio Weights The Correlation Coefficients
12. 12. Portfolio Weights and Expected Return
13. 13. Portfolio Expected Return and Risk
14. 14. Diversification of Risk <ul><li>Note that while the expected return of the portfolio is between those of CGI and DSC, its risk is less than either of the two individual securities. </li></ul><ul><li>Combining CGI and DSC results in a substantial reduction of risk - diversification! </li></ul><ul><li>This benefit of diversification stems primarily from the fact that CGI and DSC’s returns are not perfectly correlated . </li></ul>
15. 15. <ul><li>All else being the same, lower the correlation coefficient, lower is the risk of the portfolio. </li></ul><ul><ul><li>Recall that the expected return of the portfolio is not affected by the correlation coefficient. </li></ul></ul><ul><li>Thus, lower the correlation coefficient, greater is the diversification of risk. </li></ul>Correlation Coefficient and Portfolio Risk
16. 16. Consider stocks of two companies, X and Y. The table below gives their expected returns and standard deviations . Plot the risk and expected return of portfolios of these two stocks for the following (assumed) correlation coefficients: -1.0 0.5 0.0 +0.5 +1.0 Correlation Coefficient and Portfolio Risk
17. 17. Correlation Coefficient and Portfolio Risk Y Correlation Coefficient -1.0 -0.5 0.0 +0.5 +1.0 X
18. 18. Portfolios with Many Assets <ul><li>The above framework can be expanded to the case of portfolios with a large number of stocks. </li></ul><ul><li>In forming each portfolio, we can vary </li></ul><ul><ul><li>the number of stocks that make up the portfolio, </li></ul></ul><ul><ul><li>the identity of the stocks in the portfolio, and </li></ul></ul><ul><ul><li>the weights assigned to each stock. </li></ul></ul><ul><li>Look at the plot of the expected returns versus the risk of these portfolios </li></ul>
19. 19. All Combinations of Risky Assets
20. 20. Efficient Frontier <ul><li>A portfolio is an efficient portfolio if </li></ul><ul><ul><li>no other portfolio with the same expected return has lower risk, or </li></ul></ul><ul><ul><li>no other portfolio with the same risk has a higher expected return. </li></ul></ul><ul><li>Investors prefer efficient portfolios over inefficient ones. </li></ul><ul><li>The collection of efficient portfolio is called an efficient frontier . </li></ul>
21. 21.  (risk) Efficient Frontier  (expected return) F E
22. 22. Choosing the Best Risky Asset <ul><li>Investors prefer efficient portfolios over inefficient ones. </li></ul><ul><li>Which one of the efficient portfolios is best? </li></ul><ul><li>We can answer this by introducing a riskless asset . </li></ul><ul><ul><li>There is no uncertainty about the future value of this asset (i.e. the standard deviation of returns is zero). Let the return on this asset be r f . </li></ul></ul><ul><ul><li>For practical purposes, 90-day U.S. Treasury Bills are (almost) risk free. </li></ul></ul>
23. 23. Combinations of a Risk Free and a Risky Asset  (risk)  (expected return) F E N r f
24. 24. Best Risky Asset  (expected return)  (risk) F E M r f
25. 25. The Capital Market Line <ul><li>Assume investors can lend and borrow at the risk free rate of interest. </li></ul><ul><ul><li>borrowing entails a negative investment in the riskless asset. </li></ul></ul><ul><li>Since every investor hold a part of the “best” risky asset M, M is the market portfolio. </li></ul><ul><li>The Market portfolio consists of all risky assets. </li></ul><ul><ul><li>Each asset weight is proportional to its market value. </li></ul></ul>
26. 26. The Capital Market Line Sharpe Ratio
27. 27. The Capital Market Line r f  (expected return)  (risk) F E M
28. 28. <ul><li>Explain the importance of asset pricing models. </li></ul><ul><li>Demonstrate choice of an investment position on the Capital Market Line (CML). </li></ul><ul><li>Understand the Capital Asset Pricing Model (CAPM), Security Market Line (SML) and its uses. </li></ul>Next Coverage <ul><li>Understand the determination of the expected rate of return  Capital Asset Pricing Model </li></ul><ul><li>Decomposition of Risk: Systematic Vs. Unsystematic. </li></ul>
29. 29. Asset Pricing Models <ul><li>These models provide a relationship between an asset’s required rate of return and its risk . </li></ul><ul><li>The required return can be used for: </li></ul><ul><ul><li>computing the NPV of your investment. </li></ul></ul>
30. 30. Individual’s Choice on the CML  (risk)  (risk)
31. 31. The Capital Asset Pricing Model (CAPM) <ul><li>It allows us to determine the required rate of return (=expected return) for an individual security. </li></ul><ul><ul><li>Individual securities may not lie on the CML. </li></ul></ul><ul><ul><li>Only efficient portfolios lie on the CML </li></ul></ul><ul><li>The Security Market Line ( SML ) can be applied to any securities or portfolios including inefficient ones. </li></ul>
32. 32. The Security Market Line (SML) <ul><li>where </li></ul>
33. 33. What does the SML tell us <ul><li>The required rate of return on a security depends on: </li></ul><ul><ul><li>the risk free rate </li></ul></ul><ul><ul><li>the “beta” of the security, and </li></ul></ul><ul><ul><li>the market price of risk. </li></ul></ul><ul><li>The required return is a linear function of the beta coefficient. </li></ul><ul><ul><li>All else being the same, higher the beta coefficient, higher is the required return on the security. </li></ul></ul>
34. 34. Graphical Representation of the SML
35. 35. Computing Required Rates of Return <ul><li>Common stock shares of Gator Sprinkler Systems (GSS) have a correlation coefficient of 0.80 with the market portfolio, and a standard deviation of 28%. The expected return on the market portfolio is 14%, and its standard deviation is 20%. The risk free rate is 5%. </li></ul><ul><li>What is the required rate of return on GSS? </li></ul>
36. 36. Required Return on GSS First compute the beta of GSS: Next, apply the SML:
37. 37. Required Rate of Return on GSS <ul><li>What would be the required rate of return on GSS if it had a correlation of 0.50 with the market? (All else is the same) </li></ul><ul><ul><li>Beta = 0.70 and  GSS = 11.30% </li></ul></ul><ul><li>What would be the required rate of return on GSS if it had a standard deviation of 36%, and a correlation of 0.80? (All else is the same) </li></ul><ul><ul><li>Beta = 1.44 and  GSS = 17.96% </li></ul></ul>
38. 38. Estimating the Beta Coefficient <ul><li>If we know the security’s correlation with the </li></ul><ul><li>market, its standard deviation, and the standard </li></ul><ul><li>deviation of the market, we can use the </li></ul><ul><li>definition of beta: </li></ul><ul><li>Generally, these quantities are not known. </li></ul><ul><li>We therefore rely on their historical values to provide us with an estimate of beta. </li></ul>
39. 39. Interpreting the Beta Coefficient The beta of the market portfolio is always equal to 1.0. The beta of the risk free asset is always equal to 0.0
40. 40. Interpreting the Beta Coefficient <ul><li>Beta indicates how sensitive a security’s returns are to changes in the market portfolio’s return . </li></ul><ul><ul><li>It is a measure of the asset’s risk. </li></ul></ul><ul><li>Suppose the market portfolio’s risk premium is +10% during a given period. </li></ul><ul><ul><li>if  = 1.50, the security’s risk premium will be +15%. </li></ul></ul><ul><ul><li>if  = 1.00, the security’s risk premium will be +10% </li></ul></ul><ul><ul><li>if  = 0.50, the security’s risk premium will be +5% </li></ul></ul><ul><ul><li>if  = - 0.50, the security’s risk premium will be - 5% </li></ul></ul>
41. 41. Beta Coefficients for Selected Firms
42. 42. Beta of a Portfolio <ul><li>The beta of a portfolio is the weighted average of the beta values of the individual securities in the portfolio. </li></ul><ul><li>where w i is the proportion of value invested in security i, and  i is the beta of the security i. </li></ul>
43. 43. Applying the CAPM <ul><li>The CML prescribes that investors should invest in the riskless asset and the market portfolio. </li></ul><ul><li>The true market portfolio, which consists of all risky assets, cannot be constructed. </li></ul><ul><li>How much diversification is necessary to get substantially “all” of the benefits of diversification? </li></ul><ul><ul><li>About 25 to 30 stocks! </li></ul></ul>