Calculating Portfolio Returns: Expected Return, Volatility, and Systematic Risk

1. Chapter 12 Systematic Risk and the Equity Risk Premium

2. Chapter Outline 12.1 The Expected Return of a Portfolio 12.2 The Volatility of a Portfolio 12.3 Measuring Systematic Risk 12.4 Putting it All Together: The Capital Asset Pricing Model

3. Learning Objectives Calculate the expected return and volatility (standard deviation) of a portfolio Understand the relation between systematic risk and the market portfolio Measure systematic risk Use the Capital Asset Pricing Model (CAPM) to compute the cost of equity capital for a stock

4. 12.1 The Expected Return of a Portfolio In Chapter 11 we found: For large portfolios, investors expect higher returns for higher risk. The same does not hold true for individual stocks. Stocks have both unsystematic and systematic risk only systematic risk is rewarded rational investors should choose to diversify.

5. 12.1 The Expected Return of a Portfolio Portfolio weights The fraction of the total portfolio held in each investment in the portfolio: Portfolio weights add up to 100% (that is, w1 + w2 + … + wN = 100%) (Eq. 12.1)

6. 12.1 The Expected Return of a Portfolio Portfolio weights for a portfolio of 200 shares of Apple at $200 per share and 100 shares of Coca-Cola at $60 per share:

7. 12.1 The Expected Return of a Portfolio The return on a portfolio, Rp The weighted average of the returns on the investments in the portfolio: (Eq. 12.2)

8. Example 12.1 Calculating Portfolio Returns Problem: Suppose you invest $100,000 and buy 200 shares of Apple at $200 per share ($40,000) and 1000 shares of Coca-Cola at $60 per share ($60,000). If Apple’s stock goes up to $240 per share and Coca-Cola stock falls to $57 per share and neither paid dividends, what is the new value of the portfolio? What return did the portfolio earn?

9. Example 12.1 Calculating Portfolio Returns Problem (cont’d): Show that Eq. 12.2 is true by calculating the individual returns of the stocks and multiplying them by their weights in the portfolio. If you don’t buy or sell any shares after the price change, what are the new portfolio weights?

10. Example 12.1 Calculating Portfolio Returns Solution: Plan: Your portfolio: 200 shares of Apple: $200  $240 ($40 capital gain per share) 1000 shares of Coca-Cola: $60  $57 ($3 capital loss per share)

11. Example 12.1 Calculating Portfolio Returns Plan (cont’d): To calculate the return on your portfolio, compute its value using the new prices and compare it to the original $100,000 investment. To confirm that Eq. 12.2 is true, compute the return on each stock individually using Eq. 11.1 from Chapter 11, multiply those returns by their original weights in the portfolio, and compare your answer to the return you just calculated for the portfolio as a whole.

12. Example 12.1 Calculating Portfolio Returns Execute: The new value of your Apple stock is 200 × $240 = $48,000 and the new value of your Coke stock is 1000 × $57 = $57,000. So, the new value of your portfolio is $48,000 +57,000 = $105,000, for a gain of $5000 or a 5% return on your initial $100,000 investment. Since neither stock paid any dividends, we calculate their returns simply as the capital gain or loss divided by the purchase price. The return on Apple stock was $40/$200 = 20%, and the return on Coca-Cola stock was -$3/$60 = -5%.

13. Example 12.1 Calculating Portfolio Returns Execute (cont’d): The initial portfolio weights were $40,000/$100,000 = 40% for Apple and $60,000/$100,000 = 60% for Coca-Cola, so we can also compute the return of the portfolio from Eq. 12.2 as

14. Example 12.1 Calculating Portfolio Returns Execute (cont’d): After the price change, the new portfolio weights are equal to the value of your investment in each stock divided by the new portfolio value:

15. Example 12.1 Calculating Portfolio Returns Evaluate: The $3000 loss on your investment in Coca-Cola was offset by the $8000 gain in your investment in Apple, for a total gain of $5,000 or 5%. The same result comes from giving a 40% weight to the 20% return on Apple and a 60% weight to the -5% loss on Coca-Cola—you have a total net return of 5%.

16. Example 12.1 Calculating Portfolio Returns Evaluate (cont’d): After a year, the portfolio weight on Apple has increased and the weight on Coca-Cola has decreased. Note that without trading, the portfolio weights will increase for the stock(s) in the portfolio whose returns are above the overall portfolio return.

17. Example 12.1 Calculating Portfolio Returns Evaluate (cont’d): The charts below show the initial and ending weights on Apple (shown in yellow) and Coca-Cola (shown in red).

18. Example 12.1a Calculating Portfolio Returns Problem: Suppose you invest $158,000 and buy 2,000 shares of Microsoft at $25 per share ($50,000) and 3,000 shares of Pepsi at $36 per share ($108,000). If Microsoft’s stock goes up to $33 per share and Pepsi stock falls to $32 per share and neither paid dividends, what is the new value of the portfolio? What return did the portfolio earn? Show that Eq. 11.2 is true by calculating the individual returns of the stocks and multiplying them by their weights in the portfolio. If you don’t buy or sell any shares after the price change, what are the new portfolio weights?

19. Example 12.1a Calculating Portfolio Returns Solution: Plan: Your portfolio: 2,000 shares of MSFT: $25  $33 ($8 capital gain) 3,000 shares of PEP: $36  $32 ($4 capital loss)

20. Example 12.1a Calculating Portfolio Returns Plan (cont’d): To calculate the return on your portfolio, compute its value using the new prices and compare it to the original $158,000 investment. To confirm that Eq. 12.2 is true, compute the return on each stock individually using Eq. 11.1 from Chapter 11, multiply those returns by their original weights in the portfolio, and compare your answer to the return you just calculated for the portfolio as a whole.

21. Example 12.1a Calculating Portfolio Returns Execute: The new value of your Microsoft stock is 2,000 × $33 = $66,000 and the new value of your Pepsi stock is 3,000 × $32 = $96,000. So, the new value of your portfolio is $66,000 +96,000 = $162,000, for a gain of $4,000 or a 2.5% return on your initial $158,000 investment.

22. Example 12.1a Calculating Portfolio Returns Execute (cont’d): Since neither stock paid any dividends, we calculate their returns as the capital gain or loss divided by the purchase price. The return on Microsoft stock was $8/$25 = 32%, and the return on Pepsi stock was -$4/$36 = -11.1%. The initial portfolio weights were $50,000/$158,000 = 31.6% for Microsoft and $108,000/$158,000 = 68.4% for Pepsi, so we can also compute portfolio return from Eq. 11.2 as

23. Example 12.1a Calculating Portfolio Returns Execute (cont’d): After the price change, the new portfolio weights are equal to the value of your investment in each stock divided by the new portfolio value:

24. Example 12.1a Calculating Portfolio Returns Evaluate: The $12,000 loss on your investment in Pepsi was offset by the $16,000 gain in your investment in Microsoft, for a total gain of $4,000 or 2.5%. The same result comes from giving a 31.6% weight to the 32% return on Microsoft and a 68.4% weight to the -11.1% loss on Pepsi—you have a total net return of 2.5%.

25. 12.1 The Expected Return of a Portfolio The expected return of a portfolio The weighted average of the expected returns of the investments within it, using the portfolio weights: (Eq. 12.3)

26. Table 12.1 Summary of Portfolio Concepts

27. Example 12.2 Portfolio Expected Return Problem: Suppose you invest $10,000 in Boeing (BA) stock, and $30,000 in Merck (MRK) stock. You expect a return of 10% for Boeing, and 16% for Merck. What is the expected return for your portfolio?

28. Example 12.2 Portfolio Expected Return Solution: Plan: You have a total of $40,000 invested: $10,000/$40,000 = 25% in Boeing: E[RF]=10% $30,000/$40,000 = 75% in Merck: E[RTYC]=16% Using Eq. 12.3, compute the expected return on your whole portfolio by multiplying the expected returns of the stocks in your portfolio by their respective portfolio weights.

29. Example 12.2 Portfolio Expected Return Execute: The expected return on your portfolio is:

30. Example 12.2 Portfolio Expected Return Evaluate: The importance of each stock for the expected return of the overall portfolio is determined by the relative amount of money you have invested in it. Most (75%) of your money is invested in Merck, so the overall expected return of the portfolio is much closer to Merck’s expected return than it is to Boeing’s.

31. Example 12.2a Portfolio Expected Return Problem: Suppose you invest $20,000 in Citigroup (C) stock, and $80,000 in General Electric (GE) stock. You expect a return of 18% for Citigroup, and 14% for GE. What is the expected return for your portfolio?

32. Example 12.2a Portfolio Expected Return Solution: Plan: You have a total of $100,000 invested: $20,000/$100,000 = 20% in Citigroup: E[RC]=18% $80,000/$100,000 = 80% in GE: E[RGE]=14% Using Eq. 11.3, compute the expected return on your whole portfolio by weighting the expected returns of the stocks in your portfolio by their portfolio weights.

33. Example 12.2a Portfolio Expected Return Execute: The expected return on your portfolio is:

34. Example 12.2a Portfolio Expected Return Evaluate The importance of each stock for the expected return of the overall portfolio is determined by the relative amount of money you have invested in it. Most (80%) of your money is invested in GE, so the overall expected return of the portfolio is much closer to GE’s expected return than it is to Citigroup’s.

35. 12.2 The Volatility of a Portfolio Investors care about return, but also risk When we combine stocks in a portfolio, some risk is eliminated through diversification. Remaining risk depends upon the degree to which the stocks share common risk. The volatility of a portfolio is the total risk, measured as standard deviation, of the portfolio.

36. 12.2 The Volatility of a Portfolio Table 12.2 shows returns for three hypothetical stocks, along with their average returns and volatilities. Note that while the three stocks have the same volatility and average return, the pattern of returns differs. When the airline stocks performed well, the oil stock did poorly, and when the airlines did poorly, the oil stock did well.

37. Table 12.2 Returns for Three Stocks, and Portfolios of Pairs of Stocks

38. 12.2 The Volatility of a Portfolio Table 12.2 shows returns for two portfolios: An equal investment in the two airlines, North Air and West Air. An equal investment in West Air and Tex Oil. Average return of both portfolios is equal to the average return of the stocks Volatilities (standard deviations) are very different.

39. Figure 12.1 Volatility of Airline and Oil Portfolios

40. 12.2 The Volatility of a Portfolio This example demonstrates two important truths. By combining stocks into a portfolio, we reduce risk through diversification. The amount of risk that is eliminated depends upon the degree to which the stocks move together. Combining airline stocks reduces volatility only slightly compared to the individual stocks. Combining airline and oil stocks reduces volatility below that of either stock.

41. 12.2 The Volatility of a Portfolio Measuring Stocks’ Co-movement: Correlation To find the risk of a portfolio, we need to know The risk of the component stocks The degree to which they move together Correlation ranges from ‑1 to +1, and measures the degree to which the returns share common risk.

42. Figure 12.2 Correlation

43. 12.2 The Volatility of a Portfolio Correlation is scaled covariance and is defined as

44. 12.2 The Volatility of a Portfolio Stock returns tend to move together if they are affected similarly by economic events. Stocks in the same industry tend to have more highly correlated returns than stocks in different industries. Table 12.3 shows several stocks’ Volatility of individual stock returns Correlation between them The table can be read across rows or down columns.

45. Table 12.3 Estimated Annual Volatilities and Correlations for Selected Stocks. (Based on Monthly Returns, June 2002- May 2010)

46. Figure 12.3 Scatterplots of Returns

47. 12.2 The Volatility of a Portfolio Computing a Portfolio’s Variance and Standard Deviation The formula for the variance of a two-stock portfolio is: (Eq. 12.4)

48. 12.2 The Volatility of a Portfolio The three parts of Eq. 12.4 each account for an important determinant of the overall variance of the portfolio: the risk of stock 1 the risk of stock 2 an adjustment for how much the two stocks move together (their correlation, given as Corr(R1,R2)).

49. 12.2 The Volatility of a Portfolio Expected return of a portfolio is equal to the weighted average expected return of its stocks. Risk of the portfolio is lower than the weighted average of the individual stocks’ volatility, unless all the stocks all have perfect positive correlation with each other Diversification

50. Example 12.3 Computing the Volatility of a Two-Stock Portfolio Problem: Using the data from Table 12.3, what is the volatility (standard deviation) of a portfolio with equal amounts invested in Dell and Microsoft stock? What is the standard deviation of a portfolio with equal amounts invested in Dell and Target?