1. Principles of Managerial Finance 9th Edition Chapter 6 Risk and Return

2. Learning Objectives Understand the meaning and fundamentals of risk, return, and risk aversion. Describe procedures for measuring the risk of a single asset. Discuss the measurement of return and standard deviation for a portfolio and the various types of correlation that can exist between series of numbers. Understand the risk and return characteristics of a portfolio in terms of correlation and diversification and the impact of international assets on a portfolio.

3. Learning Objectives Review the two types of risk and the derivation and role of beta in measuring the relevant risk of both an individual security and a portfolio. Explain the capital asset pricing model (CAPM) and its relationship to the security market line (SML), and shifts in the SML caused by changes in inflationary expectations and risk aversion.

4. Introduction If everyone knew ahead of time how much a stock would sell for some time in the future, investing would be simple endeavor. Unfortunately, it is difficult -- if not impossible -- to make such predictions with any degree of certainty. As a result, investors often use history as a basis for predicting the future. We will begin this chapter by evaluating the risk and return characteristics of individual assets, and end by looking at portfolios of assets.

5. Risk Defined In the context of business and finance, risk is defined as the chance of suffering a financial loss. Assets (real or financial) which have a greater chance of loss are considered more risky than those with a lower chance of loss. Risk may be used interchangeably with the term uncertainty to refer to the variability of returns associated with a given asset.

6. Return Defined Return represents the total gain or loss on an investment. The most basic way to calculate return is as follows: k t = P t - P t-1 + C t P t-1 Where k t is the actual, required or expected return during period t, P t is the current price, P t-1 is the price during the previous time period, and C t is any cash flow accruing from the investment

7. Chapter Example

8. Single Financial Assets Arithmetic Average The historical average (also called arithmetic average or mean) return is simple to calculate. The accompanying text outlines how to calculate this and other measures of risk and return. All of these calculations were discussed and taught in your introductory statistics course. This slideshow will demonstrate the calculation of these statistics using EXCEL. Historical Return

9. Single Financial Assets Arithmetic Average Historical Return What you typeWhat you see

10. Single Financial Assets Variance Historical risk can be measured by the variability of its returns in relation to its average. Variance is computed by summing squared deviations and dividing by n-1. Squaring the differences ensures that both positive and negative deviations are given equal consideration. The sum of the squared differences is then divided by the number of observations minus one. Historical Risk

11. Single Financial Assets Variance Historical Risk

12. Single Financial Assets Variance Historical Risk

13. Single Financial Assets Variance Historical Risk What you type What you see

14. Single Financial Assets Standard Deviation Squaring the deviations makes the variance difficult to interpret. In other words, by squaring percentages, the resulting deviations are in percent squared terms. The standard deviation simplifies interpretation by taking the square root of the squared percentages. In other words, standard deviation is in the same units as the computed average. If the average is 10%, the standard deviation might be 20%, whereas the variance would be 20% squared. Historical Risk

15. Single Financial Assets Standard Deviation Historical Risk What you typeWhat you see

16. Single Financial Assets Normal Distribution Historical Risk R-2  R-1  R+2  R+1  R 68% 95%

17. Single Financial Assets Investors and analysts often look at historical returns as a starting point for predicting the future. However, they are much more interested in what the returns on their investments will be in the future. For this reason, we need a method for estimating future or “ex-ante” returns. One way of doing this is to assign probabilities for future states of nature and the returns that would be realized if a particular state of nature would occur. Expected Return & Risk

18. Single Financial Assets Expected Return & Risk optimistic Most likely pessimistic

19. Single Financial Assets Expected Return & Risk

20. Single Financial Assets

21. Expected Return E(R) =  piRi, where pi = probability of the ith scenario, and Ri = the forecasted return in the ith scenario. Expected Return & Risk Also, the variance of E(R) may be computed as: and the standard deviation as:

22. Expected Return & Risk Single Financial Assets

23. Expected Return & Risk

24. Risk-Averse Risk-Indifferent Risk-Seeking Averse Indifferent Seeking Risk x 1 x 2 Expected Return

25. Coefficient of Variation Single Financial Assets One problem with using standard deviation as a measure of risk is that we cannot easily make risk comparisons between two assets. The coefficient of variation overcomes this problem by measuring the amount of risk per unit of return. The higher the coefficient of variation, the more risk per return. Therefore, if given a choice, an investor would select the asset with the lower coefficient of variation. C.V.

26. Single Financial Assets Coefficient of Variation CV = Standard Deviation / Expected Return

27. Portfolios of Assets An investment portfolio is any collection or combination of financial assets. If we assume all investors are rational and therefore risk averse, that investor will ALWAYS choose to invest in portfolios rather than in single assets. Investors will hold portfolios because he or she will diversify away a portion of the risk that is inherent in “putting all your eggs in one basket.” If an investor holds a single asset, he or she will fully suffer the consequences of poor performance. This is not the case for an investor who owns a diversified portfolio of assets.

28. Portfolios of Assets Diversification is enhanced depending upon the extent to which the returns on assets “move” together. This movement is typically measured by a statistic known as “correlation” as shown in Figure 6.5 and 6.6.

29. Portfolios of Assets Negative correlation between F and G

30. Portfolios of Assets Recall Stocks A and B k A =8%k B =20% σ AB =∑Pi[ R Ai -E(R Ai )]*[ R Bi -E(R Bi )] = E{[R A -E(R A )]*[R B -E(R B )]}

31. Portfolios of Assets Portfolio AB (50% in A, 50% in B) k A =8%k B =20%K p =0.5×8%+0.5×20%=14%

32. Portfolios of Assets Portfolio AB (50% in A, 50% in B) Where the contents of cell B12 and B13 = 50% in this case. Here are cells B12 and B13

33. Portfolios of Assets Portfolio AB (50% in A, 50% in B)

34. Portfolios of Assets Portfolio AB (40% in A, 60% in B) Changing the weights K p =0.4×8%+0.6×20%=15.2%

35. Portfolios of Assets Portfolio AB (20% in A, 80% in B) And Again K p =0.2×8%+0.8×20%=17.6%

36. Portfolios of Assets Portfolio Risk & Return Summarizing changes in risk and return as the composition of the portfolio changes. ρ AB *σ A * σ B = σ AB, ρ AB =0.96

37. 若投資組合有 3 個資產，則 σ p 2 = W A 2 *σ A 2 +W B 2 *σ B 2 +W C 2 *σ C 2 +2W A *W B *σ AB 2W A *W C *σ AC +2W B *W C *σ BC 若投資組合有 N 個資產，則 σ p 2 = ∑ W i 2 *σ i 2 + ∑∑W i *W j *σ ij for i≠j,If W i =1/N = ∑(1/N) 2 *σ i 2 + ∑∑ (1/N)*(1/N)*σ ij for i≠j = (1/N)*∑[(1/N)*σ i 2 ]+[(N-1)/N]*∑∑{1/[N*(N-1)]*σ ij } for i≠j = (1/N)* σ i 2 + [(N-1)/N]* σ ij = (1/N)*(σ i 2 - σ ij )+σ ij σ p 2 =lim [(1/N)*(σ i 2 - σ ij )+ σ ij ] = σ ij (N ∞) σ ij = 系統風險, (1/N)*(σ i 2 - σ ij ) = 非系統風險

38. Portfolios of Assets Portfolio Risk & Return (Perfect Negative Correlation)

39. Portfolios of Assets Portfolio Risk & Return (Perfect Negative Correlation) Notice that if we weight the portfolio just right (50/50 in this case), we can completely eliminate risk.

40. Portfolios of Assets Portfolio Risk (Adding Assets to a Portfolio) 0 # of Stocks Systematic (non-diversifiable) Risk Unsystematic (diversifiable) Risk Portfolio Risk (SD) SD M Firm-specific risk market Market risk

41. Portfolios of Assets Portfolio Risk (Adding Assets to a Portfolio) 0 # of Stocks Portfolio of both Domestic and International Assets Portfolio of Domestic Assets Only Portfolio Risk (SD) SD M

42. Portfolios of Assets Capital Asset Pricing Model (CAPM) If you notice in the last slide, a good part of a portfolio’s risk (the standard deviation of returns) can be eliminated simply by holding a lot of stocks. The risk you can’t get rid of by adding stocks (systematic) cannot be eliminated through diversification because that variability is caused by events that affect most stocks similarly. Examples would include changes in macroeconomic factors such interest rates, inflation, and the business cycle.

43. Portfolios of Assets Capital Asset Pricing Model (CAPM) In the early 1960s, researchers (Sharpe, Treynor, and Lintner) developed an asset pricing model that measures only the amount of systematic risk a particular asset has. In other words, they noticed that most stocks go down when interest rates go up, but some go down a whole lot more. They reasoned that if they could measure this variability -- the systematic risk -- then they could develop a model to price assets using only this risk. The unsystematic (company-related) risk is irrelevant because it could easily be eliminated simply by diversifying.

44. Portfolios of Assets Capital Asset Pricing Model (CAPM) To measure the amount of systematic risk an asset has, they simply regressed the returns for the “market portfolio” -- the portfolio of ALL assets -- against the returns for an individual asset. The slope of the regression line -- beta -- measures an assets systematic (non-diversifiable) risk. In general, cyclical companies like auto companies have high betas while relatively stable companies, like public utilities,have low betas. Let’s look at an example to see how this works.

45. Portfolios of Assets Capital Asset Pricing Model (CAPM) We will demonstrate the calculation using the regression analysis feature in EXCEL. B i =σ im /σ m 2, B p =∑ W i *B i

46. Portfolios of Assets Capital Asset Pricing Model (CAPM) This slide is the result of a regression using the Excel. The slope of the regression (beta) in this case is 1.92. Apparently, this stock has a considerable amount of systematic risk

47. 在 Excel 計算 β( 可使用統計函數 ) =SLOPE(C3:C8,B3:B8) =1.92

48. Portfolios of Assets Capital Asset Pricing Model (CAPM)

49. Portfolios of Assets Capital Asset Pricing Model (CAPM)

50. Portfolios of Assets Capital Asset Pricing Model (CAPM)

51. The required return for all assets is composed of two parts: the risk-free rate and a risk premium. The risk-free rate (r f ) is usually estimated from the return on US T-bills The risk premium is a function of both market conditions and the asset itself. Portfolios of Assets Capital Asset Pricing Model (CAPM) rfrf

52. The risk premium for a stock is composed of two parts: –The Market Risk Premium which is the return required for investing in any risky asset rather than the risk-free rate –Beta, a risk coefficient which measures the sensitivity of the particular stock’s return to changes in market conditions. Portfolios of Assets Capital Asset Pricing Model (CAPM)

53. Portfolios of Assets Capital Asset Pricing Model (CAPM) After estimating beta, which measures a specific asset’s systematic risk, relatively easy to estimate variables may be obtained to calculate an asset’s required return.. E(R i ) = R f + B i [E(Rm) - R f ], where E(Ri) = an asset’s expected or required return, R F = the risk free rate of return, B i = an asset or portfolio’s beta E(Rm) = the expected return on the market portfolio.

54. Portfolios of Assets Capital Asset Pricing Model (CAPM) Example Calculate the required return for Federal Express assuming it has a beta of 1.25, the rate on US T-bills is 5.07%, and the expected return for the S&P 500 is 15%. E(Ri) = 5.07 + 1.25 [15% - 5.07%] E(Ri) = 17.48%

55. Portfolios of Assets Capital Asset Pricing Model (CAPM) Graphically E(Ri) beta r f = 5.07% 1.251.0 15.0% 17.48%

56. Portfolios of Assets Capital Asset Pricing Model (CAPM)

57. k% B 20 15 10 5 12MSFTFPL SML Portfolios of Assets Capital Asset Pricing Model (CAPM)

58. k% B 20 15 10 5 12 Shift due to change in market return from 12% to 15% FPLMSFT SML2 SML1 Portfolios of Assets Capital Asset Pricing Model (CAPM) 若市場所有投資人變得更 risk averse ，則 E(r m )-r f 會 增加，而 r f 不變，所以斜率 增加，而截距不變

59. k% B 20 15 10 5 12 Shift due to change in risk-free rate from 5% to 8%. Note that all returns will increase by 3% MSFTFPL SML2 SML1 Portfolios of Assets Capital Asset Pricing Model (CAPM) 若因 expected inflation 使 得 r f 3% ，則 E(r m ) 也會 3% ，所以 E(r m )-r f 不會變， i.e. 斜率不變，但截距

60. 歷史資料 ( 假設有 N 期資料 ) 單一資產： 平均報酬率： r = ∑ r i /N 變異數： σ 2 = ∑(r i -r) 2 / (N-1) 兩個資產 A 和 B ： 共變數： σ AB = ∑(r Ai -r A )*(r Bi -r B ) / (N-1) 相關係數： ρ AB = σ AB /(σ A *σ B ) 且 ρ 介於正負 1 之間

61. 預期未來 ( 假設未來有 N 種狀態 ) 單一資產： 平均報酬率： E (r) = ∑ P i *r i 變異數： σ 2 = ∑ [(r i -E(r)] 2 / P i 兩個資產 A 和 B ： 共變數： σ AB = ∑P i * [(r Ai - E (r A )]*[(r Bi - E (r B )] 相關係數： ρ AB = σ AB / (σ A *σ B ) 且 ρ 介於正負 1 之間

62. n 個資產所形成之投資組合： 歷史資料 r P = ∑W i * r i, 預期未來 E(r P ) = ∑ W i *E(r i ) σ p 2 = ∑ W i 2 *σ i 2 + ∑ ∑ W i * W j * σ ij, i ≠ j If n=2 →σ p 2 = W 1 2 *σ 1 2 +W 2 2 *σ 2 2 +2*W 1 *W 2 *σ 12 σ p =W 1 σ 1 +W 2 σ 2 if ρ 12 =1 σ p =W 1 σ 1 - W 2 σ 2 if ρ 12 = -1 If n=3 →σ p 2 = W 1 2 *σ 1 2 +W 2 2 *σ 2 2 +W 3 2 *σ 3 2 +2*W 1 *W 2 *σ 12 + 2*W 1 *W 3 *σ 13 + 2*W 2 *W 3* σ 23