Professor John Zietlow MBA 621

# Professor John Zietlow MBA 621

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## Professor John Zietlow MBA 621

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1. Chapter 5 Risk And Return Professor John ZietlowMBA 621

2. Chapter 5 Overview • 5.1. Introduction to Risk and Return • 5.2. Risk and Return Fundamentals • A Historical Overview of Risk and Return • Nominal and Real Returns • Risk Premium • Risk Aversion • 5.3. Basic Risk and Return Statistics • Return on a Single Asset • Arithmetic and Geometric Averages • Risk of a Single Asset • Normal Distribution

3. Chapter 5 Overview • 5.4. Risk and Return for Portfolios • Portfolio Returns • Portfolio Variance Example • Importance of Covariance • Variance of a Two-Asset Portfolio • 5.5. Systematic and Unsystematic Risk • What Drives Portfolio Risk • The Systematic Risk of an Individual Security • Limitations of Beta • 5.6. Summary

4. An Introduction To Risk & Return • Basic question in finance: “What is an asset worth?” • Valuing risky assets is fundamental to financial management • Three-step procedure for valuing a risky asset • Determining the asset’s expected cash flows • Choosing a discount rate that reflects asset’s risk • Calculating present value (PV cash inflows - PV outflows) • There is a trade-off between risk and expected return • Riskless investments (Treasury bills) offer low returns • Riskier investments (stocks) must promise higher returns • An asset pricing model attempts to expressly model the trade-off between risk & return • Benefit: Defines risk & models risk/return trade-off rigorously • Drawback: unrealistic assumptions needed to build a model

5. The Historical Trade-Off Between Risk & Return, 1926-2000 • Ibbotson Associates annually publishes “Stocks, Bonds, Bills, and Inflation” for U.S. financial assets • Investing \$1.00 in these assets in Dec 1925, then re-investing dividends/interest, would have yielded at year-end 2000: • Small-company stocks: \$6,402 • Large-company stocks: \$2,857 • Long-term corporate bonds: \$64 • Intermediate-term govt bonds: \$49 • Long-term government bonds: \$49 • Treasury bills (short-term): \$17 • Basket of goods (inflation proxy): \$10 • T-Bills (riskless, S-T investment) barely outpaced inflation • Above calculated using geometric mean returns

6. The Historical Trade-Off Between Risk & Return, 1926-2000

7. 100,000 Equities Bonds Bills Inflation Total value of reinvested returns, year-end 2000  \$ 10,000 10,000 Returns On U.S. Asset Classes, 1900-2000, In Nominal Terms 1,000 119 100 70 24 10 1 Source: Dimson, Marsh & Staunton (ABN/AMRO), Millenium Book II(2001)  Annual returns

8. Defining Financial Risk & Return • Define risk as the variability of returns associated with a given asset. • Define return as the total gain or loss experienced on an investment over a given period of time. • Return measured as the change in an asset's value plus any cash distributions (dividends or interest payments). (Eq 5.1) • Where Pt+1 = price (value) of asset at time t+1; • Pt = price (value) of asset at time t; • Ct+1 = cash flow paid by time t+1

9. Realized Return Versus Expected Return • Realized (ex post) return easily computed with equation 4.1: • Calculate yearly, monthly, daily holding period returns (HPR) • Real financial decisions, however, are based on expected (ex ante) returns, not realized returns: • Realized return (at best) useful in estimating expected return • Can specify conditional or unconditional expected returns • Conditional expected return: “If the economy improves next year, the asset’s return is expected to be 12%.” Or could be conditional on return on overall stock market. • Unconditional expected return: “The asset’s return next year is expected to be 12%.” • Usually generate expected return based on a specific asset pricing model, such as CAPM (Chapter 6).

10. Calculating Realized Returns On Two Stocks • Both stocks purchased 12/31/02 and sold 12/31/03, so calculating one-year realized return for each investment • Dynatech, bought for \$60/share (P0), pays no dividends (Ct=0) in 2003, and is sold for \$72/share (P1) 12/31/03. • Utilityco, bought for \$60/share (P0), pays \$6/share dividend (Ct=\$6) in 2003, and is sold for \$66/share 12/31/03. • Both have 20% return, one pure cap gains, one cap gains & dividends .

11. Nominal Versus Real Returns • The nominal return on a given investment has three components: the real rate of return, the expected inflation rate, and the risk premium. Nominal return = real return + E(inflation) + risk premium • Treasury bills are virtually risk free, so the nominal return on T-bills can be expressed: Nominal T-bill return = real return + E(inflation) • If the average annual rate of inflation is 3.2%, and the average nominal return on T-bills is 3.8%, the real T-bill return is just 0.6% per year. • Suppose that you expect 5% inflation next year. What nominal return would you expect on corporate bonds? Nominal corporate bond return = 0.6% + 5% + risk premium

12. Arithmetic Versus Geometric Returns • Average annual (periodic) returns can be computed as either arithmetic or geometric average returns. • Average arithmetic return is the simple average of annual returns, and is best estimate of what return to expect each year. • Geometric average return is the compound annual return earned by an investor who bought and held a stock for t years: • Geometric average return = [(1+R1)(1+R2)(1+R3)….(1+Rt)]1/t – 1 • Compute arithmetic (AAR) and geometric average returns (GAR) for series below: YearReturn 2000 -10.2% 2001 -12.5% 2002 +15.3% 2003 +8.9% AAR = [(-10.2%) + (-12.5%) + (15.3%) + (8.9%)]  4 = [-10.2% – 12.5% + 15.3% +8.9%]  4 = 0.375% GAR = [(1-0.102)(1-0.125)(1+0.153)(1+0.089)]1/4 -1 = [(0.898)(0.875)(1.153)(1.089)]0.25 -1 = -0.33%

13. Country Arithmetic mean Geometric mean Standard deviation Australia 8.5% 7.1% 17.2% Canada 6.0 4.6 16.7 France 9.9 7.5 23.8 Germany 10.3 4.9 35.3 Italy 11.0 7.0 32.5 Japan 10.0 6.8 28.0 Netherlands 7.1 5.1 22.2 Switzerland 6.1 4.3 19.4 United Kingdom 6.5 4.7 19.9 United States 7.5 5.6 19.8 The Equity Risk Premium, 1900-2000 The higher return demanded by investors to hold stocks rather than less assets is the Equity Risk Premium. Table below shows ERP, defined as stock return – bill returns for various countries.

14. Real And Nominal Rates of Return On U.S. Asset Classes, 1900-2000 Distribution of U.S. Risk PremiaArithmetic mean Geometric mean Std dev Equity risk premium vs bills 7.5% / year 5.6% / year 19.8% Equity risk premium vs bonds 6.9% / year 5.0% / year 19.9% Bond maturity premium vs bills 0.8% / year 0.5% / year 7.4%

15. Risk Preferences: Comparing Two Assets With The Same Expected Return • Stocks 1 & 2 both have an expected return of 10%. • Both offer 10% return in an average economy • Stock 2 would have higher return if economy booms • Stock 1 has lower return variability; does better in bad times • Whether an investor would consider them equally attractive depends on his/her degree of risk aversion (utility function) • Risk averse investor prefers lower variability for given R^ • Risk seeking investor prefers higher variability for given R^ • Risk neutral investor is indifferent about variability • Finance theory, common sense, and observed behavior allsuggest investors are risk averse • If two assets offer equal R^, will pick one with less variability • Must be offered higher R^ to accept higher variability

16. Two Assets With Same Expected Return But Different (Continuous) Probability Distributions Stock 1 Probability Density Stock 2 0 5 6 7 8 9 10 11 12 13 14 15 Return %

17. Risk Of A Single Asset • Can now calculate an asset’s return (expected and realized) • Next step to measure risk. • Simplest definition the likelihood of loss on an investment. • Finance defines risk in terms of the variability of returns • Measure risk based on a probability distribution (known or estimated) of expected returns. • Fig 5.1a shows histogram of returns on a portfolio of large stocks; Fig 5.1b shows this for small stock portfolio • Small stock p/f shows higher mean return, higher variability • Both show returns clustering around mean value • Following slide shows bell-shaped normal distribution • Great to use as a model of return distribution, if possible • Symmetric about mean, described fully by mean & variance (2) or standard deviation, square root of variance () • 68% of outcomes within 1 of mean; 95% within 2 

18. The Normal Probability Distribution:Area Under The Bell-Shaped Curve Normal Distribution R-2 R-1 R R+1 R+2 68% 95%

19. Calculating Variance And Standard Deviation Of Expected Returns • The variance (2)of a distribution equals the expected value of squared deviations from the mean. • Can compute expected (ex ante) or historical variance • Assume you predict that a stock has equally likelihood (p=0.167) of following six returns next year: • (-12%, -3%, 7%, 12%, 18%, 20%). Calculate expected return, E(R) • E(R) = (-12 - 3 + 7 + 12 + 18 + 20) ÷ 6 = 42 ÷ 6 = 7% • Compute variance of these expected returns using Eq. 5.3: = [(-12-7)2+(-3-7)2+(7-7)2+(12-7)2+(18-7)2 +(20-7)2]÷6 2 = [(-19)2+(-10)2+(0)2+(5)2+(11)2 +(13)2] ÷ 6 = [361 + 100 + 0 + 25 + 121+169] ÷ 6 = [776%2] ÷ 6 = 129.33%2. • Note units of variance (%-squared). Units hard to interpret, so • calculate standard deviation, square root of 2: • Standard deviation = σ =  129.33%2 = 11.37%

20. Calculating Variance And Standard Deviation Of Historical Returns • It’s rarely feasible to specify the full distribution of possible returns and expected variance. • Must know all possible outcomes & associated probabilities • Instead, analysts usually gather historical data and use these to generate expected return and variance • Historical variance computed using Eq 5.4: (Eq 5.4) • Where Rit= return on stock i during period t, R¯i = average return on stock i over sample period and N = number of periods in sample. • Denominator uses N-1 rather than N since one degree of freedom used to compute average (mean) return.

21. Month Return Month Return Jul-99 2.52% Aug-00 20.947% Aug-99 -4.105% Sep- 00 -13.402% Sep-99 24.657% Oct- 00 -16.191% Oct-99 4.533% Nov- 00 -19.697% Nov-99 42.575% Dec- 00 9.670% Dec-99 65.25% Jan-01 0.215% Jan-00 -10.847% Feb-01 -34.764% Feb-00 48.639% Mar-01 -21.158% Mar-00 5.134% Apr-01 7.877% Apr-00 2.402% May-01 -5.322% May-00 -10.086% Jun-01 24.183% Jun-00 16.956% Jul-01 -4.842% Jul-00 -10.558% Monthly Return for Oracle CorporationJuly 1999 – July 2001

22. Variance Calculation for Oracle Corp • We can compute expected return, E(R), variance, 2, std dev, , for Oracle Corp stock from Jul 1999 to July 2001: • E(R) = 4.98% per month; 2 = 534.78%2,  = 23.125% • If Oracle’s returns are approximately normally distributed, can use this to find confidence intervals for E(R): • 68% probability returns will be within +/- one  from E(R) • 95% probability returns will be within +/- two  from E(R) • Given E(R)=4.98%, =23.125%, then there is a 68% chance actual return will be between –18.145%% and +28.105% • 95% chance actual return between –41.27% and + 51.23% • Clearly, Oracle is a risky stock!

23. Calculating Expected Return For A Portfolio • Have looked only at risk and return for single assets thus far, but most investors hold multiple asset portfolios (p/fs) • Asset pricing models all assume stocks held in p/fs • Key insight of portfolio theory: Asset return adds linearly, but risk is (almost always) reduced in a portfolio • E(R) of p/f is a weighted average of individual asset E(R) • P/f variance is a non-linear function, based on covariance (defined later) between assets’ return • E(R) of a p/f calculated using Eq 5.5, where w1, w2 are weights of assets 1 & 2 in p/f: • Can be generalized to n-asset p/f using Eq 5.6:

24. Date 3M Co Praxair Inc Microsoft Berkshire Hathaway Inc 50% 3M, 50% Praxair 50% Microsoft, 50% Berkshire Jan-99 9.14% -8.33% 26.18% -7.14% 0.403% 9.520% Feb-99 -4.59% 8.12% -14.21% 9.38% 1.767% -2.415% Mar-99 -4.47% 3.22% 19.40% 0.42% -0.626% 9.911% Apr-99 25.79% 43.50% -9.27% 7.00% 34.648% -1.136% May-99 -3.65% -5.68% -0.77% -5.76% -4.664% -3.264% Jun-99 1.38% 0.26% 11.77% -4.306% 0.820% 3.734% Jul-99 1.15% -5.75% -4.85% -1.59% -2.298% -3.220% Aug-99 7.46% 1.90% 7.87% -5.317% 4.680% 1.275% Sep-99 1.65% -2.13% -2.16% -14.33% -0.237% -8.245% Oct-99 -1.04% 1.63% 2.21% 16.18% 0.295% 9.195% Nov-99 0.53% -4.55% -1.64% -10.33% -2.010% -5.983% Dec-99 2.42% 12.75% 28.23% -2.09% 7.582% 13.068% Jan-00 -4.34% -19.38% -16.17% -8.73% -11.861% -12.451% Feb-00 -5.87% -16.80% -8.68% -14.06% -11.335% -11.374% Mar-00 0.49% 23.33% 18.88% 30.00% 11.915% 24.441% Apr-00 -2.32% 6.76% -34.35% 3.67% 2.214% -15.341% May-00 -0.87% -5.49% -10.30% -1.18% -3.176% -5.743% Monthly Returns for Individual Stocks and Portfolios, 1998-2000

25. Date 3M Co Praxair Inc Microsoft Berkshire Hathaway Inc 50% 3M, 50% Praxair 50% Microsoft, 50% Berkshire Jun-00 -3.79% -10.86% 27.87% -8.191% -7.327% 9.841% Jul-00 9.17% 5.68% -12.73% 2.42% 7.421% -5.159% Aug-00 3.26% 11.85% 0.00% 4.72% 7.555% 2.359% Sep-00 -2.02% -15.54% -13.61% 11.61% -8.776% -0.998% Oct-00 6.04% -0.33% 14.20% -1.09% 2.851% 6.555% Nov-00 3.36% -3.52% -16.70% 3.45% -0.080% -6.622% Dec-00 20.65% 23.48% -24.40% 7.74% 22.065% -8.331% Jan-01 -8.17% -0.10% 40.78% -3.66% -4.138% 18.558% Feb-01 1.90% 0.61% -3.38% 2.78% 1.253% -0.300% Mar-01 -7.85% 0.11% -7.31% -6.90% -3.869% -7.104% Apr-01 14.54% 6.00% 23.89% 3.90% 10.273% 13.891% May-01 -0.36% 6.25% 2.11% 1.03% 2.946% 1.570% Jun-01 -3.78% -6.54% 5.52% 1.02% -5.160% 3.270% Jul-01 -1.95% -3.53% -9.33% -0.29% -2.739% -4.808% Aug-01 -6.95% 3.82% -13.81% 0.29% -1.569% -6.760% Sep-01 -5.48% -10.77% -10.31% 0.86% -8.123% -4.721% Oct-01 6.08% 12.33% 13.64% 1.71% 9.205% 7.678% Nov-01 9.77% 12.17% 10.42% -1.69% 10.969% 4.368% Dec-01 3.17% 4.40% 3.18% 8.00% 3.785% 5.589% Monthly Returns for Individual Stocks and Portfolios, 1998-2000 (Cont.)

26. Calculating The Expected Return Of A Two-Asset Portfolio • Table 5.3 shows monthly and average returns (mean %) & standard deviations ( %) of four stocks over 3-yr period • 3M (1.68%), Praxair (1.91%), Microsoft (1.17%), Berkshire (0.54%) • 3M (7.56%), Praxair (12.01%), Microsoft (16.51%), Berkshire (8.43%) • Also shows two p/fs with equal fractions of two stocks • p/f #1: 50% 3M, 50% Praxair ; p/f #2: 50% Microsoft, 50% Berkshire • E(R) of p/fs are weighted averages of individual stocks: E(R) pf #1 = [(0.5)(1.68%)+(0.5)(1.91%)] = 1.80% E(R) pf #2 = [(0.5)(1.17%)+(0.5)(0.54%)] = 0.86% • But actual p/f standard deviations are not equal to weighted averages of individual std devs--less in both cases:  pf #1 = 9.00%  [(0.5)(7.56 %)+(0.5)(12.01%)] = 9.78%  pf #2 = 8.95%  [(0.5)(16.51 %)+(0.5)(8.43%)] = 12.47% • Figure 5.3 shows risk/return tradeoff for 3M & Praxair Inc

27. Company or portfolio Average (mean) monthly return, % Standard deviation of monthly return, % 3M Co 1.68% 7.56% Praxair Inc 1.91 12.01 Microsoft 1.17 16.51 Berkshire Hathaway Inc 0.54 8.43 50% 3M, 50% Praxair 1.80 9.00 50% Microsoft, 50% Berkshire 0.86 8.95 Monthly Returns And Standard Deviations: Four Stocks And Two Portfolios

28. Calculating Variance And Standard Deviation Of Portfolio Expected Return • In previous table, std dev of Microsoft-Berkshire p/f below weighted average of individual std dev • Reason: returns on two stocks don’t co-move together • Microsoft & Berkshire returns have negative covariance (Cov) • 3M & Praxair have positive Cov, but don’t co-move perfectly • To compute p/f variance account for Cov between p/f assets • Calculate covariance of expected returns using Eq 5.7 • Calculate covariance of expected returns using Eq 5.7 (Eq 5.7) (Eq 5.8)

29. Calculating And Using Covariance And Correlation Coefficients • Cov measures co-movement between assets 1 and 2, 12 • Units of Cov are %-squared, same problem as variance • Shown monthly for 3M & Praxair and Microsoft-Berkshire in previous table • Positive Cov between 3M & Praxair, TP = +0.0059 • Negative Cov between Microsoft & Berkshire, MB = -0.0011 • Besides awkward measurement units, Cov also unbounded • Would like a measure normalized between –1.0 and +1.0 • Correlation coefficient, 12, is unit-less and valued –1 to +1 • Eq 5.9 is formula, then calculate for 3M/Praxair, Microsoft/Berkshire: (Eq 5.9) 3M/Praxair correlation = 0.0059 ÷ (0.0756)(0.1201) = 0.65 Microsoft/Berkshire correlation = -0.0011 ÷ (0.1651)(0.0843) = -0.079

30. Calculating And Using Covariance And Correlation Coefficients (Continued) • Forming p/fs between 3M (T) and Praxair (P) and between Microsoft (M) and Berkshire (B) yields reduction in p/f variance and std dev • Since TP>MB, combining3M and Praxair yields less risk reduction than combining Microsoft and Berkshire • Use Eq 5.10 to find variance of 50:50 3M/Praxair and 50:50 Microsoft/Berkshire p/fs using Cov (TP = +0.0053, MB = -0.0011) (Eq 5.10) 3M/Praxair: p2 =(wT2)(T2) + (wP2)(P2) + 2wTwP TP =(0.5)2(0.0756)2 +(0.5)2(0.1201)2 + 2(0.5)(0.5)(0.0059) =(0.25)(0.0057)+(0.25)(0.0144)+2(0.0015)=0.0080  p=0.08936=8.94% Microsoft/Berkshire: p2 = (wM2)(M2) + (wB2)(B2) + 2wMwBMB =(0.5)2(0.1651)2 +(0.5)2(0.0843)2 + 2(0.5)(0.5)(-0.0011) =(0.25)(0.0273)+(0.25)(0.0071)-2(0.000275)=0.008  p=0.0895= 8.95%

31. Calculating And Using Covariance And Correlation Coefficients (Continued) • Use Eq 5.12 to find variance of 50:50 3M/Praxair and 50:50 Microsoft/Berksire p/fs using Cov (TP = +0.65, MB = -0.079) • Remember that AB can be computed as AB = AB ÷ (A)(B) • Or Cov can be computed from AB : AB = AB (A)(B) (Eq 5.12) 3M/Praxair : p2 = (wT2)(T2) + (wP2)(P2) + 2wTwP TPMP =(0.5)2(0.0756)2 +(0.5)2(0.1201)2 + 2(0.5)(0.5)(0.65)(0.0756)(0.1201) =(0.25)(0.0057)+(0.25)(0.0144)+2(0.0015)=0.007986  p=0.08936 = 8.94% Microsoft/Berkshire : p2 = (wM2)(M2) + (wB2)(B2) + 2wMwB MBMB =(0.5)2(0.1651)2 +(0.5)2(0.0843)2 + 2(0.5)(0.5)(-0.079)(0.1651)(0.0843) =(0.25)(0.0273)+(0.25)(0.0071)-2(0.0003)=0.008041  p=0.0897= 8.97%

32. Perfectly Positively Correlated B Return Return B A A Time Time The Returns On Perfectly Positively and Perfectly Negatively Correlated Assets Perfectly Negatively Correlated

33. Imperfectly Correlated Assets And Portfolio Return Variability • Combining two imperfectly correlated assets into a portfolio • reduces the variability of portfolio returns Portfolio ofAsset M and N Asset M Asset N Return Return Return Time Time Time

34. Demonstrating Positive & Negative Covariance • Assume you can invest in three assets (stocks) with the same expected return, but are imperfectly correlated • Stock 3: Retailing firm, does well in expansions, 3=5.7% • Stock 4: Bankruptcy reseller, prospers in recessions, 4=5.7% • Stock 5: Wholesale distributor, does very well in expansions, very poorly in recessions, 5=10.1% • Stocks 3 & 5 have different return std dev, but historically move together (co-vary) as economy changes • Stock 5’s return follows 3’s, but with greater vigor • Stock 4 does well when other stocks do poorly & vice versa • Explain co-movement as positive or negative covariance • Stocks 3 & 4 have negative cov: 34 = -32.50 • Stocks 3 & 5 exhibit positive cov: 35 = +57.50 • Stocks 4 & 5 have negative cov: 45 = -57.50

35. Calculating Correlation Coefficients For Stocks 3, 4 And 5 • Calculate correlation coefficients between three matched pairs of stock using Eq 5.9: • Returns on stocks 3 & 5 (retailer & wholesaler stocks) are perfectly positively correlated • Stock 4's (bankruptcy reseller) returns are perfectly negatively correlated with assets 3 and 5:

36. Constructing Portfolios Based On Correlation Coefficients • Unless returns on all assets perfectly positively correlated, forming portfolios reduces p/f return variance • If =+1.0, forming p/f does not reduce return variability • For any <+1.0, forming p/f reduces variability • Can form p/f with a standard dev of 0 (thus riskless), by combining assets that are perfectly negatively correlated • Works since 34= -1.0 or 45= -1.0 , but weights must be carefully chosen • Combining assets with perfectly positively correlated returns yields a weighted-average p/f variance •  of p/f (3,5) = (0.5)(3) + (0.5)(5) = (5.7% + 10.12%) ÷ 2 = 7.91%

37. Computing And Using Correlation Coefficients In A Two-Asset Portfolio • The correlation between two assets’ returns can be used to construct an “efficient” two-asset portfolio • Minimize risk for given level of expected return & vice versa • Will ultimately expand use of correlation to include an asset’s relationship with overall market • Allows creation of efficient multi-asset portfolio • Correlation is central to all modern asset pricing models • Demonstrate using annual return data (presented on next slide) for two stocks and the S&P 500 stock index • Consolidated Consumer Corp (CCC): low-risk, low-return • Dynamic Technology Corp (DTC): high-risk, high-return • Given mean historical return series, can compute each asset’s std dev, covariance & correlation with each other

38. Stock or Index Mean historical return, % Standard deviation of return, % Consolidated Consumer Corp (CCC) 13.0% 9.5% Dynamic Technology Corp (DTC) 20.0 20.0 S&P 500 Index 15.0 12.3 Historical Returns And Standard Deviations For Two Stocks And The S&P 500 Index

39. Computing Correlation Coefficients On Two Stocks And The S&P 500 Index • Assume the following covariances are determined between: • CCC and DTC: Cov (c,d) = 112.7 • CCC and S&P 500 [market]: Cov (c,m) = 76.0 • DTC and S&P 500 [market]: Cov (d,m) = 236.9 • Can now use eq 5.9 to compute correlation coefficients and figure (next page) shows how to use these in p/f formation cd =112.7  [(9.5)(20.0)] = 0.59 cm = 76.0  [(9.5)(12.3)] = 0.61 dm = 236.9  [(20.0)(12.3)]= 0.90

40. D G F E C Portfolio Risk and Return For Combinations of CCC and DTC 25% 100% DTC 20% 15% Expected Return on the Portfolio 10% 100% CCC 5% 0% 0% 5% 10% 15% 20% Standard Deviation of Portfolio Returns

41. The Risk-Return Trade-Off For Different Correlation Coefficients • Figure (next slide) shows risk-return trade-off for p/fs of CCC & DTC stock with different jlbetween the stocks. • The straight line CD represents a p/f assuming perfectly positively correlated returns between the two shares. • Other curves represent assumed correlation coefficients of 0.00, -.50, and -1.00 • This figure shows that the lower the correlation between two assets’ returns, the greater the risk reduction from combining the assets into a portfolio • Perfectly negatively correlated assets yield the minimum possible variance for any given level of expected return.

42. D G G G F F F E E E C 0% 5% 10% 15% 20% 25% Correlation Coefficients And Risk Reduction 25% 20% 15% Expected Return on the Portfolio 10% Standard Deviation of Portfolio Returns CEFGD assumes cd is +1.0 CEFGD assumes cd is -1.0 <  <1.0 CEFGD assumes cd is -1.0

43. The Declining Importance Of Own Variance As The Number Of Assets In A Portfolio Increases • Whatever the correlation between assets, increasing the number in a p/f reduces the impact of each one’s own variance • Demonstrate with two assets, using eq , assuming equal weights of each stock (wj = wl = 0.5): p2 = wj 2j2 + (1-wj)2l2 + 2 wj(1-wj)Cov(j,l) = (0.5)2j2 + (0.5)2l2 + 2(0.5)(0.5)Cov(j,l) • Each asset’s own variance accounts for only 25% of total p/f vraiance, and both own variances together only total half

44. Declining Importance Of Own Variance (Cont) • The formula for a three-asset (including stock q) portfolio’s return variance is given : • p2 = wj2j2 + wl2l2 + wq2q2 + 2wjwl Cov(j,l) + 2wjwlCov(j,q) + 2wlwqCov(l,q) • Consider the simplest case with equal asset amounts in the portfolio (wj=wl=wq=0.333=1/3). • p2 = (0.333)2j2 + (0.333)2l2 + (0.333)2q2 + 2wjwlCov(j,l) + 2wjwqCov(j,q)+ 2wlwqCov (l,q) • = (0.111) j2 + (0.111) l2 + (0.111) q2+ 2wjwlCov(j,l) + 2wjwqCov(j,q) + 2wlwqCov(l,q) • Each asset’s own variance only represents (1/3)2= 1/9 = 0.111, or 11.1% of total p/f variance; • Three collectively represent only 33.3% of total volatility. • Summed Cov terms represent other 66.7% of portfolio var.

45. Diversifiable And Non-diversifiable Risk • As the number of assets in a portfolio increases, the importance of own variance virtually disappears • In a 10-asset p/f, each own var accounts for only 1% of total • All own var collectively account for only 10% of p/f variance • In a 25-asset p/f, each own var is only 0.16% of p/f variance • As number of assets increases, the importance of bilateral covariances also declines-- similarly to own variance • In a diversified p/f, an asset’s own var & cov matters little • Only an asset’s covariance with all other assets contributes measurably to overall p/f return variance • Investor thus only looks at asset’s covariance with “market” • Thus important to draw distinction between an asset’s total, diversifiable and non-diversifiable risk [Figure 5.7] • Diversifiable: unique, firm-specific risk (fire, flood, strike) • Nondiversifiable: systematic risk related to market or economy

46. The Impact Of Additional Assets On The Risk Of A Portfolio Diversifiable Risk Portfolio Risk, kp Total risk Nondiversifiable Risk 1 5 10 15 20 25 Number of Securities (Assets) in Portfolio