Risk and Return: Past and Prologue Chapter 5 Risk and Return: Past and Prologue We must learn how to calculate returns and how to calculate the risk associated with different investments. McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.

5.1 Rates of Return 5-2

Measuring Ex-Post (Past) Returns One period investment: regardless of the length of the period. Holding period return (HPR): HPR = where PS = Sale price (or P1) PB = Buy price ($ you put up) (or P0) CF = Cash flow during holding period Q: [PS - PB + CF] / PB Q: % returns, removes size of investment concerns Q: CF should occur at the end of the period, otherwise missing reinvestment of those cash flows. Or the CF could be the future value of any interim cashflows. Why use % returns at all? What are we assuming about the cash flows in the HPR calculation? 5-3

Annualizing HPRs Q: Why would you want to annualize returns? 1. Annualizing HPRs for holding periods of greater than one year: Without compounding (Simple or APR): HPRann = With compounding: EAR HPRann = where n = number of years held HPR/n [(1+HPR)1/n]-1 5-4

Measuring Ex-Post (Past) Returns An example: Suppose you buy one share of a stock today for $45 and you hold it for two years and sell it for $52. You also received $8 in dividends at the end of the two years. (PB = , PS = , CF = ): HPR = HPRann = The annualized HPR assuming annual compounding is (n = ): $45 $52 $8 (52 - 45 + 8) / 45 = 33.33% 0.3333/2 = 16.66% Annualized w/out compounding 2 (A good rule is to carry all calculations to at least four decimal places). (1+0.3333)1/2 - 1 = 15.47% 5-5

Measuring Ex-Post (Past) Returns Annualizing HPRs for holding periods of less than one year: Without compounding (Simple): HPRann = With compounding: HPRann = where n = number of compounding periods per year HPR x n [(1+HPR)n]-1 5-6

Measuring Ex-Post (Past) Returns An example when the HP is < 1 year: Suppose you have a 5% HPR on a 3 month investment. What is the annual rate of return with and without compounding? Without: With: Q: Why is the compound return greater than the simple return? n = 12/3 = 4 so HPRann = HPR*n = 0.05*4 = 20% HPRann = (1.054) - 1 = 21.55% (A good rule is to carry all calculations to at least four decimal places). 5-7

Arithmetic Average Finding the average HPR for a time series of returns: i. Without compounding (AAR or Arithmetic Average Return): n = number of time periods (A good rule is to carry all calculations to at least four decimal places). 5-8

Arithmetic Average AAR = 17.51% 17.51% (A good rule is to carry all calculations to at least four decimal places). 17.51% 5-9

Geometric Average With compounding (geometric average or GAR: Geometric Average Return): GAR = (A good rule is to carry all calculations to at least four decimal places). 15.61% 15.61% 5-10

Measuring Ex-Post (Past) Returns Finding the average HPR for a portfolio of assets for a given time period: where VI = amount invested in asset I, J = Total # of securities and TV = total amount invested; thus VI/TV = percentage of total investment invested in asset I (A good rule is to carry all calculations to at least four decimal places). 5-11

Measuring Ex-Post (Past) Returns For example: Suppose you have $1000 invested in a stock portfolio in September. You have $200 invested in Stock A, $300 in Stock B and $500 in Stock C. The HPR for the month of September for Stock A was 2%, for Stock B the HPR was 4% and for Stock C the HPR was - 5%. The average HPR for the month of September for this portfolio is: (A good rule is to carry all calculations to at least four decimal places). -0.9% 5-12

Measuring Ex-Post (Past) Returns Measuring returns when there are investment changes (buying or selling) or other cash flows within the period. An example: Today you buy one share of stock costing ___. The stock pays a __ dividend one year from now. Also one year from now you purchase a second share of stock for ____. Two years from now you collect a ___ per share dividend and sell both shares of stock for ___ a share. $50 $2 $53 (A good rule is to carry all calculations to at least four decimal places). $2 $54 Q: What was your average (annual) return? A: It depends. There are different ways to measure this. 5-13

Dollar-Weighted Return i. Dollar-weighted return procedure (DWR): Find the internal rate of return for the cash flows (i.e. find the discount rate that makes the NPV of the net cash flows equal zero.) (A good rule is to carry all calculations to at least four decimal places). 5-14

Tips on Calculating Dollar Weighted Returns This measure of return considers both changes in investment and security performance Initial Investment is an _______ Ending value is considered as an ______ Additional investment is an _______ Security sales are an ______ outflow inflow If different amounts of money were managed in the portfolio for each quarter it may be useful to see the Dollar weighted returns. Basically look at the inflows with respect to your wallet. Invest money. Outflow from your wallet. Remove money from the investment is an inflow into your wallet. outflow inflow 5-15

Measuring Ex-Post (Past) Returns i. Dollar-weighted return procedure (DWR): Find the internal rate of return for the cash flows (i.e. find the discount rate that makes the NPV of the net cash flows equal zero.) NPV = Solve for IRR: IRR = $0 = -$50/(1+IRR)0 - $51/(1+IRR)1 + $112/(1+IRR)2 7.117% average annual dollar weighted return (A good rule is to carry all calculations to at least four decimal places). The DWR gives you an average return based on the stock’s performance and the dollar amount invested (number of shares bought and sold) each period. 5-16

Measuring Ex-Post (Past) Returns Q: You are paying somebody to advise you which assets to buy, but you are deciding when to buy and sell shares. If you want to evaluate the quality of the investment advice you are getting, should you use dollar weighted returns to evaluate the quality of the investment advice? (A good rule is to carry all calculations to at least four decimal places). 5-17

Time-Weighted Returns ii. Time-weighted returns (TWR): TWRs assume you buy ___ share of the stock at the beginning of each interim period and sell ___ share at the end of each interim period. TWRs are thus ___________ of the amount invested in a given period. To calculate TWRs: one one independent (A good rule is to carry all calculations to at least four decimal places). Calculate the return for each time period, typically a year. Then calculate either an arithmetic (AAR) or a geometric average (GAR) of the returns. 5-18

Time-Weighted Returns Same example as before, initially buy one share at $50, in one year collect a $2 dividend, and you buy another share at $53. In two years you sell the stock for $54, after collecting another $2 dividend per share. (A good rule is to carry all calculations to at least four decimal places). TWRs assume you buy one share of the stock at the beginning of each period and sell it at the end of each period after collecting any cash flow. 5-19

Measuring Ex-Post (Past) Returns TWR Cash Flows Year 0-1 Year 1-2 1 2 -$50 $ 2 -$53 +$53 $54 Year 0-1 Year 1-2 1 2 -$50 $ 2 -$53 +$53 +$54 Same example as before, initially buy one share at $50, in one year collect a $2 dividend, and you buy another share at $53. In two years you sell the stock for $54, after collecting another $2 dividend per share. (A good rule is to carry all calculations to at least four decimal places). Year 0-1 Year 1-2 1 2 -$50 $ 2 -$53 +$53 +$54 5-20

Measuring Ex-Post (Past) Returns HPR for year 1: HPR for year 2: a) Calculating the arithmetic average TW return: Arithmetic Average Return (AAR): Calculate the arithmetic average [$53 + $2 - $50] / $50 = 10% [$54 - $53 +$2] / $53 = 5.66% Arithmetic means are the sum of returns in each period divided by the number of periods. Basically add up all of the Holding Period Returns and divide by the number of periods. Good for forecasting performance in future periods. Bad because it ignores compounding needed to represent a single quarterly rate for the year. Geometric: This gives you the single period return that would give you the same cumulative performance as the sequence of actual returns. Here multiply the 1+ HPR together and raise to the power 1/n then subtract 1. AAR = [0.10 + 0.0566] / 2 = 7.83% 5-21

Measuring Ex-Post (Past) Returns b) Calculating the geometric average TW return (GAR): GAR = HPR1 = 10% HPR2 = 5.66% Arithmetic means are the sum of returns in each period divided by the number of periods. Basically add up all of the Holding Period Returns and divide by the number of periods. Good for forecasting performance in future periods. Bad because it ignores compounding needed to represent a single quarterly rate for the year. Geometric: This gives you the single period return that would give you the same cumulative performance as the sequence of actual returns. Here multiply the 1+ HPR together and raise to the power 1/n then subtract 1. 7.81% 7.81% 5-22

Measuring Ex-Post (Past) Returns Q: When should you use the GAR and when should you use the AAR? A1: When you are evaluating PAST RESULTS (ex-post): A2: When you are trying to estimate an expected return (ex-ante return): Use the AAR (average without compounding) if you ARE NOT reinvesting any cash flows received before the end of the period. Use the GAR (average with compounding) if you ARE reinvesting any cash flows received before the end of the period. (A good rule is to carry all calculations to at least four decimal places). Use the AAR 5-23

5.2 Risk and Risk Premiums 5-24

Measuring Mean: Scenario or Subjective Returns a. Subjective or Scenario Subjective expected returns E ( r ) = p s S E(r) = Expected Return p(s) = probability of a state r(s) = return if a state occurs 1 to s states Scenario analysis: Create a list of possible outcomes s, “States of Nature.” Specify the likelihood of each state to occur p(s) What is the return if the state occurs r(s) 5-25

Measuring Variance or Dispersion of Returns a. Subjective or Scenario Variance  = [2]1/2 E(r) = Expected Return p(s) = probability of a state rs = return in state “s” 5-26

Numerical Example: Subjective or Scenario Distributions State Prob. of State Return 1 .2 - .05 2 .5 .05 3 .3 .15 E(r) = (.2)(-0.05) + (.5)(0.05) + (.3)(0.15) = 6% 2 = [(.2)(-0.05-0.06)2 + (.5)(0.05- 0.06)2 + (.3)(0.15-0.06)2] 2 = 0.0049%2  = [ 0.0049]1/2 = .07 or 7% 5-27

Expost Expected Return &  Annualizing the statistics: The variance formula is the sample variance formula that results when you multiply n / n-1 times the AVERAGE of the sum of the squared deviations 5-28

 (r - ravg)2 = Average 0.011624 0.219762458 Variance 0.003725 Stdev 0.061031 n 60 n-1 59 Annualized 0.139486 0.044697 0.211418  (r - ravg)2 = 5-29

Using Ex-Post Returns to estimate Expected HPR Estimating Expected HPR (E[r]) from ex-post data. Use the arithmetic average of past returns as a forecast of expected future returns as we did and, Perhaps apply some (usually ad-hoc) adjustment to past returns Problems? Which historical time period? Have to adjust for current economic situation Unstable averages Stable risk How much past data? How far back? How stationary is the data? 5-30

Characteristics of Probability Distributions Mean: __________________________________ _ Median: _________________ Variance or standard deviation: Skewness:_______________________________ Leptokurtosis: ______________________________ Arithmetic average & usually most likely Middle observation Dispersion of returns about the mean mode = most frequent observation These are some major characteristics of distributions. The mean is the average return we expect given all of the different possible outcomes. Variance and standard deviation give us a quantitative measurement of the level of dispersion of the possible outcomes. And skewness tells us about how evenly the possible outcomes are distributed around the mean. Skewed distributions have a mean that is different than the median. If return distributions are close to “normal”, mean and variance will do a good job at describing the expected return and the risk premium of an investment. Long tailed distribution, either side Too many observations in the tails If a distribution is approximately normal, the distribution is fully described by _____________________ Characteristics 1 and 3 5-31

Normal Distribution Average = Median  measures deviations above the mean as well as below the mean. Returns > E[r] may not be considered as risk, but with symmetric distribution, it is ok to use  to measure risk. I.E., ranking securities by  will give same results as ranking by asymmetric measures such as lower partial standard deviation. Risk is the possibility of getting returns different from expected. The normal distribution is symmetric around the mean. In a normal distribution the mean + or – one standard deviation will encompass approximately 68% of the possible outcomes. Since this is a normal distribution the mean and the median outcome will be the same. Talk about std dev as risk measure. Risk not really from returns > E[r], but symmetric so ok to use std dev. I.E., ranking securities by standard dev will give same results as ranking by lower partial standard dev. E[r] = 10%  = 20% Average = Median 5-32

Skewed Distribution: Large Negative Returns Possible (Left Skewed) Implication?  is an incomplete risk measure r = average Median Skew = (E{r – E[r]}3) / 3 Here the distribution is skewed to the left. Most of the observations are slightly above the mean but there are a few large negative return outcomes that are pulling the mean down. Example 0,8,9,10,10,11,12. The average or mean is 8.57, but the median is 10. r Negative Positive 5-33

Skewed Distribution: Large Positive Returns Possible (Right Skewed) = average Median This skewness isn’t really so bad. Most of the outcomes are slightly below the mean, but there are a few outcomes where the return is really high. Example 7,7,8,8,9,9,and 12 the mean is again 8.57 however this time the median is 8. Notice the E[r] in the sense of the most likely return is now the median. The implication is that investors are willing to accept a slightly lower E[r] for the small chance of a large positive gain. Lottery principle. Negative r Positive 5-34

Leptokurtosis Implication?  is an incomplete risk measure Kurtosis = (E{r – E[r]}4) / 4 - 3; Kurtosis for a normal distribution is 3. Thus Kurtosis above zero indicates fat tails. Note the  underestimates risk if stock returns are leptokurtotic. 5-35

Value at Risk (VaR) Value at Risk attempts to answer the following question: How many dollars can I expect to lose on my portfolio in a given time period at a given level of probability? The typical probability used is 5%. We need to know what HPR corresponds to a 5% probability. If returns are normally distributed then we can use a standard normal table or Excel to determine how many standard deviations below the mean represents a 5% probability: From Excel: =Norminv (0.05,0,1) = -1.64485 standard deviations in the Norminv function in Excel the 0.05 is the 5% probability, 0 is the mean and 1 is the standard deviation of a standard normal variate. 5-36

Value at Risk (VaR) From the standard deviation we can find the corresponding level of the portfolio return: VaR = E[r] + -1.64485 For Example: A $500,000 stock portfolio has an annual expected return of 12% and a standard deviation of 35%. What is the portfolio VaR at a 5% probability level? VaR = 0.12 + (-1.64485 * 0.35) VaR = -45.57% (rounded slightly) VaR$ = $500,000 x -.4557 = -$227,850 What does this number mean? in the Norminv function in Excel the 0.05 is the 5% probability, 0 is the mean and 1 is the standard deviation of a standard normal variate. Interpretation: The greatest annual expected loss 95% of the time is $227,85. 5-37

Value at Risk (VaR) VaR versus standard deviation: For normally distributed returns VaR is equivalent to standard deviation (although VaR is typically reported in dollars rather than in % returns) VaR adds value as a risk measure when return distributions are not normally distributed. Actual 5% probability level will differ from 1.68445 standard deviations from the mean due to kurtosis and skewness. 5-38

Risk Premium & Risk Aversion The risk free rate is the rate of return that can be earned with certainty. The risk premium is the difference between the expected return of a risky asset and the risk-free rate. Excess Return or Risk Premiumasset = Risk aversion is an investor’s reluctance to accept risk. How is the aversion to accept risk overcome? By offering investors a higher risk premium. E[rasset] – rf 5-39

5.3 The Historical Record 5-40

Frequency distributions of annual HPRs, 1926-2008 Forward looking purpose: Use Arithmetic Mean. Note relationship between means and standard deviations and the standard deviation of a large stock portfolio. Ask, which is the most risky and which is he least risky? Note apparent deviations from normality in small stocks particularly. 5-41

Rates of return on stocks, bonds and bills, 1926-2008 5-42

Annual Holding Period Returns Statistics 1926-2008 From Table 5.3 Geom. Arith. Excess Series Mean% Return% Kurt. Skew. World Stk 9.20 11.00 7.25 1.03 -0.16 US Lg. Stk 9.34 11.43 7.68 -0.10 -0.26 Sm. Stk 17.26 13.51 1.60 0.81 World Bnd 5.56 5.92 2.17 1.10 0.77 LT Bond 5.31 5.60 1.85 0.80 0.51 Geometric mean: Best measure of compound historical return Arithmetic Mean: Expected return Deviations from normality? Notice the greater divergence of the GAR and AAR for small stocks. This is because of the high variance and the higher proportion of negative returns in the small stock portfolio. These are the historical returns between 1926-1998 listed in the text. Remember the arithmetic mean would be the typical expectation for any given year, while the geometric mean is the return needed for every year to get the same cumulative returns as the sequence of historical returns. Notice that generally the higher the returns the higher the variance or SD. 5-43

Deviations from Normality: Another Measure Portfolio World Stock US Small Stock US Large Stock Arithmetic Average .1100 .1726 .1143 Geometric Average .0920 .0934 Difference .0180 .0483 .0209 ½ Historical Variance .0186 .0694 .0214 If returns are normally distributed then the following relationship among geometric and arithmetic averages holds: Arithmetic Average – Geometric Average = ½ 2 The comparisons above indicate that US Small Stocks may have deviations from normality and therefore VaR may be an important risk measure for this class. Uses data from Table 5.3 5-44

Actual vs. Theoretical VaR 1926-2008 VaR% if Normal Series World Stk -21.89 -21.07 US Lg. Stk -29.79 -22.92 US Sm. Stk -46.25 -44.93 World Bnd -6.54 -8.69 US LT Bond -7.61 -7.25 These comparisons indicate that the U.S. Large Stock portfolio, the US small stock portfolio and the World Bond portfolio may exhibit differences from normality. 5-45

Annual Holding Period Excess Returns 1926-2008 From Table 5.3 of Text Arith. Required Series Avg% Return% World Stk 7.25 US Lg Stk 7.68 US Sm Stk 13.51 World Bonds 2.17 US LT Bonds 1.85 If the risk free rate is currently 3%, then what return should an investor require for each asset class? Problems with this approach? 10.25 10.68 16.51 5.17 4.85 Problems: 1926 data? & Assumes all stocks in each portfolio are equally risky. We can do better The arithmetic mean is the typical expected return for a given year, while the geometric mean is the return needed for every year to get the same cumulative returns as provided by the sequence of historical returns. Notice that generally the higher the returns the higher the variance or standard deviation. Historical data Assumes all securities in the category are equally risky 5-46

5.4 Inflation and Real Rates of Return 5-47

Inflation, Taxes and Returns The average inflation rate from 1966 to 2005 was _____. This relatively small inflation rate reduces the terminal value of $1 invested in T-bills in 1966 from a nominal value of ______ in 2005 to a real value of _____. Taxes are paid on _______ investment income. This reduces _____ investment income even further. You earn a ____ nominal, pre-tax rate of return and you are in a ____ tax bracket and face a _____ inflation rate. What is your real after tax rate of return? 4.29% $10.08 $1.63 nominal real 6% Average inflation from 1926 to 2005 was 3.02% 1966 to 2005: real return was about 1.26%, this is pre-tax 15% 4.29% rreal  [6% x (1 - 0.15)] – 4.29%  0.81%; taxed on nominal 5-48

Real vs. Nominal Rates Fisher effect: Approximation real rate  nominal rate - inflation rate rreal  rnom - i Example rnom = 9%, i = 6% rreal  3% Fisher effect: Exact rreal = or rreal = The exact real rate is less than the approximate real rate. rreal = real interest rate rnom = nominal interest rate i = expected inflation rate [(1 + rnom) / (1 + i)] – 1 (rnom - i) / (1 + i) (9% - 6%) / (1.06) = 2.83% One issue we have ignored until now is the effect of inflation on our investments. The nominal rate is the rate we have been calculating. Not adjusted for inflation. To adjust for inflation we must calculate the real rate of return. At first glance you might think we just need to subtract the inflation rate from the nominal rate, but this is not correct. To get correct answer we must use the Fisher effect calculation. 1+r = 1+R/1+i which can be converted to r=(R-i)/(1+i) 5-49

Exact Fisher Effect Explained 1) I want to be able to buy more Quantity or Qnew = Qold x (1 + rreal) BUT 2) The Price, P, is also rising Pnew = Pold x (1 + i) i = inflation Total $ spent = Pnew x Qnew Pnewx Qnew = Pold x Qold x [(1 + rreal) x (1 + i)] or (1 + rnom)= (1 + rreal) x (1 + i) 5-50

Nominal and Real interest rates and Inflation 5-51

Historical Real Returns & Sharpe Ratios Series World Stk 6.00 0.37 US Lg. Stk 6.13 Sm. Stk 8.17 0.36 World Bnd 2.46 0.24 LT Bond 2.22 Real returns have been much higher for stocks than for bonds Sharpe ratios measure the excess return to standard deviation. The higher the Sharpe ratio the better. Stocks have had much higher Sharpe ratios than bonds. 5-52

5.5 Asset Allocation Across Risky and Risk Free Portfolios 5-53

Allocating Capital Between Risky & Risk-Free Assets Possible to split investment funds between safe and risky assets Risk free asset rf : proxy; ________________________ Risky asset or portfolio rp: _______________________ Example. Your total wealth is $10,000. You put $2,500 in risk free T-Bills and $7,500 in a stock portfolio invested as follows: Stock A you put ______ Stock B you put ______ Stock C you put ______ T-bills or money market fund risky portfolio What percentage of your holdings are in the risk-free asset? 25%. What percent are in the risky asset 75%. Your entire portfolio includes both risky and risk-free assets $2,500 $3,000 $2,000 $7,500 5-54

Allocating Capital Between Risky & Risk-Free Assets Weights in rp WA = WB = WC = The complete portfolio includes the riskless investment and rp. $2,500 / $7,500 = 33.33% $3,000 / $7,500 = 40.00% $2,000 / $7,500 = 26.67% 100.00% How much should be invested in the risky asset? 10%, 50%, 100%, 110%? 110 is taking a loan and investing (on margin) Assuming the risky portfolio has been optimized, Depending on your level of risk you must merely choose between your weights of the risk free and the risky portfolio. Wrf = ; Wrp = In the complete portfolio WA = 0.75 x 33.33% = 25%; 25% 75% WB = 0.75 x 40.00% = 30% WC = 0.75 x 26.67% = 20%; Wrf = 25% 5-55

Allocating Capital Between Risky & Risk-Free Assets Issues in setting weights Examine ___________________ Demonstrate how different degrees of risk aversion will affect __________ between risky and risk free assets risk & return tradeoff allocations How much should be invested in the risky asset? 10%, 50%, 100%, 110%? 110 is taking a loan and investing (on margin) Assuming the risky portfolio has been optimized, Depending on your level of risk you must merely choose between your weights of the risk free and the risky portfolio. 5-56

Example rf = 5% srf = 0% E(rp) = 14% srp = 22% y = % in rp (1-y) = % in rf Rf is the risk-free rate of return. Here it is 5%. Since this return is truly risk free the standard deviation of the risk-free asset is 0. Expected return on the risky portfolio is 14%. With a standard deviation of 22% Y is the percentage of your total wealth in the risky portfolio. 5-57

Expected Returns for Combinations E(rC) = yE(rp) + (1 - y)rf c = yrp + (1-y)rf E(rC) = Return for complete or combined portfolio For example, let y = ____ 0.75 E(rC) = E(rC) = .1175 or 11.75% (.75 x .14) + (.25 x .05) The percentage in the risky portfolio times the expected return of the risky portfolio. Plus the percentage in the risk-free times its return. C = yrp + (1-y)rf C = (0.75 x 0.22) + (0.25 x 0) = 0.165 or 16.5% 5-58

Complete portfolio E(rc) = yE(rp) + (1 - y)rf c = yrp + (1-y)rf Varying y results in E[rC] and C that are ______ ___________ of E[rp] and rf and rp and rf respectively. c = yrp + (1-y)rf linear combinations This is NOT generally the case for the  of combinations of two or more risky assets. The percentage in the risky portfolio times the expected return of the risky portfolio. Plus the percentage in the risk-free times its return. 5-59

Possible Combinations E(r) E(rp) = 14% P E(rp) = 11.75% 16.5% y = 1 y =.75 rf = 5% Expected return on the vertical axis, and standard deviation of total portfolio on the horizontal axis. With all of your money in the risk free you will have 7% return and no standard deviation. With all in the risky asset will have 15 expected return and 22% standard deviation. Can move along this line. F y = 0 s 22% 5-60

Possible Combinations E(r) E(rp) = 14% rf = 5% 22% P F Possible Combinations s E(rp) = 11.75% 16.5% y =.75 y = 1 y = 0 Expected return on the vertical axis, and standard deviation of total portfolio on the horizontal axis. With all of your money in the risk free you will have 7% return and no standard deviation. With all in the risky asset will have 15 expected return and 22% standard deviation. Can move along this line. 5-61

Combinations Without Leverage Since σrf = 0 σc= y σp If y = .75, then σc= If y = 1 If y = 0 E(rc) = yE(rp) + (1 - y)rf y = .75 E(rc) = y = 1 y = 0 75(.22) = 16.5% (.75)(.14) + (.25)(.05) = 11.75% 1(.22) = 22% (1)(.14) + (0)(.05) = 14.00% 0(.22) = 0% (0)(.14) + (1)(.05) = 5.00% 5-62

Using Leverage with Capital Allocation Line Borrow at the Risk-Free Rate and invest in stock Using 50% Leverage E(rc) = c = y = 1.5 (1.5) (.14) + (-.5) (.05) = 0.185 = 18.5% (1.5) (.22) = 0.33 or 33% E(rC) =18.5% 33% Expected return of your combined portfolio and the SD of your portfolio. y = 1.5 y = 0 5-63

Risk Aversion and Allocation Greater levels of risk aversion lead investors to choose larger proportions of the risk free rate Lower levels of risk aversion lead investors to choose larger proportions of the portfolio of risky assets Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations y = 1.5 y = 0 5-64

s P or combinations of P & Rf offer a return per unit of risk of 9/22. CAL (Capital Allocation Line) P E(rp) = 14% E(rp) - rf = 9% ) Slope = 9/22 rf = 5% This is the complete Capital Allocation Line. The slope says that for an increase in return of 8% we will have an increase in the SD of our portfolio by 22%. The risk premium is also shown. F s rp = 22% 5-65

Quantifying Risk Aversion E(rp) = Expected return on portfolio p rf = the risk free rate 0.5 = Scale factor A x p2 = Proportional risk premium The larger A is, the larger will be the _________________________________________ U = E[r] - 1/2Ap2 To describe how investors are willing to trade off risk for expected return. A is a version of the marginal rate of substitution between return and risk. We used to teach this with utility curves and it made more sense then. A risk premium of E(rp) – rf is required to accept some level of risk. “A” is just a measure for risk Aversion. Let’s look at an example where A=4 This says that for a standard deviation for the risky asset of 20% an investor would expect a risk premium of .005 X4X400 =8% investor’s added return required to bear risk 5-66

Quantifying Risk Aversion Rearranging the equation and solving for A Many studies have concluded that investors’ average risk aversion is between _______ Since we generally can see the historical risk premium and the historical variance we can rearrange the equation to estimate the level of risk aversion of the average investor. Think of A as the MRS or in other words the investor's desired tradeoff between return and risk. 2 and 4 5-67

Using A What is the maximum A that an investor could have and still choose to invest in the risky portfolio P? This is in the reasonable range since we know that most investors have 2 < A < 4. 3.719 Maximum A = 3.719 5-68

“A” and Indifference Curves The A term can used to create indifference curves. Indifference curves describe different combinations of return and risk that provide equal utility (U) or satisfaction. U = E[r] - 1/2Ap2 Indifference curves are curvilinear because they exhibit diminishing marginal utility of wealth. The greater the A the steeper the indifference curve and all else equal, such investors will invest less in risky assets. The smaller the A the flatter the indifference curve and all else equal, such investors will invest more in risky assets. This is not in the chapter but it should be. Note this is just one version of U, many forms are possible. 5-69

Indifference Curves I3 I2 I1 U = E[r] - 1/2Ap2 Investors want the most return for the least risk. Hence indifference curves higher and to the left are preferred. I1 U = E[r] - 1/2Ap2 5-70

s A=3 A=3 E(r) CAL (Capital P Allocation Line) Q S rf = 5% F This is the complete Capital Allocation Line. The slope says that for an increase in return of 8% we will have an increase in the SD of our portfolio by 22%. The risk premium is also shown. F s 5-71 71

s A=3 E(r) A=2 CAL (Capital P Allocation Line) T S rf = 5% F This is the complete Capital Allocation Line. The slope says that for an increase in return of 8% we will have an increase in the SD of our portfolio by 22%. The risk premium is also shown. F s 5-72

5.6 Passive Strategies and the Capital Market Line 5-73

A Passive Strategy Investing in a broad stock index and a risk free investment is an example of a passive strategy. The investor makes no attempt to actively find undervalued strategies nor actively switch their asset allocations. The CAL that employs the market (or an index that mimics overall market performance) is called the Capital Market Line or CML. 5-74

Excess Returns and Sharpe Ratios implied by the CML Excess Return or Risk Premium Time Period Average  Sharpe Ratio 1926-2008 7.86 20.88 0.37 1926-1955 11.67 25.40 0.46 1956-1984 5.01 17.58 0.28 1985-2008 5.95 18.23 0.33 The average risk premium implied by the CML for large common stocks over the entire time period is 7.86%. How much confidence do we have that this historical data can be used to predict the risk premium now? The average risk premium implied by the CML for large common stocks over the entire time period is 7.86%. However looking at the subperiod variation and the large standard deviation indicates that we cannot be very confident about using the historical data to estimate what the risk premium is likely to be in any given time period. Sharpe ratios have varied considerably as well. Notice the higher risk premium and Sharpe ratio during the time period including the Great Depression. In periods of economic uncertainty we can expect to see higher risk premiums. 5-75

Active versus Passive Strategies Active strategies entail more trading costs than passive strategies. Passive investor “free-rides” in a competitive investment environment. Passive involves investment in two passive portfolios Short-term T-bills Fund of common stocks that mimics a broad market index Vary combinations according to investor’s risk aversion. In a passive strategy the investor makes no attempt to neither find undervalued strategies nor actively switch their asset allocations. Investing in a broad stock index and a risk free investment is an example of a passive strategy. The CAL that employs the market (or an index that mimics overall market performance) is called the Capital Market Line or CML. In competitive markets active strategies that entail more information production and trading costs might not consistently perform better than passive strategies after considering those costs. As active investors trade upon their information prices will incorporate that information. This implies that passive investors are able to “free ride” upon the activities of active investors. 5-76

Selected Problems 5-77

Problem 1 V(12/31/2004) = V (1/1/1998) x (1 + GAR)7 $140,710.04 → 5-78 78

Problem 2 → a. The holding period returns for the three scenarios are: Boom: Normal: Recession: E(HPR) = 2(HPR) (50 – 40 + 2)/40 = 0.30 = 30.00% (43 – 40 + 1)/40 = 0.10 = 10.00% (34 – 40 + 0.50)/40 = –0.1375 = –13.75% [(1/3) x 30%] + [(1/3) x 10%] + [(1/3) x (–13.75%)] = 8.75% 0.031979 → 5-79 79

Problem 2 Cont. b. E(r) =  = (0.5 x 8.75%) + (0.5 x 4%) = 6.375% Risky E[rp] = 8.75% Risky p = 17.88% (0.5 x 8.75%) + (0.5 x 4%) = 6.375% 0.5 x 17.88% = 8.94% → 5-80 80

Problems 3 & 4 3. For each portfolio: Utility = E(r) – (0.5  4  2 ) We choose the portfolio with the highest utility value, which is Investment 3. Investment E(r)  U 1 0.12 0.30 -0.0600 2 0.15 0.50 -0.3500 3 0.21 0.16 0.1588 4 0.24 0.1518 → 5-81 81

Problems 3 & 4 Cont. 4. When an investor is risk neutral, A = _ so that the portfolio with the highest utility is the portfolio with the _______________________. So choose ____________. highest expected return The answer to Q8 is b., investors aversion to risk Investment 4 → 5-82 82

Problem 5 b. DWR a. TWR → 3.15% 2.33% -0.1661% a. TWR Year Return = [(capital gains + dividend) / price] 3 2 1 Explanation Cash flow Time 2007-2008 (110 – 100 + 4)/100 = 14.00% -300 Purchase of three shares at $100 per share 2008-2009 (90 – 110 + 4)/110 = –14.55% -208 Purchase of two shares at $110, plus dividend income on three shares held 2009-2010 (95 – 90 + 4)/90 = 10.00% 110 Dividends on five shares, plus sale of one share at $90 3.15% 396 Dividends on four shares, plus sale of four shares at $95 per share 2.33% -0.1661% → 5-83 83