Chapter 12 RISK & RETURN: PORTFOLIO APPROACH. Alex Tajirian

Transcription

1 Chapter 12 RISK & RETURN: PORTFOLIO APPROACH Alex Tajirian

2 Risk & Return: Portfolio Approach OBJECTIVE! What type of risk do investors care about? Is it "volatility"?...! What is the risk premium on any asset, assuming that investors are well diversified?! As a byproduct: Why should investors diversify? # Statistical Background 2. OUTLINE # Portfolio "Risk" Diversification: Why not put all your eggs in one basket? # Optimal risk-reward tradeoff: a market-based 1 approach Intuitively develop a model (theory) that tells us "what should happen to an asset's required return (price) if risk changes. ] What is the risk premium (RP) required (by an average investor) to hold the asset? Alex Tajirian

3 Risk & Return: Portfolio Approach 12-3 Objective from a financial manager's perspective:! Company Valuation: Is Company over/under-valued?! What return do shareholders require for new projects? (Ch 14)! How risky is a division, project, or the company? (Ch 14) Objective from an investor's view:! Which stock(s) is under/over-valued, i.e. mis-priced?! Why do some portfolios make sense while others do not?! Why?putting all your eggs in one basket does not make sense.! How "risky" is your portfolio?! How much return should an investor require form a given portfolio? Alex Tajirian

4 Risk & Return: Portfolio Approach 12-4 RISK & RETURN Statistical Backgound Stand-Alone Risk Porfolio Risk Variance of individual Asset Aggregate Portfolio Aggroach Assets within A Portfolio Beta Risk Financial Assets Projects, Divisions Alex Tajirian

5 Risk & Return: Portfolio Approach STATISTICAL BACKGROUND 3.1 RANDOM VARIABLE Examples: temperature, stock prices Return Return 6% 6% time time Stock A Stock B Stocks A & B have same center (average) of 6% B is more VOLATILE than A 3.2 PROBABILITY DISTRIBUTION probability = likelihood = frequency of occurrence Alex Tajirian

6 Risk & Return: Portfolio Approach SUMMARY MEASURES: ONE VARIABLE (Center & Volatility) Motivation: Need to summarize the data into few indicators 1.1 Measure of center of data: expected value Case 1a: probability of outcome is known. expected return ' p 1 k 1 % p 2 k 2 %... % p N k N N ' E p i k i i'1 p i = probability of outcome i occurring k i = value of outcome i N = number of observations Remember. We are interested in average return not average price, since price level is not very informative. Alex Tajirian

7 Risk & Return: Portfolio Approach 12-7 Case 2a: past observations are available (sample) each P i ' 1 N average ' k ' k % k 2 %... % k N N sum of actual rates of return ' number of observation Alex Tajirian

8 Risk & Return: Portfolio Approach 12-8 Example 1: Calculating Average Returns k '? ; Probability Is Given Given: Data state of economy p i k i i = 1 + 1% change in GNP.25-5% i = 2 +2% change in GNP.50 15% i = 3 +3% change in GNP.25 35% Solution: Observations (p i x k i ) i = % i = % i = % ˆ Expected return = ( ) = 15% Alex Tajirian

9 Risk & Return: Portfolio Approach 12-9 Example 2: Calculating Average Return k ZZZ '? ; Only Past Observations Given Given: Year k ZZZ, t % % % Solution: k ZZZ ' 10 % (&5) % 10 3 ' 15 3 ' 5% Alex Tajirian

10 Risk & Return: Portfolio Approach Measure of volatility: variance L Motivations: Simple example: You toss a coin, you win $1 if head, or lose $1 if tail. verage payoff ' x ' p 1 x 1 % p 2 x 2 ' 1 2 (&1) % 1 2 (1) ' What about the dispersion (deviations)?! Sum of Deviations = ( ) + (1-0) = 0 We obviously have dispersion: -1 and 1. Thus, one solution is to square deviations before you take sum. Alex Tajirian

11 Risk & Return: Portfolio Approach Case 1b: Probability of outcomes known F 2 ' variance ' p 1 (k 1 & k) 2 % p 2 (k 2 & k) 2 %... %p N (k N & k) 2 ' j N i'1 ' j N i'1 [ p i (k i & k) 2 ] [p i (i th deviation from average) 2 ] ' sum of weighted squared deviations from average F ' standard deviation ' F 2 Alex Tajirian

12 Risk & Return: Portfolio Approach Example 3: Calculating Variance (Data used in Example 1) observation k i & k (k i & k) 2 p i (k i & k) 2 i=1 ( ) x.04 =.01 i=2 ( ) 0.5 x 0 = 0 i=3 ( ) x.04 =.01 ˆ Variance = =.02 = 2% Alex Tajirian

13 Risk & Return: Portfolio Approach Case 2b: sample is available F 2 ' N j t'1 (k t & k) 2 N & 1 ' (k & 1 k)2 % (k 2 & k) 2 %... % (k N & k) 2 N & 1 sum of squared deviations from mean ' N & 1 ' Average of squared deviations from the mean F ' standard deviation ' F 2 Alex Tajirian

14 Risk & Return: Portfolio Approach Example 4: Calculate Variance of A Stock Using data From Example 2, F 2 ZZZ ' (10% & 5%)2 % (&5% & 5%) 2 % (10% & 5%) 2 3 & %.01 %.0025 ' '.015 ' F ZZZ '.0075 ' 8.68% Note. To make any intuitive sense out of the variance number (.0075) is to compare it to another stock, say that of Xerox =.01. Here, you can say that Xerox stock is more volatile than ZZZ. Alex Tajirian

15 Risk & Return: Portfolio Approach Historical Returns & Standard Deviations Series Average Annual Return Standard Deviation common stocks 12.1% 20.9% small stocks long-term corporate bonds long-term government bonds U.S. T-bills Inflation Source. R.G. Ibbotson and R.A. Sinquefield, Stocks, Bonds, Bills and Inflation.? 3 How much is the historical risk premium on stocks? Puzzle 1: (Size Effect) Even when "risk" is taken into account, small firms historically have achieved higher returns! Puzzle 2: (January Effect) Return in January have historically been higher than any other month! Alex Tajirian

16 Risk & Return: Portfolio Approach SUMMARY MEASURES: SEVERAL VARIABLES A portfolio of stocks 2.1 Central tendency (mean/average/expected value) k p ' average return on a portfolio ' w 1 k 1 % w 2 k 2 %...%w n k N N ' j ( w i k i ) i'1 (9) where, "i" N represents assets (not observations), and p represents?portfolio," which is also an asset. = number of assets in the portfolio w i ' weight of asset i in the portfolio ' proportion of total invested in stock i amount invested in asset i ' total value of investment Alex Tajirian

17 Risk & Return: Portfolio Approach Example: Calculating Portfolio Returns You have $100 to invest. Choose 50% in IBM, 50% in RCA. average returns ( k) are 15% and 20% respectively Solution: The weights are (½) each. Therefore, k p ' 1 2 k IBM % 1 2 k RCA ' % % % ' 7.5% % 10% ' 17.5% Obviously, if you invest 75% in IBM, then the weights will be (3/4) and (1/4) respectively. Alex Tajirian

18 Risk & Return: Portfolio Approach Measures of Co-Movement (no causality) (a) Absolute measure : Co-variance of x and y / cov(x,y) cov(x,y) ' (x 1 & x)(y 1 & y) % (x 2 & x)(y 2 & y) %...% (x N & x)(y N & y) N & 1 Example: Calculating Covariance X(%) Y(%) i = i = i = Average 4 Solution: cov(x,y) ' (1%&2%)(6%&4%)%(3%&2%)(2%&4%)%(2%&2%)(4%&4%) 3&1 ' &.0002&.0002%0 2 ' &.0002 ' &.02% Alex Tajirian

19 Risk & Return: Portfolio Approach Thus, If cov(x,y) is < 0, then x and y move in opposite direction If cov(x,y) is > 0, then x and y move in same direction If cov(x,y) is = 0, then x and y have no systematic co-movement ; I. 1-16, II. 1 ( Alex Tajirian

20 Risk & Return: Portfolio Approach Modern Portfolio Theory (MPT) 3.1 OUTLINE: Step 1: Diversification based on: # stocks with negative co-movement (correlation) # stock within a large portfolio Step 2: Step 3: Step 4: Step 5: Step 6: Develop a new measure of risk -- $: sensitivity of an asset to movements in the market". This measures an asset s risk relative to a benchmark or the market. Develop a market-based risk/return tradeoff model How to measure the market" in practice How to obtain estimates of $ in practice International diversification Alex Tajirian

21 Risk & Return: Portfolio Approach DIVERSIFICATION Return Stock AA Return Stock BB Return Portfolio AA + BB 2 % 2 % 2 % Return Stock AA Return Stock CC Return Portfolio AA + CC 6 % 6 % 6 % Alex Tajirian

22 Risk & Return: Portfolio Approach PORTFOLIO SIZE AND RISK Portfolio Size & Risk Naive Diversification Portfolio Standard Deviation Diversifiable risk Market Portfolio Systematic Risk Total Risk 40 Number of Stocks in the Portfolio Alex Tajirian

23 Risk & Return: Portfolio Approach Alex Tajirian

24 Risk & Return: Portfolio Approach DECOMPOSING TOTAL RISK From "portfolio size and risk" relationship, F 2 ' variance ' Total Risk ' Stand&Alone Risk ' Systematic Risk % Diversifiable Risk L systematic risk / non-diversifiable risk / market risk L diversifiable risk / idiosyncratic risk / unsystematic risk Therefore, In a large portfolio, unsystematic risk is essentially eliminated by diversification. But in practice, total risk cannot be completely eliminated by increasing the number of stock in a portfolio. Thus, the only relevant risk for investors who hold a well diversified portfolio is systematic risk, not total variance. Alex Tajirian

25 Risk & Return: Portfolio Approach Examples of factors contributing to risk: definition: Factors are sources of risk, which are outside the control of management:! systematic factors: GNP, inflation, interest rates, oil shocks! diversifiable factors: law suits, labor strikes, management luck Alex Tajirian

26 Risk & Return: Portfolio Approach SYSTEMATIC COMPONENT OF ACTUAL RETURNS In the previous section, we saw that the only relevant risk (in the context of a portfolio) is market risk. Thus, it would make sense to measure the risk of a stock in terms of how it moves with the market, i.e., in terms of how sensitive is a stock s actual return (k i, t ) to changes in the actual return of the market (k m, t ). Implication: A company's systematic component of actual return would depend only on: (a) the company s exposure to the market, denoted by $, and (b) the actual return on the market. Thus, to get a better feel for this relationship, you can:! plot the historical data t-period k i, t k m, t 6/82 5% 5% 7/82 9% 10% /94 7% 3%! The slope of the "best fit line" through the data gives you an estimate of $.! Thus, $ is a measure of relative risk. Therefore, Graphically we will have: Alex Tajirian

27 Risk & Return: Portfolio Approach Alex Tajirian

28 Risk & Return: Portfolio Approach $ AS A MEASURE OF RISK: $ value Implication $ i =1 Y if market _ by 100%, k it _ 100% on average $ i =1.5 Y if market _ by 100%, k it _ 150% on average $ i =.5 Y if market _ by 100%, k it _ 50% on average $ i =-.5 Y if market _ by 100%, k it ` 50% on average! Graphical Representation Alex Tajirian

29 Risk & Return: Portfolio Approach BETA AS A MEASURE OF RISK Actual return on stock (%) Stock A high beta Low beta Stock B Stock C negative beta K M The higher the BETA, the higher the RISK For same level of increase in market return (15-10) * Stock A increase by 100% (16-8) * Stock B increase by less * Stock C decreases Alex Tajirian

30 Risk & Return: Portfolio Approach 12-30? 4 What are possible theoretical and actual value of beta?? 5 What industry stocks tend to have high/low $?! Factors influencing $ and their direction: Amount of debt, Earnings Variability, Alex Tajirian

31 Risk & Return: Portfolio Approach Sample of Betas & Their Standard Deviations Company Beta St. Deviation AT&T % Bristol Myers Squibb Capital Holding Digital Equipment Exxon Ford Motor Co Genentech McDonald's McGraw-Hill Tandem Computer based on ? Which is riskier: Genentech or Tandem? ; I.6-13, II.3, 4 ( Alex Tajirian

32 Risk & Return: Portfolio Approach CAPITAL ASSET PRICING MODEL (CAPM) 4.1 RISK/RETURN TRADEOFF required return ' k s ' k RF % Risk Premium What is the risk premium, RP, for asset i? ] Required Return on asset "s" =? 4.2 MOTIVATION Alternatively, Alex Tajirian

33 Risk & Return: Portfolio Approach RESULT: Capital Asset Pricing Model CAPM [ pronounced "CAP-M"] Investors get rewarded only for non-diversifiable risk. Obviously they would not require a risk premium for a "bad" that they can themselves eliminate through diversification Specifically, where, k s ' k RF % RP s ' k RF % (k M & k RF ) $ s k s = required return on asset s k m = required return on the market k RF = risk-free rate = return on a T-bill! Compensation (required return) depends only on an asset's exposure to the market: $. F 2 of stock, F 2 of residuals, industry, size of firm, and inflation are not part of the equation; they are irrelevant in determining the compensation. The only reason two assets would have a different required return is a difference in their $. Alex Tajirian

34 Risk & Return: Portfolio Approach 12-34! linear (proportional) relationship between risk and return: investors require ( k M - k RF ) % compensation for each unit of $-risk. ] RP s = { (k M -k RF ) $ s } is proportional to stock's $. Illustration: Suppose ( k M - k RF ) = 8.5%. If $ s _ from 1 to 2 Y RP s increases 2 times.! RP M is independent of the security. Alex Tajirian

35 Risk & Return: Portfolio Approach Example: Calculating Required Return Given: k RF = 5%, k M = 10%, b xyz = 2 k xyz =? Solution: k xyz = 5% + (10%-5%)(2) = 15%? 6 What is risk premium of market (RP M ) in this example?? 7 What is risk premium of XYZ Inc.? Alex Tajirian

36 Risk & Return: Portfolio Approach PORTFOLIO RISK IMPLICATIONS $ p ' w 1 $ 1 % w 2 $ 2 %...%w n $ n ' weighted average of betas in the portfolio where, w i = weight of each asset i in the portfolio = proportion of total assets invested asset i. Alex Tajirian

37 Risk & Return: Portfolio Approach Example: Calculating $ p and k p Given: k RF = 3%, k M = 10%, and Asset $ i w i X 1 25% Y % Z.5 25% Solution: $ p ' (.25)(1) % (.5)(1.5) % (.25)(.5) ' k p ' k RF % (k M & k RF )$ p ' 3% % (10% & 3%)(1.125) ' % Alex Tajirian

38 Risk & Return: Portfolio Approach WHAT IS THE OPTIMAL $ p FOR AN INVESTOR? # SML / Security Market Line / Relationship between $ of any asset and k s # SML provides the "correct" tradeoff between risk & return ] The tradeoff that an average investor should get Y All positions on the SML are equally "good". Y The portfolio that an individual should choose, out of all the "good" ones on the SML, depends only on the individual's appetite for risk. # Applications:! Corporate finance: determining k project, k division, k company! Investments: Given the SML, an investor then determines which of these correct combinations of risk-return she wants to accept based on her individual appetite for risk. Alex Tajirian

39 Risk & Return: Portfolio Approach # Graphical representation: The CAPM can be written as an equation of a straight line, namely k s ' k RF % (k M & k RF )$ s y ' a % (slope)x where, a = y-intercept Alex Tajirian

40 Risk & Return: Portfolio Approach SECURITY MARKET LINE (SML) required return SML k M compensation for systematic risk k RF compensation for time value of money β SML: K i = K RF + ( k M - K RF ) * 1 i beta Risk

41 Risk & Return: Portfolio Approach UNDER/OVER REWARDED STOCKS rate of return A B SML K RF beta Stock A is OVER-REWARDED, since actual return (6%) > required return (4%), for a level of.8 beta risk. Stock B is UNDER-REWARDED, since actual return (9%) < required return (11%), for a level of 1.8 beta risk.

42 Risk & Return: Portfolio Approach 12-42? What is the slope of the above SML?? If k m = 8%, what is k RF? Dynamic Mechanism: Consider stock "A" Step 1: Suppose that stock "A" has had an average return of 6%, which is > required return. Step 2: Step 3: Suppose now people discover this stock; it looks like a great buy. However, if people start buying it, then its price _. If price _, then its actual return` until it becomes equal to required return. ˆ Historical average return will end up = required return. otherwise EMH would not hold.

43 Risk & Return: Portfolio Approach 12-43? What can you say about a financial market where you observe a number of securities like A Inc. and B Inc.?? Suppose "A" and "B" represent projects. What can you say about them?? So what might happen to the industry that "A" belongs to, i.e. what will A's competitors do?

44 Risk & Return: Portfolio Approach Simple Application 8 1 Given two mutual funds GoGo and SoSo, with respective average historical returns of 20% and 15%. Which is a better mutual fund to hold? Simple Application 2? If F IBM _ Y IBM Solution: Thus, F 2 = market risk + idiosyncratic risk Market Risk ($) Idiosyncratic Risk Total Risk (F 2 ) Required Return (k s ) _

45 Risk & Return: Portfolio Approach Application 3? Only relevant risk is ß?? For investor or manager?? What kind of investor are we assuming? Application 4? How are the values of k RF and k m determined?? Current market values?? Historical?? Other?

46 Risk & Return: Portfolio Approach HOW TO ESTIMATE BETA? # Alternative 1: Pure play as discussed earlier. This is what I am emphasizing in this course. # Alternative 2: Run the following regression, only if you have the statistical background. k i,t ' a i % $ i k m,t % e i,t Where, k i,t = actual return on stock i at time t k m,t = actual return on the "market" portfolio at time t e i,t = error in return specific to stock i $ i = slope of?best fit line! S&P500 is usually used for the market! Regression analysis is used to estimate beta (b); the best line that fits the data (observation of returns over time)! Beta measures co-movement of stock i with the "market." ] Beta measures the sensitivity of an asset to the market

47 Risk & Return: Portfolio Approach # Alternative 3: use following formula 9 : ˆ$ i ' cov(k i, k M ) F 2 M Note.?i stands for any asset. This includes individual stocks, also portfolios (p), division,... # Alternative 4: Obtain from $ service (Merrill Lynch, BARRA,...)

48 Risk & Return: Portfolio Approach INTERNATIONAL DIVERSIFICATION Motivation: Easiest way to see it is to look at each country's stocks as a portfolio. Thus, you are combining portfolios that do not necessarily have high positive correlation. Therefore, the concept of diversification is still applicable.

49 Risk & Return: Portfolio Approach SUMMARY T vocabulary CAPM, SML, $, diversifiable\idiosyncratic\non-systematic risk, market\non-diversifiable\systematic risk, variance, covariance, volatility, "best fit line" T Stock variance is not a good measure of equity risk since most of stock variance (80%) is firm specific (diversifiable). T Theoretically, according to the CAPM, the only source of equity risk is Beta. Thus, company size, idiosyncratic risk, stock volatility, and industry are irrelevant. The only risk investors care about is if it contributes to portfolio risk. T To obtain estimates of beta,! Pure play method; free-hand drawing of "best fit line".! Use beta-services: Merrill Lynch, BARRA! Run regression yourself using standard software! Or use following formula

50 Risk & Return: Portfolio Approach ˆ$ i ' cov(k i, k M ) F 2 M T Dynamic Mechanism: Stocks; projects T The SML provides the correct risk-return tradeoff.

51 Risk & Return: Portfolio Approach ' 8. IN FUTURE CLASSES ' O Tests of CAPM O Alternatives to CAPM O Limitations of CAPM O More on portfolio selection and diversification O More complicated ways to estimate beta # Anomalies! Size effect: Why do small companies have had higher returns?! January effect: Why are the historical returns in January higher than any other month?! day of the week effect

52 Risk & Return: Portfolio Approach This is what financial markets determine as a tradeoff between how much risk an investor has to accept for a specific level of desired average return. Moreover, if these markets are fair, so would the tradeoff be. There would not exist securities that are under- or over-rewarded for their inherent risk. Note, that an investor might have her own view as to what the correct tradeoff is. The issues of market fairness and its implications on investment decision fall under the rubric of Efficient Market Hypothesis (EMH). Thus, countries without any developed financial markets would have no clue as to what the tradeoff might be ENDNOTES k ZZZ,1985 ' P ZZZ,Dec. ) 85 &P ZZZ,Jan.2, ) 85 %Dividend ZZZ,) 85 P ZZZ,Jan.2, ) Actual annual risk premium for an average stock is by definition = (actual average annual return on common stocks) - (actual average annual return on U.S. T-bills) = 12.1% - 3.6% = 8.5% 4. Theoretically, they can be any number between (- infinity) and (+ infinity), as they represent the slope of a line. However, in the U.S., they tend to be between.1 and Utility companies tend to have low betas, i.e., when the market is doing very well, people tend to increase their consumption of electricity only modestly. Thus, company returns would be increase significantly. Conversely, if the market is not doing very well, consumers would cut their electricity consumption by only a small out. Thus, the performance, or return, of these companies would not suffer much. In a similar argument, entertainment stock tend to have high betas. Make sure that you distinguish between a stock s volatility and its performance relative to the Market such as the S&P RP m ' (k m & k RF ) ' 10 & 5 ' 5%

53 Risk & Return: Portfolio Approach RP xyz ' (k m & k RF ) $ xyz ' (10&5) 2 ' 10% 8. Cannot tell, since we do not know the betas. Moreover, these funds could be overvalued given their betas. 9. Formula comes from OLS regression of k i on k M.

54 Risk & Return: Portfolio Approach I. Agree/Disagree- Explain 10. QUESTIONS 1. If stocks Chombi Inc. and Xygot Inc. have the same required return, or market expected return, a rational investor should choose the one that has highest variance as it offers higher chance of attaining high returns. 2. A good measure of volatility (dispersion) is: sum of deviations from the mean. 3. Variance of a stock is a good measure of risk to investors. 4. If the historical returns on mutual funds Saddam Inc., Whoopi Inc., and the market are 20%, 10%, and 15% respectively, then Saddam Inc. is the better buy. 5. If Kumquat Inc.'s variance increases, then its required return must increase. 6. Firm managers only care about beta risk, as the rest is diversifiable. 7. H, I No one will invest in an asset that has a negative beta. 8. H, I If you (personally) believe that the stock market will rally, then you would buy the stock with the highest beta. 9. CAPM is used in determining an appropriate rate of return for regulated utility companies. [ Note, this is not discussed in the notes or the book. I put it here just to indicate another possible application of CAPM] 10. If variance of a stock increases, its beta must increase too. 11. If the beta of a stock increases, its variance must increase too. 12. The higher the beta of a stock, the riskier the returns. 13. The higher the proportion of debt to total assets, the higher the firm's beta. 14. The higher the earnings variability, the higher the beta of a firm. 15. Investors prefer to have low beta portfolios.

55 Risk & Return: Portfolio Approach Beta is a measure of variance.

56 Risk & Return: Portfolio Approach II. Numerical 1. Given the following information: Observation Return on Potato Inc. Return on S&P 500 February % 10% March April (a) (b) (c) Calculate the variance and standard deviation of Potato Inc. Calculate the beta of Potato Inc. Interpret your result in (b). 2. H Given the variances of stocks X and Y are 15% and 20% respectively, with their covariance equal to 20. (a) You are investing $100,000 of which 25% is in X. What is the variance of this portfolio? (b) Since the variance of X < variance of Y, a rational investor would increase the proportion invested in X so as to reduce the variance of the portfolio. Agree or disagree? Explain. (c) If you substitute Y by stock Z in your portfolio, which has a variance of 20% and is negatively correlated with X, what happens to your answer in (a)? (d) Can the stock Z be positively correlated with Y? 3. H Given the following rate of return (%) information on companies X and Y: i = 1 i = 2 i = 3 X Y (a) Calculate, F X, F Y, cov(x,y), r XY. (b) Is it possible to obtain a portfolio of X and Y that has a zero variance?

57 Risk & Return: Portfolio Approach Given: A portfolio of three securities A, B, & C, with: Security Amount invested Average k beta A $5,000 9%.8 B 5,000 10% 1.0 C 10,000 11% 1.2 (a) (b) (c) (d) What are the portfolio weights? What is the average return on the portfolio? What is the portfolio's beta? If k RF = 3%, k m = 12%, what is the required return on the portfolio? Is this portfolio under or over-rewarded? Explain. 5. Given: k T-Bills = 9%, ß A =.7, k A = 13.5%, and k M =15%. (a) What is k of a portfolio with equal investments in A and T-Bills? (b) If ß p =.5, what are the portfolio weights? (c) If k p = 10%, what is its ß? (d) if ß p = 1.5, what are the portfolio weights? 6. You have a portfolio of equally valued investments in two companies A & B. The beta of this portfolio is 1.2. Suppose you sell one of the companies, which has a beta of.4, and invest the proceeds in a new stock with a beta of 1.4. What is the beta of your new portfolio?

58 Risk & Return: Portfolio Approach ANSWERS TO QUESTIONS I. Agree/Disagree Explain 1. Disagree. Other things equal, you choose the one with smallest variance. Variance is "bad". Thus, you do not want to accept it if it offers you the same return (compensation), as a less risky asset. 2. Disagree. Calculation results in 0 variation, as in a coin-toss example shown below. Sum of Deviations = (-1-0) + (1-0) = 0 3. Disagree. Most of the variance is diversifiable. 4. Disagree. We cannot tell. It depends on the betas of the mutual funds, which are not provided in the question. It also depends on the risk preferences of the investor. Also see p. 38 where stock B has higher returns but also is under-rewarded. 5. Disagree. Only if the increase in variance is due to an increase in the stock's beta. See Simple Application 2 p Disagree. They care about total risk (variance of returns), since their life depends on how well the company does. 7. Disagree. Such an asset can be great when times are bad. 8. Disagree. Remember that most of a company's variance is diversifiable. Thus, you need to buy a portfolio of stocks with high betas to diversify some of the firm-specific risk. 9. Agree. The CAPM is used by regulatory agencies to figure out what a fair return should be for the utilities. This is one way to decide on how much you pay for their services. 10. Disagree. Theory tells us what happens to variance if beta changes and not the other way around.

59 Risk & Return: Portfolio Approach Agree. Remember there are two sources of variance risk: market (beta) and firm-specific. So if beta increases, then variance must increase too, other things equal. 12. Agree. Higher beta means that stock prices go up and down, in relation to the market, by larger proportions. 13. Agree. One of the factors that affects beta is the D/A ratio. Moreover, they are directly related. The higher debt makes the firm more sensitive to interest rate, which is a systematic factor. 14. Agree. Earnings variability and beta are directly proportional. High earnings variability suggests that the firm's earnings move with the market. Good times bring in high earnings, while bad times have an adverse effect on them. 15. Disagree. It depends on how risk averse the individual is. Remember the higher the beta, the higher the required return. 16. Disagree. Beta measures an asset's return (price) fluctuations with respect to a benchmark such as the S&P500.

60 Risk & Return: Portfolio Approach II. NUMERICAL 1. k ' &9%(&1)%1 3 ' & 9 3 ' &3 F 2 ' (&9%&(&3%))2 % (1%&(&3%)) 2 % (&1%&(&3%)) 2 3&1 ' (&6%)2 % (4%) 2 % (2%) 2 2 F ' F 2 '.0028 ' 5.29% ' '.0028 b) STEP 1. Calculate average returns k Potatoe ' (&9) % 1 % (&1) 3 ' &3% k SP500 ' 10 % (&2) % 4 3 ' 4% STEP 2 Calculate beta of Potato Thus,

61 Risk & Return: Portfolio Approach $ potatoe ' cov(k,k ) potato S&P500 variance(k S&P500 ) [(&9&(&3))(10&4) % (1&(&3))(&2&4) %(&1&(&3))(4&4)] / N&1 ' [(10&4) 2 % (&2&4) 2 % (4&4) 2 ] / N&1 ' &60 72 ' &.83 C) since beta is -.83, if the market (S&P500) _ 100%, then Potato Inc. tends to ` by 83%.

62 Risk & Return: Portfolio Approach (a) Note that $100,000 is irrelevant (extraneous information). F 2 p ' w 2 X F2 X % w 2 Y F2 Y % 2w X w Y cov(x,y) ' (.25) 2 (.15) % (.75) 2 (.2) % 2(.25)(.75)(20) '.0094 %.1125 % 7.5 ' 7.62 (b) (c) (d) By _ w X you would ` variance of portfolio. However, you also need to look at RETURN too. Return on the portfolio could _ or `. It would ` variance of portfolio. No. Since Z is negatively correlated with X, and (X,Y) are positively correlated, Then Z has to be negatively correlated with both.

63 Risk & Return: Portfolio Approach a) X ' 1%3%2 3 Ȳ ' 6%2%4 3 ' 2% ' 4% F 2 x ' (1%&2%)2 % (3%&2%) 2 % (2%&2%) 2 3&1 F 2 y ' (6%&4%)2 % (2%&4%) 2 % (4%&4%) 2 3&1 '.0001% '.0004% '.01% '.04% ˆ r xy ' & ' &1 b) Yes, since these stocks are negatively correlated.

64 Risk & Return: Portfolio Approach Given: Portfolio of three securities A, B, & C, with: Security Amount invested Average k beta A $5,000 9% 0.8 B 5,000 10% 1.0 C 10,000 11% 1.2 (a) (b) (c) (d) What are the portfolio weights? What is the average return on the portfolio? What is the portfolio's beta? If k RF = 3%, k m = 12%, what is the required return on the portfolio? Is this portfolio under or over-rewarded? Explain. Solution: (a) w A ' 5,000 5,000%5,000%10,000 ' 5,000 20,000 '.25 w B ' 5,000 20,000 '.25 w c ' 10,000 20,000 '.5

65 Risk & Return: Portfolio Approach (b) k p ' w A k A % w B k B % w C k C '.25(9%) %.25(10%) %.5(11%) ' 10.25% (c) $ p ' w A $ A % w B $ B % w C $ C '(.25)(.8) % (.25)(1) % (.5)(1.2) ' 1.05 (d) using CAPM, k p ' 3% % (9%)(1.05) ' 12.45% Since (required return'12.45) > (average actual return'10.25) Y under&rewarded

66 Risk & Return: Portfolio Approach Given: k T-Bills = 9%, ß A =.7, k A = 13.2%, and k M =15% (a) What is k of portfolio, with equal investment in A and T-Bills? (b) If ß p =.5, what are the portfolio weights? (c) If k p = 10%, what is its ß? (d) if ß p = 1.5, what are the portfolio weights? Solution: (a) k p ' w A k A % w T&Bill k T&Bill ' (.5)(13.2%) % (.5)(9%) (b) $ p '.5 ' w A $ A % w T&Bill $ T&Bill But w A % w T&Bill ' 1 and $ T&bill ' 0 Y.5 ' w A (.7) % (1&w A )(0) Y w A '.5.7 ' 5 7 and w T&Bill ' 1& 5 7 ' 2 7

67 Risk & Return: Portfolio Approach c) Since we do not know the weights of the assets in the portfolio, we cannot use the "formula" in (b). We need to use CAPM. k p ' 10% ' k RF % (k M &k RF )$ p.1 ' 9% % (15%&9%)$ p.1 '.09 %.06$ p Y $ p '.1& ' 1 6 (d) $ p ' 1.5 ' w A $ A % w T&bill $ T&bill 1.5 ' w A (.7) % (1&w A )$ T&bill Y w A ' ' 2.14 > 1 and w T&bill ' &1.14 Since w A > 1, then you are borrowing 114% of your investment at the T- bill rate and investing your capital + borrowed amount in asset A. Thus, the negative weight of T-bill reflects borrowing the asset.

68 Risk & Return: Portfolio Approach Given: from equation for $ of portfolio,.5$ A %.5$ B ' 1.2 suppose you sell A. Thus, $ A '.4 Y $ B ' 1.2&.5(.4).5 ' 1.5 ' 2 ˆ $ new portfolio '.5($ new ) %.5($ B ) '.5(1.4) %.5(2) ' 1.7

69 Risk & Return: Portfolio Approach ELIMINATIONS (b) Relative co-movement: more intuitive than cov(x,y) correlation between x and y = r xy r xy ' cov(x,y) F x F y such that, On average: &1 # r xy # 1 if r xy = 0; then x and y have no systematic co-movement if r xy = 1; then if one _ by 100%, the other _ by 100% if r xy = -1; if one _ by 100%, the other ` by 100% if r xy =.5; if one _ by 100%, the other _ by 50%

70 Risk & Return: Portfolio Approach Example: Calculating Correlation Given data used above (in calculation of covariance) F 2 x ' (1%&2%)2 % (3%&2%) 2 % (2%&2%) 2 3&1 F 2 y ' (6%&4%)2 % (2%&4%) 2 % (4%&4%) 2 3&1 '.0001% '.0004% '.0001 '.0004 &.0002 ˆ r xy ' ' & L Compare r xy = -1 with cov(x,y) = -.02%. Former more intuitive.

71 Risk & Return: Portfolio Approach Variance of a portfolio 1 (volatility): Special case: only two stocks/assets x & y Effect of covariance contribution on variance w 2 x F2 x % w 2 y F2 y % 2w x w y cov(x,y) ' F2 p variance effect covariance effect total No Effect _ ` L Thus, variance of a portfolio of assets depends on: 1. # of assets included 2. Weight of each asset in portfolio 3. Variance of each asset 4. Covariance of each pair of assets Note. In practice, diversification works as long as there are many assets in the portfolio which are not highly positively correlated.

72 Risk & Return: Portfolio Approach Example: Calculating Variance of a Portfolio Given: Two firms X and Y, such that variances of X and Y are 10% and 20% respectively. What is the variance of an equalweighted portfolio if cov(x,y) is 10%, 0, and -10%? Solution: sum of weighted variances = (.5) 2 (10%) + (.5) 2 (20%) = 7.5% covariance contribution: sum of weighted variances covariance contribution Portfolio Variance Effect on Variance 7.5% 2(.5)(.5)(0%) = 0 7.5% no effect 7.5% 2(.5)(.5)(10%) = 5% 12.5% _ 7.5% 2(.5)(.5)(-10%) = - 5% 2.5% ` L Thus, (the variance of the portfolio that includes assets that are negatively related) < (sum of weighted variance contribution).

73 Risk & Return: Portfolio Approach a.i LIMITATION OF PORTFOLIO VARIANCE To use F p formula:! too many items to calculate N variances and {(N 2 - N)/2} co-variances if N = 100, we need over 4,000 co-variances to calculate! Does not tell us the riskiness of individual stock/asset, i.e. cannot measure risk premium (RP) of individual stock. Y we need to make some assumptions (restrictions) about how stocks move.

74 Risk & Return: Portfolio Approach # More realistic description of historical stock returns k it ' $ i k Mt % e it where e it is firm specific return at time t Y F 2 i ' systematic risk % firm specific risk ' $ 2 F 2 M % firm specific risk

75 Risk & Return: Portfolio Approach i.1 More on firm specific return! In U.S., a company's systematic risk is on average less than 20% of its total risk. Thus, most of an individual company's total risk is firm specific.! Illustration Q Three stocks each with same beta = 2, but different idiosyncratic variances. Q Stock3 has the highest variance. In the third period, it actually went down while the market was up.

76 Risk & Return: Portfolio Approach % 20.0% 10.0% 0.0% -10.0% -20.0% -30.0% market portfolio stock1 with beta =2 stock2 with beta = 2 stock3 with beta = 2

77 Risk & Return: Portfolio Approach i.2 More Factors Influencing Actual Returns k t ' a % $k Mt % $ 1 F 1t % $ 2 F 2t %... % e t

78 Risk & Return: Portfolio Approach a.ii ASSUMPTIONS G There exists a risk-free asset (k RF ) G Investors are risk averse G Investors maximize satisfaction (utility) G All non-diversifiable factors are aggregated (incorporated) in k M G Investors hold portfolios and not individual stocks 2. Note. (compare to limitations of F p )! we only need to calculate N betas (simpler than variance)! we have risk measure for each stock (beta)

79 Risk & Return: Portfolio Approach Portfolio variance, as a measure of equity risk, has a number of shortcomings. True. See notes p. 44, Financial risk is diversifiable. 9. The higher a company's product demand variability, the higher its Business Risk. 10. The higher the fixed costs, the lower the Business Risk. Disagree. Financial risk is defined as the risk associated with a company's debt level. The higher the debt to asset ratio, the higher the beta. Thus, the higher the systematic risk. True. Demand variability is one of the sources of Business risk. The higher the variability means higher uncertainty about the firm's ability to sell its product. Thus, the higher the risk. Disagree. High fixed costs put stress on a company's CFs, as they are unavoidable cash outflows in the short-run. Thus, the higher the Business Risk. 11. International diversification cannot decrease portfolio variance since an investor is stuck with a country's non-diversifiable risk. 12. International diversification increases risk. Therefore it should be avoided. Disagree. International diversification can lower systematic risk as different countries do not have perfectly correlated systematic risks. Diversification of international systematic risk works in the same way as the diversification of domestic firm-specific risk. 13. Disagree. Although there is an additional component, foreign exchange risk, diversification principles still hold. 14. If the correlation between stocks Zart and Zed is one, then if return on Zart increases by 100%, that of Zed tends to increase by 1%. 15. If two variables are highly correlated, then a movement in one causes a movement in the other. 16. The variance of an asset can be less than 0.

80 Risk & Return: Portfolio Approach Disagree. Zed tends to increase by 100%. 18. Disagree. Correlation does not imply causality. An example would be football and stock market correlations as in "Football and Seesaw Finance." 19. Disagree. It has to be š 0, since you are squaring and summing the deviations. 20. You cannot obtain a beta estimate for a division. 21. A stock's required return (k s ) tends to change daily, just as stock prices do. 22. Actual returns (k it ) and required return (k s ) tend to move in the same direction. 23. Disagree. You can look at a division as a separate entity. Then try to obtain a beta estimate based on that of a similar firm(s) ( same line of business and size as your division). 24. Disagree. Required return does not change every day. If the beta of the company changes, then it would. Remember the actual and required returns are rarely equal. 25. Disagree. See #17 above. 26. If an asset has a beta of 1, then it must have the same variance as the market. 27. If systematic risk of a stock increases, then required return increases too. Thus, you are better off because you would be necessarily receiving higher returns. 28. Disagree. The market portfolio has only systematic risk. A stock with beta of one, has in addition an idiosyncratic components of risk. F 2 = variance = total risk = systematic risk + idiosyncratic risk If stock's beta= 1, then company systematic risk = market risk = market variance. But, since company idiosyncratic risk > 0, then company variance > market variance. 29. Disagree. See Application 2, p. 44, 74. You need to distinguish between required return and realized/actual return. 30. Low beta stocks are less volatile than high beta stocks. 31. Two stocks X&Y have the same variance but X has a higher beta. Y must have higher idiosyncratic risk.

81 Risk & Return: Portfolio Approach If the variance of the market increased, then required return on an asset increases too. 33. Disagree. Volatility, measured in terms of variance, has two components: systematic risk + idiosyncratic risk. Low beta stocks would have low systematic risk. However, such a low beta stock could have a much higher idiosyncratic risk than a high beta portfolio. Thus, low beta does not imply low volatility. Also see p.?. 34. Agree. Since total risk is the same and X has a higher beta (i.e. higher systematic risk), it must also have a lower idiosyncratic risk than Y. 35. Disagree. Look at CAPM. There is no compensation for the variance of the market.

82 Risk & Return: Portfolio Approach More Questions Agree/Disagree-Explain 36. Financial risk is diversifiable. Disagree. Financial risk is defined as the risk associated with a company's debt level. The higher the debt to asset ratio, the higher the beta. Thus, the higher the systematic risk. 37. The higher a company's product demand variability, the higher its Business Risk. Agree. Demand variability is one of the sources of Business risk. The higher the variability means higher uncertainty about the firm's ability to sell its product. Thus, the higher the risk. 38. The higher the fixed costs, the lower the Business Risk. Disagree. High fixed costs put stress on a company's CFs, as they are unavoidable cash outflows in the short-run. Thus, the higher the Business Risk. 39. International diversification cannot decrease portfolio variance since an investor is stuck with a country's nondiversifiable risk. Disagree. International diversification can lower systematic risk as different countries do not have perfectly correlated systematic risks. Diversification of international systematic risk works in the same way as the diversification of domestic firm-specific risk. 40. If the correlation between stocks Zart and Zed is one, then if return on Zart increases by 100%, that of Zed tends to increase by 1%. Disagree. Zed tends to increase by 100%. 41. If two variables are highly correlated, then a movement in one causes a movement in the other. Disagree. Correlation does not imply causality. An example

83 Risk & Return: Portfolio Approach would be football and stock market correlations 42. The variance of an asset can be less than 0. Disagree. It has to be š 0, since you are squaring and summing the deviations. 43. You cannot obtain a beta estimate for a division. Disagree. You can look at a division as a separate entity. Then try to obtain a beta estimate based on that of a similar firm(s) ( same line of business and size as your division). 44. If an asset has a beta of 1, then it must have the same variance as the market. Disagree. The market portfolio has only systematic risk. A stock with beta of one, has in addition an idiosyncratic components of risk. risk F 2 = variance = total risk = systematic risk + idiosyncratic If stock's beta= 1, then company systematic risk = market risk = market variance. But, since company idiosyncratic risk > 0, then company variance > market variance. 45. If systematic risk of a stock increases, then its required return increases too. Thus, you are better off because you would necessarily be receiving higher returns. Disagree. See Application 2, p. 44, 74. You need to distinguish between required return and realized/actual return. 46. Low beta stocks are less volatile than high beta stocks. Disagree. Volatility, measured in terms of variance, has two components: systematic risk + idiosyncratic risk. Low beta stocks would have low systematic risk. However, such a low beta stock could have a much higher idiosyncratic risk than a high beta portfolio. Thus, low beta does not imply low volatility. 47. Two stocks X&Y have the same variance but X has a higher beta. Y must have higher idiosyncratic risk.

84 Risk & Return: Portfolio Approach Agree. Since total risk is the same and X has a higher beta (i.e., higher systematic risk), it must also have a lower idiosyncratic risk than Y. 48. If the variance of the market increased, then required return on an asset increases too. Disagree. Look at CAPM. There is no compensation for the variance of the market. In terms of an equation, then the above "best fit line" would look like: nstant (sensitivity of asset))i )) to market) (return on the market pe intercept % (slope of line ) (return on the market period t )) ) % $ i k Mt where, k it / actual return observations on asset over period "t" a i / y-intercept $ i / sensitivity (exposure) of asset i to the market k Mt / actual return observations on the market over period "t" M /market portfolio, typically S&P500 ; Rest (

85 Risk & Return: Portfolio Approach In general, F 2 p ' w 2 1 F2 1 %w 2 2 F2 2 %2w 1 w 2 cov(k 1,k 2 ) % w 2 3 F2 3 %2w 1 w 3 cov(k 1,k 3 )%2w 2 w 3 cov(k 2,k 3 ) ' j w 2 i F2 i % j j 2w i w j cov(k i,k j ) 2. Why diversify? See LAT 4/19/93 p. E83.