Lecture 2 Basic Tools for Portfolio Analysis

1 Lecture 2 Basic Tools for Portfolio Analysis Alexander K Koch Department of Economics, Royal Holloway, University of London October 8, 27 In addition to learning the material covered in the reading and the lecture, students should be familiar with the concept of risk aversion and understand how investors preferences over risky portfolios may be expressed in terms of a utility function with the expected return and the volatility of the portfolio returns as arguments; feel comfortable using statistical tools such as mean, variance, and covariance for simple portfolio analysis Required reading: Bodie, Kane, and Marcus (28) (Chapters 5 [some background material] and 6) Supplementary reading: Grinblatt and Titman (22) (Chapter 4), Levy and Post (25) (Chapters 6 and 7) E-mail: AlexanderKoch@rhulacuk

2 1 Expected Value, Variance, and Standard Deviation Example 1 (from Bernoulli (1738), p31) Peter tosses a coin and continues to do so until it should land heads when it comes to the ground He agrees to give Paul one ducat if he gets heads on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled How much would you be willing to pay to play this game? Daniel Bernoulli s cousin asked him for his opinion, claiming that any fairly reasonable man would sell his chance [to play the game], with great pleasure, for twenty ducats The expected value of the payoff from this game is infinite (see table 1): E[X] = p i x i = 1/2 = i=1 i=1 Bernoulli studied this game while living in St Petersburg, from whence the St Petersburg Paradox : People accept a very small amount of money to give up a prospect with an infinite expected return i (number of tails until first heads ) p i (probability) x i (prize) p i x i 1/2 1 1/2 1 1/4 2 1/2 2 1/8 4 1/2 3 1/16 8 1/2 n 1 2 n+1 2 n 1/2 Table 1: Derivation of expected value for the St Petersburg game Remark: This example has inspired Bernoulli to lay the foundations of expected utility theory A brief introduction is given in Bodie, Kane, and Marcus (28) (Chapter 6, Appendix B) Conclusion: individuals usually evaluate an uncertain income stream not only based on its expected value but also assess its riskiness We will measure risk using the variance or standard deviation of an income stream

3 Caveat: other statistical measures, such as the skewness of an income distribution, may also be important (see Bodie, Kane, and Marcus (28) (Chapter 6, Appendix A)) 11 Definitions (single random variable) Financial decisions involve evaluating risky prospects These are described by the notion of a random variable For example, the return of a stock can take on a number of different values We shall look at discrete random variables, where there are a finite number, say n, of such possible outcomes Such a discrete random variable, which we can call X, describes a list of all possible realizations (each described by a number from the reals), denoted by x i R, i = 1,, n Let X = {x 1,, x n } be a random variable with associated probability distribution {p 1,, p n } Definition 1 (expected value) E[X] = n i=1 p i x i (variance) V ar[x] = E (X E[X]) 2 Often it is convenient to use an equivalent way of writing the variance E (X E[X]) 2 = E[X 2 ] E[X] 2 Using the definition of the expectation operator, we have V ar[x] = = n p i (x i E[X]) 2 i=1 n p i x 2 i E[X] 2 i=1 or, equivalently, (standard deviation) σ[x] = V ar[x] Some useful facts: Let X 1, X 2,, X n be random variables, and a 1, a 2,, a n be constants 1 E[a i X i ] = a i E[X i ], i = 1,, n, 2 E[a i X 1 + a 2 X 2 + a n X n ] = a i E[X 1 ] + a 2 E[X 2 ] + a n E[X n ], 3 V ar[a i X i ] = a 2 i V ar[x i], i = 1,, n, 4 V ar[a i + X i ] = V ar[x i ], i = 1,, n

4 Example 1: Coin toss Example 2: Dice x 1 =1 (dice shows 1) p 1 =1/2 x 1 =1 (heads) 1/6 1/6 1/6 x 2 =3 (dice shows 2) x 3 =5 (dice shows 3) p 2 =1/2 x 2 =-1 (tails) 1/6 1/6 1/6 x 4 =7 (dice shows 4) x 5 =9 (dice shows 5) x 6 =11 (dice shows 6) Figure 1: Examples 12 Examples Example 2 Consider the following game I flip a coin If heads come up you get 1, if tails come up you pay me 1 Example 3 Consider the following game A dice is thrown Payoffs for the different outcomes are as depicted in figure 1 Questions: 1 What is the expected payoff, the variance and the standard deviation of the lotteries described in these two examples? 2 How much would you be willing to pay to participate in each game? 3 What does this tell you about your attitude towards risk?

5 Answers: 1 First, describe the set of possible outcomes and the corresponding probabilities Recall, to correctly define a probability distribution the probabilities must sum up to one Example 2 (x 1 = 1; p 1 = 1/2), (x 2 = 1; p 2 = 1/2) Example 3 (x 1 = 1; p 1 = 1/6), (x 2 = 3; p 2 = 1/6), (x 3 = 5; p 3 = 1/6), (x 4 = 7; p 4 = 1/6), (x 5 = 9; p 5 = 1/6), (x 6 = 11; p 6 = 1/6) 2 Descriptive statistics: Example 2 Example 3 E[X] = 1/2 1 + 1/2 ( 1) =, E[X 2 ] = 1/2 1 2 + 1/2 ( 1) 2 = 1, V ar[x] = E[X 2 ] E[X] 2 = 1, σ[x] = 1 = 1 E[X] = 1/6 1 + 1/6 3 + 1/6 5 + 1/6 7 + 1/6 9 + 1/6 11 = 36 6 = 6, E[X 2 ] = 1/6 1 2 + 1/6 3 2 + 1/6 5 2 + 1/6 7 2 + 1/6 9 2 + 1/6 11 2 = 28, 6 6 4, 76667 V ar[x] = E[X 2 ] E[X] 2 1, 16667 σ[x] = V ar[x] 3415655 2 Risk aversion A fair game is one which has an expected value of zero Example 2 would be a fair game The conclusion from example 1 was that the typical individual is willing to give up the right to a risky lottery in exchange for a certain payment that is less than the lottery s expected value Such an individual is called risk averse Investments in financial assets often have an uncertain return and therefore are examples of risky lotteries The (one-period) return of an investment, which involves no payments during the period, is given by: return = end-of-period value beginning-of-period value beginning-of-period value

6 To distinguish the actual return on an investment, measured after the period ends, from the uncertain return at the beginning of the period, we adopt the following notation: The uncertain return at beginning of the period, which is a random variable, is denoted by r The actual return on an investment, measured after the period ends, which is the realization of the random variable r, is written simply as r Also, we will sometimes abbreviate the expressions for means and standard deviations as follows: r i E[ r i ], σ i σ[ r i ] For our purposes, it is convenient to think of an investor trading off the expected return of an investment against the risk involved in the investment, as measured by the variance of its returns Definition 2 Investors have mean-variance preferences over risky lotteries if preferences can be represented by the following utility function: U( r) = E[ r] 5 A V ar[ r], (1) where A is an index of the investor s risk aversion 1 Remark: While the assumption of mean-variance preferences is convenient for analytical purposes, it is important to bear in mind that it is a strong assumption However, it can be shown that mean-variance preferences can be derived from the more general expected utility theory in the case where security returns are normally distributed and investors have so-called constant absolute risk aversion 2 1 If one wants to express expected returns as percentages rather than decimals, the factor in front of A needs to be rescaled to 5 see also Bodie, Kane, and Marcus (28) p168 2 For an introduction to expected utility theory, see Copeland, Weston, and Shastri (24), Chapter 3

7 Note on returns In practice, the return on an investment can be either in terms of an increase in the price at which you can sell the asset (ie, capital gains) or in terms of income that is derived from the asset (eg, dividends for stocks or interest payments for bonds) The return we talk about here is the sum of capital gains and dividends or other income For example, you buy Stock XYZ at the beginning of 26 for 71 a share Suppose that the price is at 782 at the end of the year and that there is a dividend of 6 per share This gives total earnings per share of 87 per share, composed of a capital gain of 782 71 = 81 plus dividend 6 per share The percentage return percentage return = capital gain + dividend initial share price = 81 + 6 71 = 1241 % is the sum of percentage capital gain percentage capital gain = capital gain initial share price = 81 71 = 1155 % and dividend yield dividend yield = dividend initial share price = 6 = 86 % 71 21 Evaluating Investment Opportunities If we are dealing with risk averse investors, we can use the concept of mean-variance dominance to rank some investment opportunities even without knowing the investor s exact preferences (see figure 2) Definition 3 A portfolio A (mean-variance) dominates a portfolio B if and only if E[ r A ] > E[ r B ] and σ[ r A ] σ[ r B ], ie, portfolio A offers a higher expected return than portfolio B while it is not more risky than portfolio B, or E[ r A ] E[ r B ] and σ[ r A ] < σ[ r B ], ie, portfolio A is less risky than portfolio B while its expected return is no lower than that of portfolio B

8 Er% [ ] 2 [%] [ r% ] Er > r σ <σ P P P 1 [%] [ r% ] Er < r P σ >σ P 1 2 3 4 σ [ r% ] Figure 2: Mean-variance dominance Given the investor s utility function all investment opportunities can be ranked (see figure 3) 3 The Statistics of Portfolio Returns Portfolios are combinations of risky assets, the returns of which can be described by random variables Thus, we need to extend our definitions from section 11 to cover combinations of random variables Definition 4 The portfolio weight for a stock i, denoted w i, is the fraction of portfolio wealth invested in stock i: w i = value of portfolio position in stock i value of portfolio Note: By definition, portfolio weights sum up to one A short position in stock i means that the portfolio weight for stock i is negative A long position in stock i means that the portfolio weight for stock i is positive

9 Er% [ ] U U U 2 P 1 1 2 3 4 σ [ r% ] Figure 3: Indifference curves for an investor 31 Expected portfolio returns A portfolio s expected return is straightforward to compute, using Fact 2 from above Suppose the portfolio P consists of securities i = 1,, n with expected returns r 1,, r n and portfolio weights w 1,, w n Then, the portfolio s expected return is given by E[ r p ] = n w i r i (2) i=1 Example 4 Suppose Sher chooses to invest half of her wealth in a stock market fund with expected return r s = 1 and the other half in a risk-free investment with return r f = 2 What is the expected return of Sher s portfolio? E[ r P ] = 5 1 + 5 2 = 6 Now, suppose Sher could borrow an amount equal to 1 half of her wealth 2 twice her wealth

1 3 ten times her wealth at the risk-free rate What is the portfolio s expected return now? 1 E[ r P ] = 15 1 5 2 = 14 2 E[ r P ] = 3 1 2 2 = 26 3 E[ r P ] = 11 1 1 2 = 9 Note: By leveraging an investment (ie, using borrowed funds) one can increase the expected return on the own investment (at the cost of a lot more risk as we shall see) 32 Variance of portfolio returns 321 Portfolios composed of a risky asset and a risk-free asset Before turning to portfolios of several risky assets it is useful to first concentrate on portfolios composed of a risky asset s and a risk-free asset f Note, the risk-free asset has zero return variance by definition That is, r f is a constant (σ f = ) Using facts 3 and 4 on variances from above, V ar[w s r s + w f r f ] = V ar[w s r s ] = ws 2 σs 2 (3) Let us revisit the previous example: Example 4 (continued) Suppose that the stock market fund has standard deviation σ s = 8 w s w f E[w s r s + w f r f ] Var[w s r s + w f r f ] σ[w s r s + w f r f ] 5 5 6 16 4 15-5 14 144 12 3-2 26 576 24 11-1 9 7744 88 Table 2: Impact of leverage on portfolio expected return and variance The mean-standard deviation diagram is a useful tool to represent the trade-off between expected return and portfolio risk We have already seen the key equations for the case of a portfolio with a risky asset s and a risk-free asset f:

11 Er% [ P ] w S >1 w S <1 S r f σ S σ P Figure 4: Mean-standard deviation diagram r P = w s r s + (1 w s ) r f, (4) σ P = w s σ s (5) Equation 5 w s = σ P σs If we plug this into equation 4, we get r P = r f + r s r f σ s σ P (6) Equation 6 describes the risk-return combinations that we are interested in (see figure 4) The slope measures how much the expected portfolio return increases per unit of additional risk taken on The difference between the actually realized return on a risky asset or portfolio and that on the risk-free asset, r P r f, is called excess return Thus, the above formula relates the expected excess return of a portfolio to its standard deviation In figure 5 we can see the indifference curves for investor 1 (Joe) and investor 2 (Rebecca) Joe s optimal asset allocation is given by the tangential point of his indifference curves with the investment opportunity set (portfolio A) Correspondingly, Rebecca chooses portfolio B Who is more risk averse, Joe or Rebecca? Do Joe/Rebecca hold long or short positions in the risk-free asset?

12 Er% [ P ] U 1 (A) U 1 (B) U 2 (B) B U 2 (A) r f A σ S σ P Figure 5: Mean-standard deviation diagram 322 Portfolios composed of two risky assets We now turn to investment decisions involving two risky assets, with r 1, σ 1 and r 2, σ 2 We already know the formula for computing expected portfolio returns To determine the portfolio variance, we need one more ingredient: the covariance between the two portfolio returns r i and r j Definition 5 (covariance) Cov[ r i, r j ] = E ( r i E[ r i ]) ( r j E[ r j ]) n = p i ( r i E[ r i ]) ( r j E[ r j ]) Note: The variance is a special case of covariance: V ar[ r i, r i ] = Cov[ r i, r i ] i=1 Since the covariance depends on the unit of measurement, it is useful to express the covariation between two random variables using the concept of correlation coefficient, which always lies in the interval [ 1, +1]: Definition 6 (correlation coefficient) ρ[ r i, r j ] = Cov[ r i, r j ] σ[ r i ] σ[ r j ]

13 Often, it will be convenient to use the following abbreviations: σ ij Cov[ r i, r j ], ρ ij ρ[ r i, r j ] With all these concepts in place, we can derive the appropriate formula for the variance of a portfolio composed of two risky assets (check each step, using the definitions given in this lecture and the facts on expected values): V ar[w 1 r 1 + w 2 r 2 ] = E (w 1 r 1 + w 2 r 2 E[w 1 r 1 + w 2 r 2 ]) 2 = E (w 1 r 1 E[w 1 r 1 ] + w 2 r 2 E[w 2 r 2 ]) 2 = E (w 1 { r 1 E[ r 1 ]} + w 2 { r 2 E[ r 2 ]}) 2 = E ( w1 2 { r 1 E[ r 1 ]} 2 + w2 2 { r 2 E[ r 2 ]} 2 + 2 w 1 w 2 { r 1 E[ r 1 ]}{ r 2 E[ r 2 ]} ) = w1 2 E ( r 1 E[ r 1 ]) 2 + w2 2 E ( r 2 E[ r 2 ]) 2 + 2 w 1 w 2 E ( r 1 E[ r 1 ]) ( r 2 E[ r 2 ]) = w1 2 V ar[ r 1 ] + w2 2 V ar[ r 2 ] + 2 w 1 w 2 Cov[ r 1, r 2 ] = w1 2 σ1 2 + w2 2 σ2 2 + 2 w 1 w 2 ρ 12 σ 1 σ 2 References Bernoulli, Daniel, 1738, Exposition of a new theory on the measurement of risk, Econometrica (1954) 22, 22 36 Bodie, Zvi, Alex Kane, and Alan J Marcus, 28, Investments (Irwin McGraw-Hill: Chicago) Copeland, Thomas, J Weston, and Kuldeep Shastri, 24, Financial Theory and Corporate Policy (Addison-Wesley: Reading, Mass) Grinblatt, Mark, and Sheridan Titman, 22, Financial Markets and Corporate Strategy (Boston, Mass: Irwin McGraw-Hill) Levy, Haim, and Thierry Post, 25, Investments (FT Prentice Hall: London)