Chapter 16, Part C Investment Portfolio. Risk is often measured by variance. For the binary gamble L= [, z z;1/2,1/2], recall that expected value is

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1 Chapter 16, Part C Investment Portfolio Risk is often measured b variance. For the binar gamble L= [, z z;1/,1/], recall that epected value is 1 1 Ez = z + ( z ) = 0. For this binar gamble, z represents the gamble size, and the variance of the gamble is z. That is, the variance of this lotter is simpl the gamble size squared. How is variance measured for general lotteries? In general, gambles are more comple. Variance is generall defined as ( ) = ( n ) = i( i ). i= 1 Var X E X EX p X EX The idea is to punish large deviations from the mean, and hence the errors or deviations from the mean are squared. Variance is then the epected value of the squared errors or deviations. Eample: Probabilit Prize 40% $10 50% $100 10% $80

How is variance measured for general lotteries? In general, gambles are more comple. Variance is generall defined as ( ) = ( n ) = i( i ).

2 Mean of this lotter was $106. What is the variance of this lotter? Var( X ) =.4 (10 106) +.5 ( ) +.1 (80 106) = + + = Portfolio Choice with One Risk Asset Consider an individual investing in two assets: a risk asset (stocks) and a safe asset (Treasur bills). Let X and Y denote the returns from the risk and riskless assets, respectivel. Let σ denote the variance of X, and let α denote the fraction of the portfolio invested in the risk asset. What is the problem with risk assets? Then one can choose a conve combination of the safe and risk asset. Epected return on the risk asset is higher than that of the riskless asset, i.e., EX > Y. However, there is a small chance that one ma earn less than from the safe asset. Accordingl, the risk asset has a higher standard deviation (or variance) than the riskless asset. Let the initial asset w = 1. (If not, the total return will be wr. In the following analsis, initial wealth w does not pla a role.) The total return is a random variable, R= α X + (1 α) Y. (1) Epected return is ER = αx + (1 α) Y, () where a bar denotes epected value, and X = EX is epected return from the risk asset. The variance of the portfolio is var α σ. = (3)

$Let X and Y denote the returns from the risk and riskless assets, respectivel. Let σ denote the variance of X, and let α denote the fraction of the portfolio invested in the risk asset.$

3 If one is risk averse, he will maimize epected utilit of the portfolio. For portfolio analses, epected utilit is often epressed as a weighted average of the epected return and variance, EU = ER kασ = α X + α Y kα σ (1 ). (4) Here, k represents the penalt per unit of variance. If the investor is risk neutral, then k is 0. If he is risk averse, k > 0. Thus, k can be treated as a risk aversion measure. This is different from the Arrow-Pratt absolute risk aversion measure, U"( w) A = U'( w) or relative risk aversion measure, U"( w) R = w, U '( w ) where w is income or wealth. These two measures are widel used in the theoretical literature. For more details, see another note. I notice that the absolute and relative risk aversion measures are NOT parameters, but functions, which depend on the level of wealth. For this reason, the are impractical for everda use. Business people buing and selling risk assets need rough and read measures of risk aversion. 3

(4) Here, k represents the penalt per unit of variance. If the investor is risk neutral, then k is 0. If he is risk averse, k > 0. Thus, k can be treated as a risk aversion measure.

4 Also, the mean-variance approach is often used for practical reasons that mean and variance of risk assets are easil obtainable. Epected utilit then depends on three parameters: mean (EX), variance (σ ), and the risk aversion measure, k The first order condition is: Just differentiate epected utilit in (4) with respect to α, we can find its optimal value. You can solve for α. deu dα = = ( X Y) kασ 0. The optimal fraction of our investment in the risk asset is: X Y α* =. kσ It is interesting to note that if the two assets have identical epected rates of return, i.e., X = Y, then α* = 0. That makes sense. If two assets have the same epected rates of return, there is no point in taking risk. If EX > Y, then one will hold some risk asset and some riskless asset. If the risk of the risk asset increases, Wouldn t the investment in the risk asset decline? A risk averse investor holds a risk asset onl if it ields a higher epected return. Differentiating (6) with respect to σ, we get 4

$we can find its optimal value. You can solve for α. deu dα = = ( X Y) kασ 0. The optimal fraction of our investment in the risk asset is: X Y α* =.$

5 α * k( X Y) = < 0. σ ( kσ ) (7) That is, as the variance of the risk asset increases, the investor s holding of the risk asset declines. However, (6) shows that no matter how large the variance is, the investor alwas holds some portion of its portfolio in the risk asset, for the simple reason that it has a higher epected rate of return. If Jill is more risk averse than Jack, wouldn t she invest less in the risk asset? You can verif that b differentiating (6) with respect to k gives α k * σ ( X Y) = < 0. ( kσ ) (8) That is, a more risk averse person invests less in the risk asset. Two Risk Assets I understand now that one needs to invest in the risk assets if epected returns are higher than in safe assets. But there are man risk assets. Which ones does one choose? If one risk asset has a higher epected return than the other, should one invest all his mone into the asset with higher epected return? There is a proverb that sas Don t put all our eggs in one basket. The current financial crisis is partl caused b big banks and insurance companies that have ignored this adage. No. If one s wealth is infinite or one is risk neutral, in the long run it might be the best strateg. However, long run survival requires that the investor survive each period. 5

return. If Jill is more risk averse than Jack, wouldn t she invest less in the risk asset? You can verif that b differentiating (6) with respect to k gives α k * σ ( X Y) = < 0.

6 How does one choose among man risk assets? Suppose that one asset is riskier than another, i.e., the variance of one asset is greater than that of the other. In this case, should we abandon the riskier asset altogether? How do we choose between two risk assets? For simplicit, consider the case with two risk assets. Assume that the return on the two risk assets are independentl distributed or uncorrelated, i.e., their covariance is zero. Not if it has a higher epected return. Epected utilit of a portfolio with two risk assets can then be written as: EU = ER kα σ = αx + (1 α) Y ( ασ βσ ) k +, (9) where σ i is the variance of the risk asset i, and α and β are the fractions of the portfolio held in the risk assets, X and Y, respectivel. Thus, α + β = 1. I know how to do this. The Lagrangian function associated with this problem is: L= αx + βy ( α σ β σ ) k + + λ[1 α β]. (10) Partiall differentiating (10) with respect to α and β gives the FOCs: X kασ λ = (1) 0, Y kβσ λ = (13) 0, 6

$Epected utilit of a portfolio with two risk assets can then be written as: EU = ER kα σ = αx + (1 α) Y ( ασ βσ ) k +, (9) where σ i is the variance of the risk asset i, and α and β are the fractions$

7 1 α β = 0. I regret that I asked the question. But from (1) and (13), here is the solution: kσ + X Y α* =. k( ) σ + σ Note that if σ = 0, that is, if Y is a riskless asset, then (14) reduces to (6). If the two assets have the same return ( X = Y ), then optimal portfolio reduces to: σ α* =, (15) ( σ ) + σ which is independent of risk aversion. It simpl depends on the variances of the two risk assets. For all risk averters, investment in the riskier asset is smaller. Question: If Y has a higher variance than X, one s investment in X asset is greater than that of Y. Shouldn t he get rid of the riskier asset altogether? If the two risk assets have the same variances and same epected rates of return, then α = 1. What if the variances of the two risk assets were also No. The solution does not sa that α = 1. (This proves the wisdom of the proverb Don t put all eggs in one basket, even if one asset is riskier than the other) Just because one asset has less variance and hence less risk, it does not follow that one should get rid of the riskier asset at all. If two assets have identical rates of return and risks, then half-and-half is an optimal solution. Such a portfolio will ield the same epected return as that when all eggs are in one basket but will minimize the variance of the return of the total portfolio. Then the variance of the portfolio is: 7

It simpl depends on the variances of the two risk assets. For all risk averters, investment in the riskier asset is smaller.

8 equal? σ = α σ + (1 α) σ = α σ + (1 α + α ) σ (1.17) = σ (1 + α α). If α = 0, this reduces to σ. If α = 1, it also reduces to σ. But somewhere between both ends, variance of the portfolio is smaller. I can see that. For instance, if α = 1/3, it reduces to We can see how the variance changes as α increases as in the diagram below. σ = σ (1 + / 9 6 / 9) σ σ = (5 / 9) <. The optimal solution is 1 α * =. About the Emerging Markets If the financial assets (e.g., stocks) of two countries have the same epected rates of return and variances, shouldn t a shrewd investor should hold an even share of both assets? Yes and No. American investors generall hold more than 50% of their portfolio in domestic assets, and onl recentl began to bu stocks in emerging markets. Foreign securities ma have higher rates of return, but the are also considered riskier. 8

The optimal solution is 1 α * =. About the Emerging Markets If the financial assets (e.g., stocks) of two countries have the same epected rates of return and variances, shouldn t a shrewd investor should hold an even share of both assets?

Finance 400 A. Penati - G. Pennacchi Market Micro-Structure: Notes on the Kyle Model

Finance 400 A. Penati - G. Pennacchi Market Micro-Structure: Notes on the Kyle Model These notes consider the single-period model in Kyle (1985) Continuous Auctions and Insider Trading, Econometrica 15,