fin5121 - Outlines of Lecture 1 Objectives 1 Define...

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Unformatted text preview: Outlines of Lecture 1 Objectives 1. Define risk-averse investors and determine personal required risk premium on a risky asset 2. Discuss in which conditions the meanvariance analysis is consistent with the expected utility maximization References UtilityFunction.pdf; Quant_Review.pdf; Chapter 6 & Appendix A&B , BKM Hsiu-lang Chen Portfolio Analysis 1 Introduction Asset Allocation or “Portfolio Theory” Mean-variance portfolio analysis was developed by Harry Markowitz in the early 1960’s. This work won the Nobel prize in economics in 1990. This work is widely regarded as the first step in modern finance. An Asset Allocation Problem: – How much of your wealth should you invest in each security? Hsiu-lang Chen Portfolio Analysis 2 Introduction Before Mean-Variance Analysis A typical investment advice: – If you are young, you should be putting money into a couple of good growth stocks, maybe even into a few small stocks. Now is the time to take risks. – If you are close to retirement, you should be putting all of your money into bonds and safe stocks, and nothing into the risky stocks. Don’t takes risks with your portfolio at this stage in your life. Though this advice was intuitively compelling, it was, of course, all wrong. Hsiu-lang Chen Portfolio Analysis 3 Introduction After Mean-Variance Analysis We now know that the optimal portfolio of risky assets is exactly the same for everyone, no matter what their tolerance for risk. Investors should control risk not by reallocating among risky assets, but through the split between risky and risk-free assets. The portfolio of risky assets that investors should hold should contain a large number of assets. It should be well-diversified. Hsiu-lang Chen Portfolio Analysis 4 Introduction Hsiu-lang Chen Portfolio Analysis 5 Introduction Hsiu-lang Chen Portfolio Analysis 6 Introduction Hsiu-lang Chen Portfolio Analysis 7 Introduction Hsiu-lang Chen Portfolio Analysis 8 Why should an individual utility function be U(r)=E(r)-½Aσ2(r)? In choices under uncertainty, individuals maximize his or her expected utility U, which is a function of wealth W. Is the concept of expected utility maximization consistent with the concept of mean-variance analysis? Hsiu-lang Chen Portfolio Analysis 9 Concepts of Risk Aversion An individual is said to be risk averse if and only if his utility function is concave (i.e. U’’(W)<0). The individual will not accept “fair” gambles (i.e. E(ε̃)=0)[BKM (8th), p. 188] ][BKM (9th), p. 191] $150 P=0.5 $100 U(W) U(150) U(100) E(U) $50 U(50) Will you accept this fair gamble ε ̃? E(payoff) = 0.5 ×$50 + 0.5 ×$150 = $100 What is your expected utility level from playing the gamble? E[U($100+ε̃)] = 0.5×U($50)+0.5×U($150) W 50 WC 100 150 U($100) > E[U($100+ε̃)] No! ⇒∃ WC ∋ U(WC)= E[U($100+ε̃)] (Certainty Equivalent) ⇒ ∃ π ∋ U(W-π)= E[U(W+ε̃)] 10 WC is called the certainty equivalent of the gamble W+ε̃. The difference between 100 and WC is called the (insurance) risk premium π. To avoid a present gamble a risk-averse individual would be willing to pay an insurance risk premium. Q. What is π for an investor with log utility in this risky game? Q. Will risk premium π depend on an individual risk preference only? U ′′(W ) ≡ 1 σ ε2 A(W ) π =− σ ~ 2 U ′(W ) 1 2 2 ~ ε This is derived by taking a Taylor series approximation of the equation U(W-π)= E[U(W+ε̃)] around the point ε̃=0 and π=0 [UtilityFunction.pdf]. A(W) is the Pratt(1964)-Arrow(1970) measure of absolute risk aversion. Note that the risk premium π depends on the uncertainty of the risky asset and on the individual’s coefficient of absolute risk aversion. Since both σε̃2 and U’(W) are > 0, concavity of the utility function ensures that π>0. Q. Can you use this approximation to figure out the insurance risk premium π for an investor with log utility? Hsiu-lang Chen Portfolio Analysis 11 In finance, however, the compensatory risk premium is more useful. To induce a risk-averse individual to undertake a fair gamble, a compensatory risk premium πC would have to be offered, making the package actuarially favorable. It corresponds to the “extra” return expected on riskier assets. ∃ πC ∋ U(W)= E[U(W+ε̃+πC] In the previous example, U(100)=0.5×U(50+πC)+0.5×U(150+πC) U(W) $150 U(150) U(100) P=0.5 $100 U(50) $50 +πC +πC +πC 50 Hsiu-lang Chen 100 150 What is the compensatory risk premium required for accepting this fair gamble for an investor with log utility? Portfolio Analysis 12 Example 1 Suppose the utility function is U(W)=(W)0.5. Consider the following simple game: p $150 1-p $50 $100 1. What is the utility level at wealth levels $150 and $50? 2. What is the expected utility if p still equals to 0.5? 3. What is the certainty equivalent of the risky prospect? 4. Does this utility function also display risk aversion? 5. Does this utility function display more or less risk aversion than the log utility function? Hsiu-lang Chen Portfolio Analysis 13 Example 2 You and your friend Michael just have a trip to Las Vegas. Suppose your utility function is U(W)= – 1/W and the utility function of your friend Michael is U(W)=lnW, where W is the terminal wealth. A casino company offers a simple game described as follows: The prize of the game depends on a coin you toss. If the head appears, you get $200. If the tail appears, you get $100. To enter the game one pays an entry fee $135. 1. Who is willing to participate this game? 2. If the prize of this game changes to $250 versus $50, instead of original $200 versus $100, will your answer in part (a) change? Why? (Can you answer this without re-calculation?) 3. (Advanced) What is the maximum entry fee you are willing to pay for this fair coin game? Given a biased coin game and the entry fee of $135, what is the minimum probability of the coin head appearing so that you will play it? Hsiu-lang Chen Portfolio Analysis 14 Example 3 Suppose that your wealth is $250,000. You buy a $200,000 house and invest the remainder in a risk-free asset paying an annual interest rate of 6%. There is a probability of .001 that your house will burn to the ground and its value will be reduced to zero. 1. With a log utility of end-of-year wealth, how much would you be willing to pay for insurance (at the beginning of the year)? Assume that its end-of-year value will be $200,000 if the house does not burn down. 2. If the cost of insuring your house is $1 per $1,000 of value, what will be the certainty equivalent of your endof-year wealth if you insure your house at: i. ½ its value ii. Its full value iii. 1.5 times its value Hsiu-lang Chen Portfolio Analysis 15 Mean-Variance Analysis Consider an investor’s utility from an investment in a given portfolio of assets. Let denote the rate of return on the portfolio, so that the end of period wealth of the investor is W=W0 (1+r̃P). For convenience, we write U(W=W0(1+r̃P)) simply as U( r̃P). Take a Taylor series expansion of U( rP) around the mean of rP, E(rP): ̃ ̃ ̃ ~ U (r ) = U ( E (~ )) + (~ − E (~ ))U ′( E (~ )) + (~ − E (~ )) U ′′( E (~ )) + ⋅ ⋅ ⋅ r r r r r r r p p p p p 1 2! 2 p p p If we ignore third-order and higher terms because: i. The utility function is quadratic (U’’’=0), or ii. Portfolio rates of return are normally distributed, so that third, fourth, and all higher central moments are all a function of the mean or variance, or iii. The actual utility function and actual portfolio returns are close enough to those assumed in (i) and (ii) such that higher order terms are small and can be safely neglected Hsiu-lang Chen Portfolio Analysis 16 Then taking the expectation of both sides of the equation, we obtain E[U (~ )] = U ( E (~ )) + σ U ′′( E (~ )) r r r p p 1 2 2 p The investor’s expected utility is a function of only the mean and variance of the rate of return on the portfolio. In the real world, people’s utility functions may take a variety of forms. For example, E(U) = E(r)λσ2, E(U) = E(r)-λσ2-λTE2, U = E(r)-0.5Aσ2 [BKM, p.159 (8th) or 163 (9th)], and Utility = alpha-λσ2transaction cost-penalties in [BARRA, p.173]. An utility function is used to quantify the personal risk-return tradeoff. Hsiu-lang Chen Portfolio Analysis 17 Investor preferences can be depicted as indifference curves. Each curve traces out the combinations of {E(r),σ(r)} yielding the same level of utility. In an example of U(r)=rCE=E(r)-½A σ2(r), Hsiu-lang Chen Portfolio Analysis 18 Investors with various degrees of risk aversion in a given utility function U(r)=rCE=E(r)-½A σ2(r). Q1. What will be the certainty equivalent rate of return for this risky investment ? Q2. What will be another equivalent risky investment to this investor? Q3. Who is indifferent between this investment and an investment of (.6, .15)? How can we see that rCE=0.03 ? What is 0.25- ½ 0.78 (0.75)2 ? Hsiu-lang Chen Portfolio Analysis 19 Concept Checks Given utility function U(r)=E(r)-½A σ2(r) and the market risk-free rate of 3% Consider a risky investment (σ(r), E(r))=(75%,25%). It offers the risk premium of 22%. Will an investor with A=1.00 take this risky investment? Sol1: If investing in this, the utility level is 0.25-½ * 1.00* (0.75)2 = -0.03125 This is also the certainty equivalent rate of return for this risky investment. This is his personal risk-free rate assigned to this risky investment. The investor thinks this risky investment is worth a certain rate of -3.125%. Since his personal risk-free rate on this is below the market risk-free rate, he prefers the market risk-free rate to the risky investment. Alternatively, you might think the investor has a personal required risk premium of (25%-(-3.125%)=28.125%) for this risky investment, which is higher than what it is offered at 22%. Sol 2: If investing in the market risk-free rate, the utility level is 0.03. The investor prefers the market risk-free rate because it has higher utility level. Hsiu-lang Chen Portfolio Analysis 20 Hsiu-lang Chen Portfolio Analysis 21 As an investment advisor, how do you gauge your clients willingness to take risk? How is the utility function of an individual specified in reality? Check website. Hsiu-lang Chen Portfolio Analysis 22 Hsiu-lang Chen Portfolio Analysis 23 Hsiu-lang Chen Portfolio Analysis 24 Deal or No Deal (a TV show): 26 briefcases that contains money, varying from one cent to $1 million. The contestant picks one briefcase as his own and then begins to open the other 25, each time, by process of elimination, revealing a little more about what his own case might hold. At the end, the contestant can also trade his brief case for the last unopened one. Periodically, a “banker” offers a deal to the contestant: Stop playing now and take the money offered. An offer of $37,000 to quit winning $200,000 in a 1/5 chance. NO! An offer of $67,000 to quit winning $200,000 in a 1/3 chance. NO! An offer of $99,000 to quit winning $200,000 or $50 (1/2 chance). YES! Hsiu-lang Chen Q. Is Mr. Johnson less risk averse than an investor with log utility? Portfolio Analysis 25 ...
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