This chapter is devoted to general theory, and hence it is somewhat more abstract than other chapters, but the tools presented are quite powerful The chapter should be reviewed after reading Part 3 and again after reading Part 4
The first part of the chapter presents the basics of expected utility theory Utility functions account for risk aversion in financial decision making, and provide a more general and more useful approach than does the mean-variance framework. In this new approach, an uncertain final wealth level is evaluated by computing the expected value of the utility of the wealth. One random wealth level is preferred to another if the expected utility of the first is greater than that of the second. Often the utility function is expressed in analytic form. Commonly used functions are: exponential, logarithmic, power, and quadratic A utility function U(x) can be transformed to V(.v) = aU(x)-\-b with a > 0, and the new function V is equivalent to U for decision-making purposes, It is generally assumed that a utility function is increasing, since more wealth is preferred to less A utility function exhibits risk aversion if it is concave If the utility function has derivatives and is both increasing and concave, then U'(x) > 0 and U"(x) < 0.
Corresponding to a random wealth level, there is a number C, called the certainty equivalent of that random wealth. The certainty equivalent is the minimum (nonran-dom) amount that an investor with utility function V would accept in place of the random wealth under consideration The value C is defined such that U (C) is equal to the expected utility due to the random wealth level
In order to use the utility function approach, an appropriate utility function must be selected. One way to make this selection is to assess the certain equivalents of various lotteries, and then work backward to find the underlying utility function that would assign those certain equivalent values
Frequently the utility function is assumed to be either the exponential form —e~ax with a approximately equal to the reciprocal of total wealth, the logarithmic form In.v, or a power form yxr with y < I but close to 0 The parameters of the function are either fit to lottery responses or deduced from the answers to a series of questions about an investor's financial situation and attitudes toward risk.
The second part of the chapter presents the outline of a general theory of linear pricing In perfect markets (without transactions costs and with the possibility of buying or selling any amount of each security), security prices must be linear—meaning that the price of a bundle of securities must equal the sum of the prices of the component securities in the bundle—otherwise there is an arbitrage opportunity.
Two types of arbitrage are distinguished in the chapter: type A, which rules out the possibility of obtaining something for nothing—right now; and type B, which rules out the possibility of obtaining a chance for something later—at no cost now
Ruling out type A arbitrage leads to linear pricing. Ruling out both types A an B implies that the problem of finding the portfolio that maximizes the expected utility has a well-defined solution.
The optimal portfolio problem can be used to solve realistic investment problems (such as the film venture problem). Furthermore, the necessary conditions of this general problem can be used in a backward fashion to express a security price as an expected value Different choices of utility functions lead to different pricing formulas, but all of them are equivalent when applied to securities that are linear combinations of those considered in the original optimal portfolio problem Utility functions that lead to especially convenient pricing equations include quadratic functions (which lead to the CAPM formula) and the logarithmic utility function.
Insight and practical advantage can be derived from the use of finite state models, In these models it is useful to introduce the concept of state prices. A set of positive state prices consistent with the securities under consideration exists if and only if there are no arbitrage opportunities One way to find a set of positive state prices is to solve the optimal portfolio problem. The state prices are determined directly by the resulting optimal portfolio
A concept of major significance is that of risk-neutral pricing By introducing artificial probabilities, the pricing formula can be written as P — E{d)/R, where R, is the return of the riskless asset and E denotes expectation with respect to the artificial (risk-neutral) probabilities. A set of risk-neutral probabilities can be found by multiplying the state prices by the total return R of the risk-free asset
The pricing process can be visualized in a special space, Starting with a set of n securities defined by their (random) outcomes d-t, define the space S of all linear combinations of these securities, A major consequence of the no-arbitrage condition is that there exists another random variable u, not necessarily in S, such that the price of any security d in the space S is E(lid). In particular, for each /, we have Pi = E(vdi),. Since u is not required to be in 5', there are many choices for it One choice is embodied in the CAPM; and in this case u is in the space 5'. Another choice is u = l/R*, where R* is the return on the log-optimal portfolio, and in this case li is often not in 5. The optimal portfolio problem can be solved using other utility functions to find other i/s If the formula P = E(vd) is applied to a security d outside of S, the result will generally be different for different choices of v
If the securities are defined by a finite state model and if there are as many (independent) securities as states, then the market is said to be complete In this case the space 5 contains all possible random vectors (in this model), and hence v must be in S as well, indeed, v is unique It may be found by solving an optimal portfolio problem; all utility functions will produce the same v.
L {Certainty equivalent) An investor has utility function (J{x) = .v!/4 for salary He has a new job offer which pays $80,000 with a bonus The bonus will be $0, $10,000, $20,000, $30,000, $40,000, $50,000, or $60,000, each with equal probability What is the certainty equivalent of this job offer?
2. (Wealth independence) Suppose an investor has exponential utility function U(x) — —e~"x and an initial wealth level of W. The investor is faced with an opportunity to invest an amount w < W and obtain a random payoff .v. Show that his evaluation of this incremental investment is independent of W
3. (Risk aversion invariance) Suppose U(x) is a utility function with Arrow-Pratt risk aversion coefficient a(.v) Let V(.y) = c +bU(x). What is the risk aversion coefficient of V?
4. (Relative risk aversion) The Arrow-Pratt relative risk aversion coefficient is
Show that the utility functions U(x) = ln.v and U(x) ~ y.v5' have constant relative risk aversion coefficients
5„ (Equivalency) A young woman uses the first procedure described in Section 9 4 to deduce her utility function U(x) over the range A < x < B She uses the normalization U(A) = A, U(B) — B To check her result, she repeats the whole procedure over the range A' < x < B', where A < A' < B' < B The result is a utility function V(.v), with V(A') = A', V(B') — B' If the results are consistent, U and V should be equivalent; that is, V(.v) = aU(x) +b for some a > 0 and b Find o and b
6. (HARAo) The HARA (for hyperbolic absolute risk aversion) class of utility functions is defined by
The functions are defined for those values of .y where the term in parentheses is nonnegative Show how the parameters y, a, and h can be chosen to obtain the following special cases (or an equivalent form)
(a) Linear or risk neutral: U(x) = v (/;) Quadratic: U(x) = x - \cx2 (c) Exponential: U(x) = -e~~"x [Try y ~ -co ] id) Power: U(x) - cxy
(e) Logarithmic: U(x) = ln.v [Try U(x) = (3 ~ k)'"''«-*'' - l)/y) ] Show that the Arrow-Pratt risk aversion coefficient is of the form I/(c.v + d)
7. (The venture capitalist) A venture capitalist with a utility function U(x) = *Jx carried out the procedure of Example 9.3. Find an analytical expression for C as a function of e, and for e as a function of C Do the values in Table 9 1 of the example agree with these expressions?
8. (Certainty approximation o) There is a useful approximation to the certainty equivalent that is easy to derive A second-order expansion near x = E(.v) gives b > 0
Hence,
On the other hand, if we iet c denote the certainty equivalent and assume it is close to .v, we can use the first-order expansion
Using these approximations, show that
(Quadratic mean-variance) An investor with unit wealth maximizes the expected value of the utility function U(.x) = ci.x — b.x2/2 and obtains a mean-variance efficient portfolio A friend of his with wealth W and the same utility function does the same calculation, but gels a different portfolio return. However, changing b to // does yield the same result What is the value of //?
10. (Portfolio optimization) Suppose an investor has utility function U There are n risky assets with rates of return rt, i = 1, 2,. , », and one risk-free asset with rate of return //. The investor has initial wealth W() Suppose that the optimal portfolio for this investor has (random) payoff .v* Show that
11. (Money-back guarantee) The promoter of the film venture offers a new investment designed to attract reluctant investors One unit of this new investment has a payoff of $3,000 if the venture is highly successful, and it refunds the original investment otherwise Assuming that the other three investment alternatives described in Example 9 6 are also available, what is the price of this money-back guaranteed investment'?
12. (General positive state prices result o) The following is a general result from matrix theory: Let A be an m x n matrix Suppose that the equation Ax = p can achieve no p > 0 except p = 0 Then there is a vector y > 0 with A7y = 0 Use this result to show that if there is no arbitrage, there are positive state prices; that is, prove the positive state price theorem in Section 9 9 {Hint If there are S states and N securities, let A be an appropriate (S + 1) x N matrix ]
13, (Quadratic pricing o) Suppose an investor uses the quadratic utility function U(x) = ,.v - ^t-v- Suppose there are n risky assets and one risk-free asset with total return R Let be the total return on the optimal portfolio of risky assets Show that the expected return of any asset i is given by the formula
where = cov(R,\i, Ri)/a\, IMint Use Exercise 10 Apply the result to RAI itself j
14. (At the track) At the horse races one Saturday alter noon Gavin Jones studies the racing form and concludes that the horse No Arbitrage has a 25% chance to win and is posted at 4 to 1 odds (For every dollar Gavin bets, he receives $5 if the horse wins and nothing if it loses) He can either bet on this horse or keep his money in his pocket Gavin decides that he has a square-root utility for money
(a) What fraction of his money should Gavin bet on No Arbitrage?
(b) What is the implied winning payoff of a $1 bet against No Arbitrage?
15. (General risk-neutral pricing) We can transform the log-optimal pricing formula into a risk-neutral pricing equation From the log-optimal pricing equation we have where R* is the return on the log-optimal portfolio We can then define a new expectation operation E by
This can be regarded as the expectation of an artificial probability Note that the usual rules of expectation hold Namely:
(o) if x is certain, then E(.v) = .v. This is because E(1 /R*) — 1 JR.
(b) For any random variables .r and y, there holds E(rt.v + by) = a E(..v) + bE(y)
(c) For any nonnegative random variable ..v, there holds E(.v) > 0
Using this new expectation operation, with the implied artificial probabilities, show that the price of any security d is
This is risk neutral pricing
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