## Y Qi pi w

i = l with respect to each This gives

for i = 1,2, ,,.,«. This represents equations. The original budget constraint Yl'UifyPi — W is one more equation Altogether, therefore, there are n 4- 1 equations for the n + 1 unknowns 01( 02. ■ - -, % ancl ^ It can be shown that X > 0

These equations are very important because they serve two roles. First, and most obviously, they give enough equations to actually solve the optimal portfolio problem An example of such a solution is given soon in Example 9.5. Second, since these equations are valid if there are no arbitrage opportunities, they provide a valuable characterization of prices under the assumption of no arbitrage. This use of the equations is explained in the next section.

If there is a risk-free asset with total return /?, then (9,4) must apply when dj = R and P, = 1. Thus,

Substituting this value of X in (9.4) yields

Because of the importance of these equations, we now highlight them:

Portfolio pricing equation If x* = 's a solution to the optimal portfolio problem (9,3a), then

for i — 1, 2, . . ., n, where X > 0 If there is a risk-free asset with return R, then

Example 9,5 (A film venture) An investor is considering the possibility of investing in a venture to produce an entertainment film. He has learned that such ventures are quite risky In this particular case he has learned that there are essentially three possible outcomes, as shown in Table 9.2: (1) with probability ,3 his investment will be multiplied by a factor of .3, (2) with probability 4 the factor will be 1, and (3) with probability .3 he will lose the entire investment. One of these outcomes will occur in 2 years. He also has the opportunity to earn 20% risk free over this period. He wants to know whether he should invest money in the film venture; and if so, how much?

This is a simplification of a fairly realistic situation The expected return is ,.3 x 3 4- 4 x 1 4- -3 x 0 = 13, which is somewhat better than what can be obtained risk free. How much would you invest in such a venture? Think about it for a moment

The investor decides to use U(x) = In ..v as a utility function. This is an excellent general choice (as will be explained in Chapter 15). His problem is to select amounts 0\ and of the two available securities, the film venture and the risk-free opportunity,

Return |
Probability | |

High success |
3 0 |
03 |

Moderate success |
10 |
0 4 |

Failure |
0 0 |
03 |

Risk tree |
1 2 |
SO |

There are three passible outcomes with associated total returns and probabilities shown There is also a risk-free opportunity with total return I 2

There are three passible outcomes with associated total returns and probabilities shown There is also a risk-free opportunity with total return I 2

9.8 LOG-OPTIMAL PRICING*

each of which has a unit price of I Hence his problem is to select (0j, 82) to solve maximize L31n(30, 4- I ,202) + 4 ln(0j + 1.202) 4~ .3 ln(l 202)J subject to Ô] 4 02 = W The necessary conditions from (9,5), 01 by direct calculation, are

30| + L202 + 0! 4- 1.202 + 1.202 ~ These two equations, together with the constraint 0j + 02 = VV\ can be solved for the unknowns 0!t 02, and X. (A quadratic equation must be solved.) The result is 0i = .089W, 02 = 911W, and X = I/W. In other words, the investor should commit 8 .9% of his wealth to this venture; the rest should be placed in the risk-free security.

Example 9.6 (Residual rights) While pondering the possibility of investing in the film venture of the previous example, an investor discovers that it is also possible to invest in film residuals, which have a large payoff if the film is highly successful Each dollar invested in residual rights produces $6 if the venture has high success and zero in the other two cases. Now what should the investor do?

He must solve the portfolio optimization problem again with this new information There are now three securities: the original film venture, the risk-free alternative, and residual rights He will purchase these in amounts 0$, 02, and 03, respectively The necessary equations are

In addition there is the wealth constraint 6\ 4- 02 4- 03 — W These equations have solution 0, = -1.0IV, 02 = I 5W, 03 = 5W, and X = \/W. In other words, the investor should short the ordinary film venture by an amount equal to his total wealth in order to invest in the other two alternatives,

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