Figure 10.2 An investor's preference curve.

85trictly speaking, all we know is that the preference curve separates the area where the investor prefers the security ftom the area where he prefers the certain amount. It would be possible that above the curve he selects the certain amount. In this case he prefers less to more.

Since each is a known payoff and since C;'s of all sizes exist, we can replace each W; with a Ci of the same size. Thus, portfolio S1 can be represented as

C, with probability P, C2 with probability P2

CN with probability PN

Since for every C( there exists an equivalent lottery, we can represent an equivalent portfolio as I

b with probability h, 0 with probability 1 -ft, b with probability h^ 0 with probability 1 - k2

with probability Pl with probability P2

b with probability hN 0 with probability 1 - hN

with probability PN

The investor declared indifference between each and the lottery. Thus, it is reasonable that 5, and S2 should be equivalent.

Let us explore this in more detail. Assume that outcome i occurs. Then, if the investor selects Sj, C. is received. If the investor selects Sv then b with probability ft. and 0 with probability 1 — hj is received. However, the investor has indicated in the construction of the preference curve an indifference between C; and this lottery. Further, Wt is equal to Cr Thus the investor is indifferent between Wt and this lottery. Thus, security 2 is equivalent to security 1. Note that from axiom 3 the investor does not change preference simply because the alternatives are part of a lottery.

A tree diagram might clarify this choice further. Figure 10.3 represents the portfolio S[ and Figure 10.4 the portfolio S2. Note that while S2 is equivalent to Sv S2 has only two possible outcomes: b and 0. We could equivalently write S2 as b with probability S^P/i,- and 0 with probability 1 — Utilizing the technique just discussed, we can do lie same with any portfolio. Thus, any portfolio can be reduced to two outcomes, b and 0, with known probabilities.

Figure 10.3 Outcomes for 5,.

Figure 10.4 Outcomes for st

How do we choose between these portfolios? To decide, the individual need consider only the probability of receiving b, and the one with the higher probability is to be preferred. Define Hi = SP ./i;, where the P and h are the values appropriate for the security under question. Then if HK > Hv security K is to be preferred to security L. This leads directly to the expected utility theorem. Earlier, we replaced every Wt with a C.; associated with the Cl was an hr Thus, for each Wi there corresponds an hr Let us call the function that relates to a utility function and denote it by U{ ). Then, noting that hi is a function of Wi is equivalent to writing hi = U(W¡). Furthermore, our feelings about a gamble can be expressed as

But SjPji/iWj) is simply expected utility. Thus, expressing the feelings about an investment in terms of Hi is equivalent to expressing them in terms of expected utility. Further ranking by Hi is equivalent to ranking by expected utility.

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