马克维茨投资组合选择(2)

Hicks applies the rule to a firm rather than a portfolio.

78 The Journal of Finance The foregoing rule fails to imply diversification no matter how the

anticipated returns are formed; whether the same or different discount

rates are used for different securities; no matter how these discount

rates are decided upon or how they vary over time.3 The hypothesis

implies that the investor places all his funds in the security with the

greatest discounted value. If two or more securities have the same value,

then any of these or any combination of these is as good as any other.

We can see this analytically: suppose there are N securities; let ritbe

the anticipated return (however decided upon) at time

t per dollar invested

in security i; let djt be the rate at which the return on the ilk security at time t is discounted back to the present; let Xi be the relative

amount invested in security i .We exclude short sales, thus Xi 2 0

for all i . Then the discounted anticipated return of the portfolio is Ri= xm di, T i t is the discounted return of the ithsecurity, therefore

t-1 R = ZXiRi where Ri is independent of Xi. Since Xi 2 0 for all i andZXi= 1, R is a weighted average of Ri with the Xi as non-negative

weights. To maximize R, we let Xi = 1 for i with maximum Ri.

If several Ra,,a = 1, .. . ,K are maximum then any allocation with

maximizes R. In no case is a diversified portfolio preferred to all nondiversified

poitfolios.

It will be convenient at this point to consider a static model. Instead

of speaking of the time series of returns from the

ithsecurity

(ril, ri2) . . . ,rit, . . .) we will speak of \flow

of returns\

theithsecurity. The flow of returns from the portfolio as a whole is

3. The results depend on the assumption that the anticipated returns and discount

rates are independent of the particular investor's portfolio.

4. If short sales were allowed, an infinite amount of money would be placed in the security with highest r. Portfolio Selection 79

R = ZX,r,. As in the dynamic case if the investor wished to maximize

\return from the portfolio he would place all his funds in

that security with maximum anticipated returns.

There is a rule which implies both that the investor should diversify

and that he should maximize expected return. The rule states that the

investor does (or should) diversify his funds among all those securities

which give maximum expected return. The law of large numbers will

insure that the actual yield of the portfolio will be almost the same as

the expected yield.5 This rule is a special case of the expected returnsvariance

of returns rule (to be presented below). It assumes that there

is a portfolio which gives both maximum expected return and minimum

variance, and it commends this portfolio to the investor.

This presumption, that the law of large numbers applies to a portfolio

of securities, cannot be accepted. The returns from securities are

toointercorrelated. Diversification cannot eliminate all variance.

The portfolio with maximum expected return is not necessarily the

one with minimum variance. There is a rate at which the investor can

gain expected return by taking on variance, or reduce variance by giving up expected return.

We saw that the expected returns or anticipated returns rule is inadequate.

Let us now consider the expected returns-variance of returns

(E-V) rule. It will be necessary to first present a few elementary

concepts and results of mathematical statistics. We will then show

some implications of the E-V rule. After this we will discuss its plausibility.

In our presentation we try to avoid complicated mathematical statements

and proofs. As a consequence a price is paid in terms

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