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 val- ue, then any of these or any combination of these is as good as any We can see this analytically: suppose there are N securities; let ra be the anticipated return(however decided upon) at time t per dollar in vested in security i; let da be the rate at which the return on the i' security at time t is discounted back to the present; let X, be the rela tive amount invested in security i. We exclude short sales, thus X:>0 for all i. Then the discounted anticipated return of the portfolio is =∑X; dir rit is the discounted return of the ieh security, therefore R=EX, Ri where R is independent of X,. Since X:20 for all and XX:=1, R is a weighted average of R; with the X, as non-nega- tive weights. To maximize R, we let X= 1 for i with maximum R sever K are maximum then any allocation with ∑xa=1 maximizes R. In no case is a diversified portfolio preferred to all non portfolios. 4 It will be convenient at this point to consider a static model. In stead of speaking of the time series of returns from the ik security ) we will speak of“ the fow of returns”(t)from the i security. The flow of returns from the portfolio as a whole is 3. The results dey n the assumption that the anticipated returns and discount rates are independent of the particular investors portfo 4. If short sales were allowed, an infinite amount of money would be placed in the78 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 val￾ue, 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 in￾vested 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 rela￾tive 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 = x m di, Tit 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 and ZXi = 1, R is a weighted average of Ri with the Xi as non-nega￾tive 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 non￾diversified poitfolios. It will be convenient at this point to consider a static model. In￾stead of speaking of the time series of returns from the ithsecurity (ril, ri2) . . . ,rit, . . .) we will speak of "the flow of returns" (ri) from the ithsecurity. 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