ABSTRACT
Consider an economy operating over one period from time 0 to time 1. Suppose that there are s risky assets, i D 1; : : : ; s; the prices of these at time 0 are given by a deterministic vector S0 D .S1;0; : : : ; Ss;0/> 2 Rs and the prices at time 1 are determined by a random vector S1 D .S1;1; : : : ; Ss;1/> taking values in Rs . In addition there is a riskless asset, 0, which provides a deterministic return r1 > 0 between time 0 and time 1; the initial price of the riskless asset may be taken as 1 and here r11 is the fixed interest rate, with the price of the riskless asset at time 1 being r1. Underlying the model is a probability space .;F ;P/ on which the random vector S1 is defined. The set , which represents the set of possible states of nature ! 2 , is equipped with a -field F of measurable events, or subsets, of and P is a given probability. In this section assume thatEkS1k2 <1, that isE .Si;1/2 <1 for each i D 1; : : : ; s, and without any loss of generality assume that the covariance matrix
V D Cov .S1/ D E [ .S1 ES1/ .S1 ES1/>
] D E [S1S>1 ] .ES1/ .ES1/>
is positive definite. The .i; j / element of the matrix V is the covariance between the prices (at time 1) of assets i , j .D 1; : : : ; s/. The assumption that V is positive definite means in effect that there is only one riskless asset; it is not possible to form a new asset for which the variance of the price at time 1 is zero by taking some linear combination of the s risky assets. As pointed out in Section 1.3, this is not a serious restriction since if there are two, or more, riskless assets then investors would all choose the one with the highest return so the others could all be discarded and the risky assets relabelled, if necessary, to give the situation described here.
