Q. 8.22 - A First Course in Probability

Question

Would the results of Example$5f$change be if the investor were allowed to divide her money and invest the fraction$\alpha , 0 <\alpha < 1,$ in the risky proposition and invest the remainder in the risk-free venture? Her return for such a split investment would be$R = \alpha X + (1 - \alpha )m$.

Step-by-Step Solution

Answer

Therefore,

  we are not likely to invest in that portfolio. the answer remains the same as in the example$5 f.$

Step 1 Given Information.

the investor was allowed to divide her money and invest the fraction$\alpha , 0 <\alpha < 1$, in the risky proposition, and invest the remainder in the risk-free venture. Her return for split investment would be$R = \alpha X + (1 - \alpha )m.$

Step 2 Explanation.

Suppose that the investor invests $\alpha$a fraction of their money into the non-deterministic proposition that yields a return $X$and the remaining money he invests into the deterministic proposition with the return$\mu$. Therefore, his return is equal to

$R=\alpha X+(1-\alpha )m$

Now, Use that $E\left(X\right)=m$to obtain that

$E\left(R\right)=\alpha E\left(X\right)+(1-\alpha )m=\alpha m+(1-\alpha )m=m$

It $u$is a convex function, using the Jensen's inequality, we have that

$E\left(u\right(R\left)\right)\ge u\left(E\right(R\left)\right)=u\left(m\right)$

so we are likely to invest in that kind of portfolio. On the other hand, it $u$is a concave function, we have that

$E\left(u\right(R\left)\right)\le u\left(E\right(R\left)\right)=u\left(m\right)$

so we are not likely to invest in that portfolio. Therefore, the answer remains the same as in example 5 f.