1 20 PTS A realvalued function f is strictly conver if far 1

1. (20 PTS) A real-valued function f is strictly conver if f(ar (1 a)y) af(a) (1 f(y) for all z y e R and 0 a 1. Prove that the risk premium of an uncertain prospect that returns a with probability p and y with probability 1 p under a strictly convex utility function is always negative.

Solution

The formula for risk premium, sometimes referred to as default risk premium, is the return on an investment minus the return that would be earned on a risk free investment. The risk premium is the amount that an investor would like to earn for the risk involved with a particular investment.

Thus, risk premium = protfolio return - individual investment return

Total individual return = pf(x) + (1-p)f(y)

When combined as a portfolio, the return would be = f ( px + (1-p)y )

Now,

risk premium is strictly convex, hence:

pf(x) + (1-p)f(y) >   f ( px + (1-p)y )

Thus,

[f(px + (1-p)y)] - [pf(x) + (1-p)f(y)] < 0

thus, risk premium < 0

Hence proved.