Portfolio selection using the utility function u(w)=-exp(-bw).
The CORRECT version should read:
EU (W) = ∫U(W)dF(W).
In a portfolio selection model let W0 be initial wealth, W be final wealth, B be the amount of wealth invested in a "risky asset", M the amount of wealth held as money (money is a "sure" asset and pays zero return), and r the (random) return on the risky asset, which has a strictly positive expected value: E(r) > 0. So, if $B are placed in the risky asset, the random amount $(1 + r)B will be realised. Suppose that the investor has the von Neumann-Morgenstern utility function u(W) = − exp( - β W ) where β > 0.
(a) Graph this utility function and show that it exhibits risk aversion. Interpret β.
(b) Write out the choice problem which determines the utility-maximising division of
initial wealth between the risky asset and money.
(c) Compute the optimal level B. Interpret the result.
Hello once again,
Before going into details let's try to get an overview of what's going on in this problem.
We assume that the individual is risk-averse (later we'll show that this is true from his utility function). That means he doesn't like risk, so he will not invest all of his money in risky asset. However, he still likes more wealth, so he will try to put a little bit of money in risky asset. The objective of this problem is to find out how the amount he will put in risky asset depends on key parameters in his utility function.
To do that we'll need to setup a formal model, and this formal model will require expected utility. Expected utility is formed by taking expectation of the von Neumann-Morgenstern utility function. We'll do that in step b, which in turn will lead us to the answer of how much the individual will put in B, risky asset.
Some notation conventions that I will use:
W - the final wealth
W0 - the initial wealth
EU(W) - expected utility
' - derivative, e.g. U'(w) is the first derivative, U''(w) is second ...
Portfolio selection is emphasized using the given utility function in the solution.