Comparing Cash Equivalent of risky portfolios

Question

To compare two risky portfolios, Mean-Variance (M-V) portfolios for example, many compare their Cash Equivalent ($CE$). $CE$ is defined as the amount of cash that provides the same utility as the risky portfolio: $$U\left(CE\right)= W\left(w\right)= w'\mu-\frac{1}{2}\lambda w'\Sigma w$$ where $W(x)$ is the investor's expected utility of wealth, and basically the function to be maximized in the M-V portfolio problem. My question is why not just limit the comparison on expected utilities of the investor $W(w)$. What is the advantages behind comparing $CE$. Thank you.

Answer 1

There is a simple reason to use prefer $CE$ to pure utility: $CE$ is independent of utility units. Thus it allows direct comparison.

The cash equivalent of a risky portfolio is the certain amount of cash that provides the same utility that portfolio. So for portfolio $w$ we can define $CE$ via $U(CE)=E[U(w)]$ or $CE=U^{-1}(E[U(w)])$. Note that for risk-free portfolio the $CE$ equals to certain return. So there is one-to-one correspondence betwen expected utility and $CE$. $CE$ is not a new concept but a convinient way to express utility in different units.

Also $CE$ is used in research papers when risk premia calculation of a lottery is required, i.e. then you can just substract $CE$ from the price of the lottery.

This answer heavily borrows form the book "The Kelly Capital Growth Investment Criterion: Theory and Practice", page 251.

Answer 2

I do not see any advantage in this approach whatsoever, nor would I believe, as you suggested, that "many" use this kind of approach.

In fact I find it horribly wrong. Using a single variable (CE in this case) to represent a non-trivial risk-return construct implies the ability to map such relationship to one variable representations. Everybody values risk differently, everybody looks for a different risk/reward relationship for million different reasons. The necessary simplifying assumption that is applied here is that risk/reward utility means the same to everyone. I am not saying that utility is identical for all risk/reward mappings, but that a 90% expected return with portfolio variance of 50% has the same utility to all, and a 10% expected return with portfolio variance of 5% has another utility albeit the assumption is that everyone measures this utility equally. I find such assumption plain wrong.

It is almost as if there are no greeks in derivatives trading, only a price, buy and sell and that is it. Convenient, because now we can compare prices only, but all the fine-grained detail that makes one chose a 3 month fly over a 2 months spread gets lost because there are no more risk profiles, 2nd and higher order greeks.

Sometimes mathematicians (and economists) go overboard simply because they spend too much time in an artificially lit room instead of dwelling among mortals. I suggest this is one of those cases.