# Comparing Cash Equivalent of risky portfolios

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.

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## 2 Answers

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.

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care to back up your claim? How do you get from a risky portfolio to Cash Equivalent? To be honest I do not even know what CE is supposed to be (in the context of representing a risky portfolio), but what I know is that a single variable cannot describe nor properly represent a non-trivial risk/return construct. – Freddy Mar 7 at 14:37
@Freddy sure. updated the answer. – Alexey Kalmykov Mar 7 at 14:59
this only works if the portfolio is a risk free portfolio. As soon as you introduce risk the CE can be identical for two portfolios with entirely different risk reward profiles. Hence my criticism of the simplifying assumptions made in the application of CE. By the way could you please cite the exact paper and page you reference as your book just contains a bunch of academic papers. – Freddy Mar 7 at 15:36
I found your reference and I stand by my claim that the assumptions made are entirely unrealistic: First of all you need as input a risk tolerance metric, as I said there is no way to map a MV portfolio to CE without assumptions, in this case a risk tolerance level.More importantly, this risk tolerance level is different for every person on this planet. Thus the authors further use as input expected utility. For them this comes down to a logarithmic function. Now, how many more simplifying assumptions you want to apply to rape this poor portfolio just to squeeze out a single number? – Freddy Mar 7 at 15:56
Think about it this way: you apply for a job. You have or don't have many properties that may qualify or disqualify you from the job. Now a service company sells a new software to all hiring managers claiming a single number can be attached to each human being to rank them relative to their peers. Great idea? Well not really cause a great education may be completely worthless if the job description is about shoveling soil or picking apples on a plantation. – Freddy Mar 7 at 16:01
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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.

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Great answer. Sometimes it is surprising to see how that the academic community agreed on stuff without apparent reasons!! – Omar Mar 7 at 9:08
@Freddy "CE is suggesting we all set our utility equal" not really. – Alexey Kalmykov Mar 7 at 13:48
@AlexeyKalmykov, care to elaborate? How would you define CE as functional output of a risky portfolio? – Freddy Mar 7 at 14:21
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