# Proxy for risk in portfolio theory when return can take only two values

I'm trying to adapt tools from portfolio theory for another use, and I have a question about how I might do so.

Suppose that instead of having normally distributed returns, the return $R_i$ is either, say, 3 or 0. So, $E(R_i)=3P(R_i=3)$.

On the wiki page for Modern Portfolio, it says

Note that the theory uses standard deviation of return as a proxy for risk, which is valid if asset returns are jointly normally distributed or otherwise elliptically distributed.

I'm mainly interested in computing expected returns and risk a la wiki. I'd like to use those as metrics in comparing a relatively small number of possible portfolios. But obviously in the case I'm interested in, the returns aren't normally distributed. What are the consequences of still using standard deviation of returns as a "proxy for risk"? Is there an alternative measure that makes more sense?

How bad would it be to pretend that the returns are normally distributed, centered at the mean of the bernoulli return, with the same variance?

The stakes aren't particularly high and currently the only tool for what I'm trying to do is human judgment and experience (the application isn't in finance).

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## 2 Answers

First you need to define what you need a risk measure for. It is usually to take a decision, so you have an operational criterion that defines your risk. You should go back at this point and see what is the impact of a change of distribution on it.

Just say for instance that you need a risk measure to take decisions according to a Sharpe ratio and define it like:

$${\cal S}(R) = \frac{R-R_0}{\sigma(R)}$$

In such a case the Sharpe ratio is useful because it can be read as a straightforward proxy of the probability that your returns are greater than $R_0$ assuming that $R$ follows a Gaussian process (here we are), because if you define $\Phi$ the repartition function of a Gaussian (i.e. $\mathbb{P}(G>g)=\Phi(g)$) you have:

$$\mathbb{P}(R>R_0)=\Phi( {\cal S}(R)) = \Phi\left( \frac{R-R_0}{\sigma(R)} \right)$$

So now "what if $\tilde R$ is no more Gaussian?". For the same operational criterion (i.e. probability to be greater than a base $R_0$), you can find the answer very easily:

• if $R_0$ is greater than 3: $$\mathbb{P}({\tilde R}>R_0)=0$$

• if $R_0$ is lower than 0: $$\mathbb{P}({\tilde R}>R_0)=1$$

• else $$\mathbb{P}({\tilde R}>R_0)=1-\mathbb{P}({\tilde R}=3)$$

So you have the answer (it can be reproduced for any other distribution that your toy example):

1. go back to your operational criterion
2. write its meaning for your distribution
3. you have the answer
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Here is an example where using standard deviation as a measure of risk does not make sense:

Let's say that Case 1: Probability of getting 3 is 0.5, and probability of getting 0 is 0.5. Your expected return is 1.5, and you expected standard deviation is 1.5 (I hope my calculation is right).

Case 2: Probability of getting 5 is 0.5, and probability of getting 0 is 0.5. Your expected return is 2.5, and you expected standard deviation is 2.5 (I hope my calculation is right).

But does Case 2 really have higher risk? If so how? Would anyone ever prefer option in Case 1 because it is less riskier? In the above case increasing standard deviation when going from Case 1 to Case 2 has only positive effect and no negative effect, and option in Case 2 is clearly a better choice. Something like this will not happen in we have normal distribution.

Value at Risk is an alternative risk measure you could look at.

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OK. I get this. And I guess a similar, although less extreme, problem exists with a typical stock market application, as returns there aren't actually normal. I think I can adjust my formulation so that assuming the returns are normal is less of a stretch. How do you judge when your returns are too far from normal for the model to be useful, I wonder. – mike Jul 1 '12 at 2:46
@Mike there are different normality tests that you can do, Q-Q plot is probably the simplest one. – Akavall Jul 1 '12 at 3:47