# How to measure investors' “experienced” volatility?

In asset allocation, you usually send reports to your clients where you will report the volatility of its portfolio. Assuming you only have monthly returns, you will compute volatility over a considered period of $n$ months with the classic sample volatility estimate:

$$\sigma_s=\sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i-\bar{x})^2}, \quad \bar{x}=\frac{1}{n}\sum_{i=1}^N x_i$$

The result you will get for $\sigma_s$ depends very much on the number of months $n$ you decide to take into account. Hence, I think that this measure can become pretty abstract to unsophisticated investors and they might find it pretty different from the "feeling" the have of the volatility of their portfolio, what I call the "experienced" volatility.

My question is, has there been any research aiming to find out which $n$ is the best to make sure that the measure is closest to the experienced volatility?

I believe this is very much linked to behavioral finance and it might very well depend on the risk-aversion or the sophistication of the investor.

I tried to answer my question by proposing the following:

I assume that investors will be biased by the most recent events in the market; if they have been with you 10 years, they will remember 2008-2012 and would have forgot the quiet and lucrative early years. Hence, I took $n=36$, 3 years, as I thought is was taking a recent enough sample, yet had enough data ($n>30$, intuitively... it might be arguable) to measure a properly the estimate $\sigma_s$.

Does this make sense?

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What makes this especially difficult is the behavioral finance literature shows that investors tend to exhibit various emotional and cognitive biases. So a painful bear market might experience more "weight" than another period in time because of recallability and regret aversion for example. –  Quant Guy May 2 '12 at 15:43

Usually very good questions don't come with straight answers and I think this is the case here. I believe you have two (unfortunatly linked) problems here. One is to get a sense of investors aversion to risk and the other is to get a statistically "receivable" estimate of volatility.

1/ Different investors may be willing to take risk on significantly different time frames. This means that the answer is going to be very dependent on your client. Survey studies could seem to answer this question but will probably hide the problem that investors themselves don't really fully grasp the proper time horizon they are willing to take risk. In any cases you will need to introduce expected payoff to start getting an answer.

2/ As for the statistical estimate, you suggest n>30. It is a fairly common number economists use but is based on strong stationnary hypothesis. This THEORETICAL number (based on the convergence of student to normal) will be very dependent on the model you use to describe your data (obviously driven by your data). Unfortunatly, even returns exhibit non stationnary characterisics. Using weighting procedure (EWMA for example) might help you but will still not be parameter free and hence not fully answer the question.

Good luck with this one.

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Thanks for the answer, I agree with you, it's pretty difficult to answer... This is why is was asking for research papers addressing the subject also in the question. I couldn't find any. –  SRKX Apr 27 '12 at 5:59

1) If you want to show an unsophisticated investor with like a real-time estimate of volatility, my main suggestion would be to fit a Garch model to the returns and use those estimates of the conditional volatility. Just provide a chart of it, rather than showing the model and all the details that go into calculating it.

2) If that is too complicated, you might try checking out GIPS performance standards for an industry standard approach to reporting performance.

3) People generally call it realized volatility rather than experienced volatility.

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+1 for the different solutions proposed and the right terminology. I am wondering though if GARCH doesn't answer the question with the $p$ and $q$ parameters you choose as input. –  SRKX May 2 '12 at 19:58
Garch(1,1) is usually a good starting place. While there's evidence of assymmetry and fractionally integrated volatility, the question is whether you're trying to estimate volatility well or whether you're trying to show someone what their real-time volatility is. Garch(1,1) is probably sufficient for the latter, but not necessarily the former. –  John May 4 '12 at 21:42

There is a paper by Goldstein and Taleb (2007) which tries to address this question of what number captures investors intuitive feelings of the volatility of series of returns and whether this coincides with the standard deviation of returns.

What they found was that Median Absolute Deviation does a much better job of capturing this intuition in a small sample of investment professionals, even when the return series were sampled from a normal distribution.

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