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I am looking for a simple example which explains why variance as a risk measure can be problematic (with a long-only portfolio with no options).

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  • $\begingroup$ Variance is only a complete measure of deviation from a mean if you believe that the underlying distribution is defined by a stationary mean and variance, e.g.: $\ln(\frac{S_t}{S_{t-\Delta t}}) = \mu \Delta t + \sigma \sqrt{\Delta t}*Z $, where $Z$ is a GBM. There may already be posts on this: quant.stackexchange.com/questions/9960/…; quant.stackexchange.com/questions/96/…; quant.stackexchange.com/questions/33853/…. $\endgroup$ – David Addison Apr 25 '17 at 7:01
  • $\begingroup$ While variance/covariance risk measures are used frequently (at the very least for benchmarking) there are instances where the risk measure can be problematic. For example, consider a portfolio of positions with a collection of instruments with both normally distributed and highly skewed return/P&L distributions. In this scenario if the objective is to assess VaR contributions, then variance alone would likely not be sufficient. $\endgroup$ – NaN Apr 25 '17 at 14:08
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Although this sounds like a simple question it might not be that clear.

You say a long-only portfolio with no-options. I assume you mean a stock portfolio. As you say "no options" there should not be too much skewness. Additionally we assume that your portfolio is well diversified (no dominating weights in single stocks, countries or industry sectors).

The next question is what the aim of your risk measures is. If it is ranking portfolios in the sense portfolio A is riskier than portfolio B or my portfolio is riskier or less risky if I add/remove a tiny position in stock S then I would say:

  • Variance is (of course) as fine as standard deviation (volatility);
  • a Gaussian Value-at-Risk (VaR) or Expected Shortfall will not tell you more about your portfolio(s) as it is proportional to volatility;
  • a t-distributed VaR will not tell you more as it depends on the degree of freedom and the volatility. Your degrees of freedom could be similar for portfolio A or B - thus your choice could depend on volatility again.

We can look at other alternatives to variance but with the above stated aims and the above mentioned nature of your portfolio I would say that variance is just fine.

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Here's a simple hypothetical example:

Portfolio A = a single stock priced at 100 which can either go to 99 or 101 each with probability 0.5 in one year. The annual standard deviation is 1. The variance is 1 .

Portfolio B = a single stock prices at 100 which can go to 90 or 110 each with probability 0.005 or stay at 100 with probability 0.99. The annual standard deviation is 1 and the variance is 1 squared as before.

The difference between these two portfolios is not evident from the use of variance as a single risk measure. Of course this is a highly theoretical example.

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  • $\begingroup$ How does this answer OP's question why variance as a risk measure can be problematic ? $\endgroup$ – rbm Apr 25 '17 at 11:58
  • $\begingroup$ This cleverly constructed example shows two different distribution with the same variance but different kurtosis. Similarly you can construct two distribs with same variance but different skweness. The point is simply that variance may not tell you all about the distribution that you wish to know. $\endgroup$ – noob2 Apr 25 '17 at 19:57
  • $\begingroup$ I understand the examples, the OP's question was about why variance is problematic. dm63's example readily applies to stdev/volatility, since both sample portfolios have the same stdev, i.e. could have been written as "The difference between these two portfolios is not evident from the use of variance or standard deviation". Again, i am not disputing these examples, but it doesn't answer OP's question. The other asnwers make good points on that. $\endgroup$ – rbm Apr 26 '17 at 11:43
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I would like to echo @Richard's and @dm63's answers. I would also like to propose one scenario, in which variance is assumed to be stochastic, which demonstrates how observed variance can be problematic as a risk measure.

If we presume that returns are a random walk in the form of a normally-distributed process, then it follows that variance is a complete measure of risk -- where we define risk as a probabilistic distribution of random outcomes. For returns which are normally distributed, an arbitrary Numeraire process, $\mathbb{P}$, may evolve according to a Wiener process:

$\mathbb{E}[\frac{d\mathbb{P}}{\mathbb{P}}] \to_{\text{discretization}} \ln(\frac{\mathbb{P}_t}{\mathbb{P}_{t-\Delta t}}) = \mu \Delta t + \sigma \sqrt{\Delta t}*dZ$

where $dZ$ is a Wiener process (e.g., GBM).

If, however, we subscribe to the notion that variance is stochastic (e.g., a non-stationary, mean-reverting process), then the expectation can be re-written as such:

$\mathbb{E}[\frac{d\mathbb{P}}{\mathbb{P}}] = \mu \Delta t + \sigma_t \sqrt{\Delta t}*dZ_1$

where:

$d \sigma^2_t \propto \eta \,\sigma \sqrt{\Delta t}*dZ_2$

with:

$\langle dZ_1 \, dZ_2 \rangle = \rho \, dt$

where: $\eta$ is the volatility of volatility; and, $\rho$ is the correlation between returns and changes in $\sigma^2_t$.

This setup is commonly used to calibrate GARCH volatility models, as is detailed in Jim Gatheral's lecture on "Stochastic Volatility and Local Volatility"

The expectation of stochastic variance is not far-fetched when considering observed fat-tails in security returns due to "volatility clustering" (i.e., large moves tend to be preceded and proceeded by large moves).

If one believes that variance (and/or expected returns) are stochastic, then one should also believe that observed variation will tend to provide an incomplete measure of long-term (posterior) variation. This belief also implies that balancing the frequentist view with Bayseian inference will provide a more accurate picture of posterior realized variation.

Although this answer may not be the simple one which you requested, I believe that the ramifications of assuming non-stationarity are rather straightforward. Moreover, the implications are relevant even for a long-only portfolio without exposure to leverage or options.

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The key in our proposed methodology is a risk measure called shortfall, which we argue has conceptual, computational and practical advantages over other commonly used risk measures. It is a variation of the mean excess function and TailVaR mentioned earlier

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    $\begingroup$ This doesn't really address the "why" part of the question or provide the "simple example" that the OP was asking for. $\endgroup$ – LocalVolatility Apr 26 '17 at 11:37

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