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SUBSTANDARD
DEVIATIONS
David C. Shimko Jean-Paul St. Germain (originally published by PMA OnLine Magazine: 05/98) Your school professor didn't tell you
everything you need to know about The colossus of VaR (do we need to spell out VaR any more?) depends on a humble volatility calculation, but few seem to question how volatility is computed. We all learned at some point how to calculate return volatility in a financial instrument: How to calculate annual percent volatility, 101 1. Choose a frequency (daily, weekly, monthly) For example, the annual percent volatility of the ten-week price sequence below: Table 1: Power Prices In The US
Before you faint from electric shock, or try to plug these numbers into your Black-Scholes calculator, take a look at the historical prices:
What you might have forgotten When you calculate volatility the usual (101) way, you are implicitly making a number of assumptions: A1. The percentage change has a constant mean and volatility. A2. The percentage change is independent of previous changes. That is, the price follows a random walk. [The log change does nothing other than convert a periodic return into a continuously compounded return, and avoids the possibility of negative prices.] These assumptions are approximately true when dealing with stocks, although certainly there are many exceptions. They are less reliable for bonds, where volatility declines as a bond approaches maturity. And certainly, assumption A2 is the most often violated, especially when referring to interest rates, some currency rates and commodity prices. Why might prices not follow random walks? If the price of power jumps to $100, it will certainly not stay there; everyone knows that the price will soon revert to a more reasonable level. In statistics, this is known as mean reversion. In the context of this note, the mean price change, or expected price change is negative when the price is very high, and positive when the price is very low. In equation form, this can be represented as follows Pt+1 - Pt = k (m - Pt) + s e t which might look complicated at first. It says merely that the projected price change (Pt+1 - Pt) is proportional on average to the distance the price lies from its true long-run mean m . The term k is often called the speed of mean reversion, since the higher it is, the faster the price moves toward its long-run mean. This equation is not difficult to estimate or implement. Here's the recipe:
Table 2: Calculating the impact of mean reversion
The regression data are tabulated easily in ExcelTM, using the SLOPE, INTERCEPT and STEYX (standard error of a prediction) functions. If the slope estimate is insignificant, you needn't worry about conventional mean reversion. However, this regression results show a very high degree of mean reversion --- in fact, the mean prediction for any week (based on these very limited data) would be around $27 regardless of what it was the previous week. This is anathema to the random walk concept, where next week's expected price is equal to this week's price. The residual standard deviation of $5.11 can be divided by the mean forecast to get a percentage volatility, but any attempt to annualize that volatility is completely inappropriate --- since the price changes are not independent of each other over time. The volatility of a one-year forecasted price using these data is not much more than $5.11, or about 18% of the forecasted price level. This is a full order of magnitude below the 173% volatility estimated from percentage changes. So why is this important to VaR? Lest you think this problem is limited to the esoteric electricity market, let us caution you. Mean reversion is extremely important for all industrial commodities, for interest rates, and even for some currencies. The ways we calculate volatility are vulnerable to ignoring this important phenomenon. As a result, many of the models used to make volatility estimates may be grossly overestimating the actual voilatility one should expect as a result of holding the instrument. High VaR estimates may seem conservative and therefore appropriate at first, but overestimating risk leads to penalizing performance measures, earning subpar returns on capital, and discouraging profitable and appropriate trading activity. If you take away nothing else from this column, think twice before you calculate price volatility.
Reprinted with permission, David Shimko, Risk Magazine Advisory group. All rights reserved. Copyright 1998. |
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