Volatility and Value at Risk
Volatility is a key concept in trading, whether you trade currencies, equities or even metals. In a nutshell, volatility is the speed of the market- how fast, and how far can an underlying move be. Volatility and its close cousin Value at Risk (VaR) are great tools for understanding and managing portfolio risk.
While there are many dangers inherent in simplifying reality, price action can be usefully described by a normal distribution. We can make use of this tool in understanding the potential for price action based on historical movements, and get a pretty good idea of Value at Risk in any position. We’ll take a nice volatile pair like AUD/JPY to illustrate, using one year’s daily price action:
Before we calculate volatility, a key concept that you should be familiar with is that of log normal returns. If I have a series of prices, I can calculate the daily change using percent. If I want to aggregate a series of price changes, though, I cannot add percents. However, if I take the natural log of the ratio of prices, I can add those numbers to aggregate the returns, because you can sum logs. Here’s a small sample of AUD/JPY, using log normal returns. The sum of the log normal returns is 17.9%, which accurately reflects the total change from 55.7 to 66.7
|
daily price |
LN (n+1/n) |
|
55.765 |
|
|
62.84 |
11.94% |
|
64.963 |
3.32% |
|
67.195 |
3.38% |
|
65.42 |
-2.68% |
|
67.01 |
2.40% |
|
69.651 |
3.87% |
|
66.701 |
-4.33% |
Now back to volatility. A normal distribution is described by its mean and its standard deviation. We will take the mean to be the starting price, and the standard deviation is easily calculated. Given a series of daily returns, say for a year, we can easily calculate the standard deviation of the series. Using the trailing year’s daily returns for AUD/JPY results in a 2% Std Dev. Since I am using daily returns, I must multiple 2% by the SQRT of the number of trading days in one year (258) to get a total yearly volatility of 32%. If instead I want to calculate the volatility for one month, I multiply 2% times the SQRT of 22 (the number of trading days in a month) to get 9.38% volatility.
Now to Value at Risk, or VaR. VaR is a risk measure, the risk of a specific loss, given a specific time horizon and confidence level. There are several approaches to modeling VaR (historical, parametric, hybrid). For simplicities sake we will use historical price action as a basis for calculating VaR. Let’s examine the distribution of daily returns, and see what percent of the time they fall below our threshold. Here’s a plot of the number of days AUD/JPY log normal returns falls into each bin (0.5% bins):
They mostly resemble a normal distribution (validating our assumption), and are distributed with a slight positive skew (after all, AUD/JPY has done quite well this year).
We can use the excel function “percentile” to easily get the daily Value at Risk to any level of confidence. For example, if we want the VaR to the 95% confidence level, we set 5% in the percentile function, and get -2.84%, which is to say that 95% of the time you won’t lose more than that in one day. The VaR at a 99% confidence level is -4.77% (ie 99% of the time you won’t lose more than 4.77% in one day). To get VaR for longer periods, simply multiply by the SQRT of the number of trading days in the interval. A 95% VaR over one month would be -13.32%.
I’d like to point out the existence of a fat tail. Note the little data point out at +12%. A year ago, AUD/JPY jumped in one day from 55.76 to 62.84. Extraordinary, and a great example of how assuming a normal Gaussian distribution can hide outliers. It is a little beyond the scope of this article to examine skew and kurtosis, other than to mention that there are other, more heavy tailed distributions (such as Cauchy) which more accurately represent the real world. When I generate Monte Carlo simulations to calculate my own VaR, I use what are called Pearson IV random deviates, which lets me create skewed and fat-tailed distributions for a more accurate idea of my VaR.
I hope that this has been of some interest and help in understanding volatility and how it impacts your portfolio.
