Risk and Value-At-Risk

Risk is a statement about the likelihood and magnitude of adverse events. There are differing ways to classify 'adverse event' which leads to differing measures and ways of stating risk, such as:

• The average return deviation from the mean (standard deviation)
• The expected loss if the return over a period is less than 10%
• The probability you lose over \(X\) in a given month

To find these values, we'll need some description of how our security (or portfolio) behaves.

The behavior is described through a probability density function (PDF) which gives the likelihood that returns will be in a given interval.

We'll either have to:

1. Assume a particular PDF and estimate the parameters which describe the PDF.
2. Use an empirical PDF – that is, a PDF that fits what has happened over some past period.

Either way we must use historical data, or our own judgment, to describe future event probabilities.

The most-commonly assumed PDF is the normal distribution.

• This is partly for mathematical ease: the sum of normals is normal.

The normal distribution is completely determined by its mean and variance (or standard deviation).

• Once we determine these values, we can state the probability of a return being less than any value.
• Note, the normal distribution is symmetric about its mean value.

The standard deviation is the average squared deviation from the mean.

• Squaring means the standard deviation treats positive and negative deviations from the mean equally: \((-7\%)^2 = 7\%^2\).
• But we only care about the negative returns when calculating risk.
• Since for a symmetric distribution, the probability of a 5% or more deviation is equivalent to the probability of -5% or less deviation, it is fine to square the deviation.

What if return distributions are not symmetric? Say, for instance, that the distribution is left-skewed.

• Then the average negative return is greater than the average positive return.
• In this case, a measure like standard deviation would understate risk.

For non-symmetric distributions, the act of squaring loses important information about risk.

• Specifically, it loses the scale of the average negative return with respect to the average positive return.

Fat tails (excess kurtosis) are also a salient feature of asset returns.

• A normal distribution has a kurtosis of 3. A distribution has fat tails if the kurtosis is greater than 3.
• If the tails are fatter than the normal distribution, then there will be more large positive and negative returns (tail events) than you may expect.

In the following app we'll calculate and plot the empirical PDF of the stock's returns. We'll then use the stock's historical mean return and standard deviation to overlay a normal distribution.

• If the empirical distribution seems to lean (is skewed) relative to the normal, then standard deviation is a poor measure of risk.
• Similarly, we need to pay attention to the difference between the two distributions near the tails.

• Take a look at Exxon (Ticker: XOM) over 2020. The stock's returns exhibit both negative skewness and kurtosis. This means the standard deviation would have understated risk.
• Looking at Apple (AAPL) from 1/2015 through 4/2015 shows a symmetric, fairly normal, distribution – the standard deviation would have been a good measure of risk.
• GE from 1/2015 to 4/2015 shows positive skewness. The standard deviation would have overstated risk.

VaR is a very common way of stating the amount of risk.

• 5% VaR is the return for which, in any given period, there is a 5% chance of experiencing a worse return. So say the 5% VaR is -8%. Then there is a 5% chance of experiencing a return less than -8% over the next period.
• VaR solely focuses on the left tail of the return distribution.

VaR is widely used and is written into banking regulations.

VaR still requires an assumption concerning the returns' PDF.

• We can again assume a normal, empirical, or other distribution.
• If we assume a normal distribution, then VaR is simply a function of the mean and standard deviation.

The following app will calculate the % VaR for any stock over any time interval you choose.

• Click the Normal and Empirical tabs to see the VaR that assumes a normal PDF (using the historical mean and standard deviation) and a PDF fit to historical data.
• Can you find stocks for which the normal and empirical VaR are different? Is the difference in VaR greater if you use 1% VaR versus 5% VaR?
• VaR is most useful for portfolios, or an entire book of business. In the following app the default ticker is SPY, an exchange traded fund which tracks the S&P 500 index.

• The extent to which the parameters of the PDF change over time (and particularly during a financial crisis)
• Measures specifically designed for asymmetric distributions, such as Lower Partial Standard Deviation