### Historical Risk vs. Normal Risk - VaR/CTE/LPSD

February 03, 2009
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The rate of return on a stock portfolio is not normally distributed. Therefore standard deviation is not a good measure of risk.

This is not shocking news. Professional investors have been using other methods to measure risk that take into account the "fat tails" in stock portfolio returns.

One way that professional investors look at risk is called **Value At Risk (VaR).** VaR tells us that with a probability of x% we can expect a loss greater than the VaR.

Let's look at some historical VaR and compare it to those expected from a normal distribution.

Historically:

**10% of the time we lose 4.48% or more in any given month**

**5% of the time we lose 6.46% or more in any given month**

**1% of the time we lose 10.45% or more in any given month**

If equity portfolio returns followed a normal distribution then:

**10% of the time we lose 4.81% or more in any given month**

**5% of the time we lose 6.33% or more in any given month**

**1% of the time we lose 9.13% or more in any given month**

The excess losses at the 5% and the 1% levels seen historically as compared to what would be expected from a normal distribution are what are meant when people talk about "fat tails".

Another way to measure risk is called **Conditional Tail Expectation (CTE). CTE** tells us what the expected loss on our portfolio would be given that the terminal value of hte portfolio falls in the bottom X% of outcomes. This is a better measure than VaR since VaR only gives us the most optimistic outcomes while CTE gives us the average outcome given that we have had a bad performance. Once again historical CTE is worse than that expected by a normal distribution.

Historical CTE:

**Given that a return is in the worst 10% of monthly returns our expected loss is 7.735%**

**Given that a return is in the worst 5% of monthly returns our expected loss is 9.897%**

**Given that a return is in the worst 1% of montly returns our expected loss is 16.36%**

CTE predicted by a normal distribution is much rosier:

**Given that a return is in the worst 10% of monthly returns our expected loss is 6.791%**

**Given that a return is in the worst 5% of montly returns our expected loss is 8.057%**

**Given that a return is in the worst 1% of monthly returns our expected loss is 10.487%**

A final method of examining risk is **Lower Partial Standard Deviation (LPSD)** LPSD examines the standard deviation conmputed only from values below the mean (Left tail of the distribution). Lower partial Standard Deviation is higher than standard deviation because of the negative skew in stock returns.

Historical LPSD:

5.1335%

Normal LPSD:

4.445%

As you can see, regardless of which measure of risk you use - a normal distribution does not provide an accurate representation of what is happening.