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

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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.

## 6 Comments – Post Your Own

#1)On February 03, 2009 at 5:43 PM,Bupp (28.02)wrote:Should add that the numbers are for individual stocks not for the market as a whole.

Report this comment#2)On February 03, 2009 at 7:29 PM,anchak (99.90)wrote:Robert.....I am assuming you have some sort of an acturial background...

Were these numbers computed for a SPECIFIC stock? I know you mentioned it was not for a market ....basically to be more precise - how do you generate the specific percentiles and/or probabilities ....unless you are covering a stock which has Long Tailed history ( like IBM,GE) .....because otherwise you have to make an underlying assumption of the distribution ......Estimating Non-parametric VaR ?

Report this comment#3)On February 03, 2009 at 8:20 PM,Bupp (28.02)wrote:What they did is looked at the montly returns for large cap stocks between 1957 and 2006. Say there were 100 stocks - then there would be 100 data points for each month rather than the stocks being treated as a combined portfolio and 1 data point.

They don't really have to estimate VaR when looking at historical returns, they could just put all of the returns in ranked order and then look at the return at the 10th/5th/1st percentile.,,it's just empirical data.

Of course past returns do not guarantee future returns and past risk is not necessarily indicative of future risk.

What is interesting to me is that I read that standard deviation is not a good measure of risk but take a look at things like Sharpe ratio and information ratio and they both are using standard deviation to help rank performance.

Report this comment#4)On February 03, 2009 at 10:56 PM,anchak (99.90)wrote:They are estimating the VaR and its from Empirical evidence. We are both right :)

The sample size is big enough - whether its representative - different question altogether. As you also said - about the out-of-time/sample relevance - the esitmates will refine/change - like this bear will make all Fat-Tails very very Extreme Valued

StdDev is A measure of risk - it simply in a pictorial sense - is the fan-out or spread from the mean. The problem comes when you start doing nice multiples like 2x,3x etc and start attributing chance/probabilities to them - that is only true under the Normal distribution.

Also simple asymmetric/skewed distribution of return expectation can cause havoc.

Incidentally, the reason why Sharpe Ratio has the mean( zero risk premium) adjusted ratio - which is nothing but a tweak of the Beta from CAPM - which HAS a NORMAL assumption!

Report this comment#5)On February 04, 2009 at 5:55 AM,Bupp (28.02)wrote:Ok, that makes sense. I didn't think they needed to run a regression on the results to get the data they did, I figured they just ranked all the monthly results from lowest to highest and then for VaR stopped at the result that was at the 1%, 5%, and 10% level.

For the large-cap stocks this would not be an estimation during the time period of the study, since it contains the entire population.

So the specific numbers are applicable to large cap stocks between 1957-2006. And are less relevant now and to other classes of stocks. However the exact numbers are not relevant what is relevant is that sd underrepresents risk and that these are some methods which will better approximate it (though given recent results it appears that they also underestimate risk though not by as large a factor).

I don't have a great stats backround (university student with two stats courses and an econometrics class under his belt) - and don't know how these factors are necessarily used under real world conditions.

Report this comment#6)On February 04, 2009 at 4:23 PM,anchak (99.90)wrote:"I figured they just ranked all the monthly results from lowest to highest and then for VaR stopped at the result that was at the 1%, 5%, and 10% level." - CORRECT.

The under-representation is a TRUE statement - however not really relevant for retail investor portfolios. It can be - if you are modeling future return etc.....I am not sure how many financial advisors do that. Basically one would be looking at a VaR based Return trade-off ( Efficient frontier - as per MPT) in constructing a portfolio.

If you are looking for something - ( disclaimer: I do not do this for a living - just my overall knowledge) - Usually Stock market returns/movements are modeled under a LogNormal distribution - its a little more robust - like in Yahoo! you would see the chart on the logarithmic scale - that's why. This appx should work - I obviously have no idea of your intention.

All the best