Bill Rempel has a post up regarding the Sharpe Ratio, which I quite liked as far as it went. The only problem is that I think Bill must be a fairly advanced statistician as many of his concepts stated, but not elaborated on. For systems testers, there could be some interesting lessons hiding in there. Today will therefore be Statistics Sunday.

*When one is evaluating systems (or a manager), it’s advisable to measure both the return and the risk associated with it. Often the volatility of the return stream, which isn’t necessarily the same thing as its risk, is thought of as synonymous with risk and substituted for risk in the measurement. The Sharpe ratio is one metric often used for measuring return in relation to risk; the Sortino ratio is another.*

We have a number of concepts here, volatility, risk, Sharpe ratio, Sortino ratio. Fuzziness predominates when many of these concepts are discussed.

Volatility is defined as the **Annualized Standard Deviation.** What then is the Standard Deviation?

Standard Deviation is a measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation. Standard deviation is calculated as the square root of variance [variance being defined as measure of the dispersion of a set of data points around their mean value. Variance is a mathematical expectation of the average squared deviations from the mean]

You see, we’ve hardly even made it out of paragraph one, and already it’s getting a little fuzzy. The point of the original article was to shed light on the fact that the Sharpe ratio, that appears all over the place can be rather misleading unless you really understand the nuts & bolts. We move on.

*(1) Maximizing the Sharpe ratio maximizes compounded returns.*

*Only if the distribution of returns is symmetric. Are returns distributed symmetrically? Not in THIS universe. To paraphrase Rodney Dangerfield’s character in “Back To School,” speaking to his business professor, maybe we should base our funds in Fantasyland, home of exclusively Gaussian distributions! *

Compounded returns are also known as “Geometric Returns” which are, or can be confused with “Arithmetic Returns.” This difference plays havoc with actual dollar returns, so you need to be careful. This is all based on a chap named Bayes, who postulated this hypothesis as his PhD. The average annual returns [arithmetic] over the five years was 10% (15% + 0% + 20% – 5% + 20% = 50% ÷ 5 = 10%), but the compound annual growth rate (CAGR, or geometric return) is a more accurate measure of the realized gain, and it was only 9.49%. Volatility eroded the result, and the difference is about half the variance of 1.1%.

Notice that volatility increases as the interval increases, but not nearly in proportion: the weekly is not nearly five times the daily amount and monthly is not nearly four times the weekly. We’ve arrived at a key aspect of random walk theory: standard deviation scales (increases) in proportion to the square root of time. Therefore, if the daily standard deviation is 1.1%, and if there are 250 trading days in a year, the annualized standard deviation is the daily standard deviation of 1.1% multiplied by the square root of 250 (1.1% x 15.8 = 18.1%). Knowing this, we can annualize the interval standard deviations for the S&P 500 by multiplying by the square root of the number of intervals in a year:

Returning to Bill and Gaussian distributions, I have written many times that the Gaussian distribution in the stockmarket is pure nonsense and tantamount to potentially blowing-up your account. We also have a second component hiding in there which is critical to any Gaussian distribution, the **“Random Walk.”**

The Random Walk mandates **“independence.”** That is to say, one price change in a stock, is totally unrelated to a previous price change [in any timeframe] in the same stock. That the next price quoted is random, like the flip of a fair coin. You need only study momentum traders, or various theories on chart patterns to understand that this is pure nonsense; price change in one direction, will attract further participants to the trend developing.

Therefore, we can state that a Gaussian distribution should not exist in the financial markets. Let’s examine the data.

We can observe two differences between the normal distribution and actual returns. First, the actual returns have taller peaks – meaning a greater preponderance of returns near the average. Second, actual returns have fatter tails. [kurtosis] Taking -3 Standard Deviations as a large loss, the Gaussian curve predicts such a loss would occur approximately 3/10 years. The actual loss occured 14/10 years.

These results are within the S&P500 & NASDAQ index, thus diversification damps the volatility or standard deviations. If we look at individual stocks that make up those index, we see an increasingly non-Gaussian distribution in the 1999 data.

Total Stocks………..3SD………..4SD……….5SD……..6SD………Total……2466

Up move……………..309………..116……….44………..47………..516…….20%

Down move………….69………….29………..15………..19………..132…….5.3%

The lognormal distribution, or Gaussian distribution states that a stock really can’t move more than 3SD in any timeperiod, obviously, simply incorrect.

*(2) The Sharpe ratio measures the probability that a given return stream is significantly different from the “Risk Free” rate of return.*

A ratio developed by Nobel laureate William F. Sharpe to measure risk-adjusted performance. The Sharpe ratio is calculated by subtracting the risk-free rate – such as that of the 10-year U.S. Treasury bond – from the rate of return for a portfolio and dividing the result by the standard deviation of the portfolio returns.

Obviously there is the already highlighted problem of volatility not conforming to a standard distribution. There are additional problems however [should you still be intent on using the Sharpe Ratio]

As the above table shows, zero or constant volatility demonstrated in a falling graph can actually be reversed and show a very high Sharpe Ratio. Take as an example a Hedge Fund that “Sells Premium” as an example, in a trending market as we had from 2003 to 2007, this strategy might well have exhibited a very high Sharpe Ratio;

For example, according to Hal Lux in his article, “Risk Gets Riskier”, which appeared in Institutional Investor in 2002, Long-Term Capital Management (LTCM) had a very high Sharpe ratio of 4.35 before it imploded in 1998.

An area not touched on by Bill with regards to the Sharpe Ratio, is one of liquidity. Liquidity was an issue with LTCM, and more recently, it has been a massive problem for the Banks, Hedge Funds, Insurance Companies, Municipalities and generally anyone holding Mortgage Backed Securities [MBS]

The totally illiquid market for MBS has caused massive Mark-to-market losses for all concerned. Losses in excess potentially of the actual default rate. Of course, prior to the debacle, Sharpe ratios again will have shown high values.

The Sortino Ratio is a ratio developed by Frank A. Sortino to differentiate between good and bad volatility in the Sharpe ratio. This differentiation of upwards and downwards volatility allows the calculation to provide a risk-adjusted measure of a security or fund’s performance without penalizing it for upward price changes. It it is calculated as follows:

The Sortino ratio, as can be seen only utilises “downside volatility” in the denominator. Returning to our prior definitions of volatility, one must ask, if volatility has been discredited as a metric based on a Gaussian distribution, how accurate then is the ratio for describing risk based on a metric of volatility?

Which really opens up an entirely new discussion on how to actually define **“RISK”** as we can see volatility, while one of the numbers, does not fully describe risk accurately.