Managing Downside Risk in Financial Markets

Managing Downside Risk in Financial Markets by Frank Sortino and Stephen Satchell describes methods that are light-years ahead of traditional Mean-Variance Optimization.

In one sense, it is far from revolutionary. It retains the most important traditional assumption: the normal, Gaussian, bell shaped distribution (of return percentages) with independent monthly returns. All of the standard qualifiers related to probability distributions still apply.

The book comes with software and a limited amount of data.

The theory has been around in a developed form for more than a decade.

It strikes me as strange than anybody with access to a personal computer (even an older personal computer) would choose a Mean-Variance approach as opposed to downside risk management.

Definitions

MAR is the Minimum Acceptable Return.

The DOWNSIDE DEVIATION is similar to a standard deviation using MAR instead of mean and excluding all returns greater than the MAR.

Downside deviation is the square root of the integral (or sum) of (MAR-x)^2*f(x) for all returns x from minus infinity to the MAR, where f(x) is the probability density of x. Downside deviation EXCLUDES all values of x above the MAR.

The UPSIDE POTENTIAL is the integral (or sum) of (x-MAR)*f(x) for all returns from the MAR to plus infinity. Upside potential INCLUDES only values of x above the MAR.

Downside deviation is a root mean square (r.m.s.) calculation. It emphasizes larger deviations. Upside potential uses linear weighting.

The UPSIDE POTENTIAL RATIO is the upside potential divided by the downside deviation.

The Sortino Ratio is similar to the Sharpe Ratio. It is the amount of one’s return above that of a benchmark divided by the downside deviation.

There are additional figures of merit. The Omega excess return is the most prominent. It is awkward because it combines a variance (instead of a standard deviation) directly with a return.

The Omega excess return has the form: return-(a multiplier)*(the beta of a particular style)*(the DOWNSIDE VARIANCE related to the style), where the downside variance is the square of the downside deviation. The multiplier is estimated to be 3 from empirical observations. The multiplier depends upon what a risk adverse investor would accept.

Modeling Details

Probability Distribution

Except for a small offset, the model uses the normal, Gaussian, bell shaped probability distribution of percentage returns.

Monthly Data

The model uses monthly data.

It constructs annual returns by stringing 12 monthly returns together. Each month of these annual returns is selected at random (with an equal probability) with replacements.

The model constructs 2500 12-month sequences.

The model fits these sequences with a (slightly modified) lognormal distribution.

Curve Fit

A gain multiplier equals a final balance divided by an initial balance. Each gain multiplier equals one plus the return. When you multiply gain multipliers of two consecutive months together, the product is the balance after two months divided by the initial balance. Similarly for longer sequences: you simply include more months in the product.

The model takes the logarithm of gain multipliers plus a very small offset. The offset avoids the possibility of taking the logarithm of a negative number. It fits this with a normal, Gaussian, bell shaped probability distribution.

Because the model makes 2500 sequences, there are enough data samples to use the sample mean and sample standard deviation (along with minor adjustments for the offset).

Standard Limitations

This approach makes excellent use of a limited amount of information. Typically, the actual data cover only a few years, not even a decade. As such, you should not expect to see many of the changes that are present in longer intervals.

However, the model does extract more information than you might expect. For example, it was able to measure the downside risk throughout the Japanese bubble.

The model treats returns for each month as being independent. This is known to be in error. It has not caused dramatically obvious errors.

Expect this model to break down in those areas of well-publicized modeling difficulty. It should do a good job with confidence levels in the neighborhood of 90% (two-sided, 95% one-sided). As with all financial models based upon a normal (or lognormal) distribution, you should disregard claims of a precision of 99% (or more).

As with all stock market tools, you should consider additional areas of risk. This model tells you what is typical. It does not guard you against unusual, surprise events. Always look for failure mechanisms separate from the model.

A Misleading Endorsement

There was one endorsement that talked about guiding investment decisions as a series of small steps. It has an important element of truth. You should review and adjust your investment decisions from time to time. Yet, the nature of the endorsement sounded as if it were advocating many frequent changes. I urge caution. Pay great attention to detail. Especially, details about cost.

The real reason that the endorsement mentioned small steps stems from a deficiency in the database. The existing database is limited. You can only take small steps.

In spite of this, the endorsement does contain an important, second element of truth. The modeling approach is highly efficient in extracting information from short sequences.

Our Applications

We have data to put into the model. Professor Robert Shiller maintains a monthly database of the S&P500 index.

The model strips off the information contained in sequences and valuations. The model treats each month as completely independent of everything. This makes our task more difficult.

Overall Assessment of the Book

Managing Downside Risk in Financial Markets by Frank Sortino and Stephen Satchell provides a good overview of what is known as Post-Modern Portfolio Theory. It comes with useful software.

I am not sure how useful this will be for my own research. I am sure that this is much, much better than Mean-Variance Optimizers.

Have fun.

John Walter Russell
December 8, 2005