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Why Bother With Risk Control?

By: Dr. Robert Fernholz & Jennifer Winfield

Investors may be tempted to believe that the Sharpe Ratio does not change under leverage. Dr. Robert Fernholz and Jennifer Winfield, of Enhanced Investments Technologies (INTECH), argue this may not be the case for long-term investors.

In the 1990s, when the stock market was averaging about 20 per cent a year, anyone who talked about risk-controlled equity strategies was largely ignored. After all, who is interested in risk control under such circumstances? But things are different now. With the market having posted negative returns for three years in a row, and with the long-term return projections in the six per cent to eight per cent range, risk-controlled equity strategies are not only desirable, but perhaps necessary for serious long-term investors.

Sometimes the attractiveness of an investment is judged by its expected rate of return. However, one of the basic principles of probability theory (the strong law of large numbers) implies that the long-term performance of an investment is determined not by its expected rate of return, but by its expected compound growth rate. (The compound growth rate is sometimes called the logarithmic rate of return, or the geometric rate of return.) For riskless investments, the rate of return and the compound growth rate are the same. For risky investments, the expected compound growth rate is always less than the expected rate of return. Let us look at a simple example to see how this works.

Consider an investment that either goes up 100 per cent or down 50 per cent every year, each with a probability of one half. The average of +100 per cent and -50 per cent is +25 per cent, so the expected rate of return on this investment is 25 per cent a year. But if we compound the returns on this investment, a 100 per cent gain followed by a 50 per cent loss will leave you where you started out. Over the long-term, we can expect about half up moves and half down moves, and these will cancel each other out, so the long-term compound growth we can expect in this case amounts to zero per cent a year. For a single-period investor, the 25 per cent expected return may be accurate, but for a long-term investor, who will have to compound his/her annual returns over many years, the 25 per cent expected rate of return is illusory, and the zero per cent expected compound growth rate will become dominant.

Compound Growth Rate Vs. Rate Of Return

The mathematical relation between expected rate of return and expected compound growth rate is:

Equation 1 compound growth rate = rate of return – variance

2 where the variance is equal to the square of the standard deviation.* If we apply this equation to the 100 per cent gain, 50 per cent loss example above, we have 25 per cent expected return, and the standard deviation is about 70 per cent. With these values, the compound growth rate in Equation 1 will be about zero per cent, as determined intuitively. The volatility in the example we have considered is rather high, but for lower volatilities the negative term in Equation 1 can still have quite a disappointing effect on long-term investment results. In Figure 1, the broken line shows the 20-year expected return curve, in terms of the wealth ratio, for an investment with 10 per cent annual expected return. (The wealth ratio of an investment is its value at a given time in the future divided by its initial value). If the investment is riskless, the wealth ratio will exactly follow the expected return curve, so the value of the investment will increase by a factor of more than seven over the 20 years. However, with a standard deviation of 35 per cent a year, the negative term in Equation 1 will be -6.125 per cent, so the expected compound growth rate will be slightly less than four per cent a year, and the compound growth curve that the investment is likely to follow will attain a wealth ratio of only slightly greater than two after 20 years. A 10 per cent decrease in standard deviation, from 35 per cent to 25 per cent a year, will provide an expected compound growth of 6.875 per cent a year (using Equation 1, you can carry out the calculation yourself), and this increases the 20 year wealth ratio of the expected compound growth curve to about four. Over the long-term, the compound growth rate dominates, so the value of risk control is evident.

Leverage And Long-term Growth

Rational investors usually will judge their investments relative to some appropriate benchmark. The information ratio of an investment is frequently defined to be the ratio of the investment’s expected return over the benchmark divided by the standard deviation of the return relative to the benchmark. (Note: There exist variations of this definition.) From the point-of-view of risk, perhaps the most basic benchmark is the return on treasury bills. The information ratio of an investment versus the return on treasury bills is called the Sharpe Ratio, named after its inventor, Nobel laureate William Sharpe.

A desirable property of the Sharpe Ratio is that it does not change under leverage. If money is borrowed to lever an investment, the Sharpe Ratio of the levered investment will be the same as that of the original investment. This property may tempt us into believing that the Sharpe Ratio fully accounts for investment risk, but for longterm investors this is not the case. Let us consider an example.

Suppose we have an investment with eight per cent annual expected rate of return and 25 per cent annual standard deviation. For simplicity, let us assume here that treasury bills pay zero per cent interest. The Sharpe Ratio of this investment is .32, and its expected compound growth rate is 4.875 per cent a year, according to Equation 1. Now, if we lever this two to one by borrowing cash at the zero per cent treasury bill rate, the expected rate of return of our levered investment will be 16 per cent a year with the standard deviation at 50 per cent, so the Sharpe Ratio remains at .32, as we know it should. However, now Equation 1 tells us that the expected annual compound growth rate is 3.5 per cent. Although it may be more exciting in the short-term, the levered investment is likely to grow more slowly than its unlevered counterpart over the long term.

So be careful: in order to achieve superior long-term portfolio performance, it is essential to understand and control risk.

Dr. Robert Fernholz is chief investment officer and Jennifer Winfield is senior vice-president, client relations, at Enhanced Investments Technologies (INTECH) – a Janus Capital Group company. * For a derivation of this, see E. Robert Fernholz, Stochastic Portfolio Theory, Springer-Verlag. Nothing herein is intended to amount to investment advice.

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