By Todd Beaird (TMF Synchronicity)

October 24, 2000

The Sharpe Ratio is a widely used tool for determining return against risk. The higher the Sharpe ratio, the more return for a given amount of risk. But, it can also help you adjust the amount of risk in your portfolio.

We showed you the example of a conservative investor last week. Nervous Ned wanted the most return he could get for a lower-risk strategy. The S&P 500 was the most volatile investment Ned could stand without losing sleep. So, Ned had two choices. He could invest in the S&P 500, or he could invest in a blend of an RS-26 strategy and risk-free U.S. Treasury bills. When we ran the numbers, we discovered the straight S&P 500 offered a better return than an RS-26/T-bill strategy that was blended to match the volatility of the S&P 500.

This is the Sharpe Ratio in a nutshell. Given two strategies with similar returns, the one with the higher Sharpe Ratio will have lower volatility. But what happens when you have a strategy that has a great Sharpe Ratio, but relatively low returns? Can you take on more risk and increase your returns?

Meet Nervous Ned’s sister, Crazy Cathy. Cathy is investing for the long haul. She wants to retire early so she can hang glide and sky dive full-time. She embraces short-term volatility in the hopes of superior long-term returns.

Let’s assume that Cathy has the same three strategies available to her as Ned. One is a five-stock RS-26 strategy, rebalanced every January. This is a very volatile investment. Next is S&P 500 Depositary Receipts (AMEX: SPY) — a.k.a. Spiders — which have the same returns and volatility as the overall stock market. Her third choice is one-year U.S. Treasury bills, which are essentially risk-free. (If the U.S. government can’t pay you back, you’ve got more serious problems than worrying about your investment portfolios!) Here are the returns for each of these strategies, since 1986:

As you can guess, Cathy is a very aggressive investor who has no problems taking risks. Her first thought is to invest in the RS-26 strategy. It has a lot of volatility, but offers the highest returns. However, she notices that the S&P 500 has a higher Sharpe Ratio, meaning you get better returns per unit of risk.

Problem is, Cathy wants to maximize her return, not minimize her risk. She does not want to invest in a vanilla index fund and get lower returns than the RS-26 strategy, no matter how high the Sharpe Ratio. What can she do?

Cathy’s choice is the mirror image of Ned’s. She can buy the S&P index on margin. Here’s how it works.

Let’s assume that the margin rate is equal to the risk-free rate on T-bills. Imagine that Cathy invests $10,000 of her own money in the S&P 500, and then borrows another $10,000 at the T-bill rate and also invests that money in the S&P 500. In 1986, she would have earned 18.82% on the money she invested in the S&P 500. She invested $20,000 (the $10,000 of her own money and the $10,000 she borrowed). So, by the end of 1986, her $20,000 investment would have grown to $23,764 ($20,000 x 0.1882 = $3,764).

One problem. Cathy borrowed $10,000, and that money charged 7.21% interest. So, by the end of the year, Cathy owes $10,721. Cathy takes her $23,764, and pays off the loan. $23,764 — $10,721 = $13,043. By going on margin, Cathy has turned $10,000 into $13,043 in one year. Instead of a return of 18.82% in 1986, Cathy has earned a return of 30.43%. For the mathematically inclined out there, you can calculate the return more quickly using the formula (2 x 0.1882) — (1 x 0.0721) = 0.3043.

That’s the upside of margin. As you might know, margin has its downside. Let’s say that Cathy continues to borrow one dollar for every dollar she invests. In 1990, the S&P went down 2.97%. If Cathy had invested $10,000 of her own money and borrowed $10,000 more, her $20,000 investment will be worth only $19,406 by the end of the year. She will owe $10,738 (the $10,000 plus the interest at the T-bill rate of 7.38%). By the time she pays off her loan, Cathy will only have $8,668 left. Again, you can calculate this as (2 x -0.0297) — (1 x 0.0738). Either way, it works out to a 13.32% loss. Ouch!

Margin increases gains, but it also increases losses. In other words, using margin can increase the total overall return of a strategy, but it also increases volatility — which is exactly what Cathy was looking for. (For more on margin, particlularly its dangers, see Margin — Friend or Foe? )

She wanted a high return, and she can handle high volatility. She knows that the S&P 500 has a high Sharpe Ratio, higher than the RS-26 strategy. This means that, if she can somehow increase the volatility of the S&P 500 to match that of the RS-26, she will get higher returns than a straight RS-26 strategy. By borrowing on margin at the T-bill rate and buying the S&P 500, she can increase the volatility of the S&P 500 to match our RS-26 strategy. But how much margin does she need?

Crazy Cathy sits down with a spreadsheet and does the math. She discovers that, if she takes her initial $10,000 and borrows another $21,268.39, she’ll have returns like this:

As you can see, this strategy has the same standard deviation as our RS-26 strategy. (Yes, we planned it that way.) But, the average return is higher than the RS-26 strategy. An investor can leverage a lower-risk strategy with a high Sharpe Ratio for higher returns. This is how one can eat a risk-adjusted return.

There’s one problem. Cathy’s broker won’t let her go that far on margin! And with good reason. What this calculation doesn’t show is intra-year volatility. Remember October 1987 and, for that matter, October 1997? Had she been that far out on margin during those drops, Cathy would have been getting MailGrams inviting her to pony up some cash real quick or the broker would sell her position at the market low. That’s a sure way to lose money. Still, this does illustrate the principle of how one can adjust the risk of a portfolio using the Sharpe Ratio as a guide.

Please, don’t take out a second mortgage to invest in the market on margin. Wait until we can point out some of the shortcomings of the Sharpe Ratio. We’ll see you then — same Fool time, same Fool channel.