Balancing Risk and Return or Why AIO Does not Have Multiple Portfolio Categories

Professor Matthew Spiegel

August 17, 2005

If you look at the web sites associated with any number of newsletters you will see that they offer various recommended portfolios. These typically go by names like, “cautious,” “aggressive,” and “ultra aggressive.” Yet, Alpha Investment Opportunities only has one recommended portfolio. Why? The answer lies in knowing how a portfolio’s risk and return can be traded off.

Risk vs. Return

A Portfolio’s Expected Return

When deciding what securities should go into your portfolio you need to balance risk and return since typically higher return portfolios have higher risks. Fortunately, your portfolio’s expected return is easy to calculate. Imagine you have $100,000 in a short term government bond fund (a.k.a. the risk free asset) and another $300,000 in a stock fund. Your total portfolio has a value of $400,000. It is generally easier to work with the fraction of your portfolio invested in each asset rather than the dollars invested. The reason for this is that you can make decisions and get advice about how to allocate your funds without having to know exactly how much you have to work with. In this example you have 100,000/400,000 or .25 of your wealth in the bond fund and 300,000/400,000 or .75 of your wealth in the stock fund.

With the weights and expected returns for each asset in hand you can now calculate your portfolio’s expected return. If the bond fund returns 5% and on average the stock fund 10% then the expected return of your portfolio equals .25×.05+.75×.10 = .0875. Notice it is the weighted average of the expected return associated with each asset that you hold. The weights themselves (the .25 and .75) equal the fraction of your wealth in each asset. Also, note that the weights have to add up to one. All of your money has to be somewhere! Thus, if we know you own two assets and have 75% of your wealth in one that must mean you have 25% of your wealth in the other. In general, suppose you own two assets. If the expected return on each asset is r₁ and r₂ then the expected return on your portfolio (r_p) is

where w equals the fraction of your portfolio in the first asset.

A Portfolio’s Risk

Alas, calculating a portfolio’s risk turns out to be quite complex in general. However, it is very easy in one particular case and fortunately it is the most important case. First, though, we need to define what we mean by risk.

Risk is typically measured by the standard deviation (s.d.) of an investment’s return. Higher numbers imply greater risk. For a typical asset that has a “normally distributed” return you can expect the realized return to equal the expected return plus or minus one s.d. about 32% of the time. You can also show that for such an asset you will earn the expected return plus or minus two s.d.’s about 95% of the time.

The one case where it is easy to calculate the standard deviation of a portfolio occurs when there are two assets and one is risk free (has a standard deviation of zero). For this case the standard deviation of the portfolio (σ_p) equals wσ_s where w stands for the fraction of your portfolio invested in the risk asset (stocks) and σ_s the standard deviation of the risky asset holdings. Throughout this article we are going to assume somebody has given us σ_s so that we do not need to calculate it ourselves.

For a two asset portfolio with w in the risky asset and thus 1−w in the risk free asset we can express its return and standard deviation as:

r_p = (1−w)r_f + wr_s,

and

σ_p = wσ_s

respectively. In the first equation the r_p,r_f, and r_s stand for the expected return on the portfolio, risk free asset, and the risky asset in that order. While this may look complicated we can simplify things considerably by noticing that the second equation says that w=σ_p/σ_s. If you substitute this into the top equation and do a little rearranging you get

which is actually quite helpful. Here is what it says; the return on your portfolio is proportional to its standard deviation. How do I know that? You cannot control any of the expected returns. Each asset has some expected return and you just have to live with it. Also, you cannot control the standard deviation of the risky asset σ_s. But you can control the standard deviation of your portfolio σ_p! By changing the fraction of your portfolio in the risky asset (w) you can move this value up or down (see the second equation in this paragraph).

According to the above equations if you want a risk free portfolio you can have it. Just set w to 0. In that case you will have a σ_p of 0 and an r_p equal to r_f. In other words if you put all of your money into the risk free asset (w=0) you get the risk free return. This may not be very deep, but it is still nice to see that the equations bear out what common sense tells us. Alternatively we can produce another common sense result by noting that you can put all of your money into the risky asset by setting w to 1. In this case σ_p equals σ_s and r_p equals r_s and we find that you earn what the risky asset earns and have a portfolio with the risky asset’s standard deviation.

Why Numerous “Recommended Portfolios” are Not Very Helpful

Why go through all of this with the weights and the standard deviations and such? Because if you have followed along this far you no longer need a newsletter that offers you several different portfolios with labels like “cautious,” “aggressive,” and “ultra aggressive.” Rather, if you know a recommended portfolio’s standard deviation and expected return you can adjust the risk yourself. If you think the risk is too high simply invest a smaller amount in it and more in the risk free asset. Instant risk reduction!

Believe it or not, we can now go even further and show that offering multiple portfolios is both pointless and in a sense misleading. There is only one “right” portfolio to offer and it is the one that provides the best risk return tradeoff. Notice that in the last equation above the benefit from increasing the portfolio’s standard deviation equals (r_s−r_f)/σ_s. The bigger this ratio the better the risk-return tradeoff for your portfolio. In fact, this is such a famous ratio it has its own name! It is called the Sharpe ratio after the Noble laureate who helped found portfolio theory. Another way to look at this is that your portfolio should only contain some combination of the risky portfolio with the very largest Sharpe ratio and the risk free asset. In this way for whatever risk you are willing to take you get the largest possible return.

Now let us look at an imaginary newsletter and apply what we know to its recommendations. Suppose it offers a “cautious” portfolio that has an expected return of 9% and a standard deviation of .1 and an aggressive one with an expected return of 12% and a standard deviation of .3. The following graph displays your risk return choices with each if the risk free asset returns 5%.

Notice that you should never invest in the aggressive portfolio, no matter how risk loving you may be. For any standard deviation you always get a higher return by combining the risk free asset with the cautious portfolio. Now you know why there is no point to suggesting multiple investment portfolios. What a newsletter should do is simply suggest that its investors hold the one with the best risk return tradeoff or in mathematical terms the highest Sharpe ratio.

What is the risk return tradeoff for the Alpha Investment Opportunities model portfolio? The recommended funds are based on the Kalman filter analysis developed in the paper “Improved Forecasting of Mutual Fund Alphas and Betas” by Hong Zhang and Matthew Spiegel. Using data from 1970 to 2002 the paper found that if an investor held the Kalman filter’s top ten funds each month the resulting fund-of-funds portfolio produced a monthly return above the risk free rate of .0084 and a Sharpe ratio of .1769. If instead the fund-of-funds portfolio invested in the top 10% of all funds the average return above the risk free rate came to .0076 and had a Sharpe ratio was 0.1706 within the historical data. By comparison the market portfolio had an average monthly return in excess of the risk free rate of .0042 and a Sharpe ratio of .0883 over the same period of time. What does this mean for your portfolio? Suppose you wanted an expected return above the risk free rate equal to the market’s? You could invest .0042/.0076 or 55% of your funds in the Kalman portfolio’s top ten funds. If you did that the resulting portfolio would have had a standard deviation of only .0261 per month while those invested in the market portfolio were left with a standard deviation of .0471 per month.

Having told you what the historical data yielded the standard caution is now in order: Past performance is no guarantee of future performance. It is possible that in the future the Kalman filter portfolio will turn in a lower Sharpe ratio than the market. Also note that the fund-of-fund figures quoted above do not include transactions costs which would reduce the reported Sharpe ratios.