Measures of Portfolio Risk and How You Can Apply Them

    by James B. Cloonan

    I have often discussed the concept of risk in this column. Most often, I have analyzed the various measures of risk and looked at their strengths and weaknesses.

    While there may be better measures of risk than the ones in common use, if you are an individual investor you are really limited to popular measures if you want to be able to make useful comparisons.

    Beta, which is still used frequently, is a measure of volatility compared to the volatility of a market index—a beta of 1.0 indicates that the stock is equally as volatile as the market, while a beta greater than 1.0 indicates that the stock is more volatile and a beta less than 1.0 indicates that the stock is less volatile than the overall market.

    The S&P 500 is the most common index used in comparing a stock’s volatility. However, the S&P 500 is not a very good index of the market as we usually think of the market—and that point I will discuss further below.

    In addition, various segments of the economy move at different times in the economic cycle. This can result in a stock with a beta that is not really reflective of true long-term volatility. In other words, the stock’s beta may be low because it doesn’t move in the same direction as the overall market at the same time, yet it may still be very volatile indeed.

    Beta also doesn’t travel into foreign markets without changing the basis of comparison. And it does not allow comparison of different kinds of investments—for instance, stocks vs. bonds.

    Standard Deviation

    If you knock out beta, that basically leaves investors with the other common risk measure, standard deviation.

    Standard deviation is the most useful risk measure for individual investors—as long as you understand the limitations. The figure indicates the amount by which most actual returns varied around the average return over a period of time, and thus provides a measure of volatility—the higher the standard deviation, the greater the volatility. For example, a standard deviation of 10% indicates that most (two-thirds) of the actual returns over a particular time period varied around the average return by plus or minus 10%—in other words, if the average return was 17%, most of the actual returns ranged from 7% to 27%.

    The primary limitation of standard deviation as a measure of stock risk is that it assumes returns are “normally” distributed—in other words, actual returns follow a bell-curve above and below the average return. In the real world, this is typically not the case. Usually the fit is close enough, but occasionally stocks or markets will move way beyond what would be predicted by their standard deviation. The one-day crash of 1987 could only happen once in over a million years based on its standard deviation.

    Practical Applications

    In this column, I am more interested in looking at the practical applications of risk measures, and they are all based on standard deviation.

    First, it is important to recognize that risk measures are really only meaningful when looking at a complete portfolio. The risk of any one security should not be relevant on its own, because the risk of a single stock can be reduced by diversifying among different kinds of stocks, so that the portfolio risk is much lower. If the average stock in a portfolio has an annual standard deviation of 25%, the portfolio, if well diversified, might have a standard deviation of 10%.

    Even a portfolio that is not particularly well diversified would likely reduce standard deviation from the 25% of a single stock to 15% in the portfolio.

    In the S&P 500, the average standard deviation of the individual stocks is about 30%, but the portfolio of stocks—the index itself—has a standard deviation of about half that, or 15%—not very good diversification for so large a portfolio. This is partly due to the way the index is constructed—representation in the index is based on market capitalization [share price times number of shares outstanding], which means that the largest companies are very heavily represented. This makes the index behave more like a 50-stock equally weighted portfolio. In addition, a disproportionate percentage of the very large stocks are in the same industries and highly correlated. In fact, AAII’s Beginner’s Portfolio, our experimental portfolio of micro-cap value stocks, has only 18 micro-cap stocks yet it has a lower standard deviation than the S&P 500. (For more information on the Beginner’s Portfolio, see the August 2001 A Matter of Opinion column.)

    The important point is that the risk you should focus on is the overall risk of your entire portfolio. And the more effectively your stock portfolio is diversified, the lower the overall risk.

    Risk-adjusted returns

    Another important aspect of risk is the analysis of risk-adjusted return and the ways of measuring it.

    Some approaches are suitable only for comparing one portfolio or fund with another; other approaches provide the means to see the real value of a risk/return trade-off.

    The simplest approach divides the return over a given period by the standard deviation. For example, if a mutual fund returned 15% in a given period, and the standard deviation was 16%, then the return per unit of risk is 0.938 [15% ÷ 16%].

       TABLE 1. How to Adjust Returns for Risk
    Return per unit of risk:   Return

    Standard Deviation
    Sharpe Ratio:   Return—T-Bill Rate

    Standard Deviation
    Example: 15% Return, 16% Standard Deviation, 3% Treasury-Bill Rate
    Return per unit of risk = 15%

    = 0.150

    = 0.938
    Sharpe Ratio = 15% — 3%

    = 0.12

    = 0.75

    A more sophisticated approach, the Sharpe Ratio, points out that a riskless security (a 90-day Treasury bill) has a return, and only a return that is above the risk-free return is truly related to risk. Thus, the Sharpe Ratio takes the 15% return, subtracts the Treasury-bill rate (say 3%) from it and then divides by standard deviation: [(15% – 3%) ÷ 16%] = 0.750.

    The Sharpe Ratio is theoretically more sound, and since it is so easy to calculate, it makes sense to use it. However, return per unit of risk is satisfactory for most comparisons. Table 1 shows the calculations for these two risk-adjusted measures.

    Optimizing Return for Risk

    The problem with both of these approaches is they don’t explain trade-offs for risk and return. Why, for instance, is Portfolio A—with a return of 15% (12% above the risk-free level) and a risk of 16% and therefore a Sharpe ratio of 0.750— better than Portfolio B—with a return of 19% (16% above the risk-free level), a risk of 26% and therefore a Sharpe ratio of 0.615?

    In other words, Portfolio A has a better risk-adjusted return, but Portfolio B has a higher absolute return.

    Certainly, a 19% return seems better than 15%—and who cares about the risk?!

    But what if you want to limit risk, and still get the highest absolute return?

    The answer to this quandary takes us to the next level of risk analysis.

    In addition to the two examples used above, let us introduce two other portfolio options an investor has:

    • Buy Treasury bills, with a 3% return, for a portion of your portfolio, or
    • Buy on margin at a 3% interest rate.

    TABLE 2. Optimizing Returns for a Given Risk Level

    (Assumes 3% Treasury Bills)

      Return(%) Excess
    Portfolio A 15 12 16 0.75
    Portfolio B 19 16 26 0.615
    Portfolio B/T-Bills** 12.84 9.84 16 0.615
    Portfolio A Margined** 22.5 19.5 26 0.75

    Now let’s look at possibilities, based on your risk tolerance, illustrated in Table 2.

    What is the risk you are willing to assume?

    Let’s look at your options if your risk level is 16%.

    One possibility is to simply invest in Portfolio A and have a return of 15%.

    What if you want Portfolio B, yet want to maintain your risk at your “sleep well” level of 16%?

    You could invest only a portion of your portfolio in Portfolio B, and put the balance in risk-free Treasury bills. Dividing Portfolio B’s standard deviation of 26% into your desired 16% standard deviation tells you the percentage to invest in Portfolio B: 61.5% (your risk would then be 61.5% of 26%, or 16%).

    This Portfolio B/T-bill combination portfolio would return: 61.5% of the 19%, plus 38.5% of the 3% in Treasury bills [(61.5% × 19%) + (38.5% × 3%)] = 11.69% + 1.15% = 12.84%, clearly not as good as the first portfolio.

    In this apples-to-apples comparison where risk is held constant, the higher risk-adjusted portfolio also provides the highest absolute return.

    Let’s look at the possibilities if you are willing to assume a 26% risk level.

    You could buy Portfolio B and get a return of 19%.

    Or, to maintain the 26% risk, you can margin Portfolio A. By how much would you margin it? Again, dividing Portfolio A’s standard deviation of 16% into your desired standard deviation of 26% tells you how much: 162.5% (your risk would be 162.5% of 16%, or 26%).

    This margined Portfolio A will provide an expected return of 162.5% of the 15% return, less the margin cost of 3% on 62.5% of the portfolio [(162.5% × 15%) – (62.5% × 3%)] = 24.37% – 1.87% = 22.50%, much better than Portfolio B’s 19% return.

    Again, in this apples-to-apples comparison where risk is held constant, the better risk-adjusted portfolio provides the higher absolute return.

    In other words, either way, the better risk-adjusted portfolio also provides a higher absolute return if risk is held constant. This illustrates how, by buying on margin or by investing part of your stake in riskless securities, you can optimize return for a given level of risk.

    The important thing to remember is to hold your risk at the proper level for you. If you do, then the highest risk-adjusted return portfolio will also provide the highest absolute return.

    In most cases it is not that complicated. While margin is not appropriate for most individual investors, most investors will have an asset allocation that includes some bonds or money market funds. A lower-risk stock portfolio will permit you to allocate a higher overall percentage of your portfolio to stocks while holding risk constant.

    “Average” vs. the Indexes

    The point of portfolio risk analysis is to make sure you are effectively diversified. And that brings me back to the S&P 500. Earlier I mentioned that the S&P 500 as an index has problems when used by the individual investor. I have already mentioned that it is not very well diversified for such a large portfolio.

    There is another problem for the individual investor who uses the S&P 500 and this problem exists with almost all of the indexes.

    The major indexes are indexes of stock market wealth, not of how well the average stock is doing. As economic indicators, they do just what they are supposed to do—measure the ups and downs of total stock market wealth. But as an indicator of whether the average stock is going up or down, they fail because of the greater weighting given to larger-sized companies.

    In the S&P 500, the largest capitalization stock may be given 300 times as much weight as the smallest. Even the small-cap indexes are based on company size, and the weighting differences are huge. The largest company by market capitalization in the S&P 600 Small Cap index is $2.6 billion and the smallest is $36 million.

    Individuals don’t tend to weight their holdings by market capitalization. So what can you use to follow how well the average stock is doing?

    The closest measure is the unweighted Value Line Arithmetic index (VAY). This index is usually not in the list of common indexes, but since it is the basis for a futures contract, it can be found in the futures section of the financial pages.

    Over the last two years, this index is up about 16%, while the S&P 500 is down about 28%.

    Unfortunately there is no index fund on the Value Line Arithmetic index because of the rebalancing problems, and the large number of stocks that comprise the index. You could build the equivalent using futures or options on the index, but it would be quite complex. Nonetheless, the recent market may be bearish for large stocks but not for the typical stock.

    It should be noted that most large mutual funds tend to weight their holdings by market capitalization because it is so much easier to manage. Stock price movements don’t require rebalancing. That is why they tend to move like the market indexes.

    When comparing mutual funds, it is important to use the standard deviation of the fund, as well as the return, in making your decisions. The risk of the combination of your equity funds will tell you what percentage of total wealth can be in equities and still meet your “sleep well” risk level.

    For the most efficient and effective portfolio, the analysis of risk is just as important as the estimate of rates of return.

       Analyzing your Portfolio Risk on-line

    For those interested in the overall risk of your portfolio and the effectiveness of your diversification, I recommend going to the Web site RiskGrades ( You can enter your portfolio and determine its risk and its diversification efficiency and then compare it to several indexes. You can also include bonds and mutual funds in the portfolio mix. RiskGrades also has another interesting feature, a method that makes standard deviation more meaningful.

    First, it standardizes the somewhat meaningless levels of standard deviation by taking the average of all the world’s equities and giving that a standard deviation value of 100. All other standard deviations are expressed as a percentage of that figure. The RiskGrade of the S&P 500 is 77, implying it has a risk 77% as high as the average risk of all equities in the world.

    Second, it solves the problem of what time period we should use to measure the historical standard deviation. A period that is too short picks up random volatility and very short-term shocks; a period that is too long misses the fact that companies change through time. RiskGrades uses a time-weighted calculation that includes volatility from years back but is not given as much weight as recent moves.

    The Web site provides the mathematical details of the approach, which I found extremely valuable and use for my own analysis. It is free for individuals.


→ James B. Cloonan