For investors who are concerned about maintaining a positive return over market cycles, an efficient frontier based on negative return frequency might prove to be more useful for creating an appropriate asset allocation than standard deviation.
Negative return frequency is defined as the percentage (or frequency) of time that a monthly, quarterly or annual return is below 0%. The calculation divides the number of negative returns by the total number of returns over a specified period. For example, the negative return frequencies for 12-month returns from January 1945 through December 1996 are 0% for Treasury bills, 10% for intermediate bonds, 28% for long bonds, and 20% for the S&P 500 Index.
The relatively high negative return frequency for long bonds is notable in light of the voluminous literature directed to 401(k) participants and mutual fund shareholders that suggests bonds over long periods of time exhibit lower volatility than common stocks. For investors who perceive risk from the perspective of volatility, as measured by standard deviation, it might appear logical to assume long bonds would have fewer down years than stocks during a 50-year period. The use of negative return frequency as a risk measure helps to dispel this misconception.
Limitations of standard deviation
Efficient frontier analysis using negative return frequency as a risk proxy can avoid two limitations associated with standard deviation.
The first is that use of standard deviation as a risk proxy presupposes a symmetrical, "normal" distribution of asset returns, in that the probability and magnitude of an excess or deficit return relative to the mean (or average) return is assumed to be equal. However, recent evidence suggests investment returns might not be normally distributed, and portfolio strategies based on downside volatility measures can yield higher returns. In reviewing actual historical returns, "normal" distribution can underestimate occurrence of significant shocks, like stock market corrections, and fail to account for the skewness of returns within and between different asset classes.
A second drawback is that it is difficult to explain to many clients how standard deviation and covariance relate to a client's risk tolerance. For clients who are concerned with the possibility of either relative or absolute negative returns, or who simply wish to maintain positive investment returns over a benchmark, using negative return frequency as a risk proxy might be a more intuitive method of addressing these issues.
This is not meant to imply negative return frequency is superior to standard deviation as a risk proxy. Negative return frequency also has limitations, because it does not recognize the magnitude of the dispersion of positive returns or the size of negative returns. However, both risk measures can be extremely useful when employed together, or with other risk measures, including semideviation and average negative return.
A look at U.S. stocks and bonds
An efficient frontier based on negative return frequency can provide instructive asset allocation insights, especially for investors focused on maintaining a positive return over rate cycles. For example, we created return/risk frontiers using the average 12-month returns and negative return frequencies for different combinations of stocks and bonds. Exhibit I illustrates the historical returns and negative return frequency for weighted combinations of stocks and bonds from 1980 to 1995 (using the Standard & Poor's 500 Stock Index and the Lehman Brothers Government/Corporate Intermediate Index). Each adjacent point on the graph represents incremental weight changes of 5% between stocks and bonds, ranging from 100% bonds and 0% stocks to 0% bonds and 100% stocks.
For this same historical period, a frontier analysis using standard deviation as a risk measure would have suggested a combination of 85% to 90% bonds and 10% to 15% stocks provided the least volatility. However, the negative return frequency-based frontier graph indicates portfolios with equity weights of 20% to 30% had the lowest percentage of negative 12-month returns, while a portfolio of 100% bonds had nearly twice the frequency of negative returns. The difference between the return/risk profiles of the two frontiers can be attributed, perhaps, to the greater positive return volatility of stocks relative to bonds.
Using multiple asset classes
To depict additional benefits of risk diversification, we prepared an efficient frontier using the negative return frequency for multiple asset classes, from 1975 to 1996, including: 30-day T-bills, one-year governments, government/corporate bonds, intermediate bonds, aggregate bonds, real estate investment trusts, domestic stocks and international equities (See exhibit II).
An optimization model was used to create asset combinations that would yield the highest average return for incremental levels of risk. What if an investor were far more interested in protecting the portfolio against down years, as opposed to dampening intrayear volatility? A portfolio that had the highest possible 12-month return with a negative return frequency of 0% would have contained approximately 23% T-bills, 14% one-year governments, 19% intermediate bonds, 10% REITs, 22% domestic stocks and 11% foreign stocks. The average historical 12-month return of this portfolio was about 11.2%, while portfolios containing solely T-bills or one-year governments would have had average returns of approximately 7.2% and 8.3%, respectively.
Creating asset allocations
Exhibit II also highlights three model portfolios, ranging from aggressive (Model A), less aggressive (Model B), to conservative (Model C). Models A and B were constructed under the assumptions that the client has a relatively long time horizon, needs a minimum 5% liquidity reserve, does not want more than 20% exposure to foreign stocks, and seeks both capital growth and periodic cash flow. Models A and B were determined based on optimization routines that computed the asset mix with the highest possible average return for an incremental level of risk (negative return frequency). The asset compositions for Models A and B, respectively, are 5% and 18% T-bill; 10% and 13% 1-year governments; 15% and 16% intermediate bonds; 10% and 9% long bonds; 40% and 29% domestic stocks; and 20% and 15% foreign stocks.* Model C is a concentrated high income portfolio divided evenly between intermediate bonds and long bonds.
The dual perspective of standard deviation and negative return frequency provides an interesting insight when we compare Model A to Model B. The average historical standard deviations appear to be relatively close: 9.8% for Model A vs. 7.6% for Model B. Note, however, that the 12-month negative return frequency for Model A is 4.7%, compared with only 1.6% for Model B. In effect, the aggressive portfolio (Model A) has three times the potential for a down year than the less aggressive portfolio (Model B).
As expected, the concentrated, high-income portfolio has a considerably lower historical return (10.5%) than Models A and B (13.4% and 12%, respectively). The standard deviation of 9.3% for Model C was slightly lower than that of the aggressive Model A portfolio. What might be of most importance for risk-averse clients, however, is that Model C's 10.3% negative return frequency is more than twice that of Model A (4.7%). This is one more example of how a lower standard deviation might not necessarily correlate with a lower negative return frequency.
Summary
Negative return frequency has several attractive features as a risk measure. It can more accurately "match" the risk concerns of a client and is relatively simple to explain and calculate. In addition, it is based on a frequency distribution approach and does not assume a normal distribution with symmetrical volatility. However, as mentioned above, negative return frequency also has its limitations, especially in terms of recognizing the magnitude and dispersion of positive returns or the size of negative returns. To be most effective, negative return frequency should be used with other risk measures, including standard deviation.
Vladimir de Vassal is a vice president at CoreStates Investment Advisers, Philadelphia.
