This is my final post examining the returns for the Ben Graham conservative and aggressive portfolios. Become a Complete Fool
Post 1 � Portfolio returns
Post 2 � Market rewarded risk
Post 3 � Sharpe's Ratio
I want to return a little bit to Sharpe's ratio before a short discussion on regression analysis and my conclusion.
Some of you might find Sharpe's ratio a bit abstract or may be uncomfortable with quantifying risk.
On the last part, let me paraphrase Aristotle. We should measure things to an accuracy that the subject warrants.
Measuring risk will not be the same as measuring out medications at a pharmacy, but that doesn't mean it shouldn't be attempted. To ignore risk is essentially to assume that our portfolios have the same risk as the overall market. So even a crude model (i.e. a simple model) of risk is better than having none at all. If it grates upon you that's probably a good sign since it will not reinforce your prejudices. If it overestimates risk, that's fine too since it is better to overestimate the risk in your portfolio than to underestimate it. This is in line with the first principle of investment, which is to preserve capital. So anything that cautions us when we want to shout, "hallelujah" (unless your singing in the choir), is a good thing.
To look at the approach used in Sharpe's ratio in a slightly different way. Let's look at the total returns for the portfolios and adjust the total returns for risk.
1) Divide up the overall investment period in to a set of equal sized intervals such as months, 2 months, quarterly. We know that risk will decrease with holding period so we want the intervals as long as possible, but not so large that we don't have enough to calculate suitable statistics. As a rule, somewhere between 8 and 15 will do (keeping in mind Aristotle's advice). In the case of the 19 months, I decided to use bi-monthly intervals so that I would have 9 such intervals.
2) Determine the risk free return for these intervals. I used the same risk free return of 0.81% for all intervals. This equates to an annual risk free return of 4.96%.
3) Subtract the risk free returns from both the market (i.e. index fund in this case) and portfolio returns for each interval to determine the excess returns for each interval.
4) Calculate the standard deviation of the excess returns for both the market and portfolio.
5) Compute the ratio of the market standard deviation to the portfolio standard deviation
6) Determine the risk free return for the entire investment period. In my case I used 7.97% over 19 months (again equivalent to a 4.96% annual rate).
7) Subtract the risk free return from the portfolio return for the entire investment period. This is the excess return for the entire investment period.
8) Multiple the excess return (calculated in 7) by the ratio (computed in 5)
9) Add the risk free return for the entire period to the modified excess return computed in 8.
The results were as follows:
Portfolio Raw Return Ratio Adjusted ReturnFrom this it would appear that the Ben Graham portfolios have outperformed the market. However, there are some unanswered questions such as, "is the ratio stable over the length of the intervals chosen (i.e. whether monthly, quarterly, semi-annually, etc.?" These can't be answered yet, but I will continue to track the portfolio returns and things as the ratios of portfolio variability to market variability.
Index 29.8% n/a n/a
Conservative 67.5% 0.43 33.4%
Aggressive 106.4% 0.47 54.0%
Combined 87.2% 0.52 48.8%
There is one question that can be answered, which is "what is the confidence that the portfolio outperformed the market?" This is done through regression analysis, which is fitting a straight line to a set of data. In this case the data are the returns of the portfolio and market returns for a set of intervals of the overall investment period. The regression analysis can be looked two ways � one is that we are trying to determine our confidence that the portfolios have outperformed the market. The second is that we are testing the hypothesis that the market is efficient; if it is not then we have found bias.
The equation that is being tested here is:
Portfolio Return = Risk Free Return + B * (Market Return � Risk Free Return) + Noise
B = the slope of the straight line
Risk Free Return * (1 � B) = the intercept
Noise = industry and company related factors plus market responses to a whole lot of junk news, etc.
There are two things you should note which are that:
(1) the test is not only a test of market efficiency it is also a test of how the market rewards risk (i.e. the infamous equation above otherwise known as CAPM); and
(2) by lumping things as "noise" we are assuming that the portfolios are well diversified and that collecting more data will minimize the influence of these factors.
I'm not going to go any further into regression analysis as either you "know the math" or you "don't know the math" and an in depth discussion of regression will scare the willies out of you.
However, the results using monthly intervals showed that with the data available there is a 45% confidence that the conservative portfolio outperformed (i.e. that there is bias) and a 70% confidence that the aggressive and combined portfolios outperformed. This confidence did not change by using 2-month intervals.
The average biases (i.e. out-performance) estimated for each portfolio were:
Conservative = 0.95% per month (i.e. 12% per annum)
Aggressive = 2.14% per month (i.e. 28.9% per annum)
Combined = 1.54% per month (i.e. 20.1% per annum)
The results are certainly promising, but I have to admit that the length of time is not yet sufficient to claim a victory. In particular I would like to see if the results including a bear market. I would not expect such excessive returns to persist over time.
However, I hope I have convinced of two things: (1) value investing can be very rewarding and (2) take risk into account in some fashion when determining how well you did. Probably there's also something about the value of diversification as well.
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This is my final post examining the returns for the Ben Graham conservative and aggressive portfolios.
Become a Complete Fool