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Back to the Statistics
(And apologies to Dwaine)
by Bob Price
([email protected])
Houston, TX (December 1, 1998) -- Sorry, Dwaine, but I did promise to take a "Sharpe-er look" at statistics today.
Last week we laid out the returns and standard deviations for a number of Dow strategies. Even though the standard deviation gave some indication of how volatile a strategy was, it still wasn't perfectly clear how one could use that information to pick the best strategies. In fact, it never will be.
This is an individual choice. We're trying to give you some insight into how you might decide for yourselves (assuming you believe the Dow strategies work). This column covers the most popular one-number way of evaluating the performance of a portfolio, the Sharpe Ratio. Inventor William Sharpe shared a Nobel prize in economics, in large part due to this.
The Sharpe Ratio -- another way of looking at risk and reward
A number of readers came close to discovering this on their own. (Sorry, that Nobel Prize is taken, but you were thinking smart!) They wrote in after last week's column and suggested variations of dividing return by standard deviation. This makes good sense -- after all, we want high return and low standard deviation.
There's one more thing to consider, however. Whenever people invest money in anything other than a risk-free investment (such as a US Treasury bill), they make a choice. They choose to risk their capital in hopes of improved returns, in the full Foolish awareness of the fact that with these improved returns comes a risk of loss. (I hear you snickering out there -- but you should be thinking that!)
Suppose in a given year you get a return of 20%. The risk-free option, US Treasury bills, returned 5%. The "differential return," or return beyond the risk-free return, is 15%. This is the goal of investing -- a positive differential return. In effect, you get paid to take on the additional risk of investing in stocks instead of risk-free Treasury bills. So the Sharpe ratio measures everything in terms of this. The formula is:
(average of differential returns)
--------------------------------------------
(standard deviation of differential returns)
In case it isn't immediately obvious from just looking at the formula, a higher ratio means a higher payoff for assuming the increased risk. Think of the Sharpe ratio as a reward-to-risk ratio wherein higher=better. That is, you get more reward for the risk you take.
Below I've updated the chart from last week. As before, the numbers are based on the annual returns for the past 25 years. The table is now sorted by Sharpe ratio. (If the table isn't displaying properly, expand your window to full screen.)
Strategy CAGR Mean Std. Dev. Sharpe Ratio RP4 24.62% 26.05% 19.13% 1.0045 RP4+ 25.23% 26.84% 20.55% 0.9757 RP2 26.24% 28.42% 24.37% 0.8890 RP5 22.24% 23.67% 19.08% 0.8775 Foolish5 20.43% 21.82% 18.71% 0.7824 BTD4 22.13% 23.88% 21.25% 0.7820 BTD2-6 20.83% 22.41% 19.67% 0.7773 Foolish9 18.28% 19.32% 16.05% 0.7658 Foolish4 21.86% 23.63% 21.88% 0.7492 RP9 18.14% 19.24% 16.51% 0.7429 HY10 17.69% 18.68% 15.64% 0.7412 Foolish4+ 22.61% 24.67% 23.95% 0.7314 OldFool4 23.17% 25.48% 25.28% 0.7245 BTD5 19.62% 21.07% 19.17% 0.7205 Foolish2 23.71% 26.75% 28.63% 0.6679 HY5 18.60% 20.09% 19.89% 0.6522 PPP 25.05% 31.43% 46.39% 0.5260 Dow 30 13.83% 15.08% 16.93% 0.4697 S&P500 13.06% 14.39% 17.12% 0.4309
[Note: For year-to-date returns and a description of each strategy, see 1998 Dow Returns.]
This chart has a number of surprises. First, the Foolish 4 is way down the list. Before you bombard us with angry e-mails, the "Foolish" approaches over the longest period for which we have data (1961-1997) sort higher than shown above. The Sharpe ratio order in the full 37 year chart is: RP4, RP4+, RP5, RP2, Foolish2, Foolish 4+, Foolish 4 and others. This chart shows the past 25 years in order to be consistent with all the other charts shown in this area recently.
The chart clearly shows that all the strategies had a higher reward-to-risk ratio than the Dow 30 and the Standard & Poor's 500 Index, even the highly volatile, one-stock PPP strategy. That's a pleasant surprise.
Also, you'll notice that the Dow 30 ranks ahead of the S&P 500. This may be a surprise to some, considering the greater number of stocks in the S&P and the way the S&P 500 has beat up on the Dow in the last few years, but over longer periods, the Dow has done better. What the future holds, we can't say.
Conclusions?
"We don't need no steenking conclusions -- we're the Foolerales!" Seriously, we've laid this out to try to help you draw your own conclusions. Different people will do different things, based on their own personalities. In addition to the recent articles in this area, you may want to look at the following:
William Sharpe's comments on investments in general: Revisiting the Capital Asset Pricing Model
William Sharpe's own paper on the Sharpe ratio: The Sharpe Ratio. (Heavy reading!)
William Sharpe's home page: William F. Sharpe's Home Page
The new Foolish 4 book has returns on the main strategies from 1963 on, shown year by year.
The Dow Dividend Spreadsheet lets you manipulate the numbers with Excel, in case you've come up with a fancy new way to evaluate the strategies.
Finally, if you don't follow any other hyperlink in this column, read Elan Caspi's excellent comments on this subject.
Fool on!
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Current Dow Order | 1998 Dow Returns
12/01/98 Close
Stock Change Last -------------------- UK + 1 1/4 44.75 IP + 3/4 43.48 MO + 1 1/16 55.98 EK + 1/4 72.63 |
Day Month Year
FOOL-4 +0.34% 0.34% 14.53%
DJIA +0.19% 0.19% 15.49%
S&P 500 +1.00% 1.00% 21.11%
NASDAQ +2.78% 2.78% 27.60%
Rec'd # Security In At Now Change
12/31/97 276 Philip Mor 45.25 57.00 25.97%
12/31/97 206 Eastman Ko 60.56 72.88 20.33%
12/31/97 289 Int'l Pape 43.13 44.19 2.46%
12/31/97 291 Union Carb 42.94 43.50 1.31%
Rec'd # Security In At Value Change
12/31/97 276 Philip Mor 12489.00 15732.00 $3243.00
12/31/97 206 Eastman Ko 12475.88 15012.25 $2536.38
12/31/97 289 Int'l Pape 12463.13 12770.19 $307.06
12/31/97 291 Union Carb 12494.81 12658.50 $163.69
Dividends Paid YTD $1092.81
TOTAL $57265.75
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