Back to the Statistics
(And apologies to Dwaine)

by Bob Price

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.


"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