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August 19, 1998

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Subject: Defining Risk
Author: elann

Fellow Fools,

The recent post #9283 by nhecht (a.k.a wastingtime) with the subject "No better than chance" , caused me to spend a lot of time thinking about risk, how it's measured and what it means. I've come to some conclusions that I would like to share. Timberfool has been very helpful in pointing me to the statistical tools that allow these conclusions to be translated into numbers.

A well known measure of risk that is calculated, among others, by the Motley Fool's Dow Spreadsheet is the Sharpe Ratio. William Sharpe defined it as the ratio between an investment's average return and it's standard deviation. He states that the ratio should be measured for differential returns, i.e. the difference in return between a portfolio and a risk-free benchmark investment.

I felt that there were two problems with the Sharpe Ratio. One is that it is an abstract number that gives me no tangible sense of the risk in an investment. If the UV4, for example, has a Sharpe Ratio of 0.75 what does it mean? How risky in fact is the UV4? All I could know is that it has a better return-to-risk profile than another portfolio with a Sharpe Ratio of 0.70.

The other problem is that the ratio as it is commonly calculated doesn't reflect the relative risk of different holding periods. Tom Gardner and other Fools often write that by investing for the long haul we reduce our risk. While this is intuitively true, I found myself asking how do I measure that reduction in risk? Where is the formula that will tell me how much I reduce my risk by extending my investment horizon from, let's say, one year to ten years?

 "Risk can mean different things to different people. Some people lose sleep if their stock drops by \$2 in a day. They just can't stand the bumpy ride."

Risk can mean different things to different people. Some people lose sleep if their stock drops by \$2 in a day. They just can't stand the bumpy ride. Others' biggest fear is that they will lose everything and end up broke. I decided to define the risk of investing in the Dow Dividend Approaches (DDA) in three different ways, so that each of you can pick the definition that you think is most suitable. The three definitions are:

1. The risk (or statistical probability) of losing money. That is, what is the chance that after 1 year or N years you will have less than you started out with.

2. The risk of losing to a "risk-free" investment. That is, what is the chance that after 1 year or N years you will have less than you would, had you invested in treasury bills or bank certificates of deposit.

3. The risk of losing to the Dow index. That is, what is the chance that after 1 year or N years you will have less that you would, had you invested in all 30 Dow stocks in equal amounts (Which is what Beating the Dow is all about).

Any of us can come up with other definitions, but these seem to reflect many questions that people raise with regard to investing in the DDAs and, besides, they were easy for me to calculate.

The math I did isn't really that innovative. It is consistent with the Sharpe Ratio both conceptually and in the way it ranks various investments. (I wouldn't want to contradict a Nobel Prize winner, after all). It just translates his method into simple odds.

I used the MF Dow Spreadsheet with a history from 1961 through 1996. All the potfolio returns are calculated for one-year holding periods starting on Jan. 2 of each year. The "risk free" benchmark returns are those of the 1 year tbill, provided some time ago by Rayvt. I compared investments in the Dow 30, UV4 and RP4, with investment horizons of one year and six years. A six year period was chosen because it fits neatly into the 36 years of historic data, although that shouldn't really matter.

The Sharpe Ratios are also provided so that you can observe the consistency between the ratios and the probabilities for yourselves. The shift from the one-year probabilities to the six-year probabilities is accomplished by dividing the standard deviation by SQRT(6), as indicated to me by Timberfool and backed up by Sharpe's web page. I used the Student's T distribution, which is appropriate for small samples that are considered to be subsets of a normal distribution, to calculate the various probabilities. Note that the average returns shown below are arithmetic averages.

So here goes:

1. What is the probability of ending up with LESS than you started, if you invest in the following portfolios for one year or for six consecutive years (reinvesting every year):

Dow30   UV4     RP4
Avg. Rtn.       12.93%  19.83%  21.74%
Std. Dev.       16.16%  21.00%  20.36%
Sharpe ratio    .7998   .9445   1.0678
P(1 year)       21.46%  17.57%  14.65%
P(6 years)       5.37%   3.43%   2.37%

The UV4 numbers, for example, mean that the average return (not the compounded return) is 19.83%, the standard deviation of returns is 21%, the Sharpe Ratio for one year is .9445, the chance of ending up with less than you started after one year is 17.57%, and the chance of ending up with less than you started after 6 years is 3.43%.

Note: Sharpe says that his ratio shouldn't be used to assess an investment on it's own, but only to assess it against a benchmark. But I figure that putting your cash under a mattress is a legitimate benchmark.
2. What is the probability of making less than an investment in 1 year tbills after one year or six years. (The averages and standard deviations here are for the differential returns):

Dow30   UV4     RP4
Avg. Rtn.        6.27%  13.18%  15.08%
Std. Dev.       16.28%  21.09%  20.07%
Sharpe ratio    .3854   .6247   .7516
P(1 year)       35.11%  26.81%  22.86%
P(6 years)      19.42%   9.32%   6.25%
3. What is the probability of making less than an investment in a portfolio of all 30 Dow stocks after one year or six years. (The averages and standard deviations here are for the differential returns):
UV4     RP4
Avg. Rtn.                6.90%   8.81%
Std. Dev.               13.50%  12.05%
Sharpe ratio            .5113   .7314
P(1 year)               30.62%  23.47%
P(6 years)              13.29%   6.66%

Fool on,
Elan

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