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By Lawrence G. McMillan

This article was originally published in The Option Strategist Newsletter Volume 9, No. 19 on October 11, 2000. 

Many sophisticated traders use ‘expected returns’ to analyze the profit and loss expectations of their investment strategies. In this article, we’ll define what that entails and then point out some of the benefits and difficulties in using such statistics to predict how a strategy will perform.


The expected return of a position is the return that one can expect to make, based on theoretical statistical patterns of the investment in question. As a simple example, let’s use an example from basketball.

Example: suppose that a player is a 70% foul shooter. He is shooting two foul shots. What is the expected amount of points that he will score at the foul line? There are only four possible outcomes:

  Outcome                                  Probability
  #1) Makes the first, misses the second   0.7 x 0.3 = 0.21
  #2) Misses the first, makes the second   0.3 x 0.7 = 0.21
  #3) Makes them both                      0.7 x 0.7 = 0.49
  #4) Misses them both                     0.3 x 0.3 = 0.09
          Total Probability:                          1.000

Now, we can calculate the “expected score” for this player.

Outcome  Points  Probability  Expected
 #1        1        0.21        0.21
 #2        1        0.21        0.21
 #3        2        0.49        0.98
 #4        0        0.09        0.00
        Total Expected Score:   1.40

Thus, he can expect to score 1.40 points (on average – perhaps over the whole season) each time he gets to shoot two free throws. Obviously, he cannot score 1.40 on any one trip to the foul line (the only possible results are 0, 1, or 2 points), but 1.40 is his “statistical expectation”.

A similar method of analysis can be applied to stock market investments. It is a simple matter to determine how much money an investment will make at a given stock price – whether that investment be long stock, long calls, bull spreads, covered call writes, or whatever. However, determining the probability of the stock being at a certain price is a more difficult task. Let’s create a simple example to see how this would go:

Example: suppose that we are considering buying an XYZ July 100 call for 10 points. Furthermore suppose that we believe that the following distribution describes where the stock will be at expiration:

  Outcome    Probability
  Below 100:        45%
  At 110:           25%
  At 120:           15%
  At 130:           10%
  At 140:            5%
   Total:          100%

Note that, in reality, and in any serious calculation of expected return, one would have to account for the probability of the stock being at every possible price – not just the 5 outcomes shown above.

Returning to the example, the “expected profit” can be calculated, by multiplying the profit at each “outcome” by the probability of attaining that outcome:

Outcome     Prob     Profit     Expectation
Below 100   45%      -10             -4.5
At 110      25%        0              0.0
At 120      15%      +10             +1.5
At 130      10%      +20             +2.0
At 140:      5%      +30             +1.5
           Total Expectation:        +0.5

So, now we know that the expected profit is 0.5
points ($50), less commissions. Since this call buy
requires an investment of $1000, the expected
return is 5% ($50 / $1000). We could then
annualize this return for comparative purposes.
For example, if this was a 3-month call, then we
would say the annual expected return for buying
this call is 20%.

There is a large problem with using expected returns of this sort: the choice of distribution that one assigns to the stock movements is of utmost importance. Ostensibly, one could merely use the lognormal distribution that scientists have said describes the way that stock prices behave. However, long term readers of this newsletter know that we have proven many, many times that stock prices do not adhere to a lognormal distribution. Rather, they move farther and faster than the lognormal distribution predicts. Consider these expected returns, generated using either 1) the standard lognormal distribution, and 2) a simple “fat tail” distribution (one which at least partially allows for the type of volatile stock movements that we often see in real life). Returns such as these are dependent on the volatility of the underlying stock, but this example illustrates the point to be made:

     Stock Ownership:
Distribution   Exp. Return
Lognormal         4.3%
Fat Tail          6.4%

So, you can see that it depends heavily on which distribution you use as to what the expected return is. When comparing stocks with each other, you might say that it wouldn’t really make much difference – they’d still rank in the same relative order – and you might be right. However, if you’re comparing different strategies with one another, the choice of distribution can be even more crucial.

One reason that I bring this up is that I have seen some internet sites that are comparing investments (option strategies, actually), based on the expected returns from a lognormal distribution. This can be quite misleading – especially since it can make strategies that have unlimited or large risk appear to be more attractive than they actually are.

Fat Tails

The term that statisticians give to the distribution that more correctly describes the stock market’s movements is one that has “fat tails”. That is, the probability of an extreme move is larger than something like the lognormal distribution might predict. You don’t need to cite the Crash of ‘87 to illustrate this point. All you have to do is look at what happens to stocks on an everyday basis. Some recent examples include the following: Apple’s (AAPL) 28 point drop in a day. Micron Electronics’ (MU) 11 point drop in a day and 20 point drop in three days. Priceline.com (PCLN) and Harmonic Inc. (HLIT), both down 40% in a day, and Xilinx’s (XLNX) 17-point decline yesterday. Not all surprise moves are to the downside, of course. Takeover bids, for example, often propel a stock upward far more than the lognormal distribution would theoretically allow for, as in JP Morgan’s (JPM) recent 38-point jump in one week, or Southdown’s (SDW) 27% rise in one day, or Nat’l Discount Brokers (NDB) 86% jump yesterday. Even run-of-the-mill trading days can produce moves that are considered all but “impossible” by the lognormal distribution. As we pointed out in earlier articles this year, moves such as these are not that unusual (improbable, yes, but nowhere near as improbable as the lognormal distribution would have you believe).

So, given these facts, we thought it would be interesting to compare how various option strategies might perform when using the lognormal distribution as opposed to using the simple “fat tails” distribution. Again, it should be understood that these numbers have some problems. First of all, the volatility of the underlying is a relevant factor (these tables assume a 30% statistical volatility for the underlying). Second, these are the results if the position were held until expiration. Interim results using stops losses, partial profit strategies, and other things that humans can do to improve returns are not shown here.

Also, the options in these strategies are assumed to be priced at a 30% volatility. In reality, returns might be improved by selling “overpriced” options (for writing strategies) or by buying “underpriced” options (for option purchase strategies). In each case below, an at-the-money option is used (for the bull spread an at-the-money call is bought and one with a strike 20 points higher is sold).

Strategy     Expected Return*
              Lognormal   “Fat Tail”
Long stock       4.2%        6.3%
Covered Write    8.0%        6.0%
Call Buy       –34.1%       23.1%
Put Buy        –22.8%       12.4%
Bull Spread     –8.6%      – 5.5%
*: a $5 commission is included

A number of things jump out at me when I view this table.
! a bull spread is an inferior strategy when the options are fairly priced, no matter which distribution is assumed. This more or less agrees with observations that we have made in the past regarding the disappointments that traders often encounter when using vertical spreads.
! while covered writing might seem clearly superior to stock ownership under the lognormal distribution, the two are about equal under a “fat tail” distribution.
! most startling, though, is the disparity between option buying strategies under the two distributions. This most clearly demonstrates the “power” of the “fat tail” distribution: a limited-risk investment with unlimited profit potential can be expected to perform very well if the “fat tails” are allowed for.

Note that the lognormal column in the table more or less represents the “conventional” wisdom regarding option strategies – the one that many brokers promote: “don’t buy options, don’t mess with spreads, either buy stocks or do covered call writes”. The “fat tail” distribution column stands much of that advice on its head.

Again, remember that the figures in the above table cannot be taken literally. They are merely a statistical representation. In real life, with real investment strategies, results would be different. However, the table is useful in that it draws a reasonable comparison between the various strategies. In real life (as demonstrated by the “fat tail” distribution), strategies with limited profit potential and unlimited or large risk potential are inferior strategies.

In summary, one should be aware that the phrase “expected return” is used in many quasi-sophisticated option analyses (and even in analyses not using options). Many investors accept these “returns” on blind faith – figuring that if they’re generated by a computer, they must be correct. In reality, they may be not be representative, even for comparisons (witness the above table and how the strategies switch positions when the distribution is changed).

Is there a solution? Perhaps. At a minimum, view the returns in two ways – with a “fat tails” distribution and with a lognormal distribution (if you even want to bother). Ideally, more distributions would be allowed. Practitioners of advanced probability theory suggest using each individual stock’s distribution in analyzing returns for strategies on that stock alone. That may seem overly tedious, but that’s what’s being done in other areas of quantitative study involving things such as predicting failure rates of components on cars, satellites, and military weaponry. To this end, we are working to add more choices of distributions in our Probability Calculator 2000.

Of course, this type of analysis (using several distributions) puts an onus on the investor to choose the distribution that he wants to use in order to analyze his investment. However, such an approach should be extremely illustrative in that he can compare returns from different strategies and have a reasonable expectation as to which ones might perform the best under different market conditions.


This article was originally published in The Option Strategist Newsletter Volume 9, No. 19 on October 11, 2000.  

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