The Investing Problem
Now that we have enough knowledge of expected value to be dangerous, let's apply it to our investing problem. To review, the investment opportunity offered returns of either 30% or -10% a year, each with the probability of 1/2. We'll take a look at its expected return over 1 year and over multiple years.
What is the expected return for one year? To find out we will take the possible outcomes, multiply them by their probabilities, and add them up. The first thing we will do is stop referring to our returns in terms of percentages. From now on a 30% return is a 1.3 return and a -10% return is a 0.9 return. That is, a 30% return would leave you with 1.3 times as much money while a -10% return will leave you with 0.9 times as much money. The expected return is therefore 1.3 * 1/2 + 0.9 * 1/2 = 1.1, or a 10% return in one year.
One possible mistake is to add the percentages to get a return of 20%. To show why this is wrong lets take this line of reasoning to the level of absurdity. Lets say the odds of getting a 30% return are now 1 in 1,000,000 and the odds of getting a -10% return are 999,999 in 1,000,000. The -10% return is far more likely to occur. Therefore a 20% return would be very hard to justify in this situation. The expected value in this case is actually -9.99996%. This reinforces that we cannot add percentage returns. Incidentally, no self respecting carnival showman would offer you this game. To find a game with these odds you would have to turn to wall street.
What is the expected return for two years? We have already computed that the expected return for the first year is 1.1. Since the returns are independent (meaning the results of year one do not affect the results of year two) the expected return for the second year is also 1.1. The total expected return for the two years is therefore 1.1 * 1.1 which equals 1.21, or 21%. This equates to a return of 10% a year for two years, which only makes sense. Keep reading if you still do not believe me. We will look at a few possible mistakes, as well as see the derivation of the correct answer.
There are a number of places we can go wrong. The easiest error is to assume that we will get a return of 1.3 for one year and a return of 0.9 for the other year. This equates to a two year return of 1.17 (1.3 * 0.9 = 1.17) or 17%. This is an annualized return of 1.08166 or 8.166% (1.17^(1/2) = 1.08166, or 1.08166^2 = 1.17). This is the most likely return (also called the mode, and in this case the median), but it is not the expected return. Only in mathematics is the most likely return different from the expected return. The problem here is that we are not considering the cases of having a 1.3 return both years or a 0.9 return for both years. Please examing the following table:
| Year 1 Return | Year 2 Return | 2 Year Return | Annualized Return |
|---|---|---|---|
| 1.3 | 1.3 | 1.69 | 1.3 |
| 1.3 | 0.9 | 1.17 | 1.0816 |
| 0.9 | 1.3 | 1.17 | 1.0816 |
| 0.9 | 0.9 | 0.81 | 0.9 |
We see that a 1.17 two year return is the most likely return, happening 50% of the time. However 1/4 of the time we will get a 1.69 two year return and 1/4 of the time we will get a 0.81 two year return. The expected two year return is 1.69 * 1/4 + 1.17 * 2/4 + 0.81 * 1/4 = 1.21, or 1.1 annualized. This reinforces our earlier intuition.
A very natural mistake can be made as follows: Since there are four annualized returns in the above table (1.3, 1.0816, 1.0816, and 0.9) we can compute the expected annualized return as 1.3 * 1/4 + 1.0816 * 2/4 + 0.9 * 1/4 = 1.0908, or 9.08%. This line of reasoning makes a great deal of sense, but it is different from our other result of 1.1 or 10%. It is wrong for a very subtle reason which we will see shortly. We will call this incorrect method "Method B," and the correct method "Method A." Lets look at some more data, this time for 3 years:
| Year 1 Return | Year 2 Return | Year 3 Return | 3 Year Return | Annualized Return |
|---|---|---|---|---|
| 1.3 | 1.3 | 1.3 | 2.197 | 1.3 |
| 1.3 | 1.3 | 0.9 | 1.521 | 1.15 |
| 1.3 | 0.9 | 1.3 | 1.521 | 1.15 |
| 1.3 | 0.9 | 0.9 | 1.053 | 1.017 |
| 0.9 | 1.3 | 1.3 | 1.521 | 1.15 |
| 0.9 | 1.3 | 0.9 | 1.053 | 1.017 |
| 0.9 | 0.9 | 1.3 | 0.053 | 1.017 |
| 0.9 | 0.9 | 0.9 | 0.729 | 0.9 |
We see that there are 4 possible three year returns: 2.197 (once), 1.521 (three times), 1.053 (three times), and 0.729 (once). The expected three year return is calculated as 2.197 * 1/8 + 1.521 * 3/8 + 1.053 * 3/8 + 0.729 * 1/8 = 1.331, or 1.1 annualized (1.1^3 = 1.331). Calculating the annualized return using Method B gives us 1.3 * 1/8 + 1.15 * 3/8 + 1.017 * 3/8 + 0.9 * 1/8 = 1.087 or 8.7%. This is even less than before! In fact, Method B will give us a smaller return as the number of years we consider increases. To illustrate this fact I calculated the returns of Method A and Method B for different time periods. The following table summarizes the results:
| Number of Years | Method A | Method B |
|---|---|---|
| 2 | 1.1 | 1.0908 |
| 5 | 1.1 | 1.0853 |
| 10 | 1.1 | 1.0834 |
| 25 | 1.1 | 1.0823 |
| 50 | 1.1 | 1.0820 |
| 100 | 1.1 | 1.0818 |
| 150 | 1.1 | 1.0817 |
| 200 | 1.1 | 1.0817 |
We see that this number grows smaller until it seems to level out around 1.0817. This number should ring a bell because it is very close to another number we have seen: 1.08166, the most likely two year return. This is no accident. Method B compresses the returns into the range of 1.3 to 0.9 which makes the "tails" of 1.3 and 0.9 have less weight in the expected return calculation.
Consider the results after 200 years. There is a chance that you will get a return of 1.3 every single year. The chance is microscopic, but it is still a possibility. In this case your return will be 61,471,025,924,700,000,000,000 (1.3^200). Pretty impressive, even if your name is Warren Buffet. The true expected return after 200 years is 189,905,276 (1.1^200). This is still a pretty good return, but the optimal return is still 323,693,090,947,356 times larger. But according to Method B, the optimal return of 1.3 is only 1.18 times larger than the true expected return (1.3 / 1.1 = 1.18).
Method B has made the optimal return practically insignificant in the expected return calculation. This also holds for other returns that are unlikely to occur, such as a return of 0.9 for 150 years and a return of 1.3 for 50 years. As the number of years increases the unlikely outcomes (the tails) are dominated by the more likely outcomes around the median return of 1.0866. This causes the return of Method B to get closer and closer to the median return.
But lets stop for a second and be honest here, folks. The optimal return is practically insignificant. Only a fool would expect to gain a return of 1.3 every year for 200 years. The wise investor would expect to get a return very close to the most likely return. This return would be somewhere around the median, or what we would call the "expected" return in plain English. This "expected" return is 1.08166 or 8.16% a year. The average investor would get returns close to 8.16% a year while exceptional invesors would get a higher return.
Conclusion
This investing example has demonstrated a few important lessons:
- Probability is hard.
- Expected return is not what you expect.
- Focus on the most likely outcomes, but still consider the unlikely "tail" events.
Hopefully this article made sense. If you have any questions please send me an email. I would be happy to discuss the examples further.
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Want to learn more? Check out these links:
- The Monty Hall Problem, one of probability's greatest brain teasers.
- How to Lie With Statistics, the classic tome of statistics empowerment.
- Behavioral Finance, a great starting point for learning about the field of behavioral finance.
- Kahneman and Tversky's Prospect Theory, original research that found common pitfalls in decision making processes involving probabilities.