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  1. The Measurement of Risk
  2. Asset Allocation and Retirement
  3. Ranging the Possibilities: Monte Carlo Simulations
  4. Summary
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Ranging the Possibilities: Monte Carlo Simulations

With the advent of computers and the decline in the price of computing power, it is now easy for financial advisors to set up simulations that calculate all the possible permutations and combinations of past returns for every available asset class. In the financial world, these simulations often take the form of Monte Carlo simulations, computer calculations that take into account chance and randomness (hence the casino quality of the name). This can sound very sophisticated—and the computer spreadsheets that such models spit out indeed excite the eye—but these simulated results are nothing more than the possible combinations and permutations of past outcomes.

Although Monte Carlo simulations can be informative, they can also be hazardous. For instance, although asset classes tend to return to their means (or historical averages) over time, they do not necessarily perform in sync with market conditions on all occasions. The following example helps illuminate this discussion: During a period of sustained economic expansion and low inflation, one expects the performance of the various asset classes, as well as the level of returns of those asset classes, to be very different from those observed in a slow-growth, inflationary environment. Historically, gold has been considered a great inflation hedge as well as, for the cautious, a refuge. Thus, during inflationary times, gold and other commodities are expected to outperform not only the market, but also their historical rates of return. Separate industries, in other words, respond differently to changing economic conditions. That’s why we have a multitude of classifications for assets, such as cyclical stocks, stocks closely tied to the ups and downs of the economy; value stocks, stocks that tend to trade at a lower price relative to its fundamentals (that is, dividends, earnings, sales, and so on) and are thus considered undervalued by a value investor; growth stocks, stocks that look attractive because of the potential earnings growth of the company; and so on. Each classification is intended to capture some characteristic that relates a group of stocks to the changing economic environment. It follows that because separate economic variables (such as policy changes like tax rate cuts or shocks such as natural disasters) affect stocks differently; one must pay attention to the combination of policy changes or shocks to determine the impact of a changing economic environment on a particular asset class.

Depending on an economic shock’s nature, asset classes sometimes move together, although at other times, they travel in different directions. So, the degree of synchronization among asset-class returns depends in great part on the nature of economic policies (or the shocks they produce). This insight is different from the one a Monte Carlo simulation provides. As for the former, the reasoning goes that when times are good, asset classes should perform in an expected manner (with most rising and perhaps those that typically hedge against the bad times underperforming). This is not a bad generalization, but it leaves much to be desired. If this reasoning is actually the case, it means the separate outcomes of the individual asset classes are not truly independent of each other. Taken to the next degree, this would mean the joint occurrences of asset-class returns are not truly independent of each other either. So, if a Monte Carlo simulation can only mimic occurrences where each of the assetclass returns are independent of each other, what good is it other than to illustrate generalized fluctuations? This leaves investors with another question they must ask their financial advisors: "How good is your simulation program and how well do you use it?"

A Poor Man’s Monte Carlo Simulation

Thankfully, a simple alternative to the Monte Carlo and other simulations exists. Importantly, the alternative does not require any assumptions in addition to the original two. If one is willing to assume the past is a good guide to the future and portfolios should be annually rebalanced, one has a straightforward way to generate a range of outcomes that takes into account the joint outcomes of the actual returns of the different asset classes. Such a solution is located in the periodic table of asset returns, a feature of most asset-allocation presentations.

My version of the periodic table can be found in Table 1.1. For ease of illustration, I only consider seven asset-class returns that will be defined in later chapters. Each asset class’s annual performance is ranked in descending order, with the best performer ranked the highest and the worst the lowest. Because the returns shown on the periodic table are those the market generates at a certain point in time, it follows that the joint occurrence of the outcomes is a feasible combination because it already occurred in the past. Thus, looking at the individual returns jointly, we avoid the potential pitfall of many simulation procedures.

To begin, the periodic table can be used to calculate possible ranges of outcome for all asset classes. As shown in Table 1.2, if you chose the top-performing asset class each year for the past three decades, $1 invested in 1975 would have grown to $2,919.50 today. In contrast, $1 invested in the worst-performing categories since 1975 would have declined to $0.24 at the end of 2004. That is quite a range of possible outcomes. In the context of rates of return, the outcomes range goes from a gain of 30 percent per year during this period to a decline of 4.7 percent per year. Again, that’s quite a range. Investors, however, who required a rate of return higher than 30 percent during this period would have been out of luck—to reach their long-term objectives, they would have had to either revise their expectations or their current savings.

Table 1.1 Periodic table of asset returns.*

View Table

* The figures included in this table are percentages.

Table 1.2 Growth of $1 invested in the top-, second-, third-, fourth-, fifth-, sixth-, and seventh-ranked asset classes each year: 1975–2004.

 

Value of $1

Return

Top

$2,919.50

30.0%

Second

$365.57

21.7%

Third

$92.48

16.3%

Fourth/Median

$31.89

12.2%

Fifth

$11.43

8.5%

Sixth

$1.75

1.9%

Seventh

$0.24

–4.7%


Source: Research Insight, Morgan Stanley Capital Management and Ibbotson Associates

Likelihood of Choosing the Top-Performing Asset Class

After a range of outcomes is determined, the obvious next step is to figure a likelihood of the various outcomes. Again, using the periodic table (and some high school-level math), one can easily do this. Because we are only considering seven asset classes, the likelihood of randomly choosing the top-performing asset class in any one year is 1 in 7, or 14.29 percent. The chance we choose the top asset class for two years is a little more complicated, as we need to figure out how many possible outcome combinations exist. For the first year, seven possibilities exist: T-bills, Treasury bonds (T-bonds), large-caps, small-caps, value stocks, growth stocks, and international stocks. When you work out all the possibilities for two years, there are 49 feasible outcomes (that is, T-bills and T-bills, T-bills and T-bonds, T-bonds and large-caps, large-caps and international stocks, and so on). Hence, the chance of randomly picking the winner two years in a row is 1 in 49, or 2.04 percent. The chance of picking the winner three years in a row is 1 in 343, or 0.29 percent. So, the odds of choosing the winner declines quickly as the number of years increases.

In Table 1.3, the odds of choosing the top performer are calculated each year for 30 consecutive years (the results are listed using scientific notation). It is safe to assume most people do not achieve the 30 percent annual return that the top asset class of the last 30 years produces because the odds of doing so are small. My best guess is that people should not plan their retirements with the idea that they will hit the top-performing asset class each and every year.

Table 1.3 Likelihood of randomly selected various outcomes.

Top rank every year

4.44E-26

Top and bottom rank every year

2.01E-49

Above median every year

9.13E-12

Above median 29 out of 30 years

3.65E-10

Above median 28 out of 30 years

7.06E-09

Above median 27 out of 30 years

8.79E-08

Above median 26 out of 30 years

7.91E-07

Above median 25 out of 30 years

5.49E-06

Above median 24 out of 30 years

3.05E-05

Above median 23 out of 30 years

1.39E-04

Above median 22 out of 30 years

5.34E-04

Above median 21 out of 30 years

1.74E-03

Above median 20 out of 30 years

4.87E-03

Above median 19 out of 30 years

1.18E-02

Above median 18 out of 30 years

2.49E-02

Above median 17 out of 30 years

4.60E-02

Above median 16 out of 30 years

7.46E-02

Above median 15 out of 30 years

1.06E-01

Above median 14 out of 30 years

1.33E-01

Above median 13 out of 30 years

1.46E-01

Above median 12 out of 30 years

1.40E-01

Above median 11 out of 30 years

1.18E-01

Above median 10 out of 30 years

8.65E-02

Above median 9 out of 30 years

5.49E-02

Above median 8 out of 30 years

3.00E-02

Above median 7 out of 30 years

1.39 E-02

Above median 6 out of 30 years

5.41 E-03

Above median 5 out of 30 years

1.73 E-03

Above median 4 out of 30 years

4.44 E-04

Above median 3 out of 30 years

8.76 E-05

Above median 2 out of 30 years

1.25 E-05

Above median 1 out of 30 years

1.15 E-06


Still, this cloud—as with every cloud—has a silver lining. It can be difficult to choose the best performer every year, but it is just as difficult to choose the worst performer every time. Indeed, investors should not worry too much about the worst-case scenario presented in Table 1.3 because there is simply little chance of averaging a 4.8 percent annual decline for 30 years.

Let’s take this line of thinking a little further. The periodic data also sheds some light on the long–short strategy whereby there is money to be made on both the winners and the losers. If one has perfect foresight, one is able to pick not only the top performer each year, but also the worst performer. Shorting the latter would enhance the return (for our sample period) to 34.7 percent per year, which is a nice increase. As shown in Table 1.3, however, the chance of choosing both the winner and the loser each and every year for 30 years is almost the square of choosing the winner each year, which is an unlikely event.

So, while still using the same logic, let’s relax the performance requirements a little bit. Because seven asset classes reside in our universe, it follows that every year there will be three asset classes that come in above the median return. In other words, the chances of selecting an asset class that performs above average are 3 in 7, or 42.86 percent. This clearly is a more likely event than choosing the top- or worst-performing asset class each year. The chances of choosing an asset class that performs above the median for two years in a row are 9 in 49, or 18.36 percent. Table 1.3 shows that the chances of choosing an above-median performer for 30 years in a row are 14 orders of magnitude higher than the chances of choosing the top performer each year during this period. (An order of magnitude means that the number is 10 times larger. For example, 20 is an order of magnitude larger than 2 while 200 is two orders of magnitude larger than 2. Note that 14 orders of magnitude round out to about 200 trillion.)

Still, in spite of the huge increase in the likelihood of randomly choosing an above-median performer for 30 years in a row, the odds of doing so are still minuscule. As Table 1.3 shows, they are about 9 in 1 trillion.

Table 1.3 also illustrates the impact of relaxing the conditions on either outcome likelihood. For instance, you are 40 times more likely to choose an above-median performer in 29 out of 30 years than in 30 out of 30 years (see Figure 1.1 for a visual representation). The data show that as one reduces the requirement regarding the number of years an above-median asset has to be selected, the likelihood of choosing an above-median asset improves. For example, although it is difficult to choose the winners, it is just as difficult to choose the losers. Put another way, one can consistently choose the loser just as easily as one can consistently choose the winner. Beyond 13 out of 30 years, however, this likelihood begins to decline. Table 1.3 also shows the most likely outcome is for performance to come in near the average number of years. In fact, the likelihood of being above the median for 15 years out of the 30-year horizon is 10.6 percent. Furthermore, if events are independent, as I assume, we can calculate the intervals of likelihood. For example, the likelihood of being above the median between 11 and 15 years is 64 percent.

Figure 1.1

Figure 1.1 Likelihood of randomly choosing an above-median performer.

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