This Brief is based on a whitepaper entitled Understanding Monte Carlo Simulation, written by David Loeper in 2005. The whitepaper was written for financial advisors and contains substantially more analytical proofs than will be provided here. I will try hard to keep the industry-speak and statistical terms to a minimum.
One of life’s greatest fears is that of the unknown. When I talk to new clients and ask what concerns them most, their answers usually involve general uncertainty about the future around them and specific uncertainty about their own financial future. Of course a highly competitive and ubiquitous media bellowing out dire forecasts doesn’t help.
As wealth advisors we often find ourselves explaining how misleading the concept of returns taken on their own can be. It’s easy to understand why we rely on return to gauge progress – it seems to provide a benchmark for effective comparison: 1) relative to other investment choices and 2) as a measure of how well we are doing relative to our own goals.
The first comparison is obvious. If Fund A returned 7% for the past 10 years and Fund B returned 8% during the same period, the choice between them is made simple. The second return concept is based on the widely practiced methodology of planning with a conservative and steady rate of return. For example: If we could grow our money at say 7% a year for the next 20 years, then get a 7% return during retirement, that would yield us enough income to meet our spending goals for the rest of our life.
There are a number of problems with this simple model, the most obvious and significant is uncertainty. Everyone knows that markets are uncertain and that projecting consistent returns well into the future is at best, wishful thinking. Nevertheless, the industry and investors alike both rely heavily on returns for both selection of assets and for determining how well they are doing relative to their goals.
Mutual funds and single managed accounts use returns to tout their individual outperformance relative to their peers. The problem with this practice is that returns alone do not tell the whole story.
Consider the following example. If an investor received 7% every year, his return would obviously average 7%. Now let’s mix in a little uncertainty by adding a 21% return and a loss of 7% in the first year and the 5th year and compare it to the opposite placement.
Scenario A | Scenario B | ||||||||
Return | Return | Portfolio | Return | Return | Portfolio | ||||
Year | in % | in $ | value | Year | in % | in $ | value | ||
$100 | $100 | ||||||||
1 | 21% | $21 | $121 | 1 | -7% | ($7) | $93 | ||
2 | 7% | $8 | $129 | 2 | 7% | $7 | $100 | ||
3 | 7% | $9 | $139 | 3 | 7% | $7 | $106 | ||
4 | 7% | $10 | $148 | 4 | 7% | $7 | $114 | ||
5 | -7% | ($10) | $138 | 5 | 21% | $24 | $138 | ||
7% | Average | 7% | Average | ||||||
As you can see, it doesn’t matter when the good year and when the bad years occur, both scenarios still average 7%. But what's missing from this example? In order to reflect reality we need to add some savings or some spending. While there may be a few timecapsule investors in the world, we don’t have any clients who drop off a pile of money and say nothing more than “let it ride.” Yet this is the way mutual funds and single account managers report their value to you, but they take no responsibility for the timing of your spending or saving decisions.
Why does cash flow timing matter? Consider another example. This time we’ll take a couple who have just entered their retirement years and plan on spending 7% annually for the next 30 years. As before, we will add a great return year at 21% and a bad return year at -7% in reciprocal order to the two scenarios.
Good Return Early | Good Return Late | |||||||||
Return | Return | Contrib | Portfolio | Return | Return | Contrib | Portfolio | |||
Year | in % | in $ | (Withd) | Value | Year | in % | in $ | (Withd) | Value | |
$100 | $100 | |||||||||
1 | 21% | $21 | ($7) | $114 | 1 | -7% | ($7) | ($7) | $86 | |
2 | 7% | $8 | ($7) | $115 | 2 | 7% | $6 | ($7) | $85 | |
3 | 7% | $8 | ($7) | $116 | 3 | 7% | $6 | ($7) | $84 | |
4 | 7% | $8 | ($7) | $117 | 4 | 7% | $6 | ($7) | $83 | |
5 | 7% | $8 | ($7) | $118 | 5 | 7% | $6 | ($7) | $82 | |
6 | 7% | $8 | ($7) | $120 | 6 | 7% | $6 | ($7) | $80 | |
7 | 7% | $8 | ($7) | $121 | 7 | 7% | $6 | ($7) | $79 | |
8 | 7% | $8 | ($7) | $122 | 8 | 7% | $6 | ($7) | $78 | |
9 | 7% | $9 | ($7) | $124 | 9 | 7% | $5 | ($7) | $76 | |
10 | 7% | $9 | ($7) | $126 | 10 | 7% | $5 | ($7) | $74 | |
11 | 7% | $9 | ($7) | $128 | 11 | 7% | $5 | ($7) | $72 | |
12 | 7% | $9 | ($7) | $129 | 12 | 7% | $5 | ($7) | $71 | |
13 | 7% | $9 | ($7) | $132 | 13 | 7% | $5 | ($7) | $68 | |
14 | 7% | $9 | ($7) | $134 | 14 | 7% | $5 | ($7) | $66 | |
15 | 7% | $9 | ($7) | $136 | 15 | 7% | $5 | ($7) | $64 | |
16 | 7% | $10 | ($7) | $139 | 16 | 7% | $4 | ($7) | $61 | |
17 | 7% | $10 | ($7) | $141 | 17 | 7% | $4 | ($7) | $59 | |
18 | 7% | $10 | ($7) | $144 | 18 | 7% | $4 | ($7) | $56 | |
19 | 7% | $10 | ($7) | $147 | 19 | 7% | $4 | ($7) | $53 | |
20 | 7% | $10 | ($7) | $151 | 20 | 7% | $4 | ($7) | $49 | |
21 | 7% | $11 | ($7) | $154 | 21 | 7% | $3 | ($7) | $46 | |
22 | 7% | $11 | ($7) | $158 | 22 | 7% | $3 | ($7) | $42 | |
23 | 7% | $11 | ($7) | $162 | 23 | 7% | $3 | ($7) | $38 | |
24 | 7% | $11 | ($7) | $166 | 24 | 7% | $3 | ($7) | $34 | |
25 | 7% | $12 | ($7) | $171 | 25 | 7% | $2 | ($7) | $29 | |
26 | 7% | $12 | ($7) | $176 | 26 | 7% | $2 | ($7) | $24 | |
27 | 7% | $12 | ($7) | $181 | 27 | 7% | $2 | ($7) | $19 | |
28 | 7% | $13 | ($7) | $187 | 28 | 7% | $1 | ($7) | $13 | |
29 | 7% | $13 | ($7) | $193 | 29 | 7% | $1 | ($7) | $7 | |
30 | -7% | ($14) | ($7) | $173 | 30 | 21% | $1 | ($7) | $1 | |
7% | Average | 7% | Average | |||||||
The average returns are exactly the same, but the wealth outcomes are dramatically different. Receiving the good return early provided $172 more in wealth than the couple who ‘enjoyed’ the good return late in life. No doubt the couple on the right side spent their last few years worrying whether their nest egg would last.
Intuitively, we understand that a better return on a larger principal balance provides greater wealth. So now let’s add a twist to fool our intuition (the planning tool of choice for entirely too many people). Assume that the wife of our retired couple had a ten-year teacher’s pension that would provide not only for the couple’s living expenses, but a $6 annual surplus which could be saved.
Good Return Early | Good Return Late | |||||||||
Return | Return | Contrib | Portfolio | Return | Return | Contrib | Portfolio | |||
Year | in % | in $ | (Withd) | Value | Year | in % | in $ | (Withd) | Value | |
$100 | $100 | |||||||||
1 | 21% | $21 | $6 | $127 | 1 | -7% | ($7) | $6 | $99 | |
2 | 7% | $9 | $6 | $142 | 2 | 7% | $7 | $6 | $112 | |
3 | 7% | $10 | $6 | $158 | 3 | 7% | $8 | $6 | $126 | |
4 | 7% | $11 | $6 | $175 | 4 | 7% | $9 | $6 | $141 | |
5 | 7% | $12 | $6 | $193 | 5 | 7% | $10 | $6 | $156 | |
6 | 7% | $14 | $6 | $213 | 6 | 7% | $11 | $6 | $173 | |
7 | 7% | $15 | $6 | $234 | 7 | 7% | $12 | $6 | $191 | |
8 | 7% | $16 | $6 | $256 | 8 | 7% | $13 | $6 | $211 | |
9 | 7% | $18 | $6 | $280 | 9 | 7% | $15 | $6 | $232 | |
10 | 7% | $20 | $6 | $305 | 10 | 7% | $16 | $6 | $254 | |
11 | 7% | $21 | ($7) | $320 | 11 | 7% | $18 | ($7) | $265 | |
12 | 7% | $22 | ($7) | $335 | 12 | 7% | $19 | ($7) | $276 | |
13 | 7% | $23 | ($7) | $352 | 13 | 7% | $19 | ($7) | $289 | |
14 | 7% | $25 | ($7) | $369 | 14 | 7% | $20 | ($7) | $302 | |
15 | 7% | $26 | ($7) | $388 | 15 | 7% | $21 | ($7) | $316 | |
16 | 7% | $27 | ($7) | $408 | 16 | 7% | $22 | ($7) | $331 | |
17 | 7% | $29 | ($7) | $430 | 17 | 7% | $23 | ($7) | $347 | |
18 | 7% | $30 | ($7) | $453 | 18 | 7% | $24 | ($7) | $364 | |
19 | 7% | $32 | ($7) | $478 | 19 | 7% | $26 | ($7) | $383 | |
20 | 7% | $33 | ($7) | $504 | 20 | 7% | $27 | ($7) | $403 | |
21 | 7% | $35 | ($7) | $532 | 21 | 7% | $28 | ($7) | $424 | |
22 | 7% | $37 | ($7) | $562 | 22 | 7% | $30 | ($7) | $447 | |
23 | 7% | $39 | ($7) | $595 | 23 | 7% | $31 | ($7) | $471 | |
24 | 7% | $42 | ($7) | $630 | 24 | 7% | $33 | ($7) | $497 | |
25 | 7% | $44 | ($7) | $667 | 25 | 7% | $35 | ($7) | $525 | |
26 | 7% | $47 | ($7) | $706 | 26 | 7% | $37 | ($7) | $554 | |
27 | 7% | $49 | ($7) | $749 | 27 | 7% | $39 | ($7) | $586 | |
28 | 7% | $52 | ($7) | $794 | 28 | 7% | $41 | ($7) | $620 | |
29 | 7% | $56 | ($7) | $843 | 29 | 7% | $43 | ($7) | $656 | |
30 | -7% | ($59) | ($7) | $777 | 30 | 21% | $138 | ($7) | $787 | |
7% | Average | 7% | Average | |||||||
In this case we see that the Good Return Early did not outperform wealth-wise the Good Return Late, which is the exact opposite of what we might expect when considering the time value of money. The couple on the right was slightly better off by experiencing the loss in the first year and waiting until the last year to get the bonus return.
Now, remember that money managers and mutual funds do not take responsibility for the wealth impact on your unique savings or spending decisions, they simply generate returns in a relative vacuum where your individual goals are concerned. We have demonstrated that injecting just two years of uncertainty into a 30-year lifetime can have a dramatic impact on wealth. So we now know that the timing of returns relative to cash flows has a dramatic impact on your wealth and it is uncertain.
Advisors who know their clients well have a good idea of their expected cash flows such as when they want to spend their money and how much they want to spend over the course of our lives. The best advisors know each of these values in ranges of acceptable to ideal and they have worked with their clients to rank their goals according to priority ‘spending’ from their less important goals in order to accomplish the more important ones at their ideal levels sooner. But what the best of planners cannot know is how their clients’ cash flows will align with the uncertain timing of market returns. How can the advisor model uncertainty to help his clients be more confident and make better decisions? That’s where Monte Carlo comes in.
Quite simply, Monte Carlo is a probability analysis tool for modeling randomness of returns. The examples illustrate how dramatically injecting just two random returns into two 30-year lifetimes can alter outcomes, clearly demonstrating the need for a tool that can measure uncertainty and promote significantly more informed decisions.
Dave Loeper describes the Monte Carlo process in his whitepaper as follows. Begin with a spreadsheet of cash flows and asset levels etc. Fill a bowl with slips of paper where each one contains an historical market return. Shake the bowl as in a raffle to ensure the returns are scrambled and random. Draw a slip with a return on it, record it in my spread sheet as the return in the first year, but unlike a raffle, return the paper to the bowl, shake it and draw another slip for year two...and so on. Each slip is returned to the bowl because individual market returns have no memory. We don’t want to rule out the possibility that the same return might not occur again.
Repeat this routine for all thirty years and we would have one ‘lifetime’ result or a “trial.” The average return and risk of thirty “draws” would vary each time the exercise was repeated and the “could be higher than the average of all the numbers in the bowl (depending on if I were lucky and drew a disproportionate amount of high returns), or lower than the average (if I were unlucky and drew an excessive number of low returns.) Also there would be some outcomes that might average a higher than normal result but bad timing of when I draw a slip of paper might cause an unpleasant outcome, or I might have a lower than average result with fortunate timing that ends up exceeding the client’s goals none-the-less.” Returns alone don’t tell the story
To get a sense of confidence of the client’s odds (chances) of exceeding his goals, we need more than just one or two “trials.” Dave continues “If I had the patience to draw the numbers for 500 or 1000 trials (drawing 15,000 or 30,000 slips- one slip for each year of each 30 year trial), I would have a better understanding of how confident the client could be. Clearly with that many samples, odds are that I would have drawn Great Depression types of results, back to back with the bear market of ‘73-’74 and probably the bear market of ‘00-’02 too.”
In a basic sense the examples describe how Monte Carlo works. But there is considerably more to the procdess. For instance, even with our data going back as far as 1926, there is the chance that the markets may not have demonstrated their worst returns or therir best yet. Instead of simply filling the bowl with only actual market returns, we must add statistically “possible” results as well. Dave says that “by doing this we would be able to measure not only the uncertainty of historical returns but also of future potential returns. This additional step helps us to make sure we do not ignore the chance that we have not yet seen the worst (or the best) of what the markets might produce.”
There’s another problem to recognize. The time-frame of the data universe can have significant impact on the results of the model. Dave asks what if the bowl was filled with numbers from just the last 25 years instead of going back to 1926? The 25-year period ending in 2003 produced a compound return that fell in the top 16% of all historical 25-year periods. While there would be a lot of trials that still produced poor results, half of the trials would exceed the top 16% of all historical 25-year periods. In essence I would be telling my client that from now on there is a 50% chance the markets will do better than most historical periods. The reverse could be true as well, if for example I used just the last five years that was dominated by a bear market. If I filled the bowl with numbers like those, I would be simulating half of all the trials doing worse than the bottom 10% of all five year periods.”
Next, we need to ensure that the returns in the bowl, or the computer model, are not overly optimistic or pessimistic relative to the nature of markets. Dave suggests that “to control [outsized results] and make the confidence result materially meaningful, a great deal of diligence therefore must go into figuring out how many slips of paper have what numbers on them (or in the case of a computer, the mean, standard deviation and shape of the distribution we are asking the random number generator to create for us.)”
These inputs are known as Capital Market Assumptions or CMAs. Dave says that the CMAs “are just as critical as modeling the uncertainty of returns because if they are overly biased one way or the other we are either misrepresenting how confident the client can be, or, being overly conservative causing the client to needlessly sacrifice their lifestyle” by saving more, working longer, or sacrificing goals he might have enjoyed in life rather than passing on a much larger estate than he wished to leave.
So the model needs a random number generator that produces a population of returns shaped similarly to the potential behavior of the markets as well as a reasonable method of modeling taxes expected along the way so we can then get a sense of how confident the client can be of exceeding his and her goals.
Finally, don’t assume that Monte Carlo is used uniformly accross the industry. Like many tools in the financial industry Monte Carlo can be used effectively to improve clients’ lives or it can be misused for sales purposes. For instance, beware of a confidence level approaching 100%. While it may sound appealing to hear that you have a 98% confidence of exceeding all of your important goals, what the model is suggesting is that 980 of the 1,000 trials left money in the client’s estate. Many of those ignored trials likely provided ample estates while allowing significantly more spending or giving during the clients' lifetimes.
Alternatively, Monte Carlo can be used effectively by the wealthy to quantify surplus that almost certainly exists among their resources - in other words a shortage of goals. These clients may be able to accomplish much more with their wealth today instead of leaving it trapped in managed and trust accounts overseen by advisors whose focus may be more asset-centric than client goal-centric.
In short, uncertainty can confuse and exact a heavy toll on lifestyles or it can be continually measured to provide for more abundant and purposeful lives. Have a great weekend.
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