Wednesday, 13 February 2008

The Necessity of Monte Carlo for Financial Planning

On Monday in my review of Preet Banerjee's book RRSPs, I praised his use of Monte Carlo simulation to project investment returns. Below is a simple illustration of why this is so important in financial planning. There is growing use of Monte Carlo in retirement planning but it should apply equally in the accumulation phase since you may not end up where you thought you might after 30, 40 or 50 years of saving unless you take this into account and implement some appropriate measures in your investing strategy. Using a straight 8% annual growth rate to project your investments may be grossly misleading!

Monte Carlo simulation more closely models the way the stock market actually works, where each year's return is different from the last and where some years are negative while most are positive. For a clearly written introduction to Monte Carlo, read this series of inter-linked articles on Investopedia. This where I got the basic example, which I have expanded a bit.

Click on the spreadsheet extract above and follow along. The examples all show investing for five years using different rates of investment return, either with an initial amount of $100 left alone for the five years, or with yearly contributions of $20.

In the upper left is Case 1, the Base Case. The left hand column of returns vary each year, more like the stock market, going up and down, while the simple case of the same return every year (here 10%) is to the right. Note first that for an initial investment of $100 left alone for five years, even when the arithmetic average of returns (simply adding up each year's return and dividing by five) is the same 10%, the fact that returns go up and down through the five years means the end result is quiet different - $161.05 for the constant growth rate and a lesser $157.32 for the varying returns. The real compounded rate of return (CAGR), which is the relevant return, is obviously less than 10%, it is actually 9.49%.

Cases 2 and 3 down the left side show that when the exact same set of yearly returns occurs in a different order (in fact it is only the minus 5% year in the red box that I have moved around), the average is of course still the same 10%. An initial investment of $100 always comes out the same at $157.32. In other words, the compounded return is still the same too. The pattern doesn't matter in these instances, though the amount in each year along the way is not the same in each case.

In Case 4 at the bottom I kept the CAGR/final amount constant (an investor of $100 at the start would still end up with $157.32) but made the yearly swings more pronounced. The arithmetic average for the years now goes up to 10.2778%. So much for the helpfulness of "average rates of return".

Things become more interesting when a more realistic investing pattern is added, in these examples the $20 per year. In all cases five years of investing $20 (total $100) begets less, much less, than $100 salted away at the beginning, no matter what the sequence of returns (well, it could be more if every year was negative). This another illustration of the oft-noted powerful effects of compounded returns. The same CAGR of 9.49% every year produces only $132.34 (table in the right-most column) after five years of investing $20 per year.

Isn't it interesting how the application of the varying pattern of returns to the $20/year contributions in case 1 produces more than the CAGR rate, namely $132.98, reversing the results for the same sequence of returns applied to the single initial investment? Modeling the real way that investment returns occur begins to acquire some importance.

Now comes the really fun part. Note how Cases 2 to 4 of the $20 contributions produce markedly different end total values. Cases 2 and 3 use the exact same set of yearly returns, only in different sequences. In Case 2, when the negative year of minus 5% happens earlier, the total value is appreciably higher. in Case 3, where it happens later than in the Base, the net is much lower. When you are investing on a regular basis the pattern of returns matters and it matters a lot!

In Case 4, the very same CAGR of 9.49% ($157.32) for an initial investment but with a broader range of yearly returns (I changed the last year to minus 10% then fiddled with the preceeding year till is gave me the same CAGR) creates an even lower value of $123.75. Wider market swings can be bad for your financial health. If on the other hand, the minus 10 occurred early in the sequence, the positive end result would be greater than any of the other. Bigger year to year swings magnify the possible end results up or down. They do not necessarily cancel each other out.

Crestmont Research has published a table and graph of the Dow Jones Industrial Average called Distorted Averages that illustrates the above principle with real data. Between 1900 and 2008 the DJIA had an average annual return of 7.0% but the compounded return was only 4.6%.

Since most people would say that an unexpectedly lower net result after a lifetime of saving in the accumulation phase or many years in retirement in the drawdown phase - think of it, would you rather be forced back to work at age 80 because you have run out of money or chance the pleasure of a round the world cruise at that age? - my suggestion is that big swings are more bad than good.

What can be done about it? Certainly one cannot control the market swings, the yearly returns. But one can control and reduce volatility to a fairly significant degree by diversification, that is picking a set investments that do not move in perfect unison - that are non-correlated, or better, negatively correlated. So, to the TSX or the S&P 500 equities we add things in a portfolio like international equities, bonds, real estate/REITs, and commodities. Take a look at my model portfolio at the bottom of this blog. Most holdings have gone down but my commodity DJP and international equity VWO are up a bit, while bonds in AGG are close to breakeven (it's only because of the rising C$ and the not-shown cash distributions I have received that they are not) and my Canadian bond ladder is up considerably.

Effective diversification may reduce returns by a little but it will cut volatility/risk by a lot. Check out this chart at IFA Canada to see how their portfolio approach creates the desired effect.

Another lesson from this simple example is that to the extent one can control retirement date, it is best not to do it in the middle of a big market downswing when investment net holdings could be driven down by a lot. Turning investments into annuities at that moment could lock in 30 years of retirement at a much lower income level.

Monte Carlo in actual financial investment planning use does not give a single deterministic answer of how much your portfolio will be worth after five or fifty years. Rather, it gives a picture of a whole series of possible outcomes, depending on which of many possible sequences of returns happen in the market or in your portfolio. Varying the assumptions about the expected return and the volatility produces ranges of possible outcomes, as shown in the graphs on this post on RRSP vs Mortgage in WhereDoesAllMyMoneyGo. You don't get certainty, you get realism with Monte Carlo.

Is Monte Carlo beginning to sound useful?

Two excellent articles with examples and explanation of the reasoning behind Monte Carlo simulation can be found at William Bernstein's The Retirement Calculator from Hell and William Sharpe's Financial Planning in Fantasyland. I found the links to these articles on the MoneyChimp site, always an excellent resource for learning about investing principles.

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