Time Diversification

The concept of time diversification implies that investment risk decreases with time. Although it is true that the holding period standard deviation decreases with time, periodic standard deviation does not decrease with time, and the magnitude of the annual deviations in dollars increases with time, assuming that portfolios increase in value over their holding periods. In aggregate, these factors result in a very broad range of probable ending values for portfolios with long investment horizons. The term "time diversification" is probably a misnomer, but that's what it's called.

As they say, time is on your side, and in investing, it is, especially given the power of compounding. Time also makes it possible for you to recover from losses or periods of under-performance but, although recovery is possible, it's not guaranteed. And as your investment horizon lengthens, the probable standard deviation of your long-term average return decreases, which is the effect of what is known as time diversification. But you need to keep this in perspective. Although the long-term standard deviation of your portfolio's value may be less than the expected standard deviation of annual returns, the dollar amount of the periodic variations increases with the value of your portfolio. So, although your expected holding period standard deviation may be smaller than your expected annual standard deviation, the range of the future value of your portfolio can be quite large, and that range increases with time.

The range of the future value of your portfolio is known as terminal wealth dispersion (TWD) and this is what you should be concerned about. If you need a specific amount of money at some future date, it must fall within the range of your TWD. Assuming that the value of your portfolio at some future date can be fairly accurately predicted by the expected mean annual return is not a good assumption, as is explained below. (See Dealing with Terminal Wealth Dispersion for more on this subject.)

What's often referred to as time diversification, is commonly approximated by means borrowed from statistical sampling theory, but better approximations can be obtained with Monte Carlo simulations if you have access to the appropriate software. Sampling theory relies on the Central Limit theorem which tells us that the mean of the sample means is normally distributed and that the variance is inversely proportional to the sample size. Making a leap of faith and treating future returns as samples from a population, it is assumed that the long-term standard deviation of returns will be the historic (population) standard deviation divided by the square root of the number of years (samples) in the holding period.

S_f = S ÷ n^1/2
    Where:
       S_f= Standard Deviation of Final Portfolio Value
       S = Population Standard Deviation of Historical Returns
       n = Holding Period in Years

Using this methodology, if the expected mean and standard deviation of your portfolio's annual returns were 12% and 10%, respectively, and your investment horizon was 25 years, you could expect the standard deviation of your holding period return to be 10% ÷ √25 = 10% ÷ 5 = 2%. What that means is that there is a 96% probability that the average return on your portfolio over 25 years will be 12% +/- 4%, i.e., +/-2 standard deviations, compounded annually. So if you started with a $10,000 investment, your expected ending value at 12% would be $170,001. But the compound return could be as low as 8% or as high as 16%, which would give you ending values of $64,485 or $408,742, respectively. This range describes 96% of your TWD. So, although time diversification has diminished the standard deviation of your long-term average return, your probable ending value spans a very broad range if you base your estimate on a conservative 96% probability. If you're wondering about the 96% probability, then you should probably read the subsection on Mean & Standard Deviation.

The variation in ending value is determined by your holding period "selection," i.e., the specific period in which you are invested. In the 33 rolling 25 year periods from 1950 through 2006 the mean annual holding period return for the S&P 500 ranged from 5.31% for the period 1959 - 1983 to 13.84% for the period 1975 - 1999 with a mean of 8.28% and standard deviation of 2.32% for the 33 25-year holding period returns. The mean and standard deviation of the S&P 500's annual returns over that same period were 9.75% and 16.49%, respectively, and the geometric mean annual return was 8.10%. The geometric means for the worst, average and best 25-year holding periods were 3.94%, 8.10% and 13.04%, respectively. Therefore, if you invested $10,000 over one of the 33 holding periods, your ending portfolio value would have ranged from $26,277 to $214,192, depending on which 25 year period you were invested in. The mean ending value for the 33 holding periods is $70,088. (These ending values were calculated using the geometric mean as described in Mutual Fund Returns.)

These three series, worst, average and best 25-year periods, are plotted in the following chart:

The best 25-year period, 1975 - 1999 got a big boost over the last five years of that 25-year period. 1995 - 1999 were the years when the infamous stock market bubble formed. However, over the next three years, 2000 - 2002, the value of the S&P 500 decreased by 40.1%, wiping out 100% of the gain from mid 1997 through 1999.

Both of these are extreme but real cases. But they happened and similar periods are likely to occur in the future. Unfortunately, nobody knows when that might be and we don't get to select our holding periods after the fact. Although you may find that a bit unsettling, don't freeze up like a deer caught in your headlights. Unless you're very well-off, you can't afford not to expose yourself to this risk. But you can minimize your risk exposure by assembling a well-diversified portfolio. The S&P 500 is only one asset class...large-cap domestic stocks. You must diversify across asset classes to minimize your investment risk and the range of your TWD.

This real-life example is a good demonstration of the flip side of so-called time diversification. The assumption that by having a long holding period you could expect your realized return to be close to the geometric mean of 8.10% is obviously not a good assumption. Although the probability of being relatively close to the mean is much higher than being at either of the extremes, it's entirely possible that you would end up at one of the extremes. Terminal wealth dispersion is the flip side of time diversification.

The estimate of long-term standard deviation is actually pretty sound, but you need to remember that the annual standard deviation is unaffected by time. In the example above, the expected standard deviation in any given year remains at 10% for the full 25 years, even though the expected standard deviation for the 25-year holding period is only 2%. And, as I stated above, the dollar amount of your annual fluctuations will become greater with time as your portfolio grows in value, which is the aspect of time diversification that's often overlooked.

Many people promoting time diversification like to use a chart showing the span in holding period returns diminishing with time, with the minimum and maximum converging on the mean. This makes it appear that the variance eventually is insignificant. The folks at Political Calculations have created one of these charts, but not to promote time diversification.

The chart plots the best, worst and average returns of the S&P 500 for various holding periods over the period 1871 though 2006 using Professor Robert J. Schiller's database of historical returns. The mean annual return is 9.4% and the range for 25-year holding periods is 3.1% to 17.6%, which is 6 percentage points broader than that for the period 1950 - 2006, but leads you to the same general conclusion.

Political Calculations has made appropriate use of this chart, but beware of those who don't. The variance will only become relatively insignificant if your family holds a portfolio for a few generations and even then the timing of the initial investment and liquidation of the portfolio will have a significant impact on its final value, which is explained by the concepts of time diversification and terminal wealth dispersion.

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