4.0% 2.0% 0.0% -2.0% -4.0% 1/1/1930 1/1/1940 1/1/1950 1/1/1960 1/1/1970 1/1/1980 1/1/1990 1/1/2000 1/1/2010 Period end date

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1 I am still looking at issues around statistics of how lump sum investing behaves relative to periodic investing () which most of us do. In particular, most arguments, growth of $10,00 charts, and statistics showing the behavior of investment options made on the basis of past history are constructed using an assumed lump sum investment, leading to the question of how relevant they are to what most of us do, namely investing via periodic contributions. I m also interested in things like how does using a fixed stock/bond ratio compare to using something like age in bonds. I used 30-year rolling periods. As always we need to be very careful regarding conclusions from rolling periods. The almost 109 year period contains a bit shorter than needed to generate a mere 4 non-over lapping 30-year periods. I think we can generate some very general broad conclusions about what happened in the past, but need to be very careful not to over-generalize about what that might say about the future. Still, I found the results very interesting. I used the Global Financial Data real total returns monthly database for both the S&P 500 and 10-year US Government bonds, 1900 to October For periodic investing, I assumed that the amount invested each month grew at an annual rate of 1% real. The Age in Bonds portfolio transitions from 75% stocks to 45% stocks over 30 years. This would correspond to starting at age 25 and investing until age 55. Or, one can think of this as using an Age in Bonds Minus 10 allocation, starting at age 35 and ending at age 65. This is compared to a standard asset allocation fixed for all time. The first figure shows how compares to lump sum investing over rolling 30-year periods using the Age in Bonds method of asset allocation. There are clear differences; the most troubling is the large dip in the late 1970 s and early 1980 s. In particular it refutes the often cited fact that history has always provided a real return if you only stay in the market for X years (in this case 30 years!). We see in the table after the figures that the mean returns are similar (especially in light of the remarks above about the limitations due to the amount of data), the method is more volatile. This is not surprising given that the mean time in the market for a dollar invested in the method is about half that of a dollar invested via lump sum. That is, only a small amount invested via is in the market for the full 30 years while all the money invested as a lump sum is in the market for the full 30 years. I do not think we can conclude that will generally be worse than lump sum investing in poor market conditions as it is a simple matter to construct a plausible situation where the reverse is true. I do think we can conclude that is likely more volatile and thus more risky given these results, especially given the recognition that the method results in less time averaging (regardless of one s position on the ultimate value of time averaging, an advantage is less when using the method).

2 1 1, Age in bonds sum, Age in bonds - The next graph (note the vertical scale is different from the other graphs) takes a look at the issue of the reduced time in the market due to periodic investing. If the only difference between lump sum investing and periodic investing were the fact that the average time in the market for the method, we would expect to see statistics between lump sum investing for 15 years similar to the statistics for over 30 years. This is not the case. As seen in the next figure and the table below, lump sum investing over 15 years is much more volatile than the investing method over 30 years. This suggests the perhaps not so surprising result that there is more time smoothing going on than simply average time in the market. So while the method may be more risky than the lump sum method, it is less risky than suggested by a simple mean time in market calculation might lead you to believe. 1 1 lump 15 years 1,, 30 years -

3 The third figure compares a straight portfolio to an Age in Bonds portfolio. Graphically we see very little difference. There is a period between about 1950 and 1970 where Age in Bonds does worse than a straight portfolio. Given the limitations in the data I think the most we can say is that historically there is no evidence to support Age in Bonds vs simply holding the simpler fixed portfolio. A few years ago when I was writing Monte Carlo simulators and simple consumption smoothing models (for using in accumulation through retirement) for my own use, I could find nothing to support the superiority of any simple shifting of bonds as we age paradigm, or based on retirement dates (for example, to retirement and 40/60 afterwards). Perhaps there is value in monitoring your investment progress towards enough and shifting to a more conservative capital preservation allocation at that point, but I have yet to see such a study and did not try to do that here. In fact if that is what you want to do, one might do better to but some money into an annuity rather than shifting to more bonds. 1 1, Age in bonds, - The remaining figures are included to provide context. The next figure shows how a 100% stock portfolio fared using the two investing methods over 30-year periods. We see that historically stocks, even for people who steadily invest for 30 years, the market has been more risky than shown in studies employing a lump sum investing methodology. The only other conclusion I can draw is simply that in any given period we can expect that and lump sum investing can give rather different results. The figure after that uses a 100% bond portfolio, and raises similar issues. Note well that with the method there is an approximately 35 year period of negative or near zero real growth using a 100% bond portfolio. The last figure shows the cumulative returns for stocks and bonds for $1 invested over the period used. Shown in real dollars the results are startling. One dollar invested at the beginning of 1900 grows to $5.75 by 9/30/2008 while a dollar invested in the stock market grows to $ More startling perhaps, invested in bonds the real return is less than zero as late as 10/31/1981, after being invested continuously for nearly 82 years.

4 1 1, 100% stocks sum, 100% stocks - 1 1, 100% bonds sum, 100% bonds - Total return on $1 108 year Cumulative real returns, GFD S&P 500 total returns, / Cumulative returns stocks 800 Cumulative returns bonds

5 The table shows the real for various portfolios used above. Age for example is periodic investing using the Age in Bonds allocation, while Age is using a lump sum investment with the Age in Bond allocation. 30 years 15 years Age Age Stocks Stocks Bonds Bonds mean 4.7% 4.9% 4.9% 4.9% 6.6% 6.5% 1.6% 1.4% 5.0% std 1.6% 1.0% 1.7% 1.1% 2.3% 1.8% 2.2% 1.6% 3.5% min - 0.4% 2.2% -0.2% 1.7% -0.1% 1.1% -3.8% -1.7% -2.7% 1 2.5% 3.7% 2.2% 3.5% 3.4% 4.5% -0.8% -0.9% 0.3% 5 4.9% 4.7% 5.2% 4.7% 6.3% 6.4% 1.6% 1.7% 5.2% 9 7.1% 6.4% 7.1% 6.5% 9.6% 8.9% 4.8% 3.6% 9.1% max 8.1% 7.8% 8.5% 7.5% 10.7% 11.6% 5.7% 4.8% 13.2%