Simulations of investment strategies

Based on the ideas in a blog posting by Steven Landsburg (and the subsequent comments), I did some simulations of various investment strategies. In all cases, the basic assumptions are that we are investing for a fixed number (N) of periods, and that during each period the investment grows by a factor r which is normally distributed. For my simulations, the number of periods is N = 15 and r has mean 1.05 and standard deviation 0.20, so I'm thinking of each period as being a year and I've deliberately chosen r to be volatile to emphasize the differences between the strategies.

The first two strategies are the two described in the blog entry, which I've given descriptive names to here:

These two strategies have on average the same number of dollars invested per period, so they have the same expected growth. But in the second, the investments are spread more evenly over the periods, so the risk is reduced.

Those two strategies are a bit unrealistic in that you can't predict how much cash you need to have on hand in order to keep the investment at the desired level. For example, if N = 3 (as Landsburg took), the steady investor wants 2 dollars invested during each period. If it happens that most of the investment is lost during each period, one would need to invest 2 dollars each period for a total of 6 dollars. If you have 6 dollars available for investing, then most of the time it is better to have a larger investment, so you weren't doing the right thing. (Of course, borrowing can help make this more realistic, but I haven't included borrowing costs into this model.)

Two strategies that are more realistic are the following:

The first is easily doable by someone who has a regular pay check and can afford to invest a certain portion. The second is easily doable by someone with some cash that is available for investing.

So, how do these four compare? With 15 periods and the rate for each period independently drawn from a normal distribution with mean 1.05 and standard deviation 0.20, I ran one million trials. For each trial, I took the value of the investment and subtracted the amount invested (which in the case of step or steady depends on what happened during the simulation). The results are: 

  step:   6.00 +-  7.05  (118%)
  steady: 6.00 +-  6.20  (103%)
  dca:    7.66 +- 11.79  (154%)
  lump:   8.64 +- 13.98  (162%)

which displays the expected profit and the standard deviation of the profit (also expressed as a percentage of the profit).

The 6.00 for the steady investor is easy to understand. He or she has 8 dollars invested during each period, and so on average earns 40 cents per period (5 percent of 8 dollars). Over 15 periods, that totals to 6 dollars. As mentioned above, the step investor earns on average the same amount, but with greater volatility.

But note that the strategies that are easier to accomplish (dca and lump) in addition have a higher expected return! This is because on average they have more invested.

Here is a plot of the distributions of profits for the four strategies (with some long tails cut off). The vertical axis is an arbitrary scale related to the density of the samples in the histogram bins.
plot of distributions of profits
Note that dca and lump are more similar to each other than dca is to step or lump is to steady. So it's not appropriate to use step and steady as a way to argue whether dollar cost averaging is a reasonable strategy. Indeed, step is clearly inferior to steady (if you have the money upfront), whereas the decision between dca and lump depends on your risk tolerance and other factors. In fact, the standard deviation of dca as a percentage of the expected growth is less than that of lump, meaning that by simply increasing the amounts invested, you can get the same expected return as lump with less risk. Combined with the fact that most individuals do not have their lifetime savings available to invest early, dca comes out as a good strategy.

Similarly, one can argue that by increasing the amounts invested with step or steady, one can get the same return as dca or lump with even less risk. That is true, but it really only further reinforces the point that dollar cost averaging is reasonable, since, for example, step not only applies dollar cost averaging to the new investment made in each period, but applies it to the investments made in previous periods, either withdrawing profits or purchasing more to make up for losses. It is that rebalancing that reduces the risk.

Conclusions

Here are the conclusions I draw:

For most people, lump and steady are not realistic because they don't have a large amount of cash to draw upon, and the above shows that periodic investments of a fixed amount are a good thing to do.

If the investor does have a large amount of money to invest, and wants to decide between investing all of it now, or doing it over a fixed number of periods, then you should multiply the lump results by 2N/(N+1) = 15/8 to get your comparison (because the lump sum amount is now the sum of the periodic amounts): 

  dca:    7.66 +- 11.79  (154%)
  lump:  16.20 +- 26.21  (162%)

Now you have a tough choice between large growth with large risk or small growth with small risk...

Caveats

To be more realistic, a proper comparison would at least include an alternative (less risky?) investment, a borrowing cost, and maybe a set amount of cash on hand. Then one could try to optimize the investing strategy using these constraints.

Comments and feedback welcome: Dan Christensen <jdc@uwo.ca>

The code I used for these calculations is here: dca.py.