**Dollar Cost Averaging**

**Overview**

**Overview**

As a method of investing money into the market, dollar cost averaging (DCA) is often proposed. Dollar cost averaging involves investing a fixed proportion of your capital in each of n periods. These periods can be monthly, quarterly, etc. The premise of dollar cost averaging is that the investor wants to guard against the market losing value shortly after making his investment. Therefore, he chooses to spread his investment over a number of periods

Since the market has a positive mean rate of return, dollar cost averaging usually requires the investor to give up some expected return for the benefit of reduced variance in their eventual outcome.

The purpose of this analysis is to better understand quantitatively the trade-off between the loss in expected return versus the downside reduction provided. Further, we will explore the idea of how the decision is different depending on the relative valuation of the market at the point of entry.

**Analysis**

**Analysis**

To find answers to these issues, we have collected historical data on the monthly value of the S&P 500 from the period of 70 years (from Jan-1936 to Dec-2008) from Robert Shiller of Yale University and similar data on 3 month Treasury Bills from Global Financial Data. Against this data, we have applied 6 different investing strategies: Lump Sum, DCA over 2 quarters, DCA over 4 quarters, DCA over 8 quarters, DCA over 12 quarters, and DCA over 16 quarters. The table below shows when the investor will make investments in each strategy.

We have assumed that the investor is investing $1,000 and will invest an amount of cash at each investment period that is equivalent to (1/n) of his $1,000 where n is the number of quarters over which he is dollar cost averaging. We have chosen to make investments on a quarterly basis (every 3 months) since this is probably easier to manage for an individual investor. Our guess would be that the results based on a monthly investment schedule would be similar.

As a general method for comparing the effectiveness of each strategy, we have examined the investor’s position at the end of a 4 year horizon. Since all strategies will be fully invested at the 4 year mark and will earn the same future rates of return, this allows for a fair comparison amongst the strategies.

Bill Jones performed an analysis in 1996 (see below for a link to it) of the dollar cost averaging problem using monthly investment periods. We like some of his analysis but he fails to compare strategies over the same time horizon which we think is a major flaw in his analysis. However, he does pose the problem in an interesting way: he views the option of using dollar cost averaging as an insurance problem where the investor can buy downside insurance by giving up expected returns.

Jones uses some interesting metrics to demonstrate the value of the insurance (in terms of reduced variance and downside risk) that the investor is buying. We will employ similar metrics to illustrate the differences in the strategies.

**Metrics**

Average Gain – over the 4 year investing horizon (this is the cumulative gain)

Standard Deviation – calculated from the average 4 year gain

Down Periods – percent of periods where the investment value at the end of 4 years is less than the original $1,000

10% Down Periods – percent of periods where the investment value at the end of 4 years is down at least 10%

Average Loss – average of all 4 year losses (when their is a loss)

5% Improvement – percent of periods where the Lump Sum strategy produces a 10% down outcome where the DCA strategy would improve the outcome by at least 5%

**Assumptions**

The un-invested money earns interest at the rate of the 3-month Treasury Bills plus 1%. We add the additional percent to compensate for when the investor actually keep the money out of the market for more than one quarter. Presumably, they could buy longer term Treasury bonds which would pay more. The results aren’t terribly sensitive to this rate of return.

We assume that dividends are not re-invested but are paid as cash at the end of the 4 year period.

We assume no costs for transactions.

**General Results**

Our dataset provides 828 periods with a 4 year horizon of returns. The table below shows the results from this historical data.

As you can see from the results, the investor can reduce the average gain and standard deviation of the gain over the 4 year investing horizon by increasing the period of time over which he employs a DCA strategy. Dollar cost averaging over 16 quarters will reduce the number of down periods from 9% to 7% but will cost the investor 14% in expected return. The reduction in average gain is equivalent to 2.6% on an annual basis. The number of 10% down periods is also slightly reduced from 6% of the periods to 3% of the periods. Additionally, in 61% of the periods where the lump sum strategy will produce a loss of more than 10%, this would be alleviated by at least 5% if the 16 quarter DCA strategy was employed.

Generally, the lump sum strategy seems preferable for investors who are not exceptionally risk averse. The value of being in the market and reaping both price appreciation and dividends is fairly significant. However, the DCA strategies will reduce variance on returns. The risk preferences of the investor should lead him to choose a strategy that is appropriate.

**Smart Dollar Cost Averaging**

**Smart Dollar Cost Averaging**

We think that it is inappropriate to make a decision on whether to DCA (and over what time frame) without at least having a general view on the overall valuation of the market. While incorporating this information does take one into market timing strategies, we think that recognizing periods when the market is sufficiently under-valued or over-valued by historical standards can lead to better decisions.

As a proxy for the relative value of the S&P 500, let’s use the overall index’s 10 Year P/E ratio at the beginning of each 4 year period. By 10 Year P/E ratio, we are referring to the P/E Ratio calculated with the earnings component being the average earnings in current dollars from the last 10 years. This will smooth years where earnings may be exceptionally high or low. Graham and Dodd introduced this approach in 1934. We think that it gives a better indicator of the fairness of valuations rather than using the traditional 1 year P/E ratio. There are plenty of value metrics to use and P/E ratio is definitely not ideal but it does serve to help make decisions in this case. The table below divides the value of the P/E ratio into four intervals and displays the percentage of our total 828 investing periods which had P/E ratios in each interval at the start of the period.

Now, let’s look at the results of our general analysis when we filter by the P/E ratio of the S&P 500 at the time of making our initial investment.

**Low P/E Ratio (Less than 11) – this occurred in 159 periods**

In this case, it is clear that the investor would want to move his money into the market quickly. The risk of a negative return on your initial stake over the 4 years is very low. The decision to DCA over 16 quarters rather than invest lump sum would cost the investor 3.6% in expected annual return. That’s pretty expensive!

**Medium-Low P/E Ratio (Between 11 and 16) – this occurred in 229 periods**

The investor again will probably prefer the lump sum strategy. The reduction in standard deviation is fairly modest as the dollar cost averaging horizon is stretched to 16 quarters. But, the investor would be giving up return at a rate of about 4.3% per year.

**Medium-High P/E Ratio (Between 16 and 21) – this occurred in 218 periods**

In this scenario, the options of dollar cost averaging are more appealing. The annual lost expected return between the lump sum case and the 16 quarter DCA strategy is 0.4%. That price does buy significant reduction in standard deviation and the number of down periods is reduced from 26% to 16%. It seems like dollar cost averaging over 16 quarters would be preferable to DCA over 4, 8, or 12 quarters. Therefore, the decision can be reduced to investing quickly (lump sum or 2 quarters) versus long term DCA.

**High P/E Ratio (More than 21) – this occurred in 222 periods**

Surprisingly, the lump sum option is more appealing for the high PE ratios than it was for the Medium-High PE’s. Part of this is explained by the fact that the data doesn’t fully include the recent (late 2008 and early 2009) drop in the S&P 500 index. Thus, if the S&P were to remain at about 900 for 2009 and 2010, the 44% total return for lump summing would drop to about 36%. So, the benefits of lump summing may be overstated by the data sample. But, the 16 quarter dollar cost averaging does reduce the standard deviation by a good amount. The risk averse investor would probably prefer to DCA in these periods but there may be some small expected gains by lump summing even when the PE ratio is very high.

**High P/E Ratio (More than 21) and Moderate T-Bill Rate (> 4.5%) – this occurred in 127 periods**

One issue with dollar cost averaging is that some portion of your capital is tied up in Treasury Bills which generally yield lower returns than equities. It would seem to make sense that when Treasury rates are very low, there is less benefit to dollar cost averaging. Therefore, we have examined the results when the investor is facing a situation where dollar cost averaging would seem more attractive – when PE’s are high and Treasury rates are not low. Here are the results:

In this situation, it does seem to make sense to dollar cost average over a longer period of time. The total returns are quite similar and the standard deviation is almost halved.

**Conclusions**

**Conclusions**

Dollar cost averaging clearly offers investors the ability to limit downside and reduce variance of their returns at the cost of giving up expected returns. When making the decision of whether or over how many periods to DCA, you should ideally form an opinion (based on data) on the relative valuation of the market. However, the data indicates that it is difficult to beat the lump sum strategy in most situations even with the valuation information. Ideally, when Treasury yields are high and PE values are also high, you should strongly consider dollar cost averaging any new investment capital.

At the time of writing (May 21, 2009), the S&P 500 has an P/E ratio of about 15.5 according to Shiller’s data. Thus, the market seems relatively fairly valued according to historical averages. In these types of periods, you should probably consider moving quickly into the market unless you are particularly sensitive to volatility.

**Useful Links**

**Useful Links**

Do Not Dollar Cost Average for More than 12 Months - An article by Bill Jones found in the Efficient Frontier which poses the dollar cost averaging problem as an insurance option

Lump Sum Beats Dollar Cost Averaging - An analysis published in the Journal of Financial Planning