Home | Exchange-Traded Funds | Thrift Savings Plan

facebook | rss

Right now, subscribe to our financial advisory newsletters absolutely free! No obligation.

How to calculate returns with dollar cost averaging

The amount of money that you make in an investment or an investment strategy depends on how you put money into that investment or strategy. For instance, you could put in money as a lump sum or you could dollar cost average (DCA). The rates of return for a lump sum investment and for dollar cost averaging are different. It is important to realize that they can be very different. The returns that are typically reported are lump sum investment returns. They are not applicable to investors who are dollar cost averaging.

Many investors invest using both lump sum investment and dollar cost averaging at the same time. At the beginning of some period of time, such as a calendar year, an investor could already have some money in their account. This is the money that they put into the account before the period began, and is the lump sum portion of their investment. In addition to this, they add money to the account throughout the period of time. This is the DCA portion of their investment. The personal rate of return is a weighted average of the lump sum investment return and the DCA return. The weights in the average are proportional to the lump sum portion and the DCA portion of the investment. Because these weights are different for each person, the personal rate of return is also different for each person.

First, let's review the difference between lump sum investment and dollar cost averaging.

Returns from a lump sum investment

Before describing how to calculate returns from dollar cost averaging, let's go over the lump sum return investment calculation. To calculate the return from a lump sum investment, all you need to know is the initial price and the final price. Let pI be the initial or purchase price and pF be the final price. Then, the return from a lump sum investment is

r_L = \frac{p_F}{p_I} - 1

For example, within the Thrift Savings Plan, there is a fund, called the S Fund, that tracks the Wilshire 4500 stock index. At the beginning of 2007, the fund's price was 18.76; at the end of 2007, it was at 19.79. This means that the lump sum investment return for the fund for 2007 was 5.49%:

r_L = \frac{19.79}{18.76} - 1 = 0.0549 = 5.49%

Suppose a person put $2600 into the S Fund at the beginning of 2007, after which they made no additional contributions. This means that, in 2007, their profit was $143:

m_L = $2600 \times 5.49% = $143

Returns from dollar cost averaging (DCA): intuition

In calculating the return from dollar cost averaging, knowing the initial price and the final price is not enough. You need to know the actual path that the price took from the initial price to the final price. This is because you are putting money into the investment throughout the time period. So the prices at all points in the time period are important. In fact, two investments or strategies can have identical initial and final prices and yet have very different returns from dollar cost averaging.

Figure 1: Price steadily increasing. Effective purchase price between initial and final prices.

To see why this is so, consider the following three graphs that show the prices of three different investments. For all three investments, the price starts out at 100 and goes to 120. This means that all three investments made 20% on a lump sum investment basis. However, they made very different amounts on a dollar cost averaging basis.

The price of the first investment was going up steadily. This means that people who were dollar cost averaging into it were buying at progressively higher prices. All of their purchases were made above 100 but below 120. They thus made money, but not as much as 20%.

Figure 2: Price first decreasing then increasing. Effective purchase price below initial price.

The price of the second investment dropped below 100 at first, but then went to 120. Investors who were dollar cost averaging into this investment made all of their purchases below 100. They thus made more than 20%.

Figure 3: Price first increasing then decreasing. Effective purchase price above final price.

The price of the third investment increased quickly to above 120 and stayed above 120 for most of the time, after which it declined to 120. Thus, the people who were dollar cost averaging into this investment made most of their purchases at above 120. Because of this, the investors not only didn't make even close to 20%, they actually lost money. This is an important point that many proponents of DCA sometimes overlook. Even though "on paper", the way it would have typically been reported, the investment made 20%, because of dollar cost averaging, the people who were investing in it actually lost money.

Returns from dollar cost averaging (DCA): the formula

For more details, see the derivation of the dollar cost averaging formula.

Let pi be the daily price of a particular investment or strategy on day i, i = 1, 2, ..., n; and let \tilde{p}_I be the harmonic mean of pi. In other words,

\tilde{p}_I = \frac{n}{\sum_{i=1}^n 1/p_i}

The return from dollar cost averaging is[1]

r_D = \frac{p_F}{\tilde{p}_I} - 1

Note the similarity of this formula to the formula for the return from a lump sum investment. The only difference is that, instead of purchase price pI, the current formula uses \tilde{p}_I. Because of this, I call \tilde{p}_I the effective purchase price.

Figure 4: Price of the TSP S Fund in 2007.

For example, suppose that in 2007, an investor was putting $100 per pay period into the Thrift Savings Plan S Fund, for a total of 26 pay periods. This means that they invested a total of $2600 in the S Fund over the course of that year. As mentioned above, the price of the S Fund was 18.76 at the beginning of the year and 19.79 at the end of the year. However, the price varied a lot within the year — it dropped to a low of 18.58 and rose to a high of 21.29 before dropping again to the final price of 19.79. It turns out that, though the initial price was 18.76, the effective purchase price was 19.96, slightly above the final price. Thus, the investor actually lost about 0.87%, though the fund itself gained 5.49%:

r_D = \frac{19.79}{19.96} - 1 = -0.0087 = -0.87%

Thus, instead of making $143, the investor actually lost about $23.

Personal rate of return

The return that you actually earn, your personal rate of return, is a weighted average of the lump sum return and the DCA return. The weights in the average depend on the amount of money that you have in your account at the beginning of a year (let's call it IL) and the amount of money that you contribute throughout the year (let's call it ID). The personal return is:

r_P = \frac{I_L (1+r_L) + I_D (1+r_D)}{I_L + I_D} - 1

For example, suppose that at the beginning of 2007, an investor had $5,000 in the TSP S Fund; and that, in 2007, he contributed $100 to the S Fund per pay period, for a total 2007 contribution of $2600. Since the lump sum investment return in the S Fund was 5.49% and the dollar cost averaging return was -0.87%, the personal rate of return for this particular investor was 3.31%:

\begin{align} r_P & = \frac{$5,000 \times (1 + 0.0549) + $2,600 \times (1 + -0.0087)}{$5,000 + $2,600} - 1 \\ & = 0.0331 = 3.31% \end{align}

Note

[1] I use this formula when discussing returns from dollar cost averaging on this website. The formula is not exact — it is an approximation. In the formula, the effective purchase price is the harmonic mean of daily prices. For an exact calculation, the effective purchase price should be set equal to the harmonic mean of purchase prices.

The approximation is appropriate if the investments are made at regular intervals that are small relative to the time period under consideration. For example, for contributions made every two weeks, I would only use the formula to calculate returns for a year or more.