There’s no magic in dollar cost averaging.

**Richard Goldberg** and **James Read** are principals with The Brattle Group, in San Francisco, Calif., and Cambridge, Mass., respectively.

A new approach to hedging commodity price risk appears to be gaining popularity in the natural gas and electric utility industries. This new approach, dubbed “dollar cost averaging” (DCA) by analogy to a popular personal investment strategy, appears to offer substantial cost savings over alternative hedging approaches. However, evaluating DCA critically, to locate the source and assess the magnitude of its apparent cost advantages, reveals that DCA doesn’t reduce commodity costs compared to time-averaging—a widely used and simpler approach to hedging commodity price risks.

Like the audience at a magic show, proponents of DCA have fallen prey to misdirection; because they focus on one component of commodity procurement costs, they fail to see another. Specifically, they miss the fact that the hedge cost advantage of DCA is fully offset by the balancing cost disadvantage—*i.e.*, the cost of covering remaining commodity requirements through other forward or spot market transactions. Furthermore, not only does DCA fail to provide a cost advantage, it’s less effective than time averaging at mitigating extreme market price outcomes.

#### Dollar Cost vs. Time Averaging

DCA is similar to another approach that many public utilities have adopted to hedge commodity price risk, an approach referred to here as “time averaging” (TA). Under TA, utilities cover their anticipated commodity requirements through a series of forward transactions, the amounts and timing of which are set forth in a pre-specified schedule. These contracts could specify physical or financial settlement, but in either case they would fix the commodity price in advance of delivery. For example, a gas distribution company might follow a program under which it purchases 5 percent of its forecast gas requirements each month beginning 18 months prior to delivery. Thus, its open position would be 100 percent 18 months or more prior to delivery but then decline by 5 percent in each subsequent month, falling to 70 percent 12 months prior to delivery, 40 percent six months prior to delivery, and 10 percent one month prior to delivery.^{1} One appeal of TA is that it limits managerial discretion and thus limits the scope for speculation, conscious or unconscious.

Note that if there were no uncertainty about its gas requirements, the hypothetical utility could eliminate uncertainty about gas procurement costs by purchasing the entire requirement immediately—the lump-sum approach.^{2} However, another appeal of TA hedging is that it provides time diversification. That is, because TA hedging consists of a series of purchase or hedge transactions over an extended period of time, the cost of gas doesn’t depend entirely on the market price on a single day. In our example the cost of gas would be an average of the 18 forward purchases plus the spot purchases required to fulfill its obligation at delivery. As a result, TA carries less of the risk of regret that would accompany a one-time hedge transaction.

Whereas a TA hedging program involves a schedule of fixed hedge quantities, DCA hedging involves a schedule of fixed hedge budgets. Specifically, the DCA budget schedule is prepared based on an initial schedule of hedge quantities and an initial forecast of commodity prices. If the forward price of the commodity happens to be equal to the initial forecast price on each scheduled hedge date—an exceedingly unlikely outcome—the actual hedge quantities will equal the scheduled hedge quantities. On the other hand, if the market price differs from the forecast price, the hedge quantity is adjusted so that the cost of the hedge—the product of actual hedge price and actual hedge quantity—is equal to budget.

For example, if the market price on a given hedge date was twice the forecast price, the hedge quantity would be one-half the scheduled hedge quantity. Thus, the DCA approach can be summarized as, “hedge more if the market price turns out to be low, relative to forecast, and hedge less if the market turns out high.” Note that DCA hedging entails the possibility that the ultimate hedge target—90 percent in our example—will be reached early, *i.e.*, prior to the initial schedule date, such as one month prior to delivery. The possibility also exists that the ultimate hedge target is never achieved, so that the utility has a larger open position immediately prior to delivery than scheduled.

#### Comparing Hedge Costs

The costs and risks of commodity procurement under the DCA and TA hedging approaches can be illustrated using a simple hypothetical example. Imagine that Ace Distribution Co. is obligated to deliver 1,000 Dth of natural gas in two years’ time. Future gas prices are of course uncertain. Suppose the current forward price for delivery two years from today is $6/Dth. If Ace chose to buy 1,000 Dth today at the current forward price, it could lock in the price of gas and thus it would know for sure that its gas procurement costs would be $6,000. Ace would no longer be exposed to uncertainty about gas prices.

Assume, however, that Ace doesn’t purchase its entire requirement immediately but instead follows a TA hedging program. Specifically, Ace will purchase 50 percent of its requirements under a forward contract one year prior to delivery and purchase the balance of its requirements in the spot market at delivery. Although Ace knows the current forward price of gas, it doesn’t know what the forward price will be one year from today, nor does it know what the spot price of gas will be at the delivery date. Thus, under this TA hedge program, Ace will remain fully exposed to price uncertainty for one more year and partially exposed for the year after that.

To describe the possible outcomes for gas procurement costs, it’s necessary to describe the possible outcomes for the market price of gas over the two years remaining to delivery. Suppose the forward price one year from today will be either $2 higher or $2 lower than the current forward price, that is, either $8 or $4, and the two outcomes are equally probable, like the outcomes of a coin toss. If the forward price one year from today is $8, then the spot price one year later (two years from today) will be either $10 or $6 ($2 higher or $2 lower than $8). On the other hand, if the forward price one year from today is $4, then the spot price at delivery will be either $6 or $2 ($2 higher or $2 lower than $4). Again, each outcome is equally probable. Figure 1 summarizes the four possible price scenarios—sequences of market prices today, one year from today, and two years from today. The forward price today is $6 in each scenario. In scenario 1 the forward price rises from $6 today to $8 at the scheduled hedge date one year from today and then to $10 at delivery two years from today. In scenario 2 the price rises from $6 to $8 and then falls to $6 again. In scenario 3 the price falls to $4 one year from now and then rises to $6 at delivery. Finally, in scenario 4 the price is $4 one year from now and $2 at delivery.

The critical feature of this hypothetical distribution of possible market price outcomes is that the mean of the possible outcomes one year from today and the mean of the possible outcomes at the delivery date equal the initial forward price: $6. As a result, the expected profit associated with deferring or accelerating the purchase or hedging of commodity requirements is zero. This must be true to rule out easy profit—a.k.a. arbitrage—opportunities. Utility managers and regulators shouldn’t assume there’s a free lunch in the market for natural gas or any other commodity.

#### Hedge Costs & Total Costs

Having enumerated all possible outcomes for the spot price of gas at delivery—as well as the possible outcomes for the forward price one year prior to delivery—we can compute the cost of gas under a strategy of fulfilling all requirements in the spot market. The calculations are easy: since the spot price at delivery will be $10, $6, or $2, procurement costs for 1,000 Dth of gas will be either $10,000, $6,000, or $2,000. The mean (average) cost of gas is $6,000.^{3}

Given these four price scenarios, the possible outcomes for the TA hedging program are summarized in Figure 2. If the forward price is $8 one year from now, the cost of the hedge gas—gas purchased forward one year prior to delivery—will be $4,000, which is 50 percent of the 1,000 Dth gas requirement (500 Dth) at the $8 forward price. On the other hand, if the forward price turns out to be $4, the cost of the hedge gas will be $2,000 (500 Dth at $4). The cost of purchasing the balance of the gas requirement at delivery (spot gas) will depend on the spot price. If the spot price is $10, the spot gas will cost $5,000; if the spot price is $6, the spot gas will cost $3,000; and if the spot market is at $2, the spot gas will cost $1,000. The total procurement cost corresponding to each price scenario—the sum of the cost of the hedge gas and the cost of the spot gas—is reported in the last column. The possible outcomes are $9,000, $7,000, $5,000 and $3,000, respectively. Notice that the mean cost of hedge gas is equal to the mean cost of spot gas. There’s no commodity cost advantage *ex ante* to buying forward immediately versus buying forward one year from today or buying spot two years from today.

What about the dollar cost average (DCA) hedge program? Figure 3 summarizes the possible gas procurement costs conditional on the same four price scenarios. Here the numbers are just a bit more complicated, because the quantity of hedge gas under DCA depends on the forward price at the hedge date one year from today. The hedge budget is $3,000, which is equal to the product of a scheduled hedge quantity of 500 Dth and a price forecast of $6/Dth. If the forward price is $8, the hedge quantity is reduced to 375 Dth—versus 500 under the TA hedge program—whereas if the forward price is $4, the hedge quantity is increased to 750 Dth. The total amount spent on hedge gas is the same in each scenario. That’s what “dollar cost averaging” means: a dollar budget is set and the hedge quantity is adjusted to fit the budget. As a result, when gas prices are high in relation to the initial forecast, less gas is purchased forward, and when the gas prices are low more gas is purchased forward. The spot purchase quantity is different in each scenario, too, because spot purchases must be sufficient to meet the 1,000 Dth obligation. If the forward price of gas is high at the end of the first year so that the hedge quantity is only 375 Dth, the spot purchase quantity will be 625 Dth. If the forward price of gas is low at the end of the first period, the spot purchase quantity will be only 250 Dth.

#### Average Costs

The result that has attracted the attention of DCA advocates is shown on the last line of the fifth column in Figure 3: the mean cost of hedge gas purchases. Indeed the average cost of hedge purchases under DCA is lower than the average cost of hedge purchases under TA. Whereas the average is $6 under TA, it’s only $5.33 under DCA. That looks like a substantial cost savings. But the cost of hedge gas is only one component of total commodity costs. The average cost of spot gas—the other component of commodity costs—is $6.86 under DCA versus $6 under TA. The average total cost of gas—the sum of hedge gas and spot gas purchases—is the same under TA and DCA: $6. Thus, under DCA, the lower average cost of hedge gas is fully offset by the higher average cost of spot gas. The average total cost of gas is identical under the TA and DCA hedge programs.

What’s going on here? Look again at Figure 3. If the forward price of gas is $8 on the hedge date—the high-price outcome—the spot price of gas one year later also is high on average: either $10 or $6, an average of $8. If the forward price of gas is $4 on the hedge date—the low outcome—the spot gas a year later is low on average: either $6 or $2, an average of $4. If one considers spot costs and not just hedge costs, there is no expected gain to deferring gas purchases if forward prices are high, and no expected gain to accelerating purchases if forward prices are low. Thus, if there’s no free lunch in the underlying commodity market, then there’s no magic in DCA hedging.

Clearly, both the description of the hedging problem and the description of the commodity market in this hypothetical example are highly simplified. Nevertheless they contain the essential elements of both the hedging problem and a competitive market, and thus the bare minimum needed to understand hedging and procurement costs under DCA versus TA. Although the initial forward price was used as the price forecast for purposes of adjusting DCA hedge quantities in this example, that was neither essential nor important. The “no magic in DCA” conclusion would follow with any fixed-price forecast.

At least when viewed from the standpoint of total commodity costs, hedging by DCA doesn’t provide a cost advantage relative to hedging by TA. It also isn’t evident from this example that DCA provides an advantage from the standpoint of risk management. To the contrary, the example suggests that if anything DCA is less attractive than TA from a risk management perspective. Notice that the highest total procurement cost outcome under DCA ($9,250) is greater than the highest cost outcome under TA ($9,000). This is a consequence of deferring hedge purchases—leaving a larger exposure to market risk—in the event market prices are high. Typically, risk managers are particularly concerned with the potential for very high cost outcomes when they design hedge programs. This example suggests that DCA is less effective than TA in mitigating such outcomes. The lowest cost outcome under TA is less than the lowest cost outcome under DCA, a consequence of accelerating purchases—reducing exposure to market risk even more—in the event market prices are low. But this observation needs to be followed up with a richer description of commodity market price uncertainty.

#### Comparing Risk Reduction

We used Monte Carlo simulation to compare the impact of hedging by DCA versus TA on risk reduction. In this next hypothetical example the total commodity requirement is 1,000 Dth and the hedge target is 90 percent. The TA hedge schedule begins 12 months prior to delivery and specifies 12 equal monthly installments. The DCA budget schedule likewise starts 12 months prior to delivery but specifies 12 equal monthly budgets. We use the initial forward price curve as the price forecast for purposes of adjusting the scheduled hedge quantities. If the hedge target is reached early—because market prices are low in relation to forecast prices—further hedge transactions are suspended, so that the hedge position is never more than 90 percent of 1,000 Dth one month prior to delivery in any scenario. In all scenarios, for both the TA and DCA hedging, the uncovered commodity requirement is fulfilled in the spot market at delivery.

This second example provides a more realistic description of market price uncertainty. In particular, it models a continuous distribution of market prices at the end of each month up to and including the delivery date. The initial forward price and thus the mean of the initial distribution of market prices at each future date is $6/Dth. Future prices are simulated as log-normal random variables with levels of volatility and mean reversion that aren’t unreasonable for the natural gas market.^{4}

Figure 4 compares the probability distributions of the cost of hedge gas (4-a), the cost of spot gas (4-b), and the total procurement cost (4-c).^{5} In each chart, the mean (average) costs under both hedging approaches are indicated by vertical lines. As in the first example, the average cost of hedge gas is lower under DCA than TA hedging, but the average cost of spot gas is higher; the average total commodity cost is the same under both hedging approaches.

The chances of extremely high hedge cost outcomes under the TA and DCA approaches are very different. Under DCA the distribution is truncated, meaning that very high cost outcomes are avoided.^{6} On the other hand, the chances of high spot cost outcomes under TA and DCA are also very different. DCA tends to do worse than TA in terms of spot purchase costs. When the sum of hedge costs and spot costs—*i.e.,* total purchase costs—is considered, the Monte Carlo simulation reaffrms the finding of the first example, namely, that the DCA and TA strategies have the same expected total costs. The differences between DCA and TA results are in the risk profiles.

To see the risk of extremely high- or low-cost outcomes under the two approaches, Figure 5 shows close-ups of the lower (5-a) and upper (5-b) tails of the probability distribution of total commodity costs. TA tends to have lower possible cost outcomes because, in scenarios where prices are falling, DCA fills the hedge target early. DCA tends to have the highest-cost outcomes because, in scenarios where market prices are rising, it defers hedging, which results in higher-cost spot purchases. TA hedging appears to reduce the chances of high-cost outcomes and increase the chances of low-cost outcomes relative to DCA.

#### Why DCA?

The cost of hedge gas—gas purchased forward prior to delivery—is indeed lower on average under DCA hedging than it is under TA hedging. On the other hand, the cost of spot gas—gas purchased to cover net open positions at delivery—is higher on average under DCA than it is under TA. When total procurement costs are considered, the apparent advantage of dollar cost average hedging vanishes. To top it off, dollar cost average hedging is less effective than time average hedging at mitigating exposure to high-cost outcomes.

In light of these findings, why would a public utility choose to hedge using DCA? One answer is that DCA has become popular due to a disproportionate regulatory focus on hedging costs—due, that is, to misdirection.

Unlike a magic show, the misdirection that has prompted dollar-cost averaging is surely unintended, but with so much attention focused on hedging costs, it’s easy to miss the full implications of linking hedge quantities to price outcomes. And since DCA doesn’t entail a procurement cost disadvantage—when forward and spot commodity purchases are taken into account—perhaps there is some other aspect of DCA that has made it a useful part of some utility hedging programs. But *caveat emptor:* there’s no magic in DCA.

#### Endnotes:

1. Time average hedging has itself been referred to as a “dollar cost average” approach. In this article the term “dollar cost averaging” is used to refer to hedging based on a schedule of fixed hedge budgets and “time averaging” is used to refer to hedging based on a schedule of fixed hedge volumes.

2. See, for example, Tim Simard, “Questioning Dollar Cost Averaging” *Energy Risk*, February 2006.

3. Recall that the $6 outcome occurs in two scenarios, each of which has a probability of 25 percent. The probability of the $6 outcome is therefore 50 percent.

4. Price changes were modeled using a one-factor mean-reverting volatility model, with factor volatility equal to 100 percent/year and a reversion rate of 100 percent/year. Although the numerical results depicted in these figures depend on these parameters, any reasonable description of the underlying commodity market would lead to the same qualitative results.

5. These figures are cumulative probability distributions. The y-axis location of each point on the curve is the probability that the cost outcome will be less than or equal to the outcome on the x-axis. Graphs of this sort typically exhibit an “S” shape, starting at zero for the least likely outcomes and rising to 100 percent at the highest possible outcomes.

6. The large number of scenarios in which the hedge costs under the DCA strategy equal the expected hedge cost under the TA strategy are cases in which market prices were high enough that the hedge budget was exhausted before the hedge target was reached, with the result that the remaining commodity requirement was purchased in the spot market, generally at high costs.