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Another Look at Dollar Cost Averaging

Gary Smith Heidi Margaret Artigue

Department of Economics Department of Economics

Pomona College Pomona College

425 N. College Avenue 425 N. College Avenue

Claremont CA 91711 Claremont CA 91711

gsmith@pomona.edu hmal2015@MyMail.Pomona.edu

Another Look at Dollar Cost Averaging

Abstract

Dollar cost averaging—spreading an investor’s stock purchases evenly over time—is widely

touted in the popular press because of the mathematical fact that the average cost per share is less

than the average price. The academic press has generally been skeptical, and attributes dollar

cost averaging’s popularity to investor naiveté and cognitive errors. Yet, dollar cost averaging

continues to be recommended by knowledgeable investors as a sensible way to avoid ill-timed

purchases. We argue that dollar cost averaging is, in fact, an imperfect, but helpful strategy for

diversifying investment decisions across time.

keywords: dollar cost averaging!

Another Look at Dollar Cost Averaging

An investor following a dollar cost averaging (DCA) strategy periodically invests a constant

dollar amount in stocks, adjusting the number of shares purchased as stock prices fluctuate.

When stock prices are high, fewer shares are bought; when prices are low, more shares are

purchased. Because the average cost weights the purchase prices by the number of shares

acquired at each price, the average cost is always less than the average price.

This mathematical fact has led many to recommend dollar cost averaging. In several Barron’s

columns and multiple editions of Successful Investing Formulas, first published in 1947, Lucile

Tomlinson persuasively extolled the virtues of cost-averaging plans. In the 2012 edition, with a

preface written by Benjamin Graham, Tomlinson wrote that dollar cost averaging is “the

unbeatable formula,” because, “No one has yet discovered any other formula for investing which

can be used with so much confidence of ultimate success, regardless of what may happen to

security prices, as Dollar Cost Averaging.”

The academic literature has generally been scornful, concluding that DCA is an inferior

strategy. Constantinides (1979) finds it ironic that DCA’s fixed investment rule is considered a

benefit. He argues that DCA’s inflexibility is a fatal flaw because making portfolio decisions

based on the additional information accumulated as time passes must be superior to committing

oneself to a strategy based only on current information. One counterargument is that in an

efficient market, it may be difficult to profit from future information.

Knight and Mandell (1992/1993) argue that DCA is suboptimal because the DCA portfolio’s

riskiness increases as stock purchases are made at regular intervals, but an investor with constant

!2

relative risk aversion would not increase a portfolio’s riskiness over time.

Thorley (1994) and Milevsky and Posner (2003) criticize a comparison of average price with

average cost, since investors cannot sell the stocks they accumulate at the historical average

price; however, this does not address the issue of whether DCA allows investors to acquire

stocks at a relatively low average cost compared to an initial lump-sum (LS) investment. The

answer obviously depends on whether stock prices subsequently increase or decrease. Thus,

Grable and Chatterjee (2015) observe that, while LS does well in the early stages of a bull

market, DCA may be more suitable for risk-averse investors who are fearful of a bear market.

Luskin (2017) reports that DCA does better when initiated during historical periods when

Shiller’s cyclically adjusted price-earnings (CAPE) ratio is abnormally high.

Bierman and Hass (2004) argue that if an investor is considering an investment that has an

expected return that is higher than from not being invested, then delaying purchases must reduce

the investor’s expected return. However, Cho and Kuvvet (2015) use a two-period mean-variance

framework to argue that if the risk-return opportunity locus is concave, some investors may

prefer DCA to LS.

Milevsky and Posner (2003) show that DCA can be financially attractive if investors have a

fixed target portfolio value. In particular, DCA has a positive expected profit if the price of the

stock being purchased is the same at the end of acquisition period as at the beginning, and the

size of this profit increases with the stock’s volatility. More generally, if the terminal stock price

is known in advance, the higher the volatility, the higher the expected value of the return from

DCA, so that there is some sufficiently high level of volatility such that DCA has a higher

expected return than LS. They argue that investors do, in fact, have terminal target prices, and

!3

conclude (presumably tongue in cheek) that

perhaps the investing masses are more sophisticated than we are willing to admit. Indeed,

they may be familiar with the Ito-Markovian structure of asset prices, yet they insist on

dollar-cost averaging because they believe market prices are driven by Brownian bridges as

opposed to Brownian motions. (Milevsky and Posner 2003, p. 191)

Some academics accept the argument that DCA is inferior to LS, and ask why so many

investors continue to use and recommend it. They identify the main benefits as encouraging

regular savings, protecting investors from trend chasing and portfolio churning, and avoiding a

single ill-timed purchase that might scare them away from future stock purchases (Haley 2010).

Others (Statman 1995; Dichtl and Drobetz 2011) argue that dollar cost averaging can be justified

by Kahneman and Tversky’s (1992) prospect theory; however, Leggio and Lien (2001) present

evidence to the contrary and, in addition, argue that DCA is less successful with more volatile

stocks—contradicting Milevsky and Posner.

On the empirical front, Williams and Bacon (1993) look at DCA and LS investments in the

S&P 500 and find that LS generally has a higher return, which is unsurprising since, historically,

being in the stock market has, on average, been more profitable than being out of the market.

Leggio and Lien (2003a, 2003b) consider the average return and various risk measures and

conclude that DCA is inferior to LS. Thorley (1994) reports that LS has a somewhat higher

average return and lower standard deviation than DCA. Similarly, Rozeff (1994) concludes that,

relative to DCA, the LS strategy tends to have a higher return because it is more fully invested in

stocks and the LS strategy tends to have a lower variance because it is more uniformly invested

in stocks, as opposed to being lightly invested early on and heavily invested later. Using

!4

simulations with historically based parameters, Abeysekera and Rosenbloom (2000) conclude

that DCA generally has a lower return and less risk. Using bootstrapped CRSP index returns, Lei

and Li (2007) report that the results from DCA and LS are statistically indistinguishable over

short and long horizons, but that DCA has a higher probability of a negative return over

intermediate-term horizons.

There is some empirical evidence in support of DCA. Balvers and Mitchell (2000) and

Brennan, Li, and Torous (2005) argue that DCA may be advantageous if stock returns are

negatively autocorrelated, as in the mean-reversion literature (Poterba and Summers 1988; Fama

and French 1988). To exploit this mean reversion, Dunham and Friesen (2012) recommend an

“enhanced DCA” strategy in which stock purchases are increased after a market decline, and

reduced after a market advance. Dubil (2005a) and Trainor (2005) argue that DCA reduces

shortfall risk, the chances of wealth dropping below a specified floor during the investor’s

lifetime.

Israelson (1999) uses mutual fund data to argue that DCA is a superior strategy, especially for

low-volatility mutual funds—which contradicts the more widespread argument that DCA is best

suited for volatile investments. Atra and Mann (2001) argue that the January effect and other

seasonal patterns in stock returns affect comparisons of DCA and LS. Paglia and Jiang (2006)

report that the day of the month matters, too. These claims are examples of the more general

point made by Dubil (2005b) that a comparison of DCA with LS depends on the specific stock

price patterns that happen to occur; for example, LS will have an advantage if stock prices head

up after the plan is started, while DCA has the advantage if prices drop. Comparing strategies

that are always launched on January 1 may distort the evidence.

!5

Overall, the academic verdict is that DCA is an inferior strategy both in theory and practice,

but investors use it anyway for not particularly persuasive reasons. The unsettling thing about

this academic conclusion is that many of those who endorse DCA are, in other respects, very

sensible and experienced. Perhaps the academic verdict is too harsh and underestimates the

wisdom of experienced practitioners?

Despite the generally negative conclusions in peer-reviewed academic research, dollar cost

averaging is widely endorsed in the popular press. Damato (1994) wrote in The Wall Street

Journal that,

People who invest in stocks regularly get the benefit of “dollar cost averaging.” Because a

fixed sum is invested every month or quarter buys more shares when prices are down, the

investor’s average cost per share ends up being lower than the average price in the market

over the same period.

Sylvia Porter (1979) describes dollar cost averaging as “how to beat the stock market.” The

Investment Company Institute (1984) says that, “It takes the ups and downs of the market and

turns them to advantage.” The New York Times Financial Planning Guide (1985) calls dollar cost

averaging a “time-tested investing method” with a “seemingly magic result.”

The financial press is one thing, but it is striking that DCA is recommended by several

extremely knowledgeable observers, including Levy and Sarnat (1972), Sharpe (1978), Dreman

(1982), Loeb (1996), Bogle (2015), and Tobias (2016). In the 2015 edition of his classic book, A

Random Walk Down Wall Street, Malkiel writes that,

Dollar cost averaging can reduce the risks of investing in stocks and bonds….Periodic

investments of equal dollar amounts in common stocks can reduce (but not avoid) the

!6

risks of equity investment by ensuring that the entire portfolio of stocks will not be

purchased at temporarily inflated prices. (p. 355)

Many individuals and institutions practice what these advisors preach, ranging from Fidelity’s

Automatic Account Builder, which transfers money each month from a participant’s bank

account to a stock fund, to the MacArthur Foundation, which used dollar cost averaging in

1985-1986 (at the rate of $100 million a month) to invest $1.4 billion realized from the

liquidation of real estate and other tangible assets.

Neither unequivocal praise or scorn is warranted. The core argument in favor of dollar cost

averaging gauges performance by cost, rather than rate of return, and, when this is taken into

account, the alleged simple virtues of cost averaging vanish. On the other hand, many criticisms

of dollar cost averaging neglect the return on funds not invested in stock or don’t compare

equivalent DCA and LS strategies (for example, equally risky strategies). When these factors are

taken into account, dollar cost averaging may be a reasonable approach to investing in volatile

stocks. We first demonstrate that the claimed low-cost virtue is an illusion and then use a novel

approach to analyze how dollar cost averaging can effectively diversify investment risk across

time.

The Illusion

The presumed advantages of dollar cost averaging are invariably based on calculations similar to

those shown in Table 1. A stock currently sells for $60 a share and its price will either rise or fall

by 50 percent. If $900 is invested each period, the average cost is less than the average price, no

matter whether the stock’s price goes up or down. The intriguing implication is that, although the

expected value of the price change is zero, the expected value of the average cost is less than the

!7

expected value of the price in the second period. In Table 1, the expected value of the average

cost is 0.5($72) + 0.5($40) = $56, and the expected value of the price is 0.5($90) + 0.5($30) =

$60. DCA appears to have a positive expected profit even if changes in stock prices are

independent with a mean of zero.

Table 1 Dollar Cost Averaging Reduces Costs

Price Rises to $90 Price Falls to $30

Time period Price Shares Cost Price Shares Cost

1 $60 15 $900 $60 15 $900

2 $90 10 $900 $30 30 $900

Average $75 $72 $45 $40

This conclusion is misleading. The mathematical fact that the expected value of the average

cost is less than the expected value of the price does not imply a positive expected rate of return,

because it neglects the varying amounts invested in the each scenario. The relevant data are

shown in Table 2. DCA gains $18 a share if the price rises and loses $10 a share if it falls. The

average of these two numbers is indeed positive. But it is also irrelevant since the number of

shares is not the same. DCA gains $18 a share on 25 shares or loses $10 a share on 45 shares—a

$450 profit on an $1,800 investment or a $450 loss on an $1,800 investment. Even though the

average cost is less than the average price, if a stock’s expected return is zero, DCA’s expected

profit is also zero.

This is reminiscent of the Martingale betting system (Mitzenmacher and Upfal, 2005) in

which bets are doubled after every loss. If the outcomes are independent and each wager has a

zero expected return, the expected return from the strategy must be zero—no matter how one

!8

adjusts the size of the wagers as time passes. This is true of a Martingale betting strategy and it is

true of a DCA investing strategy. What these strategies do change is the shape of the payoff

distribution, including the variance.

Table 2 Dollar Cost Averaging Does Not Increase Expected Return

Price Rises to $90 Price Falls to $30

Average cost $72 $40

Number of shares 25 45

Total investment $1,800 $1,800

Value of portfolio $2,250 $1,350

Dollar gain $450 - $450

Rate of return 25% - 25%

A Mean-Variance Analysis

Malkiel and other supporters of cost averaging emphasize the risk of making a lump sum

investment at an unfortunate time; i.e., shortly before a big drop in prices. Cost averaging

essentially diversifies one’s investments, not across different stocks, but across different purchase

dates. We use mean-variance analysis with Tobin’s separation theorem to demonstrate that time-

diversification can be an important virtue.

An evaluation of dollar cost averaging should take into account the rate of return on funds not

invested in stocks. We assume two assets, safe Treasury bills and a risky stock. We assume that

the return R on Treasury bills is constant and that the stochastic stock return St in period t is

independent of the stock’s return in other periods and has a constant mean 𝜇 and standard

deviation 𝜎, with the stock’s expected return larger than the return on T-bills. DCA does not

!9

benefit from mean reversion in our model, and LS has the built-in advantage of a stock expected

return that is higher than the return on Treasury bills. Nonetheless, DCA has some appealing

features.

The LS strategy involves an immediate investment of all wealth in stocks. The DCA strategy

is to make regular, periodic constant stock purchases over an investment horizon of T periods.

Money not initially invested in stocks is parked in Treasury bills.

Mean-variance analysis is usually applied to a decision on how to allocate wealth among a

portfolio of different stocks or asset classes. It can be applied to dollar cost averaging by

considering each periodic investment in stocks as an initial investment in Treasury bills that is

converted into stocks at the appropriate time.

Specifically, one way to frame the timing question is to identify the delayed purchase of stock

when there are j periods left in the investment horizon as an asset that yields (1 + R) for each of

the first T - j periods and (1 + St) for the remaining j periods. The gross return over the full T-

period horizon is

! j = 0, …, T (1)

Z0 is the gross return on an investment in Treasury bills for all T periods; Z1 is the return on an

investment in Treasury bills for T - 1 periods, followed by a stock investment in the last period;

and so on. A decision to spread stock purchases over a T-period horizon is a portfolio allocation

of wealth among the T assets j = 1, 2, … T. The overall return is

! (2)

where a fraction 𝜆j of wealth is invested in asset j. For each value of 1 ≤ j ≤ T, 𝜆j is invested now

Zj=(1+R)T−j(1 +St)

t=T+1−j

T

∏

Z=

λ

jZj

j=1

T

∑

!10

in the safe asset and, after T - j periods, 𝜆j(1 + R)T-j is invested in stocks. Dollar cost averaging

corresponds to

! j = 1, …, T (3)

where C is set so that the sum of the 𝜆j is 1. For the lump-sum strategy, 𝜆T = 1, and 𝜆j = 0 for 1 ≤

j ≤ T.

If the stock returns are independent across time, then a standard mean-variance portfolio

analysis can be conducted using

! 1 ≤ j ≤ T (4)

Since asset 0 is a safe asset with return (1 + R)T, Tobin’s (1957) separation theorem applies, and

we can identify the optimal ratios 𝜆j/𝜆T for j ≥ 0, regardless of risk preferences. These can be

rescaled to give the optimal fractions 𝛾j of the risky portfolio:

! 1 ≤ j ≤ T

The optimal proportions depend on the volatility of stock prices—the more volatility, the

stronger is the case for time diversification. The time-diversification argument is consequently

more persuasive for individual stocks than for stock indexes. We assume a constant 3 percent

annual return on Treasury bills and a 7 percent expected return on stocks, and use different

values for the standard deviation.

λ

j=C

(1+R)T−j

EZj

⎡

⎣⎤

⎦=(1+R)T−j(1 +

µ

)j

Var Zj

⎡

⎣⎤

⎦=(1+R)2(T−j)(1+

µ

)2+

σ

2

( )

j−(1+

µ

)2j

( )

Cov ZiZj

⎡

⎣⎤

⎦=(1+R)T−i+T−j(1 +

µ

)j−i(1+

µ

)2+

σ

2

( )

i−(1+

µ

)2i

( )

γ

j=

λ

j/

λ

T

λ

j/

λ

T

( )

j=1

T

∑

!11

The average annual return on Treasury bills since 1926 has been approximately 3 percent and

the average annual return on large-cap stocks has been 10 percent (Ibbotson, Grabowski,

Harrington, and Nunes 2016). However, a study of the twentieth-century performance of the

stock markets in 39 different countries found that the U. S. stock market beat all the rest (Jorian

and Goetzmann 1999). It seems unlikely that investors worldwide knew that the twentieth

century would turn out to be America’s century and the U. S. stock market would turn out to be

the winner. It is more likely that the remarkable performance of the U. S. stock market was a

pleasant surprise to investors who owned U. S. stocks—and we can hardly count on pleasant

surprises indefinitely.

The equity premium seems to have declined over time and the ex ante premium is likely to be

significantly smaller in the future than the ex post premium has been in the past. Fama and

French (2002) estimate a risk premium of 2.55 to 4.32 percent for 1951 to 2000 relative to short-

term risk-free bonds. Siegel (1999) suggests it might be as low as 1 percent to 2 percent going

forward. We use 4 percent as our baseline premium (a 7 percent expected return on stocks

compared to a 3 percent return on safe Treasury securities), but also consider 2 percent and 6

percent premiums.

Table 3 shows the means, standard deviations, and correlation coefficients for annual

investments over a five-year horizon, in accordance with Equation 1, and the optimal allocation

in the risky portfolio using Tobin’s Separation theorem. The fifth asset is an initial investment in

stocks and, because its return is the least correlated with the return on the first asset (four years

of Treasury bills, followed by a stock investment), the first asset has the second largest optimal

allocation. The spread of investments across all five assets is much more dispersed than is the LS

!12

all-in strategy, which sets 𝛾5 = 1 and the other 𝛾j = 0.

Table 3 Mean-Variance Analysis with a 5-Year Horizon and R = 3%, 𝜇 = 7%, and 𝜎 = 50%

correlation coefficients optimal

Asset Mean, % SD, % 1 2 3 4 5 𝛾j

1 20.43 56.27 1.00 .67 .52 .42 .36 .229

2 25.11 87.07 1.00 .77 .63 .54 .174

3 29.96 116.86 1.00 .87 .69 .132

4 35.01 148.11 1.00 .85 .101

5 40.26 182.04 1.00 .365

Figure 1 shows the percentage of the risky portfolio invested initially in stocks for different

investment horizons and 30 percent and 50 percent annual standard deviations of the stock

return. Time diversification is most attractive for less correlated alternatives, and the correlations

among the alternatives drop for riskier stock and longer horizons. In Figure 1, as the investment

horizon lengthens or the standard deviation increases, the initial investment in stocks approaches

that recommended by dollar cost averaging.

Figure 2 shows the Markowitz frontier using the assumptions in Table 3: annual investments

over a five-year horizon with a 50 percent annual standard deviation in the stock return. The

DCA portfolio is very close to the optimal portfolio implied by Tobin’s separation theorem.

!13

!

Figure 1 Fraction Initially Invested in Stocks, annual investments

!

Figure 2 Markowitz Frontier, annual investments over a 5-year horizon with 𝜎 = 50%

Figure 3 is the analogue of Figure 1, but now with monthly investments over horizons up to

60 months and with 20 percent and 30 percent standard deviations of stock returns. Again, a

0

10

20

30

40

50

60

70

80

90

100

012345678910

Initial stock investment, percent

Horizon, years

LS

Optimal

σ = 30

DCA Optimal

σ = 50

•

•

•

•

0

5

10

15

20

25

30

35

40

45

0 20 40 60 80 100 120 140 160 180 200

Mean, percent

Standard deviation, percent

Optimal

DCA

LS

!14

longer horizon or standard deviation supports the DCA idea of diversifying stock purchases

across time. Figure 4 shows the corresponding Markowitz frontier for monthly investment over a

60-month horizon with a 20 percent standard deviation. Again, the DCA portfolio is close to the

optimal portfolio implied by Tobin’s separation theorem.

!

Figure 3 Fraction Initially Invested in Stocks, monthly investments

One might think that a higher equity premium would pull the optimal portfolio towards a

larger initial investment in stocks, because it increases the opportunity cost of investing in

Treasury bills. However, the equity premium has little effect on the curvature of the Markowitz

frontier and, looking at Figures 2 and 4, an upward shift in the Markowitz frontier slides the

optimal portfolio away from LS. Table 4 confirms this. As the equity premium increases, the

optimal initial investment in stocks moves away from the 100 percent figure used by LS towards

the 21.2 percent figure used by DCA.

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60

Initial stock investment, percent

Horizon, months

LS

DCA

Optimal

= 10

Optimal

= 20

!15

!

Figure 4 Markowitz Frontier, monthly investments over a 5-year horizon with 𝜎 = 20%

Table 4 Initial Stock Investments with a 5-Year Horizon and R = 3%

Optimal Portfolio

LS DCA 𝜎 = 30% 𝜎 = 50%

𝜇 = 5% 1.00 .212 .665 .396

𝜇 = 7% 1.00 .212 .611 .365

𝜇 = 9% 1.00 .212 .558 .335

The time-diversification value of DCA depends on the investor’s assumptions, with a longer

horizon and larger standard deviation of stock returns moving the optional portfolio away from

LS and closer to DCA. Dollar cost averaging is obviously not always a reasonable approximation

to the optimal portfolio implied by Tobin’s separation theorem. Our point is simply that Bogle,

Malkiel, Tobias, and other wise and experienced investors are neither naïve or foolish. When

contemplating an especially risky investment, there is merit in time diversification even if the

•

•

•

•

0

5

10

15

20

25

30

35

40

45

0 50 100 150 200 250 300 350 400 450

Mean

Standard Deviation

Optimal

DCA

LS

!16

stock returns are independent with a higher expected return than Treasury bills.

Empirical Evidence

We used historical data to compare the DCA and LS strategies. Even though ex post returns may

be an unreliable guide for ex ante decisions, it seems worthwhile to consider how these strategies

would have fared in the past.

We looked at the 84 stocks that have been in the Dow Jones Industrial Average from October

1, 1928, when the Dow was expanded from 20 to 30 stocks, through December 31, 2016, a total

of 23,219 trading days. These are all prominent companies, widely followed by investors, with

reliable data, though the fluctuations in the daily returns are presumably smaller than those for

many stocks that might be DCA candidates. The daily returns on Treasury bills and these 84

stocks were taken from the Center for Research in Security Prices (CRSP) data base.

To avoid biases that might arise because stocks added to the Dow generally did well before

their inclusion, we only looked at purchases of Dow stocks that were made while the stocks were

in the Dow. In order to avoid distortions that might be caused by seasonal or day-of-the-month

patterns, our portfolio simulations used every Dow stock and every starting date; for example,

IBM on April 18, 1978. When occasional gaps in the CRSP data base of returns arise, we assume

that the portfolio temporarily earns the Treasury-bill return. All initial purchases were made on

dates such that the investment horizon ended on or before December 31, 2016.

The LS strategy purchases the selected stock on the selected day and holds it until the end of

the investment horizon. The DCA strategy uses the asset allocation described by Equation 3 to

invest in the T assets that make up the DCA strategy. A specified fraction of the portfolio is

invested in the stock initially, with the remainder parked in Treasury bills. Each successive

!17

period, an additional investment (equal to the initial investment) is made in the stock. After

looking at Figures 1 and 3, we also considered a 50-50 strategy of investing 50 percent of wealth

immediately in stocks and spreading the remainder equally over the horizon.

We considered two horizons consistent with the earlier theoretical analysis—annual purchases

for five years and monthly purchases for five years. There are approximately 250 trading days in

a year, so we assumed that the annual purchases were made every 250 trading days after the

initial purchase and that the monthly purchases consisted of purchases made every 20 trading

days after the initial purchase.

Table 5 shows the results. As expected, the LS strategy had the highest average return and the

highest standard deviation, and the DCA strategy had the lowest values, with the 50-50 strategy

in between. The LS strategy had the lowest Sharpe ratios, while DCA and 50-50 were very

similar. Historically, an investment in Treasury bills and either DCA or 50-50 would have

dominated an equally risky investment in Treasury bills and LS.

Table 5 Simulated of Purchases of Dow Stocks Over a Five-Year Horizon, 1928-2016

Average Annual Return, % Standard Deviation, % Sharpe Ratio

Annual Purchases

LS 73.63 106.54 0.41

DCA 51.76 60.55 0.49

50-50 59.63 74.48 0.50

Monthly Purchases

LS 69.73 101.86 0.40

DCA 44.32 50.36 0.47

50-50 56.55 71.76 0.49

!18

Conclusion

Dollar cost averaging is not as foolish as it is sometimes portrayed. It is well known that its

mechanical nature may encourage saving and reduce the emotional anxiety associated with

making decisions. It is not well known that cost averaging can also be a valuable way of

diversifying investment decisions across time, which is particularly appealing when investing in

volatile stocks over a substantial horizon. Dollar cost averaging is not always optimal but, in

some circumstances, it may be a reasonable approximation.!

!19

References

Abeysekera, Sarath R., and E. S. Rosenbloom. 2000. A Simulation Model for Deciding Between

Lump-Sum and Dollar-Cost Averaging. Journal of Financial Planning 13 (6): 86–92.

Atra, Robert J. and Thomas L. Mann. 2001. Dollar-Cost Averaging and Seasonality: Some

International Evidence. Journal of Financial Planning, 14 (7): 98-105.

Balvers, R. J., & Mitchell, D. W. (2000). Efficient Gradualism in Intertemporal Portfolios.

Journal Of Economic Dynamics And Control, 24(1), 21-38.

Bierman, Jr., Harold, and Jerome E. Hass. 2004. Dollar-Cost Averaging. Journal of Investing, 13

(4): 21–24.

Bogle, J. C. 2015, Bogle on Mutual Funds, New York: Wiley.

Brennan, Michael J., Li, Feifei, and Walter N. Torous. 2005. Dollar Cost Averaging. Review of

Finance 9 (4): 509–35.

Cho, David D., and Emre Kuvvet. 2015. Dollar-Cost Averaging: The Trade-Off Between Risk

and Return. Journal of Financial Planning 28 (10): 52–58.

Constantinides, G.M. 1979. A Note On The Suboptimality Of Dollar-Cost Averaging As An

Investment Policy. Journal of Financial and Quantitative Analysis, 14 (2), 443-450.

Damato, Karen Slater, 1994, Long-term Investors Can Reap Rewards From Buying Stocks

During Bear Markets, The Wall Street Journal, April 20.

Dichtl, H., & Drobetz, W. 2011. Dollar-Cost Averaging and Prospect Theory Investors: An

Explanation for a Popular Investment Strategy. Journal Of Behavioral Finance, 12 (1),

41-52.

Dreman, David. 1982. New Contrarian Investment Strategy. New York: Random House.

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Dubil, R. 2005a. Lifetime Dollar-Cost Averaging: Forget Cost Savings, Think Risk Reduction.

Journal of Financial Planning, 18 (10), 86-90.

Dubil, Robert. 2005b. “Investment Averaging: A Risk-Reducing Strategy.” Journal of Wealth

Management 7 (4): 35–42.

Dunham, Lee M., and Geoffrey C. Friesen. 2012. Building a Better Mousetrap: Enhanced Dollar-

Cost Averaging. Journal of Wealth Management 15 (1): 41–50.

Fama, Eugene F. and Kenneth R. French 1988, Permanent and Temporary Components of Stock

Prices, Journal of Political Economy, 96 (2), 246-273.

Fama, Eugene, and Kenneth French, 2002. The Equity Premium, The Journal of Finance, 57 (2)

Grable, John E., and Swarn Chatterjee. 2015. Another Look at Lump-Sum versus Dollar-Cost

Averaging. Journal of Financial Service Professionals, 69 (5): 16–18.

Haley, Simon, 2010. Explaining the Riddle of Dollar Cost Averaging, Cass Business School.

Ibbotson, Roger, Grabowski, Roger J., Harrington, James P., and Carla Nunes, 2016. 2016

Stocks, Bonds, Bills, and Inflation (SBBI) Yearbook, New York: Wiley.

Investment Company Institute. 1984. Discipline: It Can’t really be Good for You, Can It?

Jorian, Philippe, and William N. Goetzmann, 1999, Global Stock Markets in the Twentieth

Century, The Journal of Finance, 54 (3),953-980.

Knight, J.R. and Mandell, L. 1992/1993. Nobody Gains From Dollar Cost Averaging: Analytical,

Numerical And Empirical Results. Financial Services Review, 2 (1), 51-61.

Leggio, Karyl B., and Donald Lien. 2001. Does Loss Aversion Explain Dollar-Cost Averaging?

Financial Services Review 10 (1-4): 117.

Leggio, Karyl B., and Donald Lien, 2003a. Comparing Alternative Investment Strategies Using

!21

Risk-Adjusted Performance Measures. Journal of Financial Planning, 16 (1), 82-86.

Leggio, Karyl B., and Donald Lien, 2003b. An Empirical Examination of the Effectiveness of

Dollar-Cost Averaging Using Downside Risk Performance Measures. Journal Of Economics

And Finance, 27 (2), 211-223.

Lei, Adam Y. C., and Huihua Li. 2007. Automatic Investment Plans: Realized Returns and

Shortfall Probabilities. Financial Services Review 16 (3), 183–95.

Levy, H. and M. Sarnat. 1972, Investment and Portfolio Analysis. New York, NY: John Wiley

and Sons, 244-248.

Loeb, M. Marshall, 1996, Loeb’s Lifetime Financial Strategies. 1996. Boston: Little, Brown, 68–

70.

Luskin, Jon M. 2017. Dollar-Cost Averaging Using the CAPE Ratio: An Identifiable Trend

Influencing Outperformance. Journal of Financial Planning 30 (1): 54–60.

Malkiel, Burton G., 2015, A Random Walk Down Wall Street, New York: W. W. Norton

Milevsky, M.A. and Posner, S.E. 2003. A Continuous-Time Re-examination of the Inefficiency

of Dollar Cost Averaging. International Journal of Theoretical & Applied Finance, 6 (2),

173-194.

Mitzenmacher, Michael and Eli Upfal, 2005, Probability and computing: randomized algorithms

and probabilistic analysis, Cambridge University Press, p. 298.

Paglia, John K., and Xiaoyang Jiang. 2006. Implementing a Dollar-Cost Averaging Investment

Strategy: Does the Date of the Month Matter?, Journal of Wealth Management 9 (2): 54–62.

Porter, Sylvia. 1979. Sylvia Porter’s New Money Book for the 80’s. New York: Doubleday, 1012.

Poterba, James M. and Lawrence H. Summers, 1988, Mean Reversion in Stock Prices, Journal of

!22

Financial Economics, 22, 27-59.

Rozeff, M.S. 1994. Lump-sum Investing Versus Dollar-Averaging. Journal of Portfolio

Management, 20 (2), 45-50.

Sharpe, W.F. 1978. Major Investment Styles, The Journal of Portfolio Management, 5, 68-74.

Siegel, Jeremy J. 1999. The Shrinking Equity Premium, The Journal of Portfolio Management,

26 (1), 10-17. as low as 1% to 2%

Statman, M. 1995, A Behavioral Framework For Dollar-Cost Averaging. Journal of Portfolio

Management, 22 (1), 70-78.

Thorley, S.R. 1994. The Fallacy of Dollar Cost Averaging. Financial Practice and Education, 4

(2), 138-143.

Tobias, Andrew. 2016. The Only Investment Guide You’ll Even Need. New York: Houghton

Mifflin Harcourt.

Tobin, James. 1958. Liquidity preference as behavior towards risk. The Review of Economic

Studies. 25 (2): 65–86.

Tomlinson, L. 1947. Successful Investing Formulas, New York: Barron’s.

Tomlinson, L. 2012. Successful Investing Formulas, Snowball Publishing.

Trainor, William J., Jr. 2005. Within-horizon exposure to loss for dollar cost averaging and lump

sum investing. Financial Services Review, 14 (4), 319-330.

Tversky, Amos; Kahneman, Daniel. 1992. Advances in prospect theory: Cumulative

representation of uncertainty. Journal of Risk and Uncertainty. 5 (4): 297–323.

Williams, R.E. and Bacon, P.W. 1993. Lump-sum Beats Dollar Cost Averaging. Journal of

Financial Planning, 6 (2), 64–67.