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Aggregation and Dollar-Weighted Returns Issues

Abstract

This study is motivated by Dichev’s (2007) finding that buy-and-hold returns are

significantly larger than dollar-weighted returns for the major stock market indexes and

Hayley’s (2014) exposition that hindsight effects not poor timing are driving these

results. In the case of domestic markets Keswani and Stolin (2008) demonstrate that

Dichev’s findings are not statistically robust due to their sensitivity to the time period

chosen. This study confirms Keswani and Stolin’s (2008) results and demonstrates that

the relative magnitude of beginning and ending balances are the principal determinants of

the differences in the estimated returns. In addition, it is shown that Hayley’s

methodology used to break Dichev’s differences between geometric mean returns and

dollar-weighted returns into hindsight and timing effects is flawed.

I. Introduction

This paper reexamines the relationship between buy-and-hold or geometric mean

returns (GM) and aggregate dollar-weighted returns (DWR) following the methodology

used by Dichev (2007). Dichev (2007) discusses how investors’ actual returns are

determined by both the returns on securities and the size and timing of investors’

investment flows. Dichev’s (2007) study explicitly recognizes this difference and

attempts to develop a new more accurate measure of stock investors’ actual historical

returns by applying the concept of DWR. DWR has been extensively discussed in the

literature (Bodie, Kane and Marcus (1996), Zweig (2002), and Johnston, Hatem and

Carnes (2010)). When cash inflows/outflows are present, the result that DWR and GM

will differ is well known. However, Dichev (2007) uses DWR in a unique way, to

systematically evaluate historical stock returns in aggregate. That is, Dichev attempts to

determine if the DWR and GM differ for investors in aggregate. To our knowledge,

Dichev (2007) is the first paper that attempts to combine aggregation and DWR.

To address the difficulty of obtaining accurate measures of capital flows, caused

by their case specificity, Dichev (2007) proposes a solution using market capitalization

and the total value weighted return. Using monthly data he calculates capital flows as:

Distributions = dt = MVt-1 (1+rt) – MVt (1)

Where MV is market capitalization and r is the return on the index. A positive

distribution means capital flows from companies in the index to investors during that

period. A negative distribution means capital flows from investors to the companies in the

index. In the DWR calculation the beginning market value of the index and capital

inflows are given a negative sign and the ending market value of the index and capital

outflows are given a positive sign.

Using this measure, Dichev concludes that DWR are systematically lower than

GM. He suggests that poor timing of investor capital flows drives these results.

However, in response Keswani and Stolin (2008) demonstrated that the results are not

statistically robust and that there is little evidence that the performance gap between GM

and DWR is of the size reported by Dichev. However, other studies using similar

methodologies have found significant differences between the GM and DWR that are in

line with the findings of Dichev. Freisen and Sapp (2007) find the GM return to be 1.6%

higher than the DWR on an annual basis for U.S. mutual funds. Examining U.K. mutual

funds, Clare and Motson (2010) find a difference of 0.8%, and Dichev and Yu (2011)

report a difference of 3.6% for hedge funds. Overall the literature suggests that in

aggregate, the effects of bad investor timing have been significant.

Hayley (2014) breaks down Dichev’s results into two components, a timing effect

and a hindsight effect. In essence he demonstrates that distributions and injections of

2

funds reweight the monthly returns in the DWR calculation. Reweighting future returns is

timing, while reweighting past returns is a hindsight effect. Hayley derives a method to

quantify and remove the hindsight effect and demonstrates that very little of the

difference in returns found in Dichev’s(2007) study is due to market timing.

This study focuses on the findings of Dichev (2007) and Hayley (2014). In

Section 2, it is demonstrated that Dichev’s results are driven principally by differences in

the beginning and ending portfolio balances. This is contrary to Dichev’s (p. 388) claim

that, “varying capital exposure over time is insufficient to create differences between

buy-and-hold and dollar-weighted returns”, and Hayley’s claim that dollar-weighted

returns are low and thus the differences (GM –DWR) are significant because aggregate

investor flows reflect past returns rather than future returns.

The third section presents simulations illustrating problems with Hayley’s

methodology. The results of these simple simulations demonstrate that Hayley’s

methodology of using a step-by-step process to determine aggregate timing and hindsight

effect is flawed. The final section contains a summary and conclusions.

II. The Empirical Model, Data and Results

Using data for NYSE/AMEX and NASDAQ value-weighted index (1973-2002),

Dichev (2007) finds a difference between the two mean returns of 1.3 percent and 5.3

percent, respectively. Keswani and Stolin (2008) divide the NYSE/AMEX data into three

sub-periods and find only the earliest sub-period has any consequential performance gap.

Expanding the NASDAQ data set from 1973-2002 to 1973-2006 they find that the

performance gap is reduced by almost half to 2.9 percent.

3

To assure consistency of our data and method, the original empirical findings

were reproduced for the NASDAQ and the NYSE/AMEX value-weighted indexes for the

each period. The monthly index data is from the Center for Research in Security Prices

(CRSP) database. For the NASDAQ 1973-2002 samples the GM and DWR are identical

to those found by Keswani and Stolin (2008) who estimated a DWR 0.1percent lower

than Dichev (2007). When Keswani and Stolin extended the NASDAQ sample to 2006

the difference remained 3 percent with a GM of 10.5 percent and a DWR of

7.5percent.Further, increasing the NASDAQ sample to include up to 2010 the difference

stays at 3 percent with a GM of 9.6 percent and a DWR of 6.6 percent. Examination of

the NYSE/AMEX returns the 1926-2002 geometric and DWR are 9.9 percent and 8.6

percent, respectively. Again these numbers match those reported by Dichev. The GM and

DWR are relatively stable with little variation when the sample size increases and the

difference between the GM and DWR varies between 1.1 percent and 1.3 percent.

In discussing the effects of aggregation Dichev (2007) states that by using indexes

and hence a high level of aggregation allows for generality and a more comprehensive

investigation. Dichev’s discussion of the downside of aggregation is limited to the

statement that DWR are likely to be more pronounced at lower levels of aggregation.

Dichev calculates the DWR for the overall period (years x months) then annualizes this

overall return. Dichev’s method of calculating the DWR over all data points and long

sub-periods) drives the empirical results as DWR is sensitive to starting and ending

market values of the index.

To demonstrate this point, the NASDAQ index is used since in the data sets used

by Dichev, it had the largest difference between the GM and DWR. Tables 1, 2 and

4

3show the NASDAQ cash flows used to calculate the DWR for the three periods. Table

1contains the results for 1973-2002 used in the Dichev’s study. Table 2presents the results

for 1973-2006 used in the Keswani and Stolin study. Table 3 updates the data to 1973-

2010. To order the cash flows by their relative impact on the estimated IRR they are

sorted by their discounted present value (Tables 1-3, Column 4).

For all three periods the two largest discounted cash flows are the beginning and

ending value of the index. The next 15 largest discounted cash flows are the same

months for all three periods. These 15 months dominate the other monthly discounted

cash flows. For example in Table 3, column 4 the sum of discounted cash flows 3 through

17 is -182,082,311 while the sum of the remaining distributions is -34,371,694.

The DWR from 1973-2002 (Table 1) is 4.2 percent, by increasing the time period

to 2006 (Table 2) the DWR increases to 7.5 percent, extending the time period to 2010

(Table 3) results in a decrease in the DWR to 6.6 percent. Since the largest discounted

distributions are the same months for each time series and thus occur at the same time,

what causes the significant difference in the DWR between the three time periods? The

results are driven by the beginning and ending market values. Tables 1-3 show the ending

market values of the index as a percentage of the beginning market value of the index

(Column 6, Tables 1-3). The actual cash flow (Column 3, Tables 1-3) is used when

calculating the percentage difference from the starting market value of the index since it

is the actual cash flow that affects the DWR. The discounted cash flow is already

adjusted for the DWR. For the time periods 1973-2002 (Table 1), 1973-2006 (Table 2)

and 1973-2010 (Table 3) these differences are -1,557 percent, -2,893 percent and -3,125

percent, respectively. These percentages are significantly larger than the sum of the

5

largest 15 distributions (cash flows 3 through 17) and hence will play the dominant role.

As expected the DWR is the smallest when the difference between the beginning and

ending market value of the index is the smallest 1973-2002. The increase in DWR, and

hence decline in the difference between GM and DWR found by Keswani and Stolin

(2008), is principally driven by the increase in the difference between the beginning and

ending market value of the index. The difference between the beginning market value and

ending market value increases from -1,557 percent to -2,893 percent as the end point of

the data changes from 2002 to 2006. In the 2010 sample the percentage increases to-

3,125 percent, one would expect the DWR to increase compared to 2006 but it decreases

from 7.5 percent to 6.6 percent. In addition, for the 1973-2006 sample in Table 2, the

sum of percent difference from the starting market value of the portfolio for data points

18 through 409 equals 230 percent (positive percentages are inflows since percentages

are calculated by dividing monthly cash flows by a negative starting value) while for the

1973-2010 sample in Table 3 the difference for data points 18 through 457 equals 98

percent. This indicates that the additional data points from expanding the sample period

from 2006 to 2010 are net negative percentages, cash outflows. Increasing inter-period

cash outflows should also increase the DWR. But as discussed earlier the difference

between the beginning market value and ending market value is the primary determinant

of DWR. The difference in the ending values (between 2006 and 2010) is not large

enough with net inter-period outflows occurring to overcome the increased discounting

effect of 48 additional months.

To test the sensitivity of Dichev’s approach to the difference between the

beginning and ending values, monthly data is used to calculate the annual DWR and GM.

6

This could moderate the impact of the difference between the beginning and ending

market values, and give an indication of the effect of distributions on returns. The annual

approach is consistent with the asset returns literature (Morningstar: Stocks, Bonds, Bills,

and Inflation 2010 Yearbook). A summary of results appears in Table 4.For the NASDAQ

index the average GM is 13.34 percent, is slightly larger than the average DWR of 13.08

percent. For the NYSE/AMEX index 11.68 percent for the GM versus 11.62 percent

DWR.

We use the Bowman-Shelton test for normality for the distribution of differences

between annual GM and DWR. An examination of the skewness and kurtosis (not shown)

indicates non-normality in all cases; the values are not near zero or three, respectively.

The test rejects normality for all distributions at the one percent level of statistical

significance. Consequently, vulnerability of the paired t-test to deviations from the

normal distribution results in it being unsuitable to test for statistical significance. To test

for normality a non-parametric analog of the paired samples t-test the Wilcoxon signed

ranks test is used. A summary of the results for the Wilcoxon test is shown in Table 4. For

all indexes the test statistic is insignificant at the one percent level. That is, the annual

GM means are not significantly greater than the annual DWR.1In aggregate investors

annual returns are not significantly influenced by the correlation between the timing of

cash flows and past and future returns.

Although not on the individual investor level, this annual approach in aggregate

takes into account that as the market value of the index grows, the beginning value of the

1For the NYSE/AMEX value weighted index the geometric mean return is significantly larger than the

dollar-weighted return at the 5% level of significance. Although statistically significant at 5%, the average

historical difference is only .6% and would not be viewed an economically significant when making

projections about future returns.

7

market index will have a smaller impact each year on the DWR. In the limit the DWR

will approximate the GM regardless of the sequence of returns and the correlation

between the timing of cash flows with past and future returns. Applying Dichev’s

methodology of using stock market index data to calculate the monthly distributions uses

an aggregation level that precludes us from finding any significant difference on an

annual basis, the buy and hold cash flows dominate the distribution cash flows. The

principal determinant of Dichev’s findings is the size difference beginning and ending

market values of the index.

III. Examination of Hayley’s Methodology to Find

Aggregate Timing and Hindsight Effects

As presented in Dichev and Yu (2011), substituting equation 1 into the IRR formula

(equation 2),

T

MV0= ∑ dt + MVT (2)

t=1 (1 +rdw)t (1+rdw )T

gives you:

T T

rdw∑ MVt-1 =∑ MVt-1x rt (3)

t=1 (1 +rdw)t t=1(1 +rdw)t

Where rdw is the monthly DWR. The distributions (dt) have been removed from equation

3 and the IRR can be regarded as the dollar-weighted average of the individual monthly

returns. As discussed in Hayley (2014), the weight that the DWR puts on the market

return in any given month is determined by the NPV of the assets that the investor holds

in this market at the beginning of the period, discounted at the DWR. Hayley

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demonstrates how distributions and injections of additional capital reweights the monthly

returns rt. Reweighting previous returns is a hindsight effect while reweighting future

returns is a market timing effect.

While acknowledging the existence of hindsight and timing effect, a simple

example in Table 5 demonstrates that Hayley’s methodology of using a step-by-step

process to determine aggregate timing and hindsight effect is flawed. Hayley’s (2014)

methodology makes a counterfactual assumption that assumed future returns are constant

and have no relationship with past returns or distributions. Hayley starts with the

expected return in each period equal to the whole sample periodic GM return (monthly)

and that each distribution is zero, therefore initially DWR equals GM. Hayley then adds

the first period’s return and the DWR is recalculated, with all future returns equal to the

periodic GM and all future distributions equal zero (this calculation has the correct

beginning value of the portfolio but the ending value will be substantially smaller since

no future distributions are included).This first calculation (the initial DWR - new DWR1)

is the incremental timing effect related to the previous distribution (in the first case the

beginning value of the portfolio). Next the first period’s distribution is added and the

DWR is recalculated with all future returns again equal to the periodic GM and future

distributions equal to zero. This calculation (new DWR1 - new DWR2) gives you the

incremental hindsight effect. Again, early on the ending value of the portfolio should be

much smaller than actual since future distributions are not included and overall net

distributions are inflows over long time horizons. Hayley continues this process month by

month, summing these differences up to get total timing and hindsight effects.

9

In Table 5, cash inflows are negative and cash outflows are positive. There is only

one return of 5% in the last period. In Panel A, all the money is taken out before the only

return. Clearly this is bad timing. Having everything distributed after poor returns (in this

case returns are zero) reweights these poor returns (higher) and gives lower weights to

future returns. This should result in a negative hindsight effect. As shown in Panel A,

both timing and hindsight effects are positive (.00928 and .0052 respectively).

In Panel B, the cash flows and returns are the same except the 4th cash flow is reduced

from .21 to .15. The DWR return is now positive. Timing should improve since less is

taken out prior to the return in period 5. Hindsight should also improve as the reweighting

of previous poor returns would be lower and higher weights to future returns compared to

Panel A. Panel B results show timing effects and hindsight effect getting worse. In Panel

C the 4th cash flow is further reduced to 0.1.Keeping everything else constant we should

see an increase in the timing and hindsight effect due to the reasons discussed above, but

again we see both effects getting worse.

Overall results indicate that this methodology is flawed. This process does not

provide consistency in that it does not cover all possible outcomes. A simple example is

provided that demonstrates where the analysis does not make sense. The results are a

mathematical artifact of the process. Regardless of the counterfactual future return

assumption made, the aggregate timing effect plus the aggregate hindsight effect will

always equal the overall difference between the GM and the DWR.

Hayley results do provide support for the size effect explanation for Dichev’s

results. He finds that when the upward trend in returns is stripped out of the data

hindsight and timing effects are very small. Hayley is calculating incremental timing and

10

hindsight effects on a month to month basis (and aggregating them). This approach in

essence is forcing the size effect to become either timing or hindsight. That is why when

the upward trend in returns is stripped out hindsight and timing effects are very small.

When you remove the upward trend in returns you eliminate the size effect.

An additional issue with Hayley’s methodology is its violation of equation 1. In

equation 1, the distribution and the monthly return are calculated simultaneously (at time

t).Hayley’s step by step process violates this. As discussed above the monthly return is

inserted first then the DWR is calculated then the corresponding distribution is added and

the DWR is recalculated, this process repeats through the entire data set. This violation of

equation 1 results in the equality in equation 3 no longer holding. Table 6 demonstrates

this point. The same data from Table 5, Panel A is used. Given the DWR is zero the left

hand side of equation 3 must be zero as the summation is multiplied by the DWR. The

right hand side of the equation equals .0105.

IV. Conclusion

This paper re-examined the relationship between GM and DWR for the individual

investor in aggregate. Applying Dichev’s (2007) methodology to derive aggregate

investor contributions and distributions, the empirical results indicate the relative sizes of

the beginning and ending value is the primary cause of the return spread. To reduce the

impact of the beginning and ending market value differences, the monthly data was used

to calculate annual DWR and the GM. On an annual basis the size of the beginning and

ending value of the indexes dominate the monthly distribution cash flows. The annual

GM are found to be insignificantly different from the DWR. These results suggest that

contrary to Dichev’s claims, the calculated aggregate DWR do not provide a stable or

11

accurate measure of the returns to equities. Consequently, the use of IRR calculations to

measure the aggregate DWR should be approached with caution.

In addition, it is demonstrated that Hayley’s methodology used to break up

Dichev’s differences between the GM return and DWR return into hindsight and timing

effects is flawed. Findings using this methodology lack consistency and violate equations

used to derive the methodology.

References

Baker, M., and J. Wurgler. “The Equity Share in New Issues and Aggregate Stock

Returns.” Journal of Finance 55.5(2000): 2219-57.

Bodie, Z., A. Kane, and A. Marcus. Investments. Chicago: Irwin (1996).

Clare, A., and N. Motson. “Do UK Investors Buy at the Top and Sell at the Bottom?”

Cass Business School Working Paper (2010).

Dichev, I., “What are Stock Investors' Actual Historical Returns? Evidence from Dollar-

Weighted Returns.” American Economic Review 97.1(2007):386-401.

Dichev, I., and G. Yu. “Higher Risk, Lower Returns: What Hedge Fund Investors Really

Earn.” Journal of Financial Economics 100 (2011):248-263.

Friesen, G., and T. Sapp. “Mutual Fund Flows and Investor Returns: An Empirical

Examination of Fund Investor Timing Ability.” Journal of Banking and Finance. 31

(2007):2796-2816.

Ikenberry, D., J. Lakonishok, and T. Vermaelen. “Market Underreaction to Open Market

Share Repurchases.” Journal of Financial Economics 39 (1995): 181-208.

Hayley, S. “Hindsight Effect in Dollar-Weighted Returns.” Journal of Financial and

Quantitative Analysis. 39.1 (2014):249-269.

Johnston K., J. Hatem, and T. Carnes. "Investor Education: How Plan Sponsors Should

Report Your Returns." Managerial Finance. 36.4 (2010): 354-363.

Keswani A., and D. Stolin. “Dollar-Weighted Returns to Stock Investors: A New

Look at the Evidence.” Finance Research Letters 5.4(2008): 228-235.

12

Loughran, T., and J. Ritter. “The New Issues Puzzle.” Journal of Finance 50.1(1995):23-

51.

Morningstar.“Stocks, Bonds, Bills, and Inflation. 2010 Yearbook.” Chicago, IL:

Morningstar, Inc. (2010).

Zweig J. “Funds That Really Make Money for Their Investors.” Money 26.4 (2002):124-

34.

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Table 1

NASDAQ Cash Flows (1973 -2002)

Dollar-Weighted

Return 4.2%

Absolute

Value Difference

Discounted of Discounted From Portfolio

Date Cash Flows Cash Flows Cash Flows Starting Value

1 20021231 1,954,673,102.27 561,483,349.27 561,483,349.27 -1556.64%

2 19721229 -125,569,671.25 -125,569,671.25 125,569,671.25 -

3 20000331 -281,956,425.75 -90,803,793.00 90,803,793.00 224.54%

4 20000229 -109,679,620.14 -35,444,818.50 35,444,818.50 87.35%

5 19991130 -100,585,630.58 -32,845,602.99 32,845,602.99 80.10%

6 20000428 -95,466,120.62 -30,638,425.45 30,638,425.45 76.03%

7 20000630 -95,422,328.59 -30,412,878.49 30,412,878.49 75.99%

8 19991029 -83,252,693.35 -27,280,003.04 27,280,003.04 66.30%

9 20000929 -78,980,608.75 -24,912,282.26 24,912,282.26 62.90%

10 19991231 -72,513,698.03 -23,596,985.42 23,596,985.42 57.75%

11 19990331 57,609,476.95 19,340,773.28 19,340,773.28 -45.88%

12 20000831 -56,433,039.71 -17,862,050.32 17,862,050.32 44.94%

13 19990930 -46,381,864.55 -15,251,029.14 15,251,029.14 36.94%

14 19980331 -41,619,220.03 -14,565,712.96 14,565,712.96 33.14%

15 19980630 -33,908,907.63 -11,744,570.32 11,744,570.32 27.00%

16 20000731 -36,141,693.52 -11,479,188.55 11,479,188.55 28.78%

17 19970131 27,694,975.63 10,174,342.60 10,174,342.60 -22.06%

18 20010531 33,160,110.41 10,173,500.28 10,173,500.28 -26.41%

19 19970331 -27,423,570.20 -10,005,060.35 10,005,060.35 21.84%

20 19960628 -26,028,866.27 -9,797,029.20 9,797,029.20 20.73%

… … … … …

360 19770228 -8,683.72 -7,302.38 7,302.38 0.01%

361 19850830 -10,450.60 -6,171.74 6,171.74 0.01%

Data

Points Sum Sum Data Points Sum

3 - 17 -1,047,037,398.68 -337,322,224.56 3 - 17 833.83%

18 – 361 -286,877,232.43 -98,591,453.46 18 - 361 228.46%

Average Average

3 – 17 -69,802,493.25 -22,488,148.30

18 – 361 -833,945.44 -286,603.06

Cash flows are the beginning and ending market values along with the monthly distributions used in the IRR

calculation to determine the dollar-weighted return over the entire period. Beginning value of index and monthly net

capital flows from investors to companies in index are given negative signs. Ending value of index and monthly net

capital flows from companies in index to investors are given positive signs. Dollar-weighted return shown is

annualized. Column 3: the actual cash flows, Column 4: cash flows discounted by the dollar-weighted return, Column

5: absolute value of discounted cash flows, Column 6: % difference from starting portfolio value using column 3. Cash

flows are sorted by their discounted present value.

14

15

Table 2

NASDAQ Cash Flows (1973 -2006)

Dollar-Weighted

Return 7.5%

Absolute Value Difference

Discounted of Discounted From Portfolio

Date Cash Flows Cash Flows Cash Flows Starting Value

1 20061229 3,632,112,981.52 312,362,235.06 312,362,235.06 -2892.51%

2 19721229 -125,569,671.25 -125,569,671.25 125,569,671.25 -

3 20000331 -281,956,425.75 -39,465,210.31 39,465,210.31 224.54%

4 20000229 -109,679,620.14 -15,444,358.86 15,444,358.86 87.35%

5 19991130 -100,585,630.58 -14,421,633.49 14,421,633.49 80.10%

6 20000428 -95,466,120.62 -13,282,204.97 13,282,204.97 76.03%

7 20000630 -95,422,328.59 -13,117,403.27 13,117,403.27 75.99%

8 19991029 -83,252,693.35 -12,008,487.92 12,008,487.92 66.30%

9 20000929 -78,980,608.75 -10,663,107.11 10,663,107.11 62.90%

10 19991231 -72,513,698.03 -10,334,442.27 10,334,442.27 57.75%

11 19990331 57,609,476.95 8,666,916.59 8,666,916.59 -45.88%

12 20000831 -56,433,039.71 -7,664,931.19 7,664,931.19 44.94%

13 19990930 -46,381,864.55 -6,730,537.65 6,730,537.65 36.94%

14 19980331 -41,619,220.03 -6,729,811.21 6,729,811.21 33.14%

15 19980630 -33,908,907.63 -5,385,030.64 5,385,030.64 27.00%

16 20000731 -36,141,693.52 -4,938,497.49 4,938,497.49 28.78%

17 19970131 27,694,975.63 4,871,595.65 4,871,595.65 -22.06%

18 19960628 -26,028,866.27 -4,775,360.37 4,775,360.37 20.73%

19 19970331 -27,423,570.20 -4,766,188.31 4,766,188.31 21.84%

20 20041130 43,971,827.45 4,395,028.83 4,395,028.83 -35.02%

… … … … …

408 20031031 46,260.84 4,999.78 4,999.78 -0.04%

409 19850830 -10,450.60 -4,189.77 4,189.77 0.01%

Data

Points Sum Sum Data Points Sum

3 – 17 -1,047,037,398.68 -146,647,144.15 3 - 17 833.83%

18 – 409 -288,528,193.91 -40,145,419.67 18 - 409 229.78%

Average Average

3 – 17 -69,802,493.25 -9,776,476.28

18 – 409 -736,041.31 -102,411.78

Cash flows are the beginning and ending market values along with the monthly distributions used in the IRR

calculation to determine the dollar-weighted return over the entire period. Beginning value of index and monthly net

capital flows from investors to companies in index are given negative signs. Ending value of index and monthly net

capital flows from companies in index to investors are given positive signs. Dollar-weighted return shown is

annualized. Column 3: the actual cash flows, Column 4: cash flows discounted by the dollar-weighted return, Column

5: absolute value of discounted cash flows, Column 6: % difference from starting portfolio value using column 3. Cash

flows are sorted by their discounted present value.

Table 3

NASDAQ Cash Flows (1973 -2010)

Dollar-Weighted

Return 6.6%

Absolute

Value Difference

Discounted of Discounted

From

Portfolio

Date Cash Flows Cash Flows Cash Flows Starting Value

1 20101231 3,924,581,638.17 342,023,676.49 342,023,676.49 -3125.42%

2 19721229 -125,569,671.25 -125,569,671.25 125,569,671.25 -

3 20000331 -281,956,425.75 -49,004,828.69 49,004,828.69 224.54%

4 20000229 -109,679,620.14 -19,164,910.77 19,164,910.77 87.35%

5 19991130 -100,585,630.58 -17,860,300.51 17,860,300.51 80.10%

6 20000428 -95,466,120.62 -16,503,732.22 16,503,732.22 76.03%

7 20000630 -95,422,328.59 -16,320,555.40 16,320,555.40 75.99%

8 19991029 -83,252,693.35 -14,861,926.07 14,861,926.07 66.30%

9 20000929 -78,980,608.75 -13,293,319.56 13,293,319.56 62.90%

10 19991231 -72,513,698.03 -12,807,043.59 12,807,043.59 57.75%

11 19990331 57,609,476.95 10,676,739.14 10,676,739.14 -45.88%

12 20000831 -56,433,039.71 -9,549,274.93 9,549,274.93 44.94%

13 19990930 -46,381,864.55 -8,324,324.33 8,324,324.33 36.94%

14 19980331 -41,619,220.03 -8,224,821.05 8,224,821.05 33.14%

15 19980630 -33,908,907.63 -6,594,386.00 6,594,386.00 27.00%

16 20000731 -36,141,693.52 -6,148,503.68 6,148,503.68 28.78%

17 19970131 27,694,975.63 5,898,876.44 5,898,876.44 -22.06%

18 19970331 -27,423,570.20 -5,778,888.77 5,778,888.77 21.84%

19 19960628 -26,028,866.27 -5,755,611.34 5,755,611.34 20.73%

20 20041130 43,971,827.45 5,663,543.35 5,663,543.35 -35.02%

… … … … …

456 19850830 -10,450.60 -4,633.35 4,633.35 0.01%

457 20090529 -37,488.61 -3,616.74 3,616.74 0.03%

Data

Points Sum Sum Data Points Sum

3 – 17 -1,047,037,398.68 -182,082,311.23 3 - 17 833.83%

18 – 457 -123,296,714.67 -34,371,694.01 18 - 457 98.19%

Average Average

3 – 17 -69,802,493.25 -12,138,820.75

18 – 457 -280,219.81 -78,117.49

Cash flows are the beginning and ending market values along with the monthly distributions used in the IRR

calculation to determine the dollar-weighted return over the entire period. Beginning value of index and monthly net

capital flows from investors to companies in index are given negative signs. Ending value of index and monthly net

capital flows from companies in index to investors are given positive signs. Dollar-weighted return shown is

annualized. Column 3: the actual cash flows, Column 4: cash flows discounted by the dollar-weighted return, Column

5: absolute value of discounted cash flows, Column 6: % difference from starting portfolio value using column 3. Cash

flows are sorted by their discounted present value.

Table 4

Comparison of Annual GM and DWR for NASDAQ and NYSE/AMEX

NASDAQ NYSE/AMEX

AVERAGE STD. DEV. AVERAGE STD. DEV.

GM 13.34% 27.82% 11.68% 20.05%

DWR 13.08% 28.10% 11.62% 20.03%

DIFFERENCE 0.26% 1.01% 0.29 0.29%

Wilcoxon 1.46 1.78

Bowman-Shelton 96.52 68.09

GM is the geometric average compounded value-weighted return from the CRSP monthly files.

DWR is the IRR from a calculation for which the beginning market value enters with a negative sign,

monthly distributions enter with their sign, and the ending market value enters with a positive sign.

Returns are calculated using the data for each year. All returns are annualized.

Table 5

Hindsight and Timing Effects

Panel A

t CFs Returns

Port

Val

DWR

CFs Timing

Hindsigh

t

0 -0.06 -0.06 1 0.00197 -0.00079

1 -0.05 0 0.11 -0.05 2 0.00216 -0.00077

2 -0.05 0 0.16 -0.05 3 0.00241 -0.00066

3 -0.05 0 0.21 -0.05 4 0.00274 0.00273

4 0.21 0 0 0.21 5 0.00000 0.00000

5 0 0.05 0 0 Sum 0.00928 0.00052

GM 0.00981 DWR 0.00000 Sum T & H 0.00981

GM - DWR 0.00981

Panel B

t CFs Returns

Port

Val

DWR

CFs Timing

Hindsigh

t

0 -0.06 -0.06 1 0.00197 -0.00079

1 -0.05 0 0.11 -0.05 2 0.00216 -0.00077

2 -0.05 0 0.16 -0.05 3 0.00241 -0.00066

3 -0.05 0 0.21 -0.05 4 0.00274 0.00176

4 0.15 0 0.06 0.15 5 -0.00398 0.00000

5 0 0.05 0.063 0.063 Sum 0.00531 -0.00046

GM 0.00981 DWR 0.00495 Sum T & H 0.00485

GM - DWR 0.00485

Panel C

t CFs Returns

Port

Val

DWR

CFs Timing

Hindsigh

t

0 -0.06 -0.06 1 0.00197 -0.00079

1 -0.05 0 0.11 -0.05 2 0.00216 -0.00077

2 -0.05 0 0.16 -0.05 3 0.00241 -0.00066

3 -0.05 0 0.21 -0.05 4 0.00274 0.00108

4 0.1 0 0.11 0.1 5 -0.00669 0.00000

5 0 0.05 0.1155 0.1155 Sum 0.00260 -0.00113

GM 0.00981 DWR 0.00834 Sum T & H 0.00146

GM - DWR 0.00146

CFs are end of period cash flows. Returns are the period returns. Port Val are the end of period portfolio

values. DWR CFs are the cash flows used in the DWR calculations. GM is the geometric mean. DWR

is the dollar-weighted return. Timing/Hindsight is the periodic timing/hindsight value calculated using

Hayley’s step-by-step methodology. Sum of T & H is the sum of the total timing and hindsight effect

which equals the difference between the GM and DWR (GM –DWR).

Table 6

Equation 3 Calculation

MVt-1x rtMVt-1

t t-1 MVt-1 rtMVt-1x rt(1 +rdw)t(1 +rdw)t(1 +rdw)t

1 0 Ko -0.06 0 0 1 0 -0.06

2 1 K1 -0.05 0 0 1 0 -0.05

3 2 K2 -0.05 0 0 1 0 -0.05

4 3 K3 -0.05 0 0 1 0 -0.05

5 4 K4 0.21 0.05 0.0105 1 0.0105 0.21

Sum 0.0105 0.0000

Eq. 3 RHS 0.0105

Eq. 3 LHS 0.0000

Data are from Table 5, Panel A.

Equation 3 is:

T T

rdw∑ MVt-1 =∑ MVt-1x rt

t=1 (1 +rdw)t t=1 (1 +rdw)t

Where MV is market capitalization, r is the return on the index and rdw is the monthly dollar-weighted return.