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VALUE AVERAGING AND THE AUTOMATED BIAS OF PERFORMANCE
MEASURES
Abstract:
Value averaging is a formula investment strategy which can be shown to achieve a lower
average cost and higher IRR than alternative strategies. However, in contrast to previous studies,
this paper shows that this does not lead to higher expected profits. Instead an “averaging down”
effect systematically biases the IRR up and the average purchase cost down. The same bias
applies to a wide class of investment strategies (including dollar cost averaging) where the
amount invested in each period is negatively correlated with the return made to date.
This version: 18 May 2010
Preliminary draft: comments welcome, but please do not cite without author’s permission
Simon Hayley
Cass Business School
00-44-(0)20-7040-0230
1

1. Introduction
Value averaging (VA) is a formula investment strategy which can be shown to achieve lower
average costs and higher IRRs than alternative strategies. However, in contrast to previous
studies, this paper will show that this does not lead to higher expected profits.
VA is in some respects similar to dollar cost averaging (DCA), which is the practice of building up
investments gradually over time in equal dollar amounts, rather than investing the desired total in
one lump sum. Table 1 compares DCA with a strategy of buying equal numbers of shares each
period (ESA). The DCA strategy invests $100 each period, whereas ESA purchases 100 shares
each period. Shares initially cost $1, but the DCA strategy buys more shares as prices fall. Thus
DCA achieves a lower average share price than ESA. Conversely, if prices rose, then DCA would
purchase fewer shares in later periods, again achieving a lower average cost.
Table 1 Equal Share Amounts (ESA) Dollar Cost Averaging (DCA) Value Averaging (VA)
Period Price
Shares
bought
Invest-
ment
($)
Portfolio
($)
Shares
bought
2
Invest-
ment
($)
Portfolio
($)
Shares
bought
Invest-
ment
($)
Portfolio
($)
1 1.00 100 100 100 100 100 100 100 100 100
2 0.90 100 90 171 111 100 180 122 110 200
3 0.80 100 80 216 125 100 240 153 122 300
Total 300 270 336 300 375 332
Average cost: 0.900 0.893 0.886
Proponents of DCA argue that as it reduces the average purchase cost, it must generate higher
returns. By contrast, previous academic research has long shown that despite its lower average
cost, DCA is a sub-optimal strategy. Nevertheless, it remains very popular among investors and
is widely recommended in the financial press and popular finance literature.
The motivation for value averaging (VA) is similar. In contrast to DCA, VA has a target increase in
portfolio value each period (assumed in Table 1 to be a rise of $100 per period). The investor
must invest whatever amount is necessary in each period to meet this target. Like DCA, VA
purchases more shares after a fall in prices, but the response is more sensitive: in this example
VA buys 122 shares in period 2, compared to 111 for DCA and 100 for ESA. As Table 1 shows,
the more aggressive response of VA to shifts in the share price results in an even lower average
purchase cost. Again, this is true whether prices rises or fall.
In contrast to DCA, VA is a relatively recent invention (first suggested by Edelson, 1988) and the
only studies assessing its performance recommend it on the grounds that it results in a higher
IRR (Edelson (1991), Marshall (2000, 2006)).
Neither strategy claims to be taking advantage of market inefficiencies. Indeed, simulations
appear to show that these trading rules bring benefits even when prices follow a random walk.
Moreover, both are fixed rules which pre-commit investors, allowing the investor no discretion
once committed to the strategy. As a result, both are subject to the criticisms set out by
Constantinides (1979), who showed that strategies which pre-commit investors must be expected
to be dominated by strategies which allow investors to react to incoming news.
DCA may also seem to improve diversification by making many small purchases but, as Rozeff
(1994) notes, the result is that overall profits are most sensitive to returns in the later part of the
period, when the investor is nearly fully invested. Earlier returns are given correspondingly little
weight, since the investor then holds mainly cash. Better diversification is achieved by investing in
one initial lump sum, and thus being equally exposed to the returns in each sub-period. Milevsky

and Posner (1999) show that it is always possible to construct a constant proportions
continuously rebalanced portfolio which will stochastically dominate DCA in a mean-variance
framework, and that for typical levels of volatility and drift there will be a static buy and hold
strategy which dominates DCA.
Studies based on historical data have found that investing in one lump sum has generally given
better mean-variance performance than DCA. These include Knight and Mandell (1992/93),
Williams and Bacon (1993), Rozeff (1994) and Thorley (1994). This inefficiency may seem at
odds with DCA’s lower average costs, but Hayley (2009) shows that comparing the average cost
achieved by DCA with the average price is misleading: it implicitly compares DCA with a strategy
which uses perfect foresight to invest more when prices are about to fall and less when they are
about to rise. It is only because of this bias that DCA appears to offer higher returns.
Proponents of VA tend to focus not on its lower average cost, but on the fact that it achieves a
higher IRR than alternative strategies. Higher IRRs might seem to imply higher expected profits,
but we demonstrate here that VA systematically biases up its IRR without increasing expected
profits.
The structure of this paper is as follows: section 2 demonstrates that in contrast to proponents’
claims, VA cannot expect to generate excess returns when prices follow a random walk. This is
confirmed by the Monte Carlo simulations presented in Section 3, which show that DCA and VA
generate lower average purchase costs and higher IRRs, but do not increase average profits. We
then investigate why average purchase costs (Section 4) and IRRs (Section 5) are systematically
biased by these formula strategies. Section 6 briefly considers cashflow management and risk
issues. Conclusions are drawn in the final section.
2. Expected Profits
In the analysis below we assume that investors do not believe that they can forecast market
prices - in effect they assume that prices follow a random walk. However, we should stress that
this is a statement about investors’ ex ante expectations, and does not imply any presumption
that markets are in fact weak form efficient. The key point in this context is that VA, like DCA, will
only ever be an attractive strategy for investors who do not believe that they can forecast shortterm
price movements. These strategies commit investors to invest a specified amount no matter
what they expect in the coming period – those who feel that they can forecast short-term price
movements will reject this and follow other strategies instead.
We also assume that this random walk has zero drift. This too is a statement about investors’ ex
ante expectations rather than about markets themselves. Investors presumably believe that over
the medium term their chosen securities will generate an attractive return, but they must also
believe that the return over the short term (while they are using DCA or VA to build up their
position) is likely to be small. Investors who expect significant returns over the short term would
clearly prefer to invest immediately in one lump sum rather than delay their investments by
following a strategy which invests gradually. Marshall (2000) suggests that VA boosts returns
even in a random walk with zero drift.
The assumption of zero drift need not imply a loss of generality, since drift could be incorporated
into this framework by defining prices not as absolute market prices, but as prices relative to a
numeraire which appreciates at a rate which gives a fair return for the risks inherent in this asset
(pi*=pi/(1+r) i , where r reflects the cost of capital and a risk premium appropriate to this asset). We
could then assume that pi* has zero expected drift since investors who use VA or DCA will not
believe that they can forecast short-term relative asset returns: those who do would again reject
trading strategies which forced them to delay their purchases. The results derived below would
3

continue to hold for pi*, with profits then defined as excess returns compared to the risk-adjusted
cost of capital. 1
We consider investing over a series of n discrete periods. The price of the asset in each period i
is pi. The alternative investment strategies differ in the quantity of shares qi that are purchased in
each period. We evaluate profits at a subsequent point, after all investments have been made. If
prices are then pT, the expected profit made by any investment strategy is:
n
n
piqi pTqi i1
i1
Our assumption of a random walk implies that future price movements (pT/pi) are independent of
the past values of pi and qi. This gives us:
n
p
iqi
piqi
i1
p T
p
i
4
n
i1
But the random walk has zero drift, so E[pT/pi]=1 for all i and expected profits are zero. The
amount piqi which is invested in each period is irrelevant: expected profits are zero for all the
investment strategies that we consider here. Total expected profit is the expected percentage
capital gain between each period i and period T, multiplied by the amounts invested in each
period. But our assumption of a driftless random walk implies that the expected gain is zero in all
periods, so the timing of investments makes no difference. Against this background, the claim
that VA and DCA can generate excess profits even in the absence of market inefficiencies is
surprising.
3. Monte Carlo Simulations
Tables 2 and 3 shows the results of 10,000 simulations in which prices movements are
distributed uniformly within the range +/-5% in each of five consecutive periods. The share price
is initially $10. The four strategies compared here are:
- ESA: buy 40 shares each period.
- DCA: invest $400 each period.
- VA: increase the portfolio value by $400 each period 2
- Lump sum: purchase 200 shares in the first period.
1 We must also assume that funds not yet needed for the VA strategy can be held in assets with the same
expected return. This assumption is clearly generous to VA – if instead cash is held on deposit at lower
expected return, then VA’s expected return is clearly reduced by delaying investment.
2 The VA strategy has one more parameter than DCA and ESA, since it requires us to specify both the
expenditure in the initial period and the target increase in portfolio value each period. In order to make the
DCA and VA results here as comparable as possible, we have set both these figures to $400. Thus if prices
remain constant the three strategies would have identical cashflows, with each investing $400 in every
period. As shown in the previous section, this makes no difference to expected profits.
(1)
(2)

Table 2: Strategy Costs
and Returns
ESA DCA VA Lump Sum
Avg.Cost ($) Mean 10.0009 9.9943 9.9842 10.0000
Std. Error 0.0032 0.0032 0.0032 0.0000
IRR (%) Mean -0.0159% -0.0034% 0.0133% -0.0182%
Std. Error 0.0158% 0.0158% 0.0158% 0.0145%
Profit ($) Mean 0.8706 0.8659 0.8648 1.0572
Std. Error 0.6337 0.6331 0.6326 1.1589
It might seem odd to use IRR to compare performance instead of more conventional measures
such as the Sharpe ratio. In looking at IRRs we are following the methodology used by Edelson
(1991) and Marshall (2000, 2006). Moreover, some of the key problems normally associated with
the use of IRR (notably in real estate applications) are absent here: traded securities are likely to
be highly divisible, in contrast to large real estate projects. Furthermore, the cashflows generated
by real estate projects might have to be re-invested at very different yields, but our null
hypothesis here (that formula investment strategies generate no excess profits) would imply that
surplus cashflows could be invested using different strategies at the same expected yield.
Ultimately, we will argue that the IRR is a poor measure of profitability in this context, but this is
for a more subtle reason. Phalippou (2008) notes that IRRs can be biased where the cashflows
are endogenous to the IRR achieved to date, and that investment managers could thus
manipulate their cashflows in order to boost their recorded IRRs. We show below that this bias is
inherent in the VA and DCA strategies.
Sharpe ratios are not used in this comparison because DCA and VA claim to achieve their
benefits by strategically varying the cashflows involved. Thus if these strategies do bring benefits,
they can only be assessed using a dollar-weighted performance measure such as IRR. By
contrast, the time-weighted rates of return which are conventionally used in calculating Sharpe
ratios (notably in GIPS methodology) deliberately strip away any cashflow effects, leaving only
the relative performance of the assets involved. This would remove the effect of DCA and VA, so
this is not a useful measure here. We will ultimately conclude that DCA and VA give zero
expected excess profits, so we can infer that the Sharpe ratio will be zero in all the strategy
simulations shown here, but in the meantime we investigate the IRR to avoid pre-judging the
issue and to show the nature of the bias involved.
Table 3: Differences
DCA -
VA - Lump
Between Strategies DCA - ESA Lump Sum DCA - VA VA - ESA Sum
Avg.Cost ($) Mean -0.00668 -0.00574 0.01001 -0.01669 -0.01576
Std. Error 0.00005 0.00317 0.00007 0.00011 0.00316
IRR (%) Mean 0.0125% 0.0148% -0.0167% 0.0292% 0.0315%
Std. Error 0.00013% 0.00646% 0.00018% 0.00028% 0.00646%
Profit ($) Mean -0.00469 -0.1913 0.0011 -0.00579 -0.1924
Std. Error 0.0149 0.636 0.0149 0.0279 0.637
Table 3 shows that VA and DCA achieve highly significant reductions in average cost and
increases in IRR compared to both ESA and lump sum investment strategies. But there are no
significant differences in the profits made. As a robustness check, simulations were also run with
price movements substantially more volatile (-25% to +25% per period) or less volatile (-1% to
+1% per period) than those shown here. In each case DCA and VA recorded significantly higher
IRRs and lower average costs, but no significant change in profits. We find the same if we use
5

Marshall’s random investing strategy as our non-dynamic benchmark strategy (in place of the
lump sum and ESA strategies).
Even on the assumption of a driftless random walk, where we know that the ex ante expected
return must be zero, our simulations show VA and DCA generating higher IRRs than other
strategies. Thus IRRs appear to be biased measures of expected profit. We look at the reasons
for this in section 5. But first we investigate the very similar mechanism whereby these trading
strategies can expect to achieve low average purchase costs without improving their expected
profits.
4. The Bias In Purchase Cost
To illustrate the bias in the average purchase cost, we contrast the outcomes in comparable
DCA, ESA and VA strategies. We initially consider the outturns for these strategies where the
share price declines, as shown in Table 1 (replicated below).
Table 1 Equal Share Amounts (ESA) Dollar Cost Averaging (DCA) Value Averaging (VA)
Period Price
Shares
bought
Invest-
ment
($)
Portfolio
($)
Shares
bought
6
Invest-
ment
($)
Portfolio
($)
Shares
bought
Invest-
ment
($)
1 1.00 100 100 100 100 100 100 100 100 100
2 0.90 100 90 171 111 100 180 122 110 200
3 0.80 100 80 216 125 100 240 153 122 300
Total 300 270 336 300 375 332
Average cost: 0.900 0.893 0.886
We initially compare ESA and DCA. With the share price at $1, both strategies invest $100 in the
first period. The price subsequently falls to $0.90. The ESA strategy buys 100 shares in the
second period, but DCA again invests $100 so it buys more shares (111). By buying more shares
when they are relatively cheap, DCA will achieve a lower average purchase cost than ESA.
However, this does not alter the expected profits of the two strategies, since the ex ante expected
profit from investment in each period is zero. The expected profit on the 100 shares purchased in
period one was zero at the time they were purchased. When the price falls to $0.90 in period 2
the investor suffers a loss of $10. The assumption of a driftless random walk means that this loss
must be expected to persist. It also means that the expected profit on any shares purchased at
the lower price in period two is zero. Thus the fact that the DCA and ESA strategies purchase
different numbers of shares in period two cannot affect their expected profit levels.
The larger purchase made by DCA in the second period reduces the average purchase price
achieved (it will then have spent $200 to purchase 121 shares, giving an average cost of $0.947,
compared to $0.95 for ESA), but the ex ante expected profit of each strategy remains the same.
In other contexts investors refer to doubling down: trying to make a virtue of a price fall after their
initial investment by using it as an opportunity to acquire more shares at the new lower price. This
reduces the average share price at which they entered the trade. Both ESA and DCA can be
regarded as doubling down in the second period, but DCA is more aggressive at doubling down,
buying more shares than in period one, and thus achieving a larger reduction in average cost.
But this cannot alter their expected losses, which remain at $10 for each strategy at this stage.
Portfolio
($)

VA doubles down even more aggressively than DCA. It seeks to increase its portfolio value by
$100 each period so, just like DCA, a lower share price means that VA will buy more shares in
period 2. But in order to achieve its target portfolio value, VA must also make up for the $10 loss
suffered on its earlier investment by investing an additional $10 in period 2. By buying 122 shares
this period, VA achieves an even larger reduction in its average purchase cost, but again this
makes no difference to expected profits.
Chart 1 shows that the number of shares purchased in period 5 of our simulations by VA and
DCA are highly correlated, but the range of variation is much larger for VA. This illustrates its
greater tendency to average down, and hence to achieve a lower average cost.
DCA shares purchased
60
55
50
45
40
35
30
25
Chart 1: Total Number Of Shares Purchased
Under DCA and VA
25 30 35 40 45 50 55 60
VA shares purchased
Table 4 gives a different example which shows that these effects also apply when prices rise. In
period two prices have risen to $1.10, so purchases made in period one now look cheap. The
best way to achieve a low average purchase cost is then to buy very few shares at this higher
price. ESA does badly here, buying another 100 shares, and thus “doubling up” the average
purchase price to $1.05. DCA buys 91 shares, and VA buys only 82 as the capital gain on its
initial investment helps to achieve its target portfolio value without needing further investment.
Thus VA again achieves the lowest average purchase price, then DCA, with ESA highest again.
But none of this alters expected profits.
Table 4 Equal Share Amounts (ESA) Dollar Cost Averaging (DCA) Value Averaging (VA)
Period Price
Shares
bought
Invest-
ment
($)
Portfolio
($)
Shares
bought
7
Invest-
ment
($)
Portfolio
($)
Shares
bought
Invest-
ment
($)
1 1.00 100 100 100 100 100 100 100 100 100
2 1.10 100 110 231 91 100 220 82 90 200
3 1.20 100 120 396 83 100 360 68 82 300
Total 300 330 274 300 250 272
Average cost: 1.10 1.094 1.087
Portfolio
($)

Difference in average cost ($)
0.00
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
Chart 2: Comparative Average Purchase Cost Achieved
By Different Strategies
DCA avg. cost - ESA avg. cost
VA avg. cost - ESA avg. cost
8 9 10 11 12
Terminal share price ($)
The simulations confirm these points. Chart 2 compares the average costs achieved by these
three strategies. We can see that:
(i) DCA and VA always achieve lower average purchase costs than ESA (and VA
always achieves a greater cost reduction than DCA because of its more
aggressive response to share price movements).
(ii) The difference is largest where the terminal price PT has moved a long way from
the starting value. Volatile share prices allow DCA and VA greater scope to
benefit from buying more shares at low prices. Conversely, if prices do not vary
at all, the three strategies will be identical.
Proponents of DCA almost invariably assume that a strategy which buys at lower average cost
must lead to higher profits. The continued popularity of DCA (in spite of academic studies which
show it to be a sub-optimal strategy) suggests that investors generally find this argument highly
persuasive.
All else equal, lower average costs must indeed lead to higher profits, but all else is not equal.
This can be shown by comparing the total amount invested by the different strategies with the
share price at the end of the simulation (Chart 3). The ESA strategy naturally invests more in
simulations where prices end up falling, and less when they are falling. By contrast, DCA invests
a fixed total dollar amount ($2000 for these simulations). The fact that DCA invests at a lower
average cost is balanced by the fact that it tends to invest a smaller amount in periods of rising
prices and more in periods of falling prices. These factors tend to cancel out: as we saw earlier,
the expected profits of the two strategies are identical.
This bias is even more pronounced for VA, which tends to invest less during periods of rising
prices (since capital gains help achieve the investor’s desired portfolio value without the need for
substantial additional investment). Thus a VA strategy tends to invest far less than ESA during
periods of rising prices and far more in periods of falling prices. This offsets the fact that VA
achieves a lower average cost. Once again, expected profits are identical for the two strategies.
8

Total amount invested ($)
2200
2100
2000
1900
1800
Chart 3: Total Amount Invested By Each Strategy
ESA
DCA
VA
8.0 9.0 10.0 11.0 12.0
Terminal share price ($)
Ingersoll et al. (2007) show that cynical investment managers can manipulate conventional
performance measures (including the Sharpe ratio, Jensen’s alpha etc.). The ‘gaming’ of these
performance measures is achieved by reducing risk exposure following a good performance and
increasing exposure after a poor performance. This strategy could be characterized as applying
an element of “quit while you are ahead, gamble more when you are behind”. DCA and VA in
effect use this strategy automatically, since by construction they invest more following price falls
and less after price rises. This allows them to achieve low average purchase costs without
achieving any increase in profitability. The following section shows that the same bias applies to
the IRRs achieved by these strategies.
5. The Bias In IRRs
The section above showed that DCA and VA achieve lower average costs, but the same
expected profits as ESA. However, VA is a more complex strategy than DCA, with varying
cashflows, so proponents of VA focus instead on the fact that it tends to generate a higher
internal rate of return than either ESA or DCA. Our simulations confirm (Table 3 and Chart 4) that
IRRs are generally higher for VA than for DCA, with both strategies giving higher average IRRs
than ESA 3 . It might appear intuitive that a higher IRR will imply higher expected profits, but in this
section we show that these IRRs are misleading, since the same bias is at work as we found for
average costs.
3 Marshall (2000) calculates which strategy gives the highest IRR for each simulated path. In 73.5% of cases
VA was best, with DCA best in only 3.9% of cases. Chart 4 helps explain this: where the simulated price path
tends to mean-revert, the aggressive VA strategy generally gives the highest IRR. Where prices make
sustained movements, strategies with no dynamic component fare better (ESA here, or Marshall’s random
investing strategy). The less aggressive DCA strategy is almost always outperformed by one of the other two
strategies, but frequently comes in second place, consistent with the results in Table 3 showing that it
records a higher average IRR than ESA.
9

IRR differential
Chart 4: Comparative IRRs Achieved By Different
Strategies
0.15%
0.10%
0.05%
0.00%
-0.05%
8.0 9.0 10.0 11.0 12.0
Terminal share price ($)
10
VA IRR - ESA IRR
DCA IRR - ESA IRR
We can illustrate this by comparing the investments made by each strategy in the second period.
Again we initially consider the scenario of falling prices shown in Table 1. By period 2 each of the
strategies has made a $10 loss on the $100 invested in the first period and the assumption of a
driftless random walk means that this loss must be expected to persist, leading to a negative
overall expected IRR.
However, the ex ante expected IRR of any sum we invest in period 2, taken in isolation, is zero.
The overall IRR on our combined investment in periods 1 and 2 will depend on the IRRs of these
investments taken separately, and the relative amounts invested in each of these periods. This
relationship is polynomial, but the direction of the relationship is intuitive: if prices have fallen in
period 2, the expected IRR on the amount invested in period 1 (IRR1) is now negative, but the
expected IRR on the amount that we are about to invest in period 2 (IRR2) is still zero. The more
that we invest in the second period, the more the expected IRR on the two investments combined
(IRRc) is likely to move away from IRR1 and towards IRR2 (ie. zero) 4 . Thus if our objective is to
maximize IRRc, our best response in period 2 to the loss made already would be to dilute the
negative IRR1 with a large new investment which carries a zero expected IRR.
As we have seen, this is exactly what DCA does by automatically investing more following a fall
in prices. VA does the same, but more aggressively. This process is very similar to the doubling
down we saw in the previous section, the only difference being that the more complex arithmetic
of the IRR means that this is no longer a simple averaging, but a more complex dilution effect.
Conversely, if prices rise after our initial investment, then our expected IRR1 is positive, whilst our
expected IRR2 is still zero. The best strategy for maximizing our expected IRRc would thus be to
invest relatively little in the second period, to avoid diluting the positive expected IRR1 with the
zero expected IRR2.
4 The polynomial arithmetic of IRRs means that there may be exceptions to this. Indeed, VA can entail the
return of cash to investors in some periods (where a large price rise results in a capital gain that is greater
than the target increase in portfolio value). Thus there may be multiple swings from positive to negative
cashflow, so we cannot rule out multiple roots in our IRR calculation. However, as long as it is generally true
that the aggregate IRR is biased towards zero by investing larger amounts in the current period, then there
will be a bias in the average IRR. Our simulations confirm that this is indeed the case.

Phalippou (2008) notes that the IRRs recorded by private equity managers can be manipulated
by adjusting the cashflows involved: returning cash to investors rapidly for projects with high IRRs
and extending the exposure of poorly-performing projects. This is similar to the mechanism noted
by Ingersoll et al. (2007) for biasing other performance measures. The bias in each case is
achieved by reducing exposure following a good outturn and increasing exposure following a bad
outturn. By doing this automatically, DCA and VA achieve IRRs which are better than ESA, whilst
their ex ante expected profits remain zero.
Difference in profit ($)
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
Chart 5: Comparative Profits
DCA Profit - ESA Profit
VA Profit - ESA Profit
8.0 9.0 10.0 11.0 12.0
Terminal share price ($)
Chart 5 shows the increase or decrease in profits generated by each of the formula trading
strategies compared to a static ESA strategy. As we saw in Table 3, the differential averages
zero, but the formula trading strategies outperform when the share price ends up relatively close
to its starting value ($10). These strategies buy more shares at relatively low prices, and when
prices mean-revert this lower average cost does translate into higher profits (with none of the
offsetting downside illustrated in Chart 3). This advantage is larger for VA, with its more
aggressive response to changing share prices, than it is for DCA. Conversely, DCA and VA do
poorly in sustained price trends, since they purchase more shares than ESA in downtrends and
fewer shares than ESA in uptrends.
Thus VA can be profitable where investors correctly anticipate mean-reversion (implying that
markets are to some extent forecastable). But this is a dramatic contrast to the claim made by
proponents of VA that the strategy increases expected returns in any market, even when
investors have no ability to forecast returns. Furthermore, even where there is an element of
mean reversion, other dynamic strategies (e.g. based on calibrated filter rules) are likely to be
more efficient mechanisms for profiting from such forecastable price movements.
6. Risk And Cashflow
We have established that VA, like DCA, cannot expect to generate excess profits in markets
where price movements cannot be forecast. We now consider briefly the effects that VA has on
cashflow management and risk levels.
11

DCA generates perfectly stable cashflows by construction, with a fixed dollar amount invested
each period. Thus even though the strategy is mean-variance inefficient, it can claim the
incidental benefit of encouraging regular savings. Just as for DCA, the case for VA is based on a
misleading claim of superior returns, but VA comes with the added drawback of highly uncertain
investor cashflows, since the amount which must be invested each period depends on the most
recent movements in the market price (VA can even imply an unexpected return of cash to the
investor following a large price rise). Indeed, these cashflows are likely to become increasingly
volatile over time as the existing portfolio increases in size relative to the target increase in value
each period. Edelson (1991) envisages investors holding a ‘side fund’ containing liquid assets
sufficient to meet these needs.
As for risk, Table 2 shows that the profits recorded by the gradual investment strategies (ESA,
DCA and VA) have almost identical standard deviations, but profits on the lump-sum strategy are
more volatile. However, it would be a mistake to conclude that this is an advantage to using
gradualist strategies. DCA, VA and ESA all keep a large proportion of the available funds in cash,
and so will naturally record a lower level of volatility. By contrast, the lump-sum strategy is always
fully exposed. In our example of a random walk with zero drift, this cash allocation is not
penalized, but if instead long-term expected returns on the chosen asset are higher than the risk
free rate, then delayed investment will come at the cost of lower expected returns.
In normal circumstances we might turn to performance measures such as the Sharpe ratio to
assess whether the resulting trade-off between lower risk and lower expected return is
worthwhile, but such measures are systematically biased here. However, the intuition of the point
made by Rozeff (1994) applies: DCA is mean-variance inefficient because it gives insufficient
time diversification, concentrating the risk into later periods. We should expect VA to be similarly
inefficient compared to lump-sum investment.
7. Conclusion
The small amount of previous academic work on VA concludes that it generates higher expected
profits than alternative strategies even when price movements are unforecastable. This paper
shows that DCA and VA do indeed achieve lower average costs and higher IRRs than alternative
strategies, but they do not give higher expected profits. Instead an “averaging down” effect
systematically biases the IRR up and the average purchase cost down.
We first noted that where price movements are unforecastable no formula investment strategy
can expect to generate excess returns. The expected excess return is zero for each period, so
strategies which alter the amount invested in each period cannot change this expectation. We
then presented simulations which confirmed that VA achieves a higher expected IRR and lower
average purchase cost than alternative strategies, but it does not generate higher profits.
Previous studies have shown that investment managers can bias IRRs and Sharpe ratios by
manipulating future risk exposures in the light of the returns already achieved. DCA and VA
automatically manipulate their exposures in this way, by buying more after a fall in prices, and
less after a price rise. It is only for this reason that they appear to outperform other strategies.
The same bias will apply to a wide class of investment strategies where the amount invested in
each period is negatively correlated with the return made to date.
In conclusion, VA has little to recommend it. Contrary to the claims made by previous studies it
does not improve expected profits unless there is systematic mean reversion. Moreover, it
causes unpredictable cashflows and requires a large holding of liquid assets which is likely to
result in portfolios which are mean/variance inefficient.
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