507~~Simonhayley Value Averaging and the Automated Bias of Performance

7/31/2019 507~~Simonhayley Value Averaging and the Automated Bias of Performance

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VALUE AVERAGING AND THE AUTOMATED BIAS OF PERFORMANCEMEASURES

Abstract:

Value averaging is a formula investment strategy which can be shown to achieve a loweraverage 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 downeffect systematically biases the IRR up and the average purchase cost down. The same biasapplies to a wide class of investment strategies (including dollar cost averaging) where theamount 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Preliminary draft: comments welcome, but please do not cite without authors permission

Simon HayleyCass Business School00-44-(0)20-7040-0230


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1. Introduction

Value averaging (VA) is a formula investment strategy which can be shown to achieve loweraverage costs and higher IRRs than alternative strategies. However, in contrast to previousstudies, 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 upinvestments gradually over time in equal dollar amounts, rather than investing the desired total inone lump sum. Table 1 compares DCA with a strategy of buying equal numbers of shares eachperiod (ESA). The DCA strategy invests $100 each period, whereas ESA purchases 100 shareseach period. Shares initially cost $1, but the DCA strategy buys more shares as prices fall. ThusDCA achieves a lower average share price than ESA. Conversely, if prices rose, then DCA wouldpurchase 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 PriceSharesbought

Invest-ment($)

Portfolio($)

Sharesbought

Invest-ment($)

Portfolio($)

Sharesbought

Invest-ment($)

Portfolio($)

1 1.00 100 100 100 100 100 100 100 100 1002 0.90 100 90 171 111 100 180 122 110 2003 0.80 100 80 216 125 100 240 153 122 300

Total 300 270 336 300 375 332Average cost: 0.900 0.893 0.886

Proponents of DCA argue that as it reduces the average purchase cost, it must generate higherreturns. By contrast, previous academic research has long shown that despite its lower averagecost, 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 inportfolio value each period (assumed in Table 1 to be a rise of $100 per period). The investormust invest whatever amount is necessary in each period to meet this target. Like DCA, VApurchases more shares after a fall in prices, but the response is more sensitive: in this exampleVA 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 averagepurchase 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 theonly studies assessing its performance recommend it on the grounds that it results in a higherIRR (Edelson (1991), Marshall (2000, 2006)).

Neither strategy claims to be taking advantage of market inefficiencies. Indeed, simulationsappear 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 discretiononce committed to the strategy. As a result, both are subject to the criticisms set out byConstantinides (1979), who showed that strategies which pre-commit investors must be expectedto 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 theperiod, when the investor is nearly fully invested. Earlier returns are given correspondingly littleweight, since the investor then holds mainly cash. Better diversification is achieved by investing inone initial lump sum, and thus being equally exposed to the returns in each sub-period. Milevsky


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and Posner (1999) show that it is always possible to construct a constant proportionscontinuously 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 holdstrategy which dominates DCA.

Studies based on historical data have found that investing in one lump sum has generally givenbetter 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 atodds with DCAs lower average costs, but Hayley (2009) shows that comparing the average costachieved by DCA with the average price is misleading: it implicitly compares DCA with a strategywhich uses perfect foresight to invest more when prices are about to fall and less when they areabout 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 ahigher 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 expectedprofits.

The structure of this paper is as follows: section 2 demonstrates that in contrast to proponentsclaims, VA cannot expect to generate excess returns when prices follow a random walk. This isconfirmed by the Monte Carlo simulations presented in Section 3, which show that DCA and VAgenerate lower average purchase costs and higher IRRs, but do not increase average profits. Wethen investigate why average purchase costs (Section 4) and IRRs (Section 5) are systematicallybiased by these formula strategies. Section 6 briefly considers cashflow management and riskissues. 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 marketprices - in effect they assume that prices follow a random walk. However, we should stress thatthis is a statement about investors ex ante expectations, and does not imply any presumptionthat markets are in fact weak form efficient. The key point in this context is that VA, like DCA, willonly ever be an attractive strategy for investors who do not believe that they can forecast short-term price movements. These strategies commit investors to invest a specified amount no matterwhat they expect in the coming period those who feel that they can forecast short-term pricemovements 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 exante expectations rather than about markets themselves. Investors presumably believe that overthe medium term their chosen securities will generate an attractive return, but they must alsobelieve that the return over the short term (while they are using DCA or VA to build up theirposition) is likely to be small. Investors who expect significant returns over the short term wouldclearly prefer to invest immediately in one lump sum rather than delay their investments byfollowing a strategy which invests gradually. Marshall (2000) suggests that VA boosts returnseven in a random walk with zero drift.

The assumption of zero drift need not imply a loss of generality, since drift could be incorporatedinto this framework by defining prices not as absolute market prices, but as prices relative to anumeraire which appreciates at a rate which gives a fair return for the risks inherent in this asset(p i *=p i /(1+r)

i , where r reflects the cost of capital and a risk premium appropriate to this asset). Wecould then assume that p i * has zero expected drift since investors who use VA or DCA will notbelieve that they can forecast short-term relative asset returns: those who do would again rejecttrading strategies which forced them to delay their purchases. The results derived below would


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continue to hold for p i * , with profits then defined as excess returns compared to the risk-adjustedcost of capital. 1

We consider investing over a series of n discrete periods. The price of the asset in each period i is p i . The alternative investment strategies differ in the quantity of shares q i that are purchased ineach period. We evaluate profits at a subsequent point, after all investments have been made. Ifprices are then p T , the expected profit made by any investment strategy is:

n

iii

n

iiT q pq p

11

(1)

Our assumption of a random walk implies that future price movements ( p T /p i ) are independent ofthe past values of p i and q i . This gives us:

n

iii

n

iii

i

q pq p p

pT

11

(2)

But the random walk has zero drift, so E[ p T /p i ]=1 for all i and expected profits are zero. Theamount p i q i which is invested in each period is irrelevant: expected profits are zero for all theinvestment strategies that we consider here. Total expected profit is the expected percentagecapital gain between each period i and period T , multiplied by the amounts invested in eachperiod. But our assumption of a driftless random walk implies that the expected gain is zero in allperiods, so the timing of investments makes no difference. Against this background, the claimthat VA and DCA can generate excess profits even in the absence of market inefficiencies issurprising.

3. Monte Carlo Simulations

Tables 2 and 3 shows the results of 10,000 simulations in which prices movements aredistributed uniformly within the range +/-5% in each of five consecutive periods. The share priceis 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 sameexpected return. This assumption is clearly generous to VA if instead cash is held on deposit at lowerexpected return, then VAs 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 theexpenditure in the initial period and the target increase in portfolio value each period. In order to make theDCA and VA results here as comparable as possible, we have set both these figures to $400. Thus if pricesremain constant the three strategies would have identical cashflows, with each investing $400 in everyperiod. As shown in the previous section, this makes no difference to expected profits.


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Table 2: Strategy Costs

and Returns ESA DCA VA Lump SumAvg.Cost ($) Mean 10.0009 9.9943 9.9842 10.0000

Std. Error 0.0032 0.0032 0.0032 0.0000IRR (%) 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 measuressuch 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 withthe use of IRR (notably in real estate applications) are absent here: traded securities are likely tobe highly divisible, in contrast to large real estate projects. Furthermore, the cashflows generatedby real estate projects might have to be re-invested at very different yields, but our nullhypothesis here (that formula investment strategies generate no excess profits) would imply thatsurplus 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 isfor a more subtle reason. Phalippou (2008) notes that IRRs can be biased where the cashflowsare endogenous to the IRR achieved to date, and that investment managers could thusmanipulate their cashflows in order to boost their recorded IRRs. We show below that this bias isinherent in the VA and DCA strategies.

Sharpe ratios are not used in this comparison because DCA and VA claim to achieve theirbenefits 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. Bycontrast, the time-weighted rates of return which are conventionally used in calculating Sharperatios (notably in GIPS methodology) deliberately strip away any cashflow effects, leaving onlythe relative performance of the assets involved. This would remove the effect of DCA and VA, sothis is not a useful measure here. We will ultimately conclude that DCA and VA give zeroexpected excess profits, so we can infer that the Sharpe ratio will be zero in all the strategysimulations shown here, but in the meantime we investigate the IRR to avoid pre-judging theissue and to show the nature of the bias involved.

Table 3: DifferencesBetween Strategies DCA - ESA

DCA -Lump Sum DCA - VA VA - ESA

VA - LumpSum

Avg.Cost ($) Mean -0.00668 -0.00574 0.01001 -0.01669 -0.01576Std. 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 -0.1924Std. 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 andincreases in IRR compared to both ESA and lump sum investment strategies. But there are nosignificant differences in the profits made. As a robustness check, simulations were also run withprice 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 higherIRRs and lower average costs, but no significant change in profits. We find the same if we use


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Marshalls random investing strategy as our non-dynamic benchmark strategy (in place of thelump sum and ESA strategies).

Even on the assumption of a driftless random walk, where we know that the ex ante expectedreturn must be zero, our simulations show VA and DCA generating higher IRRs than otherstrategies. Thus IRRs appear to be biased measures of expected profit. We look at the reasonsfor this in section 5. But first we investigate the very similar mechanism whereby these tradingstrategies can expect to achieve low average purchase costs without improving their expectedprofits.

4. The Bias In Purchase Cost

To illustrate the bias in the average purchase cost, we contrast the outcomes in comparableDCA, 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)


Invest-ment($)

Portfolio($)

Sharesbought

Invest-ment($)

Portfolio($)

Sharesbought

Invest-ment($)

Portfolio($)

1 1.00 100 100 100 100 100 100 100 100 1002 0.90 100 90 171 111 100 180 122 110 2003 0.80 100 80 216 125 100 240 153 122 300Total 300 270 336 300 375 332Average cost: 0.900 0.893 0.886

We initially compare ESA and DCA. With the share price at $1, both strategies invest $100 in thefirst period. The price subsequently falls to $0.90. The ESA strategy buys 100 shares in thesecond period, but DCA again invests $100 so it buys more shares (111). By buying more shareswhen 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 expectedprofit from investment in each period is zero. The expected profit on the 100 shares purchased inperiod one was zero at the time they were purchased. When the price falls to $0.90 in period 2the investor suffers a loss of $10. The assumption of a driftless random walk means that this lossmust be expected to persist. It also means that the expected profit on any shares purchased atthe lower price in period two is zero. Thus the fact that the DCA and ESA strategies purchasedifferent 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 theirinitial investment by using it as an opportunity to acquire more shares at the new lower price. Thisreduces the average share price at which they entered the trade. Both ESA and DCA can beregarded 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.


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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 losssuffered on its earlier investment by investing an additional $10 in period 2. By buying 122 sharesthis period, VA achieves an even larger reduction in its average purchase cost, but again thismakes no difference to expected profits.

Chart 1 shows that the number of shares purchased in period 5 of our simulations by VA andDCA are highly correlated, but the range of variation is much larger for VA. This illustrates itsgreater tendency to average down, and hence to achieve a lower average cost.

25

30

35

40

45

50

55

60

25 30 35 40 45 50 55 60

D C A s

h a r e s p u r c

h a s e

d

VA shares purchased

Chart 1: Total Number Of Sh ares PurchasedUnder DCA and VA

Table 4 gives a different example which shows that these effects also apply when prices rise. Inperiod two prices have risen to $1.10, so purchases made in period one now look cheap. Thebest way to achieve a low average purchase cost is then to buy very few shares at this higherprice. ESA does badly here, buying another 100 shares, and thus doubling up the averagepurchase price to $1.05. DCA buys 91 shares, and VA buys only 82 as the capital gain on itsinitial 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)


Invest-ment($)

Portfolio($)

Sharesbought

Invest-ment($)

Portfolio($)

Sharesbought

Invest-ment($)

Portfolio($)

1 1.00 100 100 100 100 100 100 100 100 1002 1.10 100 110 231 91 100 220 82 90 2003 1.20 100 120 396 83 100 360 68 82 300Total 300 330 274 300 250 272Average cost: 1.10 1.094 1.087


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-0.01

0.00

8 9 10 11 12

D i f f e r e n c e

i n a v e r a g e c o s

t ( $ )

Terminal share price ($)

Chart 2: Comparative Average Purchase Cost AchievedBy Different Strategies

DCA avg. cost - ESA avg. cost

VA avg. cost - ESA avg. cost

The simulations confirm these points. Chart 2 compares the average costs achieved by thesethree strategies. We can see that:

(i) DCA and VA always achieve lower average purchase costs than ESA (and VAalways achieves a greater cost reduction than DCA because of its moreaggressive response to share price movements).

(ii) The difference is largest where the terminal price P T has moved a long way fromthe starting value. Volatile share prices allow DCA and VA greater scope tobenefit from buying more shares at low prices. Conversely, if prices do not varyat all, the three strategies will be identical.

Proponents of DCA almost invariably assume that a strategy which buys at lower average costmust lead to higher profits. The continued popularity of DCA (in spite of academic studies whichshow it to be a sub-optimal strategy) suggests that investors generally find this argument highlypersuasive.

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 insimulations where prices end up falling, and less when they are falling. By contrast, DCA investsa fixed total dollar amount ($2000 for these simulations). The fact that DCA invests at a loweraverage cost is balanced by the fact that it tends to invest a smaller amount in periods of risingprices 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 risingprices (since capital gains help achieve the investors desired portfolio value without the need forsubstantial additional investment). Thus a VA strategy tends to invest far less than ESA duringperiods of rising prices and far more in periods of falling prices. This offsets the fact that VAachieves a lower average cost. Once again, expected profits are identical for the two strategies.


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1800

1900

2000

2100

2200

8.0 9.0 10.0 11.0 12.0

T o t a l a m o u n

t i n v e s

t e d ( $ )


Chart 3: Total Amoun t Invested By Each Strategy

ESADCAVA

Ingersoll et al. (2007) show that cynical investment managers can manipulate conventionalperformance measures (including the Sharpe ratio, Jensens alpha etc.). The gaming of theseperformance measures is achieved by reducing risk exposure following a good performance andincreasing exposure after a poor performance. This strategy could be characterized as applyingan 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 fallsand less after price rises. This allows them to achieve low average purchase costs withoutachieving any increase in profitability. The following section shows that the same bias applies tothe IRRs achieved by these strategies.

5. The Bias In IRRs

The section above showed that DCA and VA achieve lower average costs, but the sameexpected profits as ESA. However, VA is a more complex strategy than DCA, with varyingcashflows, so proponents of VA focus instead on the fact that it tends to generate a higherinternal rate of return than either ESA or DCA. Our simulations confirm (Table 3 and Chart 4) thatIRRs are generally higher for VA than for DCA, with both strategies giving higher average IRRs

than ESA3

. It might appear intuitive that a higher IRR will imply higher expected profits, but in thissection we show that these IRRs are misleading, since the same bias is at work as we found foraverage costs.

3 Marshall (2000) calculates which strategy gives the highest IRR for each simulated path. In 73.5% of casesVA was best, with DCA best in only 3.9% of cases. Chart 4 helps explain this: where the simulated price pathtends to mean-revert, the aggressive VA strategy generally gives the highest IRR. Where prices makesustained movements, strategies with no dynamic component fare better (ESA here, or Marshalls randominvesting strategy). The less aggressive DCA strategy is almost always outperformed by one of the other twostrategies, but frequently comes in second place, consistent with the results in Table 3 showing that itrecords a higher average IRR than ESA.


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-0.05%

0.00%

0.05%

0.10%

0.15%

8.0 9.0 10.0 11.0 12.0

I R R d i f f e r e n

t i a

l


Chart 4: Comparative IRRs Ach ieved By Different

Strategies

VA IRR - ESA IRRDCA 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 thestrategies has made a $10 loss on the $100 invested in the first period and the assumption of adriftless random walk means that this loss must be expected to persist, leading to a negativeoverall 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 theseinvestments taken separately, and the relative amounts invested in each of these periods. Thisrelationship is polynomial, but the direction of the relationship is intuitive: if prices have fallen inperiod 2, the expected IRR on the amount invested in period 1 (IRR 1) is now negative, but theexpected IRR on the amount that we are about to invest in period 2 (IRR 2) is still zero. The morethat we invest in the second period, the more the expected IRR on the two investments combined(IRRc) is likely to move away from IRR 1 and towards IRR 2 (ie. zero)

4. Thus if our objective is tomaximize IRR c, our best response in period 2 to the loss made already would be to dilute thenegative IRR 1 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 fallin prices. VA does the same, but more aggressively. This process is very similar to the doublingdown we saw in the previous section, the only difference being that the more complex arithmeticof 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 IRR 1 is positive, whilst ourexpected IRR 2 is still zero. The best strategy for maximizing our expected IRR c would thus be toinvest relatively little in the second period, to avoid diluting the positive expected IRR 1 with thezero expected IRR 2.

4 The polynomial arithmetic of IRRs means that there may be exceptions to this. Indeed, VA can entail thereturn of cash to investors in some periods (where a large price rise results in a capital gain that is greaterthan the target increase in portfolio value). Thus there may be multiple swings from positive to negativecashflow, so we cannot rule out multiple roots in our IRR calculation. However, as long as it is generally truethat the aggregate IRR is biased towards zero by investing larger amounts in the current period, then therewill be a bias in the average IRR. Our simulations confirm that this is indeed the case.


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Phalippou (2008) notes that the IRRs recorded by private equity managers can be manipulatedby 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 notedby Ingersoll et al. (2007) for biasing other performance measures. The bias in each case isachieved by reducing exposure following a good outturn and increasing exposure following a badoutturn. By doing this automatically, DCA and VA achieve IRRs which are better than ESA, whilsttheir ex ante expected profits remain zero.

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0

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8.0 9.0 10.0 11.0 12.0

D i f f e r e n c e

i n p r o

f i t ( $ )


Chart 5: Comparative Profits

DCA Profit - ESA Profit

VA Profit - ESA Profit

Chart 5 shows the increase or decrease in profits generated by each of the formula tradingstrategies compared to a static ESA strategy. As we saw in Table 3, the differential averageszero, but the formula trading strategies outperform when the share price ends up relatively closeto its starting value ($10). These strategies buy more shares at relatively low prices, and whenprices mean-revert this lower average cost does translate into higher profits (with none of theoffsetting downside illustrated in Chart 3). This advantage is larger for VA, with its moreaggressive response to changing share prices, than it is for DCA. Conversely, DCA and VA dopoorly in sustained price trends, since they purchase more shares than ESA in downtrends andfewer shares than ESA in uptrends.

Thus VA can be profitable where investors correctly anticipate mean-reversion (implying thatmarkets 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 wheninvestors have no ability to forecast returns. Furthermore, even where there is an element ofmean reversion, other dynamic strategies (e.g. based on calibrated filter rules) are likely to bemore 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 marketswhere price movements cannot be forecast. We now consider briefly the effects that VA has oncashflow management and risk levels.


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DCA generates perfectly stable cashflows by construction, with a fixed dollar amount investedeach 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 amisleading claim of superior returns, but VA comes with the added drawback of highly uncertaininvestor cashflows, since the amount which must be invested each period depends on the mostrecent movements in the market price (VA can even imply an unexpected return of cash to theinvestor following a large price rise). Indeed, these cashflows are likely to become increasinglyvolatile over time as the existing portfolio increases in size relative to the target increase in valueeach period. Edelson (1991) envisages investors holding a side fund containing liquid assetssufficient 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 aremore volatile. However, it would be a mistake to conclude that this is an advantage to usinggradualist 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 alwaysfully exposed. In our example of a random walk with zero drift, this cash allocation is notpenalized, but if instead long-term expected returns on the chosen asset are higher than the riskfree 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 toassess whether the resulting trade-off between lower risk and lower expected return isworthwhile, but such measures are systematically biased here. However, the intuition of the pointmade by Rozeff (1994) applies: DCA is mean-variance inefficient because it gives insufficienttime diversification, concentrating the risk into later periods. We should expect VA to be similarlyinefficient compared to lump-sum investment.

7. Conclusion

The small amount of previous academic work on VA concludes that it generates higher expectedprofits than alternative strategies even when price movements are unforecastable. This papershows that DCA and VA do indeed achieve lower average costs and higher IRRs than alternativestrategies, but they do not give higher expected profits. Instead an averaging down effectsystematically biases the IRR up and the average purchase cost down.

We first noted that where price movements are unforecastable no formula investment strategycan expect to generate excess returns. The expected excess return is zero for each period, sostrategies which alter the amount invested in each period cannot change this expectation. Wethen presented simulations which confirmed that VA achieves a higher expected IRR and loweraverage 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 bymanipulating future risk exposures in the light of the returns already achieved. DCA and VAautomatically manipulate their exposures in this way, by buying more after a fall in prices, andless 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 ineach 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 itdoes not improve expected profits unless there is systematic mean reversion. Moreover, itcauses unpredictable cashflows and requires a large holding of liquid assets which is likely toresult in portfolios which are mean/variance inefficient.


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References

Constantinides, G.M. 1979. A note On The Suboptimality Of Dollar-Cost Averaging As AnInvestment Policy. Journal of Financial and Quantitative Analysis , vol.14, no. 2 (June): 443-450.

Edleson, M.E. 1991 Value Averaging: The Safe and Easy Strategy for Higher InvestmentReturns. Wiley Investment Classics, revised edition 2006.

Edleson, M.E. 1988 Value Averaging: A New Approach To Accumulation. American Association of Individual Investors Journal vol. X, no. 7 (August 1988)

Hayley, S. 2009. Explaining the Riddle of Dollar Cost Averaging, Cass Business School workingpaper.

Ingersoll, J; Spiegel, M; Goetzmann, W and Welch, I. (2007) "Portfolio Performance Manipulation

and Manipulation-proof Performance Measures," Review of Financial Studies 20-5, September2007, 1503-1546.

Knight, J.R. and Mandell, L. 1992/93. Nobody Gains From Dollar Cost Averaging: Analytical,Numerical And Empirical Results. Financial Services Review , vol. 2, issue 1: 51-61.

Marshall, P.S. 2000. A Statistical Comparison Of Value Averaging Vs. Dollar Cost AveragingAnd Random Investment Techniques. Journal of Financial and Strategic Decisions , vol. 13, no.1 (Spring) 87-99.

Marshall, P.S. 2006 A multi-market, historical comparison of the investment returns of valueaveraging, dollar cost averaging and random investment techniques. Academy of Accountingand Financial Studies Journal, Sept 2006.

Milevsky, M. A. and Posner, S. E. 2003 "A Continuous-Time Re-examination of the Inefficiency ofDollar-Cost Averaging" International Journal of Theoretical & Applied Finance, Mar 2003, Vol. 6Issue 2.

Phalippou, L. 2008 The Hazards of Using IRR to Measure Performance: The Case of PrivateEquity. Journal of Performance Measurement, Fall issue.

Rozeff, M.S. 1994. Lump-sum Investing Versus Dollar-Averaging. Journal of Portfolio Management , vol. 20, issue 2 (winter): 45-50.

Thorley, S.R. 1994. The fallacy of Dollar Cost Averaging. Financial Practice and Education , vol.4, no. 2 (Fall/Winter): 138-143.

Williams, R.E. and Bacon, P.W. 1993. Lump-sum Beats Dollar Cost Averaging. Journal of Financial Planning , vol. 6, no.2 (April): 6467.

Documents

507~~Simonhayley Value Averaging and the Automated Bias of Performance