a tournament model of fund management

8/3/2019 A Tournament Model of Fund Management

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A Tournament Model of Fund Management

Daniella AckerNigel W. Duck1

Working Paper No. 01/529December 2001

Department of EconomicsUniversity of Bristol

8 Woodland RoadBristol BS8 1TN

UK

Corresponding Author: Nigel W. DuckE-mail: [email protected]: +44 (0)117 928 8406

Fax: +44 (0)117 928 8577

Abstract

We develop a two-period model of portfolio management where managers aim

for a higher-valued portfolio than their rivals whilst making a respectable return.It predicts that losing managers will show a greater tendency to adopt extremeportfolios; this will be more marked the further behind the fund is and the nearerthe final ranking period; and, when taking extreme positions, losing managers willchoose high /low market exposure depending upon whether they (a) expect a risingor falling market, and (b) have sufficient assets to take advantage of a rising market.Tests, using UK investment trusts data, broadly support these predictions.

Classification Code: G1, G10, G11

Key Words: Tournament, Fund Management, Investment Trusts, Mutual Funds

1 We are grateful to John Core, David Demery, Kees Jan van Garderen, Ian Jewitt, Hedley Rees,Suro Sahay, and Ian Tonks for helpful comments. Any errors are entirely our responsibility.


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A Tournament Model of Fund Management 1

1 Introduction

There is at present considerable interest in whether decisions taken by man-agers of pooled investment vehicles are optimal for the individuals on whosebehalf those decisions are made. One area of research, such as Brown, Goet-zmann and Park (2001) and Carpenter (2000), is the analysis of the impacton risk-taking of a convex managerial reward system. Other work inves-tigates whether the ranking of a mutual fund has consequences for futureinows of investment funds (see, for example, Chevalier and Ellison (CE)(1997) and Goetzmann and Peles (1997)). Our paper concentrates on thetournament aspect of the fund management sector and its eect on port-folio choice, taking as given that ranking is an important consideration forfund managers. We develop a model of portfolio management in which fund

managers aim to achieve a higher-valued portfolio than their rivals whilstmaking a respectable return.1 We draw out from the model a set of predic-tions which we test using monthly data on UK investment trusts coveringthe four years ending March 31, 2001.

Our model extends the literature in three main ways. First, it is agenuine two-period model. Existing work on tournaments and fund man-agement is based on the intuition derived from theoretical models that es-sentially relate to one period only (see, for example Taylor (2000)). In suchmodels, the portfolio values are inherited at the beginning of the period,and fund managers take one set of decisions to maximise the probability ofwinning at the end of that period. They predict that inheritors of an un-

derperforming portfolio will take more risk. Our model explicitly allows fortwo sequential sets of decisions that are taken to maximise the probabilityof winning at some given future date. Implications can be loosely drawnfrom it about a more prolonged decision process. Secondly, our denition ofrisk-taking is dierent from the denition other models employ. Empiricalpapers, such as Brown, Harlow and Starks (BHS) (1996) and CE (1997),dene risk primarily in terms of variability of portfolio value. For example,BHS compare portfolio standard deviations at year-ends with those at themid-way stage. In contrast, we categorise portfolio positions by how ex-treme they are: an extreme portfolio consists mainly of cash or mainlyof shares, that is, very low or very high market exposure. Which of these

extreme positions is adopted depends primarily, but not entirely, on whetherthe fund manager believes the market will rise or fall. In other words, weincorporate the idea of taking bets, either with or against the market. 2

1 If we assume that fund managers are exclusively driven by the aim of beating theirrivals, then the reward to a fund manager who made heavy losses would be exactly thesame as if he made exceptional prots, provided all other fund managers made even worselosses, even if share prices were rising.

2 CE decompose the portfolio variance into a specic risk component and a componentreecting the variance caused by moving the portfolios beta away from 1. The latter


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Finally, we set the problem within a signal-extraction framework which

allows us to recognise the market timing aspect of fund managers activi-ties: they try to assess whether the market is falling or rising, and adoptappropriately defensive or aggressive investment strategies (see, for exam-ple, Pesaran and Timmermann (2000) for a recent UK study and Granger(1992) for a summary of related results for the US).

The testable hypotheses we derive from our model are:

Managers of losing funds will show a greater tendency than averageto adopt extreme portfolios; this tendency will be more marked thefurther behind the fund is.

This tendency to adopt extreme portfolios is likely to increase as the

nal ranking period approaches.

When taking an extreme position, the manager of a losing fund willchoose between high market exposure or low market exposure accord-ing to (a) whether he expects the market to rise or fall;3 and (b)whether he has sucient assets to warrant adopting a high marketexposure when he expects the market to rise. This last conditionis one not found elsewhere in the literature, and arises because ourmodel is based on absolute share price movements rather than per-centage returns, as is more conventional in the nance literature. Thisunorthodox approach may help explain the anomaly in Elton et al.(2001), who nd that funds often adopt betas less than one when they

might expect a higher rate of return from higher beta strategies.

Our empirical procedures dier from those generally used in US studiesinto tournament models of portfolio management. US studies tend to exam-ine changes in the risk-taking strategy of loser funds between the mid-pointand the end of the tournament at 31 December, and have, on the whole,generated inconclusive results. But, in the UK at least, the date of the nalranking period may be less clear-cut. Furthermore, losers may have beenlosers for some time when they are sampled; and, nally, losers under oneranking criterion - for example, performance over the previous year - maynot be losers under another - for example, performance over the last three

years. For all these reasons, changes in strategy may well not be observedbetween one period and another. In this paper we examine instead cross-sectional dierences between losers and winners, where losers and winners

enters their regressions as a dependent variable in the form j-1j, so that no distinctionis made between decisions to increase the portfolio beta above 1, or to reduce it below1. CE describe such risk as variance from implicit bets with or against the market as awhole (p.1187). We distinguish between the two types of bet.

3 We use the term he throughout to denote the more cumbersome he or she.


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are dened in three dierent ways, following the ranking criteria of the Fi-

nancial Times newspaper. We nd strong support for the three hypothesesmentioned above.The paper is in two broad sections. In the rst we develop our model

and draw out its main implications; and in the second we explain our dataand the results of our tests. We end with a set of conclusions and a briefdiscussion of the implications of our ndings.

2 A Model of Fund Management

2.1 Assumptions and Notation

We consider two representative, risk-neutral traders, denoted A and B. B

manages the average sector fund against which As performance will bemeasured: specically, A will win if, in the nal period, his portfolio isworth more than Bs.4 As decisions are the focus of our model.

We assume three periods. At the beginning of the rst period, t1 , thetwo traders are endowed with initial portfolios consisting of a combination ofa risky asset (shares) and a risk-free asset (cash). Borrowing is precluded.5

Bs portfolio (the average portfolio) consists of NB shares and cash, CB.Cash gives a zero return. The number of shares in Bs portfolio is assumed tobe known to trader A and does not change over the three periods, althoughthe value of Bs portfolio changes as the share price changes. The shareprice, P, is assumed to follow a random walk, so that

Pi+1 = Pi + "i+1 (1)

where "i+1 is the shock to the share price between periods i and i + 1(i = t 1; t), is i.i.d, and is known to be distributed as N(0; 2"). Weassume that trader A has no information about the size of the shock to theshare price more than one period ahead. His expectation of "i+1 formed inperiod i 1 will therefore be zero with an associated variance of 2". But,importantly for what follows, we assume that trader A obtains news duringperiod i which allows him to make an inference about the shock to the shareprice in period i+1.This news can be viewed as information that is specic toA, or as As interpretation of more widely available information. Formally,we decompose the shock to the share price into trader As inference plus anerror that is uncorrelated with that inference:

"i+1 = "Ai+1 + v

Ai+1 (2)

where "Ai+1 is trader As inference of the price shock in period i + 1after hearing this news in period i; and the vAi+1 are i.i.d. and known to be

4 We assume that As portfolio is suciently small within the sector to allow the per-formance of the average fund to be exogenous.

5 The predictions of the model hold as long as borrowing is restricted to some degree.


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distributed as N(0; 2vA). The inverse of2vA can be seen as a measure of the

precision of As beliefs - the greater is 2

vA, the less accurate are trader Asinferences about future share price movements. Note that 2vA 2" (that

is, receipt of the news does not decrease the precision of As inference aboutthe shock to the share price6).

We adopt the following other notation:NAi is the number of shares held by trader A at the beginning of period

i, where i = t 1; t ; t + 1;NB is the constant number of shares held by trader B;CAi is the amount of cash held by trader A at the beginning of period i;CB is the constant amount of cash held by trader B; andP FJi is the value of trader Js portfolio in period i; J = A; B.Figure 1 shows a timeline of events, together with the decisions that A

must take.

FIGURE 1 ABOUT HERE

2.2 Trader As Decision Process

The strategy underlying trader As decision process is the same for bothperiods: his aim is to maximise the probability that the value of his portfolioin period t + 1 will exceed that of Bs portfolio. He attempts to achieve thisaim by arranging his portfolio in period i (i.e. selecting the number of shareshe will be holding at the beginning of period i + 1) after hearing the news

relating to the share price in period i + 1. His portfolio choice depends onwhether the news leads him to expect the price to rise or to fall.The precise decisions A makes depend on the period in which he makes

them. We rst analyse As decision in period t, the period immediatelybefore the nal period when the tournament is decided. We then analysehis decision in period t 1.

2.2.1 As decision in period t

The value of trader Js portfolio in period i can be written,

P FJi = PiNJi + C

Ji (3)

And each traders budget constraint can be written

PiNJi+1 = PiN

Ji + C

Ji C

Ji+1 (4)

6 This a commonly-used denition of information. A notable exception is Beaver (1968),who argues that a decision maker may be more uncertain about a given event afterreceiving a message about the event than he was b efore he received the message. To useTheils terminology, the entropy may increase as a result of a message, yet the messagehas information content. (footnote 8, citing Theil 1967 Ch.1)


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Updating equation (3) by one period and using equations (1) and (4) we

can write

P FJi+1 = P FJi + "i+1N

Ji+1 (5)

This states that the value of trader Js portfolio in any period equalsits value in the previous period, adjusted by the shock to the share pricemultiplied by the number of shares in the portfolio when the shock occurs.Three implications follow directly from this equation: (i) if As portfoliovalue were higher than Bs at period t then A would be certain to winthe tournament if he held the same number of shares as B since any shareprice movement would then aect the two portfolios equally;7 (ii) for thesame reason, if A were behind at period t and held the same number of

shares as B then he would be certain to lose; and (iii) there are two ways inwhich A might overtake B: by holding more shares than B when the shareprice rises suciently; or holding fewer shares when the share price fallssuciently. The rst two of these implications are straightforward. In therest of this subsection we consider the more complex third implication andthe decision which it leads A to make. We therefore impose the conditionthat As portfolio value at the beginning of period t is less than Bs, that isP FAt < P F

Bt .

Under this condition As problem in period t, formally stated, is

maxNAt+1 hPrP F

At+1 > P F

Bt+1i (6)

Given that A is maximising this probability after hearing the news re-lating to the share-price movement between period t and t + 1, his problemcan be conveniently recast as that of selecting NAt+1 to,

maxNAt+1

ZL

1

f(x)dx maxNAt+1

N(L) (7)

where f(x) is the standard Normal distribution density function; N(:)is the corresponding cumulative density function; and the limit L can bewritten,

L =1

Av

"P FBt P F

At "

At+1(N

At+1 N

B)NAt+1 N

B

#(8)

From the properties of the Normal distribution it is clear that the prob-ability that A wins is maximised when L is minimised. If A chooses NAt+1 >NB then the limit L will be negatively related to NAt+1. This implies that if

7 This is analogous to covering in a two-boat sailing race, or to indexing in the nancialmarkets.


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A decides to hold more shares than B he should hold the maximum number

of shares he can, which is P FA

t =Pt , and the limit L becomes

LS1

Av

"P FBt P F

At

P FAt =Pt NB "At+1

#(9)

If NAt+1 < NB then L is positively related to NAt+1and A will want to

hold as few shares as possible, which is zero. In this case L becomes

LC1

Av

"P FBt P F

At

NB+ "At+1

#(10)

The two limits, LS and LC, allow us to identify "

t+1, the value of "At+1

which will trigger the movement from an all-cash to an all-shares strategy.Formally, it will be where LS = LC and so,

"t+1

P FBt P F

At

2

!1

P FAt =Pt NB

1

NB

!(11)

If "At+1 exceeds "

t+1, As optimum strategy is to choose shares, hopingthat the price rises suciently to close the gap between his portfolio and Bs.If"At+1 is less than "

t+1, his optimum strategy is to choose cash and hope thatthe price falls suciently to eliminate Bs advantage. Notice that "t+1 will

be zero if the maximum number of shares that A can hold P FAt =Ptequalstwice the number that B holds. The reason for this is that a rise in priceof $x when A is holding 2NB shares, and a fall in price of $x when he isholding none, will cause exactly the same closure of the gap between Asand Bs portfolios. The probability of winning with the all-shares strategyand the probability of winning with the all-cash strategy are, in this senseand under this special condition, symmetric around a zero value of "At+1. Onthe other hand, if A were holding more than 2NB shares, then a price riseof $x would mean a larger closing of the gap than would a price fall of $x ifhe held no shares. Hence, if A has sucient funds to hold more than 2NB

shares "t+1 is negative and As optimal strategy may be to hold all shareseven though he expects a (slight) fall in the share price. The opposite is

also true when A cannot hold more than 2NB shares.As decision process in period t can now be summarised as follows:

1. If A is in front at t, he should hold NB shares and will win at t + 1with probability 1.

2. If A is behind at t and his portfolio is not suciently valuable toenable him to hold more shares than B he should hold all cash andthe probability that he wins will be N(LC).


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3. If A is behind at t, but his portfolio is suciently valuable to enable

him to hold more shares than B, he will, depending upon the valueof "At+1, either hold all shares (in which case the probability that hewins is N(LS)), or all cash (with a probability of winning equal toN(LC)) .

As decision at period t clearly has an all-or-nothing quality: provided Ainherits a portfolio in period t which is lower in value than Bs A will holdeither all cash or all shares; only if he inherits a portfolio which is higher invalue than Bs will A hold a diversied portfolio. As we shall now show, thesame is not true of period t 1.

2.2.2 As decision at t 1

As problem at t 1 is to choose the number of shares to hold in periodt, NAt , to maximise the probability that he will win in period t + 1. Thisprobability, w, can be written,

w f + (1 f)wb (12)

wheref is the probability that A gets in front at t;8 andwb is the probability that A will win at t + 1 even if he is behind at t.

If As portfolio in period t 1 exceeds Bs then his problem is trivial -he should hold NB shares in periods t and t + 1 and will then be certain

to win. So we consider only the case where A is behind at t 1. In thesecircumstances, A can no longer act simply to maximise f, the probabilityof having a higher portfolio value than B in the next period. Were he todo so, he would, by the logic of the previous section, choose either an all-cash or an all-shares strategy in period t 1 and thereby run the risk offalling so much further behind (if the share price moved the wrong way)that he would have no chance of making up the gap in the nal period. Insome circumstances an extreme strategy might still be optimal, for exampleif he were so far behind that only successive extreme strategies with twofavourable share price movements oer him a chance of overtaking B. Butsince in this two-period model he will still have another chance to win, he

may sometimes be willing at t 1 to sacrice some probability of gettingahead immediately, in exchange for reducing the probability of being too farbehind at t to win. More formally, to maximise w he must consider theeects of his decision on both f and wb, the probability that he will winat t + 1 despite being behind at period t. To show that this may lead toA adopting neither an all-shares nor an all-cash strategy in period t 1 weneed to work out the various probabilities in equation (12).

8 Note that once A does get in front, he simply needs to hold NB shares and will thenbe certain to win.


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The calculation off This is simply the probability derived in the previ-

ous section lagged one period, and it can be derived by rolling back equations(7) and (8) one period to give:

1. IfNAt > NB f = N(L1) where L1

1Av

PFB

t1PFA

t1

NAtNB

"At

2. IfNAt < NB f = N(L1)

3. IfNAt = NB f = 0

The calculation of wb This probability depends on As assessment att 1 of his probable strategy at t. He knows that he will either choose tocarry an all-shares portfolio into period t + 1, in which case he will win if

"t+1 is suciently high; or he will choose all cash, in which case he will winif "t+1 is suciently low. Calculating wb therefore involves the calculation

of three separate probabilities: (i) the probability that A will choose an all-shares strategy at period t;9 (ii) the probability that "t+1 is suciently highto ensure that this strategy allows A to overtake B in period t + 1; and (iii)the probability that "t+1 is suciently low to ensure that the alternative all-cash strategy allows A to overtake B in period t + 1: The detailed derivationof these probabilities, all of which are assessed in period t 1, is given inthe appendix. Here we give the main results.

First, As assessment (in period t 1) of the probability that he willchoose an all-shares strategy for the nal period will depend upon the rela-tive shareholdings that A and B carry into period t. This is because theseshareholdings will determine the relative portfolio values after the period tprice change, and hence the strategies available at the end of that period.The precise relationships are as follows:

1. If A chooses NAt > 2NB the probability, assessed in period t

1, that he will choose an all-shares strategy for the nal periodequals N(L2):N(L3)

2. IfA chooses NB < NAt 2NB it equals N(L2):N(L3)

3. IfA chooses NAt NB it equals N(L2):N(L3)

where:L2 =

1Av

2NBPt1PFAt1

NAt2NB

"At

L3 =1Av

NBPt1PFAt1

NAtNB

"At

9 The probability that A will choose an all-cash strategy is, as the previous section

implied, simply one minus the probability that A selects an all-shares strategy.


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Secondly, the probability (evaluated in period t 1) that "t+1 is su-

ciently high to ensure that an all-shares strategy allows A to overtake B inperiod t + 1 can be written as

H = Pr

8 0, is dened as ITj P

j=P

j0 , where P

j is the share price of the

investment trust at date . Similarly, the normalised value of the averagefund, ITav , is the relative price of the relevant sector index. Losing fundsare then identied as funds for which ITav IT

j is positive: the higher this

measure is, the further behind is the fund:14

3.2 Extreme portfolios

In our model, managers have a choice between one risky and one risk-freeasset. Clearly in reality there are many risky assets to choose from, butinvestment trusts tend to track a market index so we use an appropriateindex to represent the risky asset. Although the Association of InvestmentTrust Companies publish comprehensive UK monthly statistics, there is in-sucient detailed data on fund composition, so we proxy the proportionof the fund held in the form of shares by the trusts market model beta. 15

Funds with extreme portfolios (i.e. extreme betas) in any period are thendened as those whose betas are, in that period, more than 1 standard devi-ation away from the average beta of their sector. We discuss below how wemeasure betas, and the way in which we classify funds and hence extremebetas.

3.3 The nal period

The empirical counterpart to the models nal period (period t + 1) isless clear. US work on mutual funds, such as Busse (2001) and Brown etal. (1996), takes the nal rankings to be at the end of the calendar year(31 December). UK investment trusts are ranked weekly in the FT and,furthermore, the rankings are based on one-, three- and ve-year perfor-mance so each fund has three dierent rankings in the FT alone. (Thereare also other sources of rankings, but the FT is the most widely read -nancial paper.) Consequently there are many intermediate ranking days

and no obvious nal ranking day, though one might conjecture that thereare periods in the UK when rankings may be more important than at other

1 4 An alternative measure would base the fund value on the value of the investments inthe fund, the net asset value (NAV). However, it is well-established that an investmenttrusts shares trade at a discount to its NAV and we are interested in the value of theportfolio from the point of view of an investor in the fund, not an investor in the underlyingassets of the fund.

1 5 Using a funds beta as our risk measure addresses the problem noted in Busse (2001),who stresses the importance of distinguishing between risk changes that are due tochanges in common factors and changes attributable to the fund managers. (p.72)


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times. The most likely of these are the end of December, when there are

many reviews assessing fund managers performance, and the month or soleading up to the end of the tax year (31 March for companies and 5 Aprilfor individuals) when investors are disposing of surplus funds and may wellbe looking at rankings to help them decide where to invest.

3.4 Maximum shareholdings

While it is more usual to discuss portfolio composition in terms of propor-tions rather than absolute values, our model shows that this obscures anissue that is important when borrowing is precluded or restricted16. Whilethe loser can always hold the same proportion of shares as the average fund,he cannot always expose the same total asset value to the market. It may

be not be possible in a rising market for a loser fund to stake a sucientlylarge bet to win.17 Indeed, this may explain the anomaly in Elton et al.(2001, Abstract): Surprisingly, funds on average have a beta less than onewhen a beta greater than one would have provided a higher expected returnwith potentially the same tracking error.

To proxy a funds maximum possible shareholding we use the NAV of afund - a measure of the total assets available for exposure to the market. Aswith the normalised value of investment trust j, we can dene a normalisedNAV: NAVj NAV

Uj =NAV

Uj0 , where NAV

Uj is the unadjusted netasset value of investment trust j. As discussed above, is the proxy forthe proportion of assets held in the form of shares, so the value of these

assets at time is j

NAVj . The maximum number of shares that fundJ can hold relative to the average fund, MAXN, is therefore MAXNj

(NAVj )=(av NAV

av ).

3.5 Anticipated market movements

Observations on each fund managers news about future share price move-ments are not available. To capture any common element in the news in any

1 6 Investment trusts can borrow within the limits laid down by their Articles of Associa-tion. As with any company there are other limits to their borrowing that are a function ofthe value of their assets. It is interesting to note that the split-capital investment trust(a form of fund which was introduced into the UK relatively recently and is often designed

to increase volatility and expected returns by gearing up) has, at the time of writing, beensubject to much adverse comment in the nancial press: Analysts are concerned thatmany of these new trusts ... have heavy borrowings [and] are now close to breaching theirasset-to-loan agreements... (Financial Times, 11 August, 2001).

1 7 Taylor (2000), which most closely resembles our paper, avoids this issue by implicitlyassuming in his equation (1) that the nal portfolio value is the sum of the performancesin the rst and the second halves of the year, rather than the result of a compoundedreturn on the portfolio value at the end of the rst half. This is equivalent to assumingthat there are two independent tournaments, the winner being the fund with the bestperformance overall.


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particular month we have made use of the monthly summaries produced by

the UK Treasury of the forecasts of the rate of growth of UK GDP overthe forthcoming year made by (generally) 35 independent bodies such asBarclays Bank, Goldman Sachs and Merill Lynch. We take the average ofthese forecasts for month i as a indicator of each funds "Ai+1; specically weassume that a high value for this average suggests that each fund is morelikely to expect a rise in share prices; a low value suggests that each fund ismore likely to expect a price fall.

3.6 Data

We used the categories of investment trusts designated by the Associationof Investment Trust Companies as at May 31, 200118 and, since this makes

the estimation of expected market movements easier, we chose those whospecialise in UK companies. Trusts are either grouped by style such ashigh income or growth, or by sector specialism such as biotechnology andproperty. There were 47 sector specialist funds in total, specialising in 11dierent sectors, giving sample sizes that were too small. The four stylecategories were high income (31 companies), high growth (30), growth andincome (45) and smaller companies (35). To carry out our tests we need toestimate betas for each fund and this of course requires some appropriatemarket index. It was dicult to construct a meaningful market index forthe rst two categories, so our nal sample consisted of the 80 companies inthe last two categories. Of these, 19 had insuciently long price histories,

so the sample was reduced to 61 companies, 33 in growth and income and28 in smaller companies.19

Other than the UK Treasury data, all data were collected from Datas-tream, which provides prices and NAVs for investment trusts within eachsector and for the sector index (the average fund). The sample period isthe 48 months ended February 28, 2001. Treasury forecasts were availableonly from September 1997, so tests involving estimates of future marketmovements were limited to 42 months of observations.

At each month end the sample of investment trusts was divided intothose who performed worse than the sector index, the losers, and those whoperformed better than the sector index. The winner category excluded thetop four funds in each sector at each month end, as they are likely to be

1 8 Since the tests use historic data, there is a risk of survivorship bias. Past loser fundsmay have been wound up, resulting in a sample that is disproportionately weighted infavour of successful funds. However, the performance index of the average fund, whichis used to categorise sampled funds as winners or losers, is a historic sector index thattakes account of all funds within the sector at the time the index is calculated.

1 9 Not all companies sampled had sucient data to be included in the earlier months,particularly when tests involved 5-year performance. The maximum nal sample size was2,486 rm-months.


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competing for top ranking and therefore may be operating a strategy that

is dierent from the other funds, as suggested in CE (1997).20

As the FT publishes rankings based on one-, three- and ve-year fundperformance, we constructed separate groups based on performance overeach of these periods. All tests were carried out separately under eachperformance measure.

3.7 Betas

We measured betas using the market model (the FTSE All-Share indexproxied the market for the growth-and-income funds, the FTSE Small-Capfor the smaller companies), based on log changes in NAV rather than changesin price, since the latter will be aected by variations in discount. (Using

NAVs also removed the estimation problems caused by thin trading.)Since we are eectively treating each months observations as indepen-

dent (see below for further discussion of this point), we would be dealing withoverlapping data were we to use historic monthly returns to estimate the be-tas, as is standard. We therefore based our estimates on daily NAV returnsover the month starting on the observation day. This clearly reduces thenumber of data points used in the beta estimation. One compromise wouldbe to halve the sample size, taking observations only every two months andmeasuring the betas over two-month intervals. Although this would improvethe reliability of the betas, it would increase the standard errors in the testsof the model.

We used both approaches. The betas based on one-month returns didthrow up a number of negative betas, which one would not expect to nd,while such anomalies were considerably fewer with the two-month betas.On the other hand, the t-statistics of both sets of betas were, in general,very high, with an average value of 8.7 for one-monthly betas and 12.3 fortwo-monthly ones. Furthermore the patterns of test results were robust tothe choice of beta, except for the last set of tests (see below), for whichthe sample sizes were extremely small when using two-month betas. Wetherefore report results only for tests based on one-month betas.

To dene extreme betas we treated the two sectors in our sample sep-arately, as the small-company sector had a higher mean beta with a largerstandard deviation than the income-and-growth sector. We measured thestandard deviation of each sectors betas around the mean beta of the sec-tor index over the sample period (the standard deviations were 0.39 for thegrowth-and-income funds, 0.63 for small-company funds). We dened anextreme beta as one lying more than one standard deviation away from therelevant sector index beta in the month of observation. This classied 22%

2 0 All the tests reported below were repeated using the whole sample but with appropri-ate zero/one dummies for these top funds. The results of the tests were not qualitativelyaltered.


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of the growth-and-income funds and 21% of the small-company funds as

extreme.

3.8 Descriptive statistics and tests

Descriptive statistics are given in table 2. The mean and median betas areslightly lower than 1, as found in Elton et al (2001), and losers appear to havelower betas than winners. Returns are positive overall and the dierencebetween the performance of the sector index and that of an individual fundis greater when the fund is a loser than when it is a winner, as one wouldexpect when the top funds are omitted.

TABLE 2 ABOUT HERE

Our tests consisted of a series of probit equations. We rst tested thehypothesis that losers are more likely to have extreme betas than are winnersby estimating the following probit equations:

PrfEXTBETAj;g = a + bLOSEBj; + j; (18)

PrfEXTBETAj;g = a0 + b0W INBj; +

0

j; (19)

where:EXTBETAj; takes the value 1 if fund j has an extreme beta in month

, and 0 otherwise;

LOSEBj; = BEHINDj; if observation j is a losing fund in month ,and 0 otherwise, where BEHINDj; = IT

av IT

j (positive for losers and

negative for winners); andWINBj; = BEHINDj; if observation j is a winning fund in month

:

We predict that losers are more likely to have extreme betas than arewinners, and that this tendency will be more marked the further behindthey are. We would therefore expect the coecient on LOSEBj; in the rstequation to be signicantly positive. Conversely, the coecient on W INBj;in the second equation should not be signicantly dierent from zero. Totake account of the fact that we are repeatedly sampling the same funds inconsecutive months we use a Huber and White robust variance estimatoradjusted for clustering of observations within company. We also report theresults of adding a lagged dependent variable, PrfEXTBETAj;1g, toallow for any inertia in the movement of a funds beta.

Tables 3 and 4 report the results for losers and winners respectively, withand without the lagged dependent variable.

TABLES 3 and 4 ABOUT HERE


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The tables show that in all cases the coecient on LOSEBj; is highly

positively signicant, as expected, while the coecient on WINBj; is notsignicantly dierent from zero. Panel B of each table shows that there issignicant inertia in the beta chosen each period, but the results are robustto inclusion of the lagged dependent variable.

Having established that losing funds are indeed associated with moreextreme betas, we proceed to investigate in more detail the eect of timeand the losers performance on the choice of beta. We estimate equation(20) for losers only (under each of the three performance criteria).

PrfEXTBETAj;g = a00 + b00BEHINDj; +XT

c00TQTj; (20)

+XT

d00TQTBEHj; +

00

j;

where:QTj; is a quarterly dummy, taking the value 1 if = T and 0 otherwise

(T = f1; 2; 4g); andQTBEHj; are slope dummies equal to Q

Tj;BEHINDj;.

Our model suggests that the incentive for losers to choose high betasis greater as the ranking period approaches. We chose quarter 3 (July-September) as the base period, as this is the least likely to be a nalranking period. We therefore expect at least one of the quarter dummiesto be signicantly positive. Positive coecients on Q1 (January-March) orQ4 (October-December) would support our earlier argument that the end ofthe tax year or the end of the calendar year respectively are candidates fornal ranking periods, . The interaction terms between BEHINDj; and thequarter dummies test the prediction of our model, that the degree of under-performance becomes less important as the ranking period approaches: inthe period just before the end of the game, simply being a loser is sucientto ensure that an extreme portfolio is chosen. We would therefore expectthat when the coecient on the quarterly dummy is positively signicant,suggesting that the nal ranking day is close, the coecient on the interac-tion term should be signicantly negative, indicating that the choice of an

extreme beta is less sensitive to the degree of under-performance.The results are given in tables 5 and 6, with and without the lagged

dependent variable respectively.

TABLE 5 and 6 ABOUT HERE

As expected, the coecient on BEHINDj; is strongly signicantly pos-itive in all cases: the more behind the fund is, the more likely is the man-ager to choose an extreme beta. In table 6 the lagged dependent variable


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is again highly signicantly positive and the remaining coecients are not

signicantly aected by its inclusion.All the quarterly intercept dummies are positive and all the interactionslope dummies are negative. The results suggest that quarter 1 is the mostlikely candidate as the nal ranking period, although quarter 2 is also a con-tender. In both tables these quarters have signicantly positive coecientsunder all three performance measures, and quarter 1 has particularly strongnegative coecients on the interaction term, indicating that for this quarterthe decision to adopt an extreme beta is more inuenced by the date thanthe degree of the funds under-performance. This supports the hypothesisthat March, the end of the tax year, is likely to be seen as a nal rankingperiod.

Our predictions apply to losers only, so we repeated the tests on winnerfunds, to conrm that the same results do not apply to them. The resultsare given in tables 7 and 8, with and without the lagged dependent variablerespectively. There are very few signicant coecients at all in these results,other than the lagged dependent variable and the constant. In particular,performance (BEHINDj;) is still not signicant. The quarterly interceptand slope coecients tend to have the same sign on them as in the loserregressions, but only the 5-year winners have any signicant coecients.

TABLE 7 and 8 ABOUT HERE

For funds that have adopted an extreme position, we now investigate the

choice between a high and a low beta. As discussed above, the variablesthat are important here are the funds maximum possible market exposurecompared with average, MAXN, and anticipated market movements. Ac-cording to our model, a losing fund that is expecting the market to fall willgo into cash (adopt a low beta). Conversely, a losing fund that is expectingthe market to rise will only adopt a high beta if it can expose a sucientlyhigh asset value to the market compared with the average (in fact, if it hasa MAXN above 1). A losing fund that has a MAXN below 1 may choosea low beta in the hope that the market might fall, even if it is expected torise. Consequently, our probit equation must reect the fact that a neg-ative "A will always decrease the probability of choosing a high beta, buta positive "A will only increase that probability if MAXN is also above 1.

The independent variable in the probit equation is therefore an interactionterm between "A and a high MAXN.

We estimate equation (21) for losing funds with extreme betas.

PrfHIBETAj;g = + HIMAXN"j; + j; (21)

where:HIBETAj; takes the value 1 if observation j has an extreme beta that

is higher than the sector index beta in month , and 0 otherwise; and


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HIMAXN"j; = HIMAXNj; "Aj;, where MAXNj; is a proxy for

fund js maximum possible market exposure in month compared withaverage, HIMAXNj; is a dummy that equals 1 if MAXNj; exceeds 1,and "Aj; is a proxy for the expected market change between months and + 1 (the proxies are discussed in detail above).

HIMAXN"j; will be positive if "A is positive and MAXN exceeds 1,negative if "A is negative and MAXN exceeds 1, and zero otherwise. Ourmodel predicts that for losing funds, the coecient on HIMAXN"j; willbe positive. As our model suggests that only losing funds are constrainedin this way, we also estimated it separately for winning funds, to conrmthat the coecient for these funds is not signicantly dierent from zero.

The results excluding the lagged dependent variable are presented in ta-ble 9. Panel A applies to losers only and panel B to winners. For losers,all coecients on HIMAXN" are positive, but only the one-year loser co-ecient is signicant. The constant term is signicantly negative, showingthat losers adopting extreme positions are more likely to choose low betasthan high. For winners there is no evidence that HIMAXN" plays any rolein the choice of a high beta. The constant is also not signicant, indicatingthat winners are equally likely to choose high and low betas when adoptingextreme positions.

TABLE 9 ABOUT HERE

Inclusion of a lagged dependent variable is more problematic here. We

cannot include only HIBETAj;

1, since this will assign a zero both to thosefunds with extremely low betas in the previous period and to those withmiddling, but not extremely high, betas. To capture these dierences weincluded both HIBETAj;1 and MIDBETAj;1, which takes the value1 if observation j had a non-extreme beta in month 1, and 0 otherwise.It turned out that HIBETAj;1 was a perfect predictor ofHIBETAj; infour out of the six regressions, and was extremely highly signicant in theother two. This is not surprising, given the inertia in beta choice which wasidentied earlier, and tells us that rms choosing a high beta in one monthare unlikely to change that decision in the following month, regardless offorecasts of market movements. We therefore dropped the observations forwhich HIBETAj;1 was a perfect predictor and re-estimated the equation

with only MIDBETAj;1 as the lagged dependent variable. Results arepresented in table 10.

TABLE 10 ABOUT HERE

For these observations, HIMAXN" becomes more signicant for allthree types of loser, but remains not signicantly dierent from zero forwinners. For both losers and winners MIDBETAj;1 is signicantly pos-itive as one might expect.


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We can conclude that loser funds adopting extreme positions are unlikely

to change from a high beta to a low one, or vice versa, between one periodand the next. However, if a rm adopts an extreme position for the rsttime, a loser will be strongly inuenced by the expected market movementand the maximum exposure it can achieve relative to the average, whilethese will not be important considerations for winners.

4 Conclusions

We have developed a model of fund management in which fund managerssee themselves as, to some extent, engaged in a tournament, but also asrequired to make a reasonable return. A number of predictions follow fromour model, notably that managers of losing funds will increasingly adopt

extreme market positions as the nal date of the tournament approaches.The particular position adopted will generally be related to the bullishnessor bearishness of the market, but fund managers of very low-value funds mayelect to bet against the market (adopt low betas when the market is expectedto rise) if they cannot expose sucient assets to the market to allow theirrelative position to benet from a rise in share prices. Our tests of thesepredictions, using recent UK data and cross-sectional beta distributions, aregenerally very supportive of the model. Funds worth less than average aremore likely to select extreme betas and this tendency seems to be morepronounced around the end of the tax year. Losing funds generally increasemarket exposure by adopting high betas when the market is expected to

rise, but are also subject to constraints imposed by the low value of theirtotal assets. They may even adopt low betas when the market is expected torise, if they cannot improve their position suciently to beat the averagein a rising market.

The additional risk-taking caused by the tournament aspect of fund man-agement has been identied by other writers as a possible source of distortionin decision-making in this market. However, we show that the unwarrant-ed adoption of extreme positions by loser funds is likely to be a signicantproblem only at certain times of the year. Conversely, our model suggestsan additional distortion: where it may be optimal for loser funds to choosehigh market exposure there can be a disincentive for them to do so if their

fund value is too low.

5 Appendix

5.0.1 The probability that A chooses shares at t

From the analysis in section 2.2.1, A will choose shares at t if:

his portfolio is valuable enough to allow him to hold more than NB

shares; and


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he believes that "t+1 will exceed "

t+1 in equation (11)

In period t 1, A has no information about "t+1 and "t+1 N(0; 2").

The required probability is therefore Prn

0 > z andPFA

t

Pt> NB

o, where z =

PFBtPFA

t

2

1

PFAt=PtNB

1NB

.

This can be written as Prn

0 > z jPFA

t

Pt> NB

oPr

nPFA

t

Pt> NB

o. We

take each part in turn.

Prn

0 > z jPFA

t

Pt> NB

o

By assumption P FAt < P FBt , so the required probability equals Prn

PFAt

Pt 2NB > 0o,

which, from equations (1) and (5) of the main text, can be recast as:

Prn

"t

NAt 2NB

> 2NBPt1 P FAt1

o.

Since this probability is being assessed at t 1, after news about the

price shock in period t, and since"t"AtAv

N(0; 1), then if NAt > 2NB this

probability will be N(L2) where L2 =1Av

2NBPt1PF

At1

NAt2NB

"At

.

If NAt 2NB this probability will be N(L2).

PrnPFA

tPt > NBo

This is equivalent to PrnPFA

t

Pt 2NB > 0

owith 2NB replaced by NB.

Hence, when NAt > NB this probability will be N(L3) where L3 =

1Av

NBPt1PF

At1

NAtNB

"At

. And ifNAt < N

B the probability will be N(L3).

Therefore at t 1 the probability that A will choose shares at t, if hisportfolio value is less than Bs in period t, is given by:

1. N(L2)N(L3) when NAt > 2NB;

2. N(L2)N(L3) when NB < NAt 2NB;

3. N(L2)N(L3) when NAt N

B

5.0.2 The probability that A chooses cash at t

This probability equals (1 the probability that A chooses shares at t) andhence can be derived directly from the previous analysis.


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5.0.3 Probability that "t+1 will be suciently high for A to

win if he adopts an all-shares strategy for the nalperiod

If A chooses all shares at t, his portfolio at t + 1 will be worth NAt+1Pt+1,where NAt+1 = P F

At =Pt. Using equations (1) and (5) again to substitute for

Pt, Pt+1 and P FAt , we can establish that:

P FAt+1 =PFA

t1+"tNAt

Pt1+"t (Pt1 + "t + "t+1)

Similarly, P FBt+1 = P FBt1 + N

B ("t + "t+1). The required probability istherefore

PrPF

A

t1+"tN

A

t

Pt1+"t (Pt1 + "t + "t+1) > P FBt1 + NB ("t + "t+1)

Collecting terms, we have,

Pr H = Pr

8

a tournament model of fund management

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