market selection in an evolutionary market with ... · market selection in an evolutionary market...
TRANSCRIPT
Market Selection in an EvolutionaryMarket with Nonstationary
Dividends
Urs Schweri∗†
Swiss Banking InstituteUniversity of Zurich
July 7, 2006
Which strategies will survive in the long run and which will diminish? –These are important questions in evolutionary finance. Up to now we onlyknow answers in the case of stationary dividends. But the evolutionary per-spective to the financial market is considering very long time horizons. Insuch a setting the creation and the vanishing of firms becomes a high rel-evance. If companies do not live infinitely long this implies automaticallynonstationary dividends. Since the creation and the vanishing can be ob-served one time in the life of a company, long time series do not help tolearn the process of dividends. Thus the dividend process must be learnedfrom similar companies. This happens even if the dividend processes of thecompanies are independent. To price a firm the observed dividend processof one company is not enough. Information from the cross-section must beincluded to get an investment strategy, which is competitive in the marketselection process.
∗Urs Schweri, University of Zürich, Swiss Banking Institute, Plattenstrasse 32, 8032 Zürich,www.isb.unizh.ch, [email protected], ++41 (0)44 634 48 22
†The author acknowledges the data from CRSP R©, which were obtained through Wharton ResearchData Services (WRDS) and sponsored by the University Research Priority Program Finance andFinancial Markets, Swiss Banking Institute, University of Zurich.
1. Introduction
1. Introduction
In analyst reports there is usually a section which compares the analyzed company withsimilar ones. A usual explanation of this fact is, that the value of similar companiesare correlated. This paper offers an alternative explanation: Many events in the liveof companies for example the first blockbuster of a biotech company, the developmentof the first Mac from apple or the default can only be observed one time in the wholelive of a company, but such events can change the future of a firm tremendously. Theonly way to learn something about such events is to learn it from the cross-section.How often are other biotech companies able to release a blockbuster? Or how many ofsimilar companies made default in the past? Investors who incorporate cross sectionalinformation will perform better than competitors who are not doing that.
Friedmann [1953] and Fama [1965] came up with the idea that the market selectsinvestor who use all available information and act rationally. After them irrationalinvestors will earn lower returns and in the long run disappear. But what is meantwith rationality: Rational investors maximize discounted expected utilities and theybelieve in the correct probabilities about the future states of the world. But Long et al.[1990] showed in a partial equilibrium model that irrational investors can produce somuch noise in the stock price that rational investors cannot correct this because of theirrisk aversion. Also Blume and Easley [1992] showed in a complete market setting, whereevery investor has the same saving rate, that a rational investor who does not maximize alogarithmic utility function can be driven out of the market by some irrational investors.For exogenous asset prices Kelly [1956] developed a theory how to invest the money inthe long run to become rich as fast as possible. This rule is equivalent to maximizea log-utility function. Therefore the fastest growing strategy takes over the market inBlume and Easley [1992]. Samuelson [1979] criticized that it does not make happy tomaximize the wealth (and to survive), but people should maximize their utility and behappy. But this paper focuses on the question which strategies survive and not how canwe make people happy, therefore this critique does not hurt. Because of the exogenouslygiven saving rate the outcome of the model of Blume and Easley [1992] is not paretooptimal and not a general equilibrium. Sandroni [2000] and Blume and Easley [2001]looked at market selection in a general equilibrium setting with complete markets: ifall investors have the same discount rate then rational investors survive, but this is notthe case if markets are incomplete. The case of incomplete market was addressed byEvstigneev et al. [2006]. If a stationary Markov chain describes the state of the world,there is a simple portfolio strategy λ∗: If λ∗ dominates the market there is no invadingstrategy which can earn higher returns than λ∗.
Most of the previous mentioned work is based on a stationarity assumption: This isproblematic because observed dividends could contain a unit root and companies canbe born or die. This paper will concentrate on the latter case, because an evolutionaryperspective looks on very long time horizons. Therefore this case seems to be morerelevant. A first task is to look for a simple but realistic setting of a dividend process.
2
A first question is what do we know about the living and dying of companies? –Because of the relevance for credit risk there exists a huge literature about default. Onthe empirical side Altman [1968] and Beaver [1968] were the first who used accountingdate to estimate the default likelihood. Duffie et al. [2006] argue that the probabilityof default is time dependent in a way that cannot be explained by other explanatoryvariables like interest rates. The opposite effect of default is the foundation of newcompanies. Audretsch [1991], Audretsch and Mahmood [1995] and Hensler et al. [1997]were looking at the survival chance of an IPO. The chance of survival is small in anindustry with high economies of scale and high setup costs. A solid balance sheet and ahigh percentage of insider owning the firm is an advantage for surviving.
It is often assumed that dividends are driven by one and the same process over thewhole life of a company. This is a very strict assumption: Why should a small startuphave the same risk and expected returns like a big concern? – Mueller [1972] suggeststhat small firms are more risky but more profitable than big old firms. The idea isthat the management of big companies is less flexible than the one in small companies.Furthermore, the management of big companies has (in contrast to the shareholders)an interest to grow even if the projects are not profitable. Gort and Klepper [1982] andKlepper [1996] see the changes of companies over their life time driven by a product lifecycle. At the beginning of a new industry there are many entrances and a lot of productinnovations and the companies can work profitable. Over the time the concurrenceincreases and the prices drop. Exits occur and at the end everything converges to astable number of firms. Empirical evidence that firms change their characteristics overlife time comes form Evans [1987] who showed that the growth of firms decreases withthe age of firms.
Up to now most of the discussed ideas were of theoretical nature or not directly relatedto the dividend process. The empirical literature aboute dividends should help us todetermine which ideas are good and important to model. The most famous empiricalmodel for dividends is the model form Lintner [1956]. For example Benartzi et al. [1997]conclude that the Lintner model "remains the best description of the dividend settingprocess available". Lintner models the current dividends as a linear combination of thepast dividends and the current earnings after taxes plus an error term. Therefore we donot know much more about the dividends, if we do not now the process of the earnings.
Another debate is the question if dividends contain a unit root. There was a verylong debate about this topic and the answer is still not clear.1 Overall it seems to bemore difficult to maintain the hypothesis of a unit root in dividends than in stock prices.Furthermore a unit root in the dividends is in Harris and Tzavalis [2004] rejected andseems to be not very plausible in DeJong and Whiteman [1991].
Over all there is not much literature about life cycle in dividends and the default ofcompanies. This is in some sense surprising because discounted expected dividends are
1Papers in the debate about a unit root in dividends are: Shiller [1981], Kleidon [1986], Campbelland Shiller [1987], John Y. Campbell [1988], DeJong and Whiteman [1991] and Harris and Tzavalis[2004].
3
2. Empirical Evidence on dividends
needed to determine asset prices. This paper cannot give an answer to the questionwhat is the true process of dividends. But it wants to take a simple dividend processwhich contains the following aspects: Living and dying of companies and the fact thatfirms are going through a life cycle: A company starts as an IPO. Then it becomes astartup. After some time the startup can die or become a concern. The concern willpay higher dividends than the startup but also the concern will default at one point oftime, but the default probability in one year is much lower than in the case of a startup.
Section 2 will show some empirical evidence that the dividend process proposed insection 3 is in line with the data. The market selection model chosen for this paper will bepresented in section 3. Afterward different investment strategies for the simulations arepresented. In section 5 will be some simulations and in the last section some concludingremarks.
2. Empirical Evidence on dividends
CRSP data over the time horizon of 1973 to the end of 2005 were used for this sec-tion. In the sample were 23’956 North American companies listed at NYSE, AMEX orNASDAQ. The data were chosen from 1973 because from this point of time the datafrom AMEX and NASDAQ were also available. The number of active companies werebetween 5’246 and 9’694 per year. On average there were 7’913 active companies peryear. The big difference between the active companies per year and the number of to-tal companies indicates, that many firms were new founded and also a big number ofcompanies disappeared.
The average time a company was in the sample was 9.8 years. The median is with 7years even lower. Figure 1 shows the distribution of the time the companies were in thesample. Many companies were in sample for a short time. Therefore it is very difficultto determine the value of a company based on its past dividends, because the dividendrecord is much to short. Therefore it is important to use cross-sectional information todetermine the value of a firm.
The biggest part of the movement can be explained by merger and acquisitions. 8’670of 16’953 delistings reported by CRSP came from this source. Table 1 shows that thenext important source for deletion is the category dropped from the stock exchange. Thenumber of dropped companies is much higher than the number of liquidated companies.The big stock exchanges seem to delist companies with financial problems before theworst things can happen. Table 2 gives detailed reasons for the droppings: 1’168 com-panies were delisted because of insufficient capital, 883 because of a too low price, 579because of insolvency and 933 because of not paying the fees of the exchange. In this waywe get over 3’500 companies which were dropped from the stock exchange because of bigfinancial problems. Therefore the fraction of default companies can be estimated to atleast one eighth of all companies and this for a time period of 33 years. The number ofnew companies is also impressive: 18’426 companies or over 550 per year. These figures
4
Year Active Mergers Liquidation Dropped1973 5’930 103 5 3541974 5’534 99 11 1681975 5’376 83 10 801976 5’391 101 17 531977 5’366 155 18 611978 5’321 191 11 701979 5’246 209 16 511980 5’405 164 24 831981 5’762 155 15 881982 5’968 172 21 1591983 6’548 173 11 1641984 6’797 210 13 2291985 6’911 248 16 3251986 7’287 225 26 3191987 7’522 185 5 2261988 7’530 352 12 3121989 7’277 273 13 3181990 7’102 197 9 3401991 7’129 119 12 3671992 7’397 135 9 3871993 7’982 177 3 1701994 8’551 279 3 1991995 8’923 358 9 2381996 9’460 437 10 1741997 9’694 511 7 2571998 9’549 604 5 4331999 9’215 613 11 4052000 8’884 632 12 3342001 8’171 463 7 4842002 7’478 257 18 4012003 7’062 262 11 3092004 6’895 265 17 1532005 6’868 263 7 178Total 8’670 394 7’889
Table 1: Active Companies and reasons for delisting (CRSP data from 1973 to 2005)
5
2. Empirical Evidence on dividends
0 10 20 30
010
0020
0030
0040
00
Time of a Company in CRSP
Num
ber
of C
ompa
nies
Figure 1: Time a company (CRSP data from 1973 to 2005) was in the sample.
demonstrate impressively that long run investment strategies should not neglect the factthat firms have only a finite live.
The number of deletions (and their subcategories) in table 1 is not constant over thetime. Typically everything happens in waves. For example between 1991 and 1997many new companies were founded and from 1996 to 2001 there was a merger wave.The number of droppings and liquidations increased tremendously between 1998 and2003. As a consequence the number of companies also moves in waves.
To obtain dividends data from CRSP the number of outstanding shares at the daybefore the ex-distribution date must be multiplied with the dividend amount and thenaggregated over one calendar year. For these calculations only cash dividends were takeninto account (no subscription rights etc.). In figure 2 the dividends aggregated over thesectors are shown. In the lower graph a shift in the sectors which are paying the dividendscan be observed. The financial sector increased its dividends over time more than themanufacturing industry and the transport and telecommunication sector. Therefore weobserve a shift in the relative weight of dividends (i.e. the relative dividends of thefinancial sector increased). Such shifts are quite natural, because hundred years agorailroads or the textile industry would have had a much bigger weight than today.
In over 56.7% of the company years (each year a company is active is defined as acompany year) no dividends are paid. In 5.6% of all company years dividends changedfrom zero to a positive amount and in 5.1% of the cases dividends were lowered tozero. Large jumps in dividends are therefore a very characteristic part of dividend time
6
Number ofReason for Dropping CompaniesIssue stopped trading on exchange - reason unavailable. 967Issue stopped trading current exchange - to Mutual Funds. 18Issue stopped trading current exchange - to Boston Exchange. 33Issue stopped trading current exchange - to Midwest Exchange. 2Issue stopped trading current exchange - to Pacific Stock Exchange. 17Issue stopped trading current exchange - to Philadelphia Stock Ex-change.
3
Issue stopped trading current exchange - to Toronto Stock Ex-change.
3
Issue stopped trading current exchange - trading Over-the-Counter. 176Delisted by current exchange - insufficient number of market mak-ers.
464
Delisted by current exchange - insufficient number of shareholders. 152Delisted by current exchange - price fell below acceptable level. 883Delisted by current exchange - insufficient capital, surplus, and/orequity.
1’168
Delisted by current exchange - insufficient (or non-compliance withrules of) float or assets.
715
Delisted by current exchange - company request (no reason given). 406Delisted by current exchange - company request, deregistration(gone private).
58
Delisted by current exchange - bankruptcy, declared insolvent. 579Delisted by current exchange - company request, offer rescinded,issue withdrawn by underwriter.
15
Delisted by current exchange - delinquent in filing, non-payment offees.
933
Delisted by current exchange - failure to register under 12G of Se-curities Exchange Act.
112
Delisted by current exchange - failure to meet exception or equityrequirements.
156
Delisted by current exchange - denied temporary exception require-ment.
10
Delisted by current exchange - does not meet exchange’s financialguidelines for continued listing.
806
Delisted by current exchange - protection of investors and the publicinterest.
121
Delisted by current exchange - corporate governance violation. 13Conversion of a closed-end investment company to an open-endinvestment company.
47
Delisted by current exchange - delist required by Securities Ex-change Commission (SEC)
31
Table 2: Reasons why companies were dropped from their exchange (CRSP data from1973 to 2005)
7
3. The Model
73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 0 1 2 3 4 5
050
100
200
300
Dividends over Sectors
Bill
ion
Dol
lars
Agriculture / Mining / ConstructionManufacturingTransp. / Comm. / ElectricTradeFinance / Real Est.ServicesPublic Administration
73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Relative Dividends
Rel
ativ
e D
ivid
ends
Figure 2: Dividends and relative dividends over Sectors (CRSP data from 1973 to 2005)
series. Furthermore the 5% biggest dividend payer are distributing in average 78% of alldividends. Over the years this fraction moved between 70.4% and 84.7%. The maximumwas in 2001 and the minimum in 1979, therefore there was an upward trend over time.In any case these figures show that there is an enormous concentration in the dividendpaying firms.
A long run dividend model should contain the following ingredients: The number ofcompanies must be dynamic, in the number of companies must be some waves, there aresome large jumps in dividends, dividend payments are concentrated on a small fractionof the firms and shifts of dividends between different sectors must be possible. Thesestylized facts will be included in the model in the next section.
3. The Model
The economy is consisting of three types of companies IPOs, startups and concerns.IPOs are new firms entering the market. In the period they enter the investors pay acertain amount for the IPO and do not get a dividend. In the second period an IPOautomatically becomes a startup. Startups are paying low dividends but they have thegrowth opportunity to become a concern. The concerns are paying much more dividendsbut they cannot grow further. Both type of companies can default. In every periodthere is a probability that a company changes its state. A startup can default, becomea concern or stay a startup and a concern can default or stay a concern. A default
8
Figure 3: Development of a company over time. An IPO becomes a startup, a startupcan become to a concern or disappear after several periods and a concern willlive several periods and then disappear.
9
3. The Model
0 200 400 600 800 1000 1200 1400 1600 1800 20000
2
4
6
8
10
12
14
16
18Number of Companies
ConcernStartupIPO
Figure 4: Number of companies over time (parameters see table 3)
company is dead for ever. The transition probabilities in figure 3 must be understood asfollows: pSD is the probability that a startup makes default in one period. The startupstays a startup with probability pSS. But if the startup survives there is in the nextperiod again the probability pSD that the startup defaults. These change of states areindependent between companies and over time. If pSD and pCD are bigger than 0, everycompany will default at one point of time (with a probability converging to 1). Thereforennew IPOs are born every year.
The state of world in t can be characterized by the number of startups nSt and the
number of concerns nCt . Since companies of one type are identical the only relevant
information is their number. For notational convenience the state of nature in t isωt =
(nS
t , nCt
)′. The history of states is written as ωt = (ω0, . . . , ωt). The number ofIPOs does not matter because it is constant.
Each startup pays every year a fix dividend DS > 0 and each concern pays DC > DS.The dividend process of a startup is the constant DS from the foundation up to the pointwhere the startup becomes a concern or dies. In the first case there is a huge upwardjump in the later case the dividend falls to zero for ever. As a concern the companypays constantly DC up to the death. The dividend of all startups together is then nS
t DS
and all concerns together pay nCt DC . The total dividend paid out in t is the sum of the
dividends of the concerns and the startups. IPOs do not pay dividends.
The model for the dividends is not completely new Hurley and Johnson [1994] used
10
a similar trinomial model to price individual stocks. But this paper considers manycompanies, the idea of concerns and startups is new and only one upward jump isallowed during the live of a company.
This model covers most of the features mentioned in section 2: The number of com-panies is dynamic and moves in waves (see figure 4 for a simulation). The dividendpayments can be parametrized such that the concerns are paying most of the dividendsand the startups are only paying a small fraction of them. With this parametrizationthe dividend payments are concentrated on the concerns. For simplicity the number ofIPO is kept constant, which is not in line with the observations. Mergers are not in themodel, because they do not matter, if the dividends and the portfolio weights of thenew firm are the sum of the merging companies. As a whole the model is very simplebut it includes many elements which are important in the long run. The next step is toinclude this dividend process in an evolutionary model to see which are the implicationson the market selection.
There are I investment strategies. λi0,t (ωt) is defined as the fraction of wealth which
investment strategy i ∈ {1, . . . , I} consumes in t given the history of states ωt. λi0,t (ωt)
is assumed to be constant over time and identical for all strategies. This can be justifiedby the focus of this paper: The performance of the investment strategies should becompared and not the influence of the saving rate analyzed. Therefore λi
0,t (ωt) can bewritten as λ0.
Every strategy can invest in three asset classes: IPOs, startups and concerns. Fornotational convenience the numbers 1, 2 and 3 are assigned to them. Each investorinvests a fraction of his wealth into these asset classes: a fraction λi
1,t (ωt) into IPOs,λi
2,t (ωt) into startups and λi3,t (ωt) into concerns. Short selling is not allowed, therefore
0 ≤ λik,t (ωt) ≤ 1 and the budget constraint implies
∑3k=0 λi
k,t (ωt) = 1. Since for theinvestor every startup is the same, he will invest an equal of money into every startup.The same is true for the concerns. Therefore we are only interested in the fraction θi
k,t
an investor i holds on asset class k.
θik,t =
{λi
k,t(ωt)wit
qk,tif nk
t > 0
0 otherwise(1)
wit is the wealth of investor i in t and qk,t is the total market capitalization of all companies
of asset class k in t. Therefore the price of a single startup is given by q2,t/nSt . The total
market capitalization of asset k is the amount which all investors together pay for theasset of type k in t:
qk,t =I∑
i=1
λik,t
(ωt)wi
t =: λk,t
(ωt)wt (2)
From t to t + 1 every company can change the state. For example from t to t + 1 allIPOs become startups. A strategy which holds 10% (θi
k,t = 0.1) of all IPOs at the end
11
3. The Model
of t will get a fraction nnew
nSt+1
θik,t of all startups at the beginning of t + 1. For all asset
classes this can be written as a change in portfolio weights:
θ̃it+1 =
θ̃i1,t+1
θ̃i2,t+1
θ̃i3,t+1
=
0 0 0
nnew
nSt+1
nSt+1−nnew
nSt+1
0
0nSC
t+1
max(1, nCt+1)
nCt+1−nSC
t+1
max(1, nCt+1)
θi
1,t
θi2,t
θi3,t
=: Nt+1
(ωt+1, nSC
t+1
)′θi
t
nSCt+1 is the number of startups becoming a concern between t and t + 1. nC
t+1 canbecome zero, therefore the maximum is used to make the result well defined. It isimportant to mention that the rows of Nt+1
(ωt+1, nSC
t+1
)add up to 1. In contrast to the
concerns the number of startups is always positive, because in every period nnew IPOsbecome startups. Therefore the first row of Nt+1
(ωt+1, nSC
t+1
)must be zero. With the
definition of θ̃it+1, the wealth dynamic can be written as:
wit+1 =
3∑k=1
(nk
t+1Dk + qk,t+1
)θ̃i
k,t+1 (3)
Inserting equation 2 we obtain:
wit+1 =
3∑k=1
(nk
t+1Dk +
I∑i=1
λik,t+1
(ωt+1
)wi
t+1
)θ̃i
k,t+1 (4)
To write down everything in matrix notation we define: wt =(w1
t , . . . , wIt
)′, Dt (ωt) =
(0, nSt DS, nC
t DC)′, Θ̃t+1 =(θ̃1
t+1, . . . , θ̃It+1
)′, Θt =
(θ1
t , . . . , θIt
)′ and the portfolio weights
of the investment strategies Λt+1 (ωt+1) = (λ1,t (ωt) , λ2,t (ωt) , λ3,t (ωt))′. The wealth of
the investors can be written as:
wt+1 = Θ̃t+1Dt+1
(ωt+1
)+ Θ̃t+1Λt+1
(ωt+1
)wt+1 (5)
= ΘtNt+1
(ωt+1, nSC
t+1
)Dt+1
(ωt+1
)+ ΘtNt+1
(ωt+1, nSC
t+1
)Λt+1
(ωt+1
)wt+1 (6)
Solving this equation results in
wt+1 =[Id−ΘtNt+1
(ωt+1, nSC
t+1
)Λt+1
(ωt+1
)]−1ΘtNt+1
(ωt+1, nSC
t+1
)Dt+1
(ωt+1
)(7)
=:[Id−ΘtNt+1
(ωt+1, nSC
t+1
)Λt+1
(ωt+1
)]−1At+1
(ωt+1, nSC
t+1
)(8)
12
At+1
(ωt+1, nSC
t+1
)can be interpreted as the payoffs the investors get from their in-
vestment in t + 1. In the next step it is shown that this system is well behaved. Inother words wt+1 must be well defined and nonnegative. To guarantee this we need twoassumptions:
Assumption 1. Consumption takes place and does not exhaust any portfolio rule’swealth, i.e. 0 < λi
0,t (ωt) < 1 for all i, t ans ωt.
Assumption 2. There is at least one completely diversified portfolio rule: There is a isuch that λi
k,t (ωt) > 0 for all k = 1, . . . , K, t and ωt. Unless nCt = 0 then λi
3,t (ωt) canbe zero.
Proposition 1. Suppose w0 > 0 and assumption 1 and assumption 2 are satisfied forsome portfolio rule with wj
0. Then the evolution of wealth (7) is well-defined in all periodsof time.
The proof of proposition 1 can be found in appendix A. Every investor with wit > 0
and λi1,t (ωt) > 0 for all ωt has a strict positive wealth in t+1. This results from the fact,
that a startup founded in t pays in t + 1 for sure DS. But every investment in startupsor concerns can be lost, because all companies of one type can theoretically default atone point of time. In the case of the investment in a new startup (k = 1), this cannothappen.
Even if the model looks very similar to the one of Evstigneev et al. [2006] it differs inone central point: Every time an IPO is founded the investors pay some money to getit. This are the resources used to found the IPO. This amount is not constant over time.The effect would be in terms of Evstigneev et al. [2006] that the consumption rate isnot constant over time. Therefore it is not possible to prove nice analytical results likelocal evolutionary stability. Since the whole model is a complex stochastic system moreresults can only be provided by simulations.
4. Strategies
Which strategies should be chosen in this model? – A good starting point seems to beEvstigneev et al. [2006]: They showed for asset k and dk,t, the dividends of asset k in tdivided by the sum of the dividends of all assets in t, that
λ∗k,t =λ0
1− λ0
∞∑m=1
(1− λ0)m E
(dk,t+m(ωt+m) | ωt
)is locally evolutionary stable. That means, if the strategy λ∗ dominates the market
there is no other single strategy which can invade the market. But this result is onlyvalid for infinitely long lived assets and if the relative dividends dk,t+m are a stationaryMarkov chain. If companies can be born
∑k λ∗k,t ≤ 1, because the new born companies
13
4. Strategies
will pay positive dividends and therefore some investor will assign positive portfolioweights to these assets. In other words the investor does not invest his whole wealth.To get a reasonable starting point for the setting of this paper, λ∗k,t must be interpretedas the portfolio weight of a sector and the portfolio weights must be normed to one bythe adjustment factor c:
λ1k,t =
c nnew
λ0
1−λ0
∑∞m=1 (1− λ0)
m E (dIPO in t,t+m(ωt+m) | ωt) if k = 1 (IPO)c nS
tλ0
1−λ0
∑∞m=1 (1− λ0)
m E (dstartup in t,t+m(ωt+m) | ωt) if k = 2 (startup)c nC
tλ0
1−λ0
∑∞m=1 (1− λ0)
m E (dconcern in t,t+m(ωt+m) | ωt) if k = 3 (concern)
The expected value can be determined by brute force simulations: Taking the actualnumber of the different company types and simulate the next 150 time steps. Do this 500times and calculate the mean of the relative dividends and then λ1
k,t can be determined.To get an idea how important the assumption of stationarity in Evstigneev et al. [2006]is, investor 2 assumes wrongly that the underlying process is stationary and the assetsare infinitely lived. He assumes that no new companies are born and companies donot die but become an IPO. With this wrong specified process the conditional expectedrelative dividends are determined.
The next group of investors have to take their decision only on the observed variablesnot knowing the true dividend process. In the real world exists an uncertainty about theunderlying model, therefore it is an interesting question how good a bit wrong specifiedmodel performs. One possible strategy is to invest the same fraction of wealth into everycompany. This strategy may seem very crude but DeMiguel et al. [2005] showed thatsuch a strategy can perform astonishing well. Formally this strategy can be written as:
λ3k,t =
nnew
nnew+nSt +nC
tif k = 1 (IPO)
nSt
nnew+nSt +nC
tif k = 2 (startup)
nCt
nnew+nSt +nC
tif k = 3 (concern)
A different strategy is to invest after the current relative dividends, neglecting growthopportunities and the possibility of default.
λ4k,t =
0 if k = 1 (IPO)
nSt DS
nSt DS+nC
t DC if k = 2 (startup)nC
t DC
nSt DS+nC
t DC if k = 3 (concern)
The most sophisticated strategy on the empirical side is to determine something likethe conditional expected value in λ1
k,t out of the data. This is done as following: The last50 periods of relative dividend data were taken. Out of this sample all companies of one
14
type were taken (for example startups). Then the relative dividends m periods aheadare taken and then the average of that is the estimator for E (dstartup in t,t+m(ωt+m) | ωt).This defines λ5
k,t.
5. Simulations
Knowing the investment strategies the wealth dynamic can be simulated via equation(7). In table 3 are the parameters for the simulation. These parameters produce a largeconcentration of dividend payers (a few companies are paying most of the dividends andthe rest of the companies pay practically no dividends) and the number of companiesshows some waves (see figure 4). Therefore the model covers some of the features men-tioned in section 2. The aim of this model is not to use the true dividend process; theidea is to use some stylized facts about dividends and to show the implications to marketselection.
Startup Concern DiversedS 1 dC 200 nnew 1pSD 9% pCD 2% λ0 5%pSS 90% pCC 98%pSC 1%
Table 3: Parameters of the simulation
How strong is the effect nonstationarity assumption? – 1’000 simulations over 2’000time periods show in figure 5 that the strategy based on the correct model gets in thelong run in average almost 90% of the total wealth. But even over this long time periodthe correct process is not able to eliminate the strategy based on the wrong stationaryassumption completely.
In the next step the three empirical strategies are compared. Figure 6 shows theresult of 1’000 simulations. First of all the confidence intervals are very huge. Eventhough the differences in the mean of the relative wealth are very huge, it should bechecked if the differences between the strategies are statistically significant. Due to thesimulations 1’000 drawings of the relative wealth are available at each point of time andit can be tested if the difference between the two strategies is significantly different fromzero. To check this a t-test for the mean and a sign test for the median were applied.The results were absolutely clear: The mean and the median of all strategies after 2’000time periods are different from every other strategy with a significance level over 0.1%.Thus the ranking of the different strategies is clear: First the strategy based on thecross-sectional estimation of the conditional expected value, then the strategy based onthe current relative dividends and then the naive 1/n strategy. This demonstrates nicelythat it makes sense to use the cross-sectional information.
In the last step all strategies were put into one market. For the overview the confidencebands were omitted. But even with confidence bands the strategy which uses the true
15
5. Simulations
200 400 600 800 1000 1200 1400 1600 1800 20000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Relative Weahlth
Mean and 95% Confidence Interval
true dividend processstationary div. process
Figure 5: Fraction of wealth of the different strategies: One investment strategy bases itsconditional expectation on the true model and the other strategy on the wrongstationary model. Mean and 95% confidence intervals out of 1’000 simulationsover 2’000 time periods.
200 400 600 800 1000 1200 1400 1600 1800 20000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Relative Weahlth
Mean and 95% Confidence Interval
cond. exp. rel. div.naivecur. rel. dividends
Figure 6: Fraction of wealth of the conditional expected value investor, the current rel-ative dividends investor and the naive investor over a time horizon of 2’000.Mean and 95% confidence intervals out of 1’000 simulations.
16
200 400 600 800 1000 1200 1400 1600 1800 20000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Relative Weahlth (Mean)
true dividend processstationary div. processcond. exp. rel. div.naivecur. rel. dividends
Figure 7: Fraction of wealth of all suggested strategies. Means out of 1’000 simulationsover a time horizon of 2’000.
dividend process to determine the conditional expected value outperforms all the otherstrategies. The wrongly specified stationary process shows the next best performancefollowed by the empirical estimation of the conditional expected value and the currentrelative dividends and on the last place the naive strategy.2 One implication of thissimulation is that a slight adaption of the λ∗ strategy from Evstigneev et al. [2006]works also well in this nonstationary setting.
Can this model be used to describe existing asset prices? – Probably not: Existingstock-prices contain typically a unit root and a drift term. But because of the meanstationarity of the aggregated dividend process the asset prices seem to be also meanstationary. This can be seen in figure 9 where the averages of the asset prices are constantover time. Also the optical inspection of one simulation of the asset prices in figure 8indicates the same. The unrealistic features of the asset prices are normal for this kindof models and can also be observed in Hens et al. [2002] and Evstigneev et al. [2006].One way to obtain more realistic asset prices would be to integrate a unit root in thelogarithm of the dividend process.
2Also here the formal test on the mean and median were done and the relative wealth is after 2’000periods significantly different between all strategies.
17
5. Simulations
0 200 400 600 800 1000 1200 1400 1600 1800 20000
500
1000
1500
2000
2500
3000
3500Asset Price
ConcernStartupIPO
Figure 8: Asset prices of the different asset categories (one simulation) in a market withall types of investors over 2’000 time periods.
0 500 1000 1500 20000
200
400
600IPO: Stock Price
Mean and 95% Confidence Interval0 5 10 15 20
0.2
0.4
0.6
0.8
1
Mean and 95% Confidence Interval
IPO: SACF of Stock Price
0 500 1000 1500 20000
200
400
600Startup: Stock Price
Mean and 95% Confidence Interval0 5 10 15 20
0.2
0.4
0.6
0.8
1
Mean and 95% Confidence Interval
Startup: SACF of Stock Price
0 500 1000 1500 20001500
2000
2500
3000
3500Concern: Stock Price
Mean and 95% Confidence Interval0 5 10 15 20
0.2
0.4
0.6
0.8
1
Mean and 95% Confidence Interval
Concern: SACF of Stock Price
Figure 9: Mean and 95% confidence intervals of the asset prices and the sample auto-correlation function out of 1’000 simulations in the market with all investors
18
6. Conclusions
The paper showed that the living and dying of companies is a very important part ofthe dividend process as well as large jumps in dividends. The very idealized model ofthis paper shows the influence of these facts to the performance of different investmentstrategies. The main impact is that it is not enough to look at one time series and thento determine the portfolio weights out of that. To estimate the model cross-sectionalinformation must be incorporated. For the evolutionary finance this is a new insight andfor the rest of the finance this is an often neglected fact. The bad side of this messageis that this makes a lot of things more complicated.
If no cross-sectional information is included, the performance is the worst one. In-cluding some cross-sectional information out of the sample improves the performance.Knowing and applying the true model generates the best results and a slightly misspec-ified model is also doing a good job. At the end the message is simple: if we knew thetrue dividend process, it would be no problem to find an optimal strategy. But we donot live in such an ideal world. Thus it is a good idea to incorporate cross-sectionalinformation to the investment process.
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A. Proof of Proposition 1
Proof of Proposition 1. Evstigneev et al. [2006] provide practically the same proof, butin this case the matrix Nt+1
(ωt+1, nSC
t+1
)is new and it must be checked that Nt+1
(ωt+1, nSC
t+1
)does not matter. With w0 > 0 it suffices to show that the step from t to t + 1 is wellbehaved.
First it is shown that the matrix C := Id−ΘtNt+1
(ωt+1, nSC
t+1
)Λt+1 (ωt+1) is invertible
by proving that it has a column dominant diagonal (Murata [1977], Corollary p. 22).Defining the nt+1
ij as the ij-th element of Nt+1
(ωt+1, nSC
t+1
)then C has entries:
Cjj = 1−3∑
k=2
3∑h=1
λjk,t+1
(ωt+1
)nt+1
hk
λjh,t (ωt) wj
t
λh,t (ωt) wt
Cij = −3∑
k=2
3∑h=1
λjk,t+1
(ωt+1
)nt+1
hk
λih,t (ωt) wi
t
λh,t (ωt) wt
(i 6= j)
Notice that the first column of Nt+1
(ωt+1, nSC
t+1
)contains only zeros. Because of that
there is no need to add up over k = 1. If the number of concerns is at one point of
21
A. Proof of Proposition 1
time zero, also the sum over k = 3 is zero. The completely diversified portfolio ensuresthat the market capitalizations pk,t = λk,t (ωt) wt are strictly positive (unless nC
t = 0).Therefore Cjj and Cij are well defined. In the special case of nC
t = 0 equation (1) makessure that θi
h,t = 0, therefore everything is well defined.
A particular condition for a column dominant matrix is, if for every j = 1, . . . , I:
|Cjj| >∑i6=j
|Cij| . (9)
The non-diagonal elements Cij (i 6= j) are non-positive. The diagonal elements Cjj
are positive: If there is a positive number of companies of asset type k and since 0 ≤λj
h,t(ωt)wjt
λh,t(ωt)wt≤ 1, it can be written:
Cjj = 1−3∑
k=2
λjk,t+1
(ωt+1
) 3∑h=1
nt+1hk
λjh,t (ωt) wj
t
λh,t (ωt) wt
≥
{1−
∑3k=2 λj
k,t+1 (ωt+1) if nCt+1 > 0
1− λj2,t+1 (ωt+1) if nC
t+1 = 0
> 0
The last bigger as sign follows from the fact that due to assumption 1 the consumptionrate is bigger than zero and all portfolio weights add up to 1. Equation (9) can berewritten as
1 >I∑
i=1
3∑k=2
3∑h=1
λjk,t+1
(ωt+1
)nt+1
hk
λih,t (ωt) wi
t
λh,t (ωt) wt
The right hand side of this equation is:
3∑k=2
λjk,t+1
(ωt+1
) 3∑h=1
nt+1hk
I∑i=1
λih,t (ωt) wi
t
λh,t (ωt) wt
=3∑
k=2
λjk,t+1
(ωt+1
) 3∑h=1
nt+1hk
=
{∑3k=2 λj
k,t+1 (ωt+1) if nCt+1 > 0
λj2,t+1 (ωt+1) if nC
t+1 = 0
< 1
The last inequality is following from the fact that the consumption rate must bepositive (assumption 1) and from the budget constraint. Thus matrix C is invertible.
22
C has strictly positive diagonal entries and non-positive off-diagonal entries. If addi-tionally At+1
(ωt+1, nSC
t+1
)≥ 0 then Murata [1977] (Theorem 23, p. 24) implies that
wt+1 ≥ 0. At+1
(ωt+1, nSC
t+1
)= ΘtNt+1
(ωt+1, nSC
t+1
)Dt+1 (ωt+1) ≥ 0 holds if all ele-
ments of the three matrices are nonnegative. This holds by definition in the case ofNt+1
(ωt+1, nSC
t+1
)and Dt+1 (ωt+1). w0 > 0 and the completely diversified investor (as-
sumption 2) ensures a positive market capitalization (unless in the case nCt = 0 where
nobody holds stocks of the concern). Due to the positive market capitalization Θt isnonnegative as long the diversified investor has a strictly positive wealth. θj
k,t > 0 for allk ≤ 2 implies that θ̃j
2,t+1 > 0. Because of that and DS > 0 the diversified investor getsa positive payoff in t + 1 and gets via (4) a strictly positive wealth in t + 1.
23