the capital gain lock-in effect and perfect substitutes

www.elsevier.com/locate/econbase

Journal of Public Economics 88 (2004) 2765–2783

The capital gain lock-in effect and perfect substitutes

Peter Klein*

Faculty of Business Administration, Simon Fraser University, Burnaby, BC, Canada V5A 1S6

Received 15 April 2000; received in revised form 15 January 2004; accepted 15 January 2004

Available online 13 April 2004

Abstract

This paper analyses the effect of investors’ accrued capital gains on optimal portfolio composition

and equilibrium returns under the assumption that investors are able to re-balance with perfect

substitute securities. No-dominance arguments are used to show that pricing differences because of

accrued capital gains do not arise among securities which are perfect substitutes. These arguments

are insufficient, however, to prevent pricing differences because of accrued capital gains among

securities which are not perfect substitutes. Trading rules are developed which outline the conditions

necessary for the realisation of accrued capital gains and the deferral of capital losses. These trading

rules also provide guidance on which securities investors should sell, given their tax basis, when re-

balancing their portfolios.

D 2004 Elsevier B.V. All rights reserved.

Keywords: Capital gain taxes; Lock-in effect; Perfect substitutes

1. Introduction

The effect of taxation on investor behaviour is an important theoretical and empirical

issue. The taxation of capital gains is particularly interesting because investors can choose

to defer their tax liability by not selling shares in which capital gains have accrued. If

doing so prevents investors from re-balancing their portfolios as they would have in the

absence of these accrued capital gains, portfolio risk, and hence equilibrium asset prices,

may also be affected.

A number of articles in the public economics literature have analysed this effect on

investor behaviour in a partial equilibrium. For example, Feldstein and Yitzhaki (1978)

find that the switching and selling of common stock is sensitive to capital gains taxation.

0047-2727/$ - see front matter D 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.jpubeco.2004.01.002

* Tel.: +1-604-291-5605.

P. Klein / Journal of Public Economics 88 (2004) 2765–27832766

Poterba (1987) presents evidence that investors do, in reality, pay capital gains taxes.

Balcer and Judd (1987) show that investors will accept a lower rate of return before tax on

shares in which they have accrued capital gains and Landsman and Shackelford (1995)

find empirical evidence that supports this result. Poterba and Weisbenner (2001) find that

January returns depend on capital gains tax rates.

General equilibrium models of the effect of capital gains taxation on investor

behaviour have traditionally obtained results which are very different. Constantinides

(1983), for example, assumes that when an investor owns too much of a security with an

accrued capital gain, the portfolio can be re-balanced by frictionlessly short selling either

the same security ‘‘against the box’’, or a tax exempt perfect substitute. As a result, the

timing of capital gains realisations is independent of portfolio risk. This finding implies

that investors should defer perpetually the realisation of their capital gains. It also

implies that equilibrium security prices are unaffected by investors’ accrued capital

gains.1

The portfolio composition and equilibrium pricing results of Constantinides (1983) are

certainly correct based on the assumptions which are made. Under different assumptions,

however, results more similar to those in the public economics literature can be obtained.

For example, Klein (1998, 1999) develops general equilibrium models under the

assumption that short selling is not allowed. This assumption is justified in this context

because of the considerable costs associated with short selling in order to defer accrued

capital gains for extended periods of time.2 Based on this assumption, investors are

forced to make a trade-off between portfolio re-balancing and capital gains realisation.

They skew their portfolios in favour of stocks in which they have accrued capital gains

and away from stocks in which other investors have accrued capital gains. As a result,

equilibrium returns are lower for stocks which are held by investors with accrued capital

gains. Klein (2001) tests this pricing result using a proxy for investors’ accrued capital

gains and finds that the model is well supported by the cross-section of long horizon

stock returns. Klein (2003) conducts additional empirical tests and finds investors’

accrued capital gains are largely responsible for return seasonality, in particular, the

January effect.

It is important to note that Klein’s models not only prohibit short selling for extended

periods of time, but also assume that perfect substitute securities do not exist. This second

assumption is controversial because perfect substitutes may allow investors to re-balance

their portfolios without realising their gains. As a result, the pricing differences caused by

2 This assumption is in contrast with typical asset pricing models when investors are assumed to be able to

short sell frictionlessly for limited periods of time in order to force a realignment of securities which are

temporarily mis-priced. For example, see Sharpe (1964); Lintner (1965, 1969); Mossin (1966), as well as the

after-tax CAPM models of Brennan (1970); Elton and Gruber (1978). Ross (1977); Jarrow (1980); Sharpe (1991),

among others, show that portfolio composition and equilibrium returns are affected when short sales are

constrained. A discussion of the practical difficulties and costs associated with short selling can be found in Brent

et al. (1990); Kwan (1995) as well as most textbooks on investments, such as Sharpe et al. (1995). See Duffie

(1996) for an analysis of the costs of short selling fixed income securities in the institutional market.

1 A tax timing option also arises if there is a difference between the short and long term capital gains tax

rates, which has been shown by Constantinides (1984); Dammon and Spatt (1996) to be valuable to investors.

P. Klein / Journal of Public Economics 88 (2004) 2765–2783 2767

accrued capital gains identified in Klein’s models may disappear when perfect substitute

securities exist.

The purpose of this paper is to show that riskless trading among substitutes does not

eliminate the effect of accrued capital gains on equilibrium prices. This finding is

significant because it confirms that Klein’s pricing results are robust to this important

change of assumptions. No-dominance arguments are used to show that price differences

because of accrued capital gains do not arise among securities which are perfect

substitutes. These arguments are insufficient, however, to prevent such pricing differences

among securities which are not perfect substitutes.

This paper also shows that the presence of perfect substitutes does not necessarily

allow investors to re-balance their portfolios without realising at least part of their

accrued capital gains. Rules are developed which provide guidance on which securities

should be sold in order to re-balance without undue realisation of accrued capital gains.

Although these rules are already well known in practice, this paper makes a contribution

by confirming their applicability in a general equilibrium when perfect substitute

securities exist.

The organisation of this paper is as follows. Section 2 develops an asset pricing model

which will be used to demonstrate the major contributions of this paper. Section 3 shows

that pricing differences because of accrued capital gains should not arise among securities

which are perfect substitutes. Section 4 develops rules for optimal gains realisation and

loss deferral among securities which are perfect substitutes. Section 5 discusses the effect

of accrued capital gains on optimal portfolio composition and Section 6 confirms that

equilibrium returns are affected by investors’ accrued capital gains. Section 7 illustrates

the model through a few numerical examples, and Section 8 provides a brief conclusion.

2. The model

Assume there are I investors in a pure exchange economy in which K firms have issued

shares. Investors are endowed at time t=1 with heterogeneous non-negative shareholdings

Hik in firms 1 to K and Hi0 in the riskless asset. The endowed shareholding of investor i in

security k (Hik) has tax basis Bik. These tax bases are also heterogeneous across stocks as

well as investors. As in Dyl (1979); Milne and Smith (1980); Constantinides (1983, 1984);

Stiglitz (1983); Dammon and Spatt (1995, 1996), this heterogeneity is assumed with no

formal justification, but could be due to differing times at which investors entered the

market, for example. In order to focus the analysis on the effect of accrued capital gains, it

is assumed that the number of investors and securities is constant, and that investors have

homogeneous expectations.

Assume a given security k pays a stochastic terminal cash flow Dk which is taxed as a

capital gain; note Dk can be considered to be the ex-dividend price of security k at t=2. The

terminal cash flows of all securities are jointly distributed with positive expected value.

Each security is also assumed to pay a non-stochastic cash dividend dk at terminal time

t=2. Without loss of generality, assume the first KI securities are independent (the ‘‘basis

securities’’), i.e. there are KI independent sources of risk. Assume that the total cash flow

on the remaining KD=K�KI securities (the ‘‘dependent securities’’) can be expressed as a


linear combination of the total cash flow on the basis securities. For example, if security k

is dependent, its total cash flow can be written as:

Dk þ dk ¼ ½DI þ dI �Aik ð1Þ

where DI and dI represent row vectors of terminal cash flows and dividends corresponding

to the basis securities, A is a KI by KD matrix representing the manner in which the

dependent securities are related to the basis securities (the ‘‘dependency matrix’’) and ik is acolumn vector of zeros with a one in position k. This specification of security interdepen-

dence is sufficiently general to allow a dependent security to be replicated by a single basis

security, or by a linear combination of the basis securities. It also allows the existence of

dependent securities which are perfect substitutes for each other, i.e. which can be

replicated by the same linear combination of the basis securities. Groupings of dependent

securities which are perfect substitutes for each other, as well as the linear combination of

the basis securities which replicate them, will be termed ‘‘replicating groups’’.

In order to provide further intuition to the manner in which securities may be

interdependent, consider a simple economy where there are six securities. Assume the

first three securities span the state space and thus are the basis securities. Thus KI=3 and

there are three independent sources of risk. Assume the manner in which the three

dependent securities are related to the three basis securities can be represented by the

following matrix:

A ¼

1 1 �1

1 1 1

0 0 1

266664

377775

Note the first and second columns of the dependency matrix are identical; this indicates

the first and second dependent securities are perfect substitutes for each other since they

can be replicated by the same linear combination of the basis securities. The third column

of the dependency matrix is different; this implies that the third dependent security can

also be replicated by a linear combination of the basis securities but that linear

combination is different than for the first two dependent securities. Thus KD=3 in this

example but there are only two replicating groups.

This paper also assumes that short sales, either outright or against the box, are not

allowed.3 This assumption is not realistic when modelling the effect of short selling for

limited periods of time in order to take advantage of mis-aligned prices, but it is an

appropriate reflection of the prohibitive costs of short selling for extended periods of time

in order to defer the realisation of capital gains.4 This assumption is also intended to

3 The assumption that short sales are not allowed does not affect the composition of replicating groups, but it

may prevent some investors from trading to take advantage of mis-aligned prices. This issue will be discussed in

Section 3.4 Note that due to the constructive realisation rules of Section 1259 of the US tax code, adopted in 1997, it is

no longer possible to defer capital gains by short selling.


prohibit derivative securities which are designed to circumvent the prohibition on short

sales. In reality, derivative securities which provide an indirect short interest to a given

underlying security for a limited period of time are often available to investors. The costs

to the derivative provider of hedging such a position need to be built-in to its price, but are

not significant because the costs of short selling the underlying security for a short period

of time are small. In contrast, in order to offer a long term short exposure via a derivative,

the derivative provider would need to hedge by short selling the underlying asset for an

extended period of time. In doing so, the derivative provider would incur substantial costs

which would need to be passed on to the derivative user or embedded in the price of the

derivative. In both cases, whether directly or indirectly via a derivative, such costs

effectively prohibit short selling to defer capital gains taxes for extended periods of time,

which justifies the absolute prohibition against short sales which is assumed in this paper.

Assume there are no transaction costs but that interest income, dividends and realised

capital gains are taxed at constant rates Tb, Td and Tg, respectively. The assumption of a

single tax rate on capital gains eliminates the effect of asymmetric capital gains taxation and

allows clearer focus on the tradeoff between diversification and capital gains realisation.5

Investors’ accrued capital gains are taxed at t=1 if they are realised, or at the terminal

time t=2. This assumption is conservative because it may understate the effect of accrued

capital gains on investor behaviour. In practice, investors compare an immediate tax

liability with the present value of a tax liability at a more distant time in the future. Thus

investors may optimally choose to be much more undiversified in order to defer the taxation

of capital gains than is implied by this paper.6 If an investor sells all or part of the endowed

shareholding in stock k at time t=1 the amount of capital gains tax payable at that time is:

ðPk � BikÞaikTgwhere aik represents the number of shares sold at t=1.7

Investor i’s total capital gains tax payable or recoverable at t=1 is:

ðP � BiÞ � aiTgwhere P and Bi are K-vectors consisting of the corresponding elements for each stock.8

Investor i’s accrued capital gain in stock k at terminal time t=2 is:

ðPk � BikÞðHik � aikÞ þ ðDk � PkÞSikwhere Pk is the equilibrium price of k and Sik is investor i’s shareholding in k at time t=1.

Investor i’s gain in security k at t=2 depends on how much of the gain or loss was not

realised at t=1 (i.e. Hik�aik), as well as the capital gain from t=1 to t=2 on the shareholding

5 Thus the tax timing strategies of Constantinides (1984) and Dammon and Spatt (1996) based on

asymmetric taxation are not applicable. The constant tax rate also eliminates the possibilities of tax arbitrage as

outlined in Dammon and Green (1987); Ross (1987); Dybvig and Ross (1986).6 See Dammon and Spatt (1995) who assume gains untaxed at t=1 remain untaxed at terminal time t=2 as well.7 Strictly speaking, aik represents the net number of shares that are sold at t=1, which prevents the excessive

capital gains realisation which would arise in the course of trading to establish equilibrium prices. Alternatively, it

is assumed that a Walrasian auction market governs trading. See Klein (1998) for more discussion on this point.

This assumption also prevents wash sales.8 Note throughout this paper that the use of bold face and the absence of the k subscript indicates K-vectors of

the corresponding elements for each stock k=1, . . ., K.


in k that was selected at t=1.9 Note that purchases by i in security k can be expressed as

time t=1 shareholding less endowment less sales, i.e. Sik�Hik�aik.Each investor i chooses a K-vector of shareholdings Si and allocation Si0 to the riskless

asset to solve the following problem:

maxSi;Si0Ui;t¼1ðci;t¼1Þ þ E½Ui;t¼2ðci;t¼2Þ�subject to short sales constraints Sikz0 for k=1 to K, as well as budget constraints:

ci;t¼1 ¼ Wi � Si0 � P � Si � ðP � BiÞ � aiTgand

ci;t¼2 ¼ Si0Rþ ðDþ dð1� TdÞÞ � Si � ððD� PÞ � Si þ ðP � BiÞ � ðHi � aiÞÞTgwhere

Wi ¼ Hi0 þ P � Hi

R ¼ 1þ ð1� TbÞrfand rf is the risk-free rate of interest, Hik and Bik represent the endowed shareholding and

basis of investor i in security k, and UV()>0 and UW()<0.As shown in Klein (1998), investor i’s first order condition for a given stock k is:

Pk ¼ E½Ui;t¼2V ðci;t¼2ÞðDk þ dkð1� TdÞ � TgðDk � PkÞÞ�=Ui;t¼1V ðci;t¼1Þ þ dik ð2Þ

where dik=wik (Tg (1�1/R)(Pk�Bik)+Eik), Eik represents the Lagrange multiplier for the

short sales constraint and wik is an indicator variable which takes a value between 0 and 1

inclusive as will be discussed below. This First Order Condition states that the investor

will benefit from trading in stock k unless its price equals the sum of the terms on the right

hand side of Eq. (2). The first term on the right hand side is straightforward: it depends on

the expectation of the product the ratio of the marginal utilities at both points in time

(Ui,t=2V (ci,t=2)/Ui,t=1V (ci,t=1)) and the after-tax cash flow on the security (Dk+dk (1�Td)�Tg(Dk�Pk)). Since UV()>0 and UW()<0 by assumption, this first term decreases as the

investor’s shareholding in a given stock increases.

The second term on the right hand side of the First Order Condition (dik) arises becauseof investor i’s accrued capital gains in stock k. This term is specific to investor i and will be

referred to as the ‘‘individual deferral’’ term throughout this paper. This term takes

different values depending on whether investor i buys, sells or does not trade in stock k.

Each of these three cases is considered in the following paragraphs.

First, if the price of the security is sufficiently low, it will be optimal for i to buy more

shares in k. In this case, no capital gains or losses are being realised and the short sales

constraint does not bind. This case is represented in the model by setting the indicator

variable wik equal to zero. This in turn implies that the deferral term equals zero, i.e. that

the First Order Condition with respect to this stock is unaffected by the investor’s accrued

capital gain.

9 This method of calculating the gain on a share sold represents a simplification of the US code where a first-

in-first-out method of accounting usually applies. The first-in-first-out method of accounting would tend to

increase the capital gain lock-in effect as compared to the model developed in this paper, since shares which are

purchased the earliest would tend to have the highest gains.


Second, if the price of the security is sufficiently high, it will be optimal for i to sell some

of k. If the investor has a gain in k and sells some but not all of the endowment, dik ispositive and depends on the nominal capital gain tax rate (Tg), the size of the gain (Pk�Bik)

and a timing factor representing the net present value of deferring the capital gain tax

liability for one period (1�1/R). The investor may also be prevented from selling k by the

short sales constraint which implies that the deferral term is positive because of the

Lagrange multiplier. Both of these cases are indicated by setting wik equal to one in Eq. (2).

A third case applies when the price is above the price that would motivate a purchase,

but below the price that would motivate a sale. In other words, the investor does not trade,

even though the allocation to the security may be larger than would be optimal in the

absence of his or her accrued capital gain. In this case a value for wik between zero and one

can be selected in order that Eq. (2) continues to hold. This value depends on the

endowment and on the gain on the endowment, i.e. wik is closer to zero if the endowment

is not much in excess of what the optimal position would be without the accrued gain or

short sales constraints, and is closer to one if the endowment is well in excess of what

would otherwise be optimal but not sufficiently large to outweigh the benefits of deferring

the capital gain.

When there are no perfect substitutes these three cases cover the entire range of

possibilities for an investor’s optimal trading with respect to a given security k. Trading in

a different security may mitigate the extent to which an investor wishes to purchase or sell

security k but it is not always able to eliminate the need to re-balance holdings in k because

each security provides a unique source of risk. As in previous work by Klein, this implies

that equilibrium prices will depend on investors’ accrued capital gains. When there are

perfect substitutes, however, an investor may have the opportunity to re-balance a non-

optimal shareholding in a given stock by trading one of its substitutes. If investors are

always able to re-balance in this way, no capital gains would ever need to be realised and

thus the deferral terms for all investors would always equal zero. This result would imply

that portfolio risk and equilibrium prices are not affected by investors’ accrued capital

gains, which is the conclusion in Constantinides (1983).

The next two sections of this paper address this important question of whether investors

are always able to re-balance with perfect substitutes. Section 3 starts by outlining pricing

relationships which must hold among members of a given replicating group. These pricing

relationships imply that investors’ deferral terms on the members of a given replicating

group must also be related. Section 4 uses these results to demonstrate that it is not always

optimal to defer capital gains on overly large endowments, and that it is not always

optimal to realise immediately all capital losses. Thus investors’ individual deferral terms

do not always equal zero, which will be shown to imply in Sections 5 and 6 that investors’

accrued capital gains do indeed affect optimal portfolio risk and equilibrium returns, even

when perfect substitute securities exist.

3. Pricing relationships among perfect substitute securities

In perfect markets with no short sales constraints or taxes, riskless trading will eliminate

any pricing differences that might arise among perfect substitutes. This result can be


expressed in the notation of this paper if tax rates and the Lagrange multiplier are set equal

to zero. In this case the First Order Condition (Eq. (2)) and the definition of security

interdependence in Eq. (1) imply:

Pk ¼ PIAik ð3aÞ

i.e. the price of a given dependent security must equal the linear combination of the

prices of the basis securities which replicate it. The analysis in this paper is more

complicated, however, for three reasons: the difference between the tax rates on

dividends and capital gains, the prohibition against short sales, and the effect of investors’

accrued capital gains. The effect on Eq. (3a) of these three imperfections will be

considered sequentially.

The first complication is merely the well-known dividend yield effect. If dividend

income and capital gains are taxed at different rates, a pricing difference may arise among

securities with identical pre-tax cash flows because the after-tax cash flows may differ.

This dividend yield effect also implies that more than KI securities may be needed to span

the state space on a post-tax basis. This complication has been well studied by other

authors and is not the focus of this paper. Accordingly, the following assumption will be

imposed in order to simplify the analysis:

dk ¼ dIAik

This assumption requires that the cash dividends on a given dependent security equals

the cash dividends on the linear combination of the basis securities which replicate the pre-

tax cash flow in Eq. (1). With this additional assumption, Eq. (3a) will continue to hold on

a post-tax basis, i.e. there will be no pricing differences among members of a given

replicating group because of the difference in tax rates on dividends and realised capital

gains.

The second complication arises because of the constraint on short sales. If prices of

perfect substitutes do not satisfy Eq. (3a), an investor would wish to buy the inexpensive

security and sell the expensive security, but may be prevented from doing so because of

the constraint on short sales. In this case Eqs. (1) and (2) imply:

Pk ¼ PIAik þEIi Aik � Eik

Tg=R� 1ð3bÞ

where the Lagrange multiplier on a given dependent security k and the linear combination

of the Lagrange multipliers on the basis securities which replicate it do not necessarily

equal zero. It should be noted, however, that the short sales constraint does not prevent a

given investor from selling his or her holdings of the expensive security and purchasing

more of the inexpensive security. Further, new investors with no exposure to any of the

securities in this replicating group would clearly have a preference to purchase only the

inexpensive security because its payoff would dominate the payoff on the expensive

security. Henceforth it is assumed there is sufficient trading by unconstrained investors in

order to eliminate dominance among substitutes; this implies that the second term on the


right hand side of Eq. (3b) must equal zero in equilibrium. Thus Eq. (3a) still applies even

if some investors are bound by the short sales constraint.10

The third complication arises because of investors’ accrued capital gains. In this case,

Eqs. (1) and (2) imply:

Pk ¼ PIAik þdIi Aik � dikTg=R� 1

ð3cÞ

The second term on the right hand side of this equation is the same as the corresponding

term in Eq. (3b) where the Lagrange multipliers have been replaced by the investor’s

deferral terms. If this second term could be shown to be non-zero, it would immediately

demonstrate the principal goal of this paper, i.e. that security prices are affected by

investors’ accrued capital gains even when perfect substitute securities exist. Unfortu-

nately such a hasty conclusion is not realistic because no-dominance arguments similar to

those in the preceding paragraph should apply to this case as well. As noted in Dammon

and Spatt (1995), it is realistic to assume that if prices are mis-aligned, any investor who

does not have gains in the over-priced security would desire to sell it in exchange for a

lower priced substitute. Further, new investors, for whom the deferral terms equal zero by

definition, will have a preference for the underpriced substitute and will not purchase the

over-priced substitute. Assuming there is sufficient trading activity to eliminate dominance

among perfect substitutes implies once again that Eq. (3a) must hold in equilibrium. This

means that the second term on the right hand side of Eq. (3c) equals zero for all investors,

but it does not imply that the deferral terms for all investors equal zero.

To recap, standard no-dominance assumptions concerning the presence of uncon-

strained or new investors are sufficient to eliminate pricing differences among substitutes

because of accrued capital gains. Fortunately for this paper, this result applies only to the

relative prices of securities within a given replicating group and does not imply that the

absolute prices of all securities in the group are unaffected by accrued capital gains. In

order to analyse this latter issue, it is necessary to determine whether the price of the

replicating linear combination of the basis securities is affected by investors’ accrued

capital gains. Accordingly, a second pricing equation will be derived in Section 6 for the

basis securities. Since each basis security provides exposure to a unique source of risk,

however, no-dominance arguments can not be applied to eliminate potential pricing

differences because of accrued capital gains among the basis securities.

4. Optimal gains deferral and loss realisation within replicating groups

This section develops trading rules which provide guidance on which securities within

a given replicating group should be sold in order to re-balance without undue gains

realisation and loss deferral. Although these trading rules are well known in practice, it is

important to demonstrate that they apply in a general equilibrium when perfect substitute

10 See Dammon and Spatt (1995) who also rely on the assumption of no-dominance in order to eliminate

pricing differences among perfect substitutes.


securities exist. These rules also provide the conditions under which an investor’s deferral

term is non-zero, which will be useful in developing an equilibrium pricing equation for

the basis securities.

The formal basis for these trading rules starts by recognising that Eqs. (3a) and (3c)

must simultaneously hold for all investors. This implies:

dik ¼ dIi Aik ð4Þi.e. that each investor’s individual deferral terms must equal the linear combination of the

deferral terms on the basis securities which replicate it. This result leads to the following

rule which governs the optimal selling of securities with accrued capital gains.

Trading rule I: An investor will realise a gain only if all other member securities of the

replicating group that are already owned by that investor also have accrued gains.

Within a replicating group, an investor will sell member securities with the smallest

gains first.

This trading rule can be demonstrated by analysing the possible values of the

components of the deferral term for each of the members of a given replicating group.

Recall that the deferral term depends on the indicator variable (wik), the magnitude of the

capital gain (Pk�Bik), and the Lagrange multiplier. If investor i realises a gain in

dependent security k, the indicator variable equals one and the deferral term dik will bepositive. In order to prevent dominance, Eq. (4) requires that the linear combination of the

individual deferral terms of the independent securities which replicate security k must

equal dik. Eq. (4) also implies that individual deferral terms for any other members of the

replicating group must equal dik as well. This condition can hold only if these other

members also have accrued gains, or if the investor has a zero shareholding in a given

substitute and is bound by the short sales constraint. Further, since the value of the

indicator variable wik cannot exceed one by definition, Eq. (4) can hold simultaneously for

all members of a replicating group only if the member with the smallest accrued capital

gain is sold first. Members with larger accrued gains will not be traded, in which case wik

will be between zero and one in order that Eq. (4) is satisfied.11

Trading rule II: An investor will defer losses only if all other members of the replicating

group also have accrued losses.

This trading rule will be demonstrated by discussing three cases. First consider an

investor who has losses on at least one, but not all members of a replicating group. In this

case, the investor is able to sell all members with losses and purchase more of one of the

members which has no loss in order to re-balance. The investor’s deferral term on the

member which is purchased equals zero by definition, thus, according to Eq. (4), the

investor’s deferral terms for all other members of that group must also equal zero. In the

second case, assume the investor has losses on all members of the replicating group. In this

11 A numerical example of optimal gains realisation is provided in Table 1.


case the investor can realise losses on most of the members of the replicating group, but is

unable, due to the constraint on wash sales, to realise the loss on the last member if a

positive shareholding in the replicating group is to be maintained. If it is optimal to

purchase more of this remaining member of the replicating group, the loss on this security

is deferred but the investor’s deferral term on this last member would equal zero by

definition. Eq. (4) implies that the deferral terms for all other securities in the replicating

group must also equal zero for this investor. The third case is similar to the second, but

assumes that after realising losses on all other members in the replicating group, the

endowment in the remaining security is still too large. In this case, the investor would

optimally realise some but not all of the loss on the last member and the deferral term

would be negative. Since wik can take a maximum value of one, Eq. (4) will hold only if

the member with smallest accrued loss is the one in which the loss is only partially

realised.12

To recap, investors are not always able to re-balance their portfolios without realising

capital gains or deferring capital losses, even when perfect substitute securities exist. Thus

investors’ deferral terms do not necessarily equal zero. This result is based on no-

dominance arguments which are assumed to eliminate pricing differences among securities

within a given replicating group. If this assumption is not made, similar trading rules could

be developed but they would need to take into account the potential difference in prices

among securities within a given replicating group.

5. Optimal portfolio composition among basis securities

The preceding section outlined rules governing the trading of securities within a given

replicating group. These rules can also be interpreted as providing weak guidance on how

portfolio allocations should be made among members of that replicating group. This

section considers a related portfolio allocation issue, i.e. how large should the allocation be

to each of the basis securities, which will lead to a pricing equation for the basis securities

in the following section.

Insight to optimal portfolio weights for the basis securities can be gained on a general

level by examining the first order condition in Eq. (2). This condition was developed

without the typical restrictions on distributions or preferences that would allow an exact

expression for optimal portfolio weights. It states that an investor will purchase or sell a

given security if the price does not satisfy Eq. (2). If the deferral term in this equation does

not equal zero, then the investor’s decision to hold more or less of a given security depends

on his or her accrued capital gains. This result applies not only for the dependent securities

but for the basis securities as well. If the deferral term is positive on a given basis security

because of accrued capital gains, Eq. (2) implies that at a given price, an investor will hold

more of that security than would otherwise be the case. In other words, investors tend to be

‘‘locked-in’’ to stocks in which they have accrued capital gains. This result in turn implies

that investors who are not ‘‘locked-in’’ hold less of a given stock.

12 A numerical example of optimal loss deferral is provided in Table 1.


Further insight to this general result can be gained if additional assumptions are

imposed in order to allow a solution for optimal portfolio weights. Although these

assumptions are not needed for the other results in this paper, assume for the remainder

of this section that the terminal cash flows of the K securities have covariance matrix

V. Since KD of these securities are dependent, V is assumed to be singular. The

covariance matrix for the KI basis securities, VI, is assumed to be non-singular but not

necessarily diagonal. In order to invoke a mean-variance framework, assume that the

terminal cash flows are joint-normally distributed. Accordingly, the first order condition

can be written as:13

Pk ¼ R�1½E½Dk � � ðE½Dk � � PkÞTg þ dkð1� TdÞ þ dik �Q�1i Si*Vik � ð5Þ

where

Qi ¼ �E½Ui;t¼2V ðci;t¼2Þ�=E½Ui;t¼2W ðci;t¼2Þ�

and the deferral term has been re-scaled by R and put inside the square brackets.

Following the approach in Booth (1983), according to the definition of an equivalence

transformation and the nature of the stochastic dependence among securities as

represented by A, it is possible to rewrite V as follows:

V ¼VI VIA

AVVI AVVIA

24

35 ¼ E

I 0

0 0

24

35F�1

where

E ¼B1 0

AVB1 I

24

35

F�1 ¼B2 B2A

0 I

24

35

and

VI ¼ B1B2

Substituting this expression into Eq. (5) and post-multiplying by F, rearranging

and post-multiplying by B1�1, then imposing the market clearing condition and re-

13 Note the kink in the budget set needs to be replaced with a smoothly-pasted spline in order for the Stein’s

lemma to be applicable. See Klein (1998) for further discussion of this point.


substituting in the usual way allows optimal portfolio composition to be expressed

as:

½Sik þ SDi AVik �* ¼ Qi

Qm

½1þ 1AVik � þQi½dIi � d̄I �½VI ��1ik ð6Þ

The left hand side of this equation represents the optimal allocation of investor i to the

risk provided by basis security k. This allocation consists of the direct weight in basis

security k (Sik) as well as the contribution to this source of risk by the holdings in the

dependent securities (SiD). The right hand side of this equation contains two terms. The

first term represents the risk-tolerance weighted allocation (Qi/Qm) to the entire sources of

exposure to the kth independent source of risk (1+1AVik ). This result is analogous to the

basic CAPM result except that optimal allocations are expressed in terms of the

independent sources of risk instead of individual securities.14

The second term on the right hand side of Eq. (6) demonstrates that optimal weights

depend on accrued capital gains. This term depends on the vector of investor i’s deferral

terms (dIi ) and the covariance matrix of the basis securities (VI). In general, this term

implies that an investor holds more of a given security when his or her accrued capital gain

on that security is positive, which is consistent with the more general result obtained

above. The second term on the right hand side of Eq. (6) also contains a vector

representing the risk-tolerance weighted average of the deferral terms of all other investors

(d̄I ). The presence of this average deferral term indicates that investors hold less of a

security for which other investors have positive deferral terms.

Thus far, this paper has provided guidance, in the form on trading rules, of how

portfolio allocations should be made among securities within a given replicating group. It

has also developed insight, and a more precise expression under additional assumptions,

for an investor’s optimal allocation to the risk provided by a given basis security. These

two results imply a form of portfolio separation whereby the investor first solves for the

composition of the replicating portfolios, and then finds the optimal weights for each

independent source of risk.15

6. Equilibrium prices and returns

This section demonstrates how accrued capital gains affect the prices of the basis

securities, even when there is assumed to be no pricing differences among perfect

substitutes. Intuition to the pricing of the basis securities can be provided by reconsidering

Eq. (2) in the absence of accrued gains or losses. Before trading at t=1, each investor is

either over-endowed, optimally endowed, or under-endowed. For a price Pk* on a given

basis security k to be an equilibrium price, the number of shares sold by over-endowed

investors must equal the number of shares purchased by under-endowed investors. In the

absence of accrued gains and losses, Eq. (2) without the individual deferral term dik appliesfor all investors after completion of optimal trading.

14 This analogy is attributed to the insightful comments of a reviewer.15 The comments of a reviewer with respect to this point are gratefully acknowledged.


Consideration of accrued capital gains changes this equilibrium relationship. Eq. (2)

without the deferral term continues to apply for investors who are under-endowed

because wik equals zero for any under-endowed investor who buys. Thus at the same

price Pk* the demand for shares is unaffected by the accrued gains of investors who are

under-endowed. This result does not hold for over-endowed investors; if they are unable

to re-balance with substitutes, their individual deferral terms do not equal zero as

discussed above. When the individual deferral term is added, Eq. (2) does not hold at Pk*

which indicates that the supply of shares is decreased. As discussed in Section 4, the

supply of shares of other members of the replicating group is decreased as well. The

balance between supply and demand is restored by increasing the price above Pk*. This

higher price induces more investors to sell. It also decreases demand because buyers will

not be willing to purchase as many shares in k because of the higher price. Note that the

no-dominance assumptions invoked to eliminate pricing differences among perfect

substitutes do not apply because each basis security provides an independent source

of risk.

Further insight to equilibrium pricing can be gained under the additional assumptions

that lead to Eq. (6) in the preceding section. Summing Eq. (6) across all investors in order

that the market clears results in the following equilibrium pricing equation which is similar

to the result in Klein (1998, 1999):

Pk ¼ R�1½E½Dk � � ðE½Dk � � PkÞTg þ dkð1� TdÞ þ d̄k �Q�1m 1Vik � ð7aÞ

This equation is a general expression which applies for the basis securities as well as

the dependent securities. It states that the price of a given security k equals the discounted

sum of its expected after-tax cash flows, less the adjustment for risk contained in the final

term on the right hand side. The deferral term in this expression is the risk-tolerance

weighted average of all investors’ individual deferral terms.

For the basis securities, Eq. (7a) can be re-written in terms of the risk on only the basis

securities as follows:

Pk ¼ R�1½E½Dk � � ðE½Dk � � PkÞTg þ dkð1� TdÞ þ d̄k �Q�1m ½1þ 1AV�VI ik � ð7bÞ

This equation confirms that investors’ accrued capital gains affect the prices of the basis

securities, because the average deferral term for each basis security may be different.

If security k is dependent, its equilibrium price can also be written in terms of the

equilibrium pricing vector of the independent securities and the dependency matrix as

follows:

Pk ¼ R�1½E½DI � � ðE½DI � � PI ÞTg þ dI ð1� TdÞ þ d̄I �Q�1m ½1þ 1AV�VI �Aik ð7cÞ

Eq. (7c) confirms that relative prices of perfect substitutes are unaffected by investors’

accrued capital gains because the prices of all members of a given replicating group are

influenced by the same linear combination of the average deferral terms of the independent

set.

The difference in security prices that is caused by investors’ accrued capital gains

implies that additional terms need to be added to factor models of equilibrium returns. In


the case of the single factor CAPM, for example, the expected return on a given stock k

can be expressed as:

E½rk � ¼ rf þ bk ½E½rm� � rf � ymsd þ rf sb þ d̄m� þ yksd � rf sb � d̄k ð8Þ

where

bk ¼Covðrm;rkÞVarðrmÞ

sd ¼Td � Tg

1� Tg; sb ¼

Tb � Tg

1� Tg

terms gk and yk represent the portions of return on security k that arise from capital gains

and dividends respectively; rk=yk+gk denotes total pre-tax return on security k; subscript m

denotes the analogous terms for the market portfolio; and d̄k and d̄m have been re-scaled.

Note that the terms containing sd and sb are analogous to the terms in the post-tax CAPM

of Brennan (1970). These terms relate to the difference in tax rates on interest, dividend

and realised capital gains and disappear if these rates are equal.

7. Numerical examples

In order to provide further intuition to the model and to illustrate the potential

magnitude of the effect of accrued capital gains, some simple numerical examples are

presented in Table 1. These examples are based on three investors (A, B and C) and four

securities which pay no dividends but which pay terminal cash flows which are taxed as

capital gains. In order to be able to obtain an exact expression for portfolio weights, a

mean-variance framework is assumed as in Section 5 of this paper. It is assumed that there

are two replicating groups. Securities 1a and 1b are perfect substitutes as are securities 2a

and 2b. The mean and variance of the terminal cash flows are identical in order to isolate

the effect of differing endowments and accrued gains across investors. There are a total of

three shares in each security. The three investors are identical except for initial endow-

ments and accrued gains or losses. As a result, optimal shareholdings for each investor

would consist of one share in each of the four securities, if there were no accrued capital

gains or losses. Constant risk tolerance is assumed in order to eliminate the effect of

differing degrees of risk tolerance as optimal portfolios change. Parameter values are

arbitrarily selected, but these values are consistent with a 1 year investment horizon.

The first panel of Table 1 shows the effect on optimal portfolio composition and

equilibrium returns when both investors A and B have accrued gains on one of the

securities in the first replicating group, but they do not own the perfect substitute. The sum

of the endowments in securities 1a and 1b are greater than what is optimal for investors A

and B, thus both sell to C even though both realise some of their accrued capital gains.

These gains realisations are consistent with Trading rule I, and imply that investor A and

B’s individual deferral terms are non-zero on securities 1a and 1b. For both A and B, the

Table 1

Example calculations of equilibrium prices and portfolio weights

Stock Investors’

endowments

(Hik)

Endowed

gains

( Pk�Bik)

Individual

deferral terms

(dik)

Average

deferential

term (dk̄)

Optimal

shareholdings

(Sik*)

Equilibrium

A B C A B C A B C A B C Price

( Pk)

E[rk]

I: Base case–optimal realisation of capital gains

1a 2.60 0.00 0.40 5 0 0 0.090 0.090 0.000 0.060 2.24 0.00 0.761 $8.63 15.9%

1b 0.00 2.60 0.40 0 5 0 0.090 0.090 0.000 0.060 0.00 2.24 0.761 $8.63 15.9%

2a 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0.941 0.941 1.121 8.55 17.0

2b 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0:941 0:941 1:121 8.55 17:0

4.12 4.12 3.76 �1.1%

II: Optimal realisation of smaller gains first

1a 2.50 0.50 0.00 5 0 0 0.090 0.000 0.000 0.030 2.02 0.24 0.741 $8.59 16.4%

1b 0.10 1.70 1.20 6 5 0 0.090 0.000 0.000 0.030 0.10 1.70 1.201 $8.59 16.4%

2a 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0.971 1.021 1.011 8.55 17.0

2b 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0:971 1:011 1:021 8.55 17:0

4.06 3.97 3.97 �0.6%

III: Optimal deferral of capital losses

1a 2.50 0.10 0.00 �5 �6 0 �0.090 0.000 0.000 �0.030 1.76 0.00 1.241 $8.45 18.3%

1b 0.10 2.50 1.20 �6 �5 0 0.000 �0.090 0.000 �0.030 0.00 1.76 1.241 $8.45 18.3%

2a 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 1.061 1.061 0.881 8.55 17.0

2b 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 1:061 1:061 0:881 8.55 17:03.88 3.88 4.24 1.3%

IV: Base case with smaller accrued capital gain

1a 2.60 0.00 0.40 2.5 0 0 0.045 0.045 0.000 0.030 2.12 0.00 0.881 $8.59 16.4%

1b 0.00 2.60 0.40 0 2.5 0 0.045 0.045 0.000 0.030 0.00 2.12 0.881 $8.59 16.4%

2a 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0.971 0.971 1.061 8.55 17.0

2b 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0:971 0:971 1:061 8.55 17:04.06 4.06 3.88 �0.6%

V: Base case with lower capital gains tax rate (Tg=0.15)

1a 2.60 0.00 0.40 5 0 0 0.045 0.045 0.000 0.030 2.12 0.00 0.881 $8.82 13.3%

1b 0.00 2.60 0.40 0 5 0 0.045 0.045 0.000 0.030 0.00 2.12 0.881 $8.82 13.3%

2a 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0.971 0.971 1.061 8.79 13.8

2b 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0:971 0:971 1:061 8.79 13:84.06 4.06 3.88 �0.5%

VI: Base case with lower risk tolerance (Qi=0.75)

1a 2.60 0.00 0.40 5 0 0 0.090 0.090 0.000 0.060 2.12 0.00 0.881 $7.97 25.2%

1b 0.00 2.60 0.40 0 5 0 0.090 0.090 0.000 0.060 0.00 2.12 0.881 $7.97 25.2%

2a 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0.971 0.971 1.061 7.89 26.7

2b 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0:971 0:971 1:061 7.89 26:7

4.06 3.97 3.97 �1.5%


Table 1 (continued)

Stock Investors’

endowments

(Hik)

Endowed

gains

( Pk�Bik)

Individual

deferral terms

(dik)

Average

deferential

term (dk̄)

Optimal

shareholdings

(Sik*)

Equilibrium

A B C A B C A B C A B C Price

( Pk)

E[rk]

VII: Base case with lower covariance (j12=0.25)1a 2.60 0.00 0.40 5 0 0 0.090 0.090 0.000 0.060 2.19 0.00 0.811 8.74 14.4%

1b 0.00 2.60 0.40 0 5 0 0.090 0.090 0.000 0.060 0.00 2.19 0.811 8.74 14.4%

2a 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0.981 0.981 1.041 8.66 15.5

2b 1.00 0.70 1.30 0 0 0 0.000 0.000 0.000 0.000 0:981 0:981 1:041 8.66 15:5

4.15 4.15 3.70 �1.1%

Calculations are for investors i=A, B, C and securities k={1a, 1b, 2a, 2b} based on Eqs. (6) and (7a) in the text.

Parameter values are: R=1.06; Tg=0.3; E[Dk]=10, dk=0 for all k; Qi=1.5 (constant risk tolerance) for i=A, B, C.

There are two replicating groups: securities 1a and 1b, as well as 2a and 2b are perfect substitutes; the after-tax

elements of VI are r12=r2

2=1 and r12=0.5.1 For securities indicated by this footnote, only the sum of shareholdings in the replicating group can be

precisely determined. In these cases, the division of shareholdings indicated in the table between the securities in

the replicating group is arbitrary.


optimal sum of the shareholdings in the first replicating group is higher than would be the

case in the absence of the accrued capital gains, and lower for the second replicating

group. In total, A and B own hold larger portfolios of risky assets than would be optimal if

there were no accrued capital gains. Investor C optimally holds less of 1a and 1b, and more

of 2a and 2b, but holds a smaller risky portfolio than would be the case if A and B had no

gains. These gains also affect the equilibrium price of securities 1a and 1b. The average

deferral term d̄k equals zero for securities 2a and 2b thus their equilibrium prices are

unaffected. Since the parameter values for all of the securities are identical except for the

accrued gains of their shareholders, the effect of those gains can be determined by

comparing the prices of the two replicating groups. The $0.08 price differential between

replicating groups implies an expected return differential of 1.1% over the 1 year horizon.

Panel II in Table 1 provides a similar example but when investor A has gains in both

securities of the first replicating group. In this example, investor B has a gain on security 1b,

but is able to re-balance by selling security 1a in which there is no accrued capital gain. As a

result, investor B’s individual deferral terms for securities 1a and 1b are zero. Investor A is

unable to re-balance without realising a capital gain, and pursuant to Trading rule I, sells 1a

which has the smaller accrued capital gain. The average deferral term for the first replicating

group is not zero, but is smaller than in the previous panel. The effect on equilibrium

shareholdings and returns is also similar to that in the previous panel, but is less pronounced.

Panel III presents an example of optimal loss deferral. Both A and B have losses on

securities 1a and 1b. Investor A realises all of his losses on security 1b since the loss is

larger than on security 1a but does not optimally realise all of his losses on 1a in order to

retain at least some exposure to the risk in the first replicating group. These results are

consistent with Trading Rule II. Note investor A’s overall exposure to the first replicating

group is lower than would otherwise be the case. The effect of B’s capital losses is similar.

Both A and B hold more of securities 2a and 2b because of their accrued capital losses and


relative under-weight on securities 1a and 1b. As a result of A and B’s capital losses,

investor C holds more exposure to the first replicating group, and less exposure to the

second replicating group. The equilibrium price of the securities 1a and 1b is lowered

because of A and B’s accrued capital losses.

Panels IV–VII present results under assumptions which are slightly modified from Panel

I. Panel IV demonstrates that when the accrued capital gain is half as large as in Panel I, the

deferral terms are also halved, and the extent to which investors A and B are willing to skew

their portfolios towards the stocks in which they have gains is reduced. The effect of the

accrued capital gains on the relative prices of the two replicating groups is also smaller.

Panel V demonstrates the effect when the capital gains tax rate is half of what it was in

Panel I. In general, the results are the same as when the gain is halved as in Panel IV,

except for the equilibrium price, which is higher. This is because the lower capital gains

tax rate applies not only to each investor’s accrued capital gains, but also to the capital gain

which is yet to accrue in the next period.

Panel VI demonstrates the effect under lower risk tolerance. The deferral terms are the

same as in Panel I, but the extent to which investors A and B are willing to skew their

portfolios towards the stocks in which they have gains is less. The equilibrium prices are

also lower than in Panel I because of the lower tolerance for risk.

Panel VII provides results when the covariance between the two replicating groups is

reduced by half. As a result, investors A and B are less willing to skew their portfolios

towards stocks 1a and 1b because the benefit of being diversified (i.e. holding more of 2a

or 2b) is greater. The equilibrium prices are higher than in Panel I because of the lower

overall risk of the entire universe of stocks.

8. Conclusion

This paper makes a contribution by analysing the effect of investors’ accrued capital

gains on optimal portfolio composition and equilibrium returns under the assumption that

investors can re-balance with perfect substitute securities. No-dominance arguments are

sufficient to prevent pricing differences among securities which are perfect substitutes, but

are insufficient to eliminate pricing differences because of accrued capital gains among

securities which are not perfect substitutes. This paper also develops trading rules which

provide guidance on which securities should be sold in order to re-balance without undue

realisation of accrued capital gains.

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