Portfolio Decisions with Taxable and Tax-Deferred Accounts ... · 1For example, IRA, Roth-IRA,...

Portfolio Decisions with Taxable and Tax-Deferred

Accounts: A Tax-Arbitrage Approach

Jennifer Huang ∗

Current version: April 7, 2003

Abstract

We analyze an intertemporal portfolio problem with both taxable and tax-

deferred (retirement) accounts when investors are allowed to borrow and short-sell

in taxable accounts. Using a tax-arbitrage argument, we show that the optimal lo-

cation decision (of where to place an asset) is separable from the allocation decision

(of portfolio composition among assets). Investors choose asset location to maximize

the effective tax subsidy received for investing in tax-deferred accounts, and they

derive the optimal allocation by replacing tax-deferred assets with taxable repli-

cating portfolios and reducing the two-account problem to a taxable-account-only

problem.

∗The author thanks John Cox, Stephen Ross, and especially Jiang Wang for advice and constantencouragement. The author also appreciates comments from Kenneth French, Lorenzo Garlappi, JimPoterba, Laura Starks, Sheridan Titman, and seminar participants at Asset Location Conference atStanford, Boston College, Cornell, Duke, MIT, New York University, Ohio State University, Penn StateUniversity, Stanford, University of British Columbia, University of Chicago, University of North Carolina,University of South California, University of Texas at Austin, Yale, and all Ph.D students at MIT financelunch. Address correspondence to Jennifer Huang, Department of Finance, B6600, McCombs Schoolof Business, University of Texas at Austin, Austin, TX 78712. Email: [email protected],Phone: (512)232-9375, Fax: (512)471-5073, and Web: //www.bus.utexas.edu/Faculty/Jennifer.Huang.

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Tax-deferred retirement accounts1 have grown considerably in popularity and become

a crucial tool for retirement planning. Data from the 1998 Survey of Consumer Finances

shows that 45.7% of households participate in self-directed saving plans, up from 30.7%

in 1989. As reported by Bergstrasser and Poterba (2002), “in 1998, the median household

with both tax-deferred and taxable financial assets had 57.1 percent of its financial assets

in a tax-deferred account. At the 25th percentile this value was 28 percent, while at the

75th percentile it was 85.6 percent.”

Given the relevance and popularity of tax-deferred accounts, the allocation of wealth

between taxable and tax-deferred accounts and the portfolio composition within each are

obviously very important. Following Shoven (1999), we use the terms asset allocation for

the decision of overall portfolio composition among different assets and asset location for

the decision of whether to hold an asset in taxable or tax-deferred accounts.

Not surprisingly, the decisions are complex. Even with only a taxable account, there

is no consensus among either academics or practitioners regarding the life-cycle portfolio

allocation decision. The presence of tax-deferred accounts doubles the size of the invest-

ment opportunity set, since the same asset held in taxable or tax-deferred accounts have

different after-tax return distributions. Moreover, the taxable and tax-deferred returns for

the same asset are highly correlated, inducing complicated interactions between the tax-

able and tax-deferred portfolio decisions. For example, if investors have views about asset

returns, say they believe the stock market is going to outperform bonds,2 they can take

advantage of this view either by increasing their overall exposure to stocks (i.e., changing

the allocation decision) or by placing their stock holdings in tax-deferred accounts in order

1For example, IRA, Roth-IRA, 401(k) and 403(b), etc.2Their views could be either rational (e.g., due to private information) or irrational (e.g., due to

behavioral biases).

1

to reduce their expected capital gains taxes (i.e., changing the location decision). The

interaction between taxable and tax-deferred portfolio decisions makes it a daunting task

to isolate the impact of tax-deferred accounts on overall portfolio decisions.

While both academics and practitioners are grappling with the task of identifying the

impact of tax-deferred accounts, they take rather different approaches. On one hand,

academics focus on properly capturing the interaction between taxable and tax-deferred

decisions by solving the two decisions simultaneously in general frameworks that, up to

now, rely heavily on numerical procedures.3 On the other hand, practitioners often rely on

simple rules of thumb and in general ignore the interaction between the two decisions. We

bridge the gap between the two approaches by identifying the conditions under which the

two decisions are indeed separable, and based on the conditions, quantifying the impact

of tax-deferred accounts on overall portfolio decisions.

Our first result is that the location and allocation decisions can be solved separately

if investors are allowed to borrow and short-sell in their taxable accounts with full tax

credits on interest and dividend payments.4 Specifically, the optimal tax-deferred portfolio

can be determined without any knowledge of the overall portfolio composition. The asset

location decision is therefore fully specified by the tax-deferred portfolio, and is separable

from the overall portfolio allocation decision.

The main intuition of the separability result can be explained in three steps: First, tax-

deferred portfolio strategies can be dynamically replicated by portfolios of taxable assets.5

3See, for example, Dammon, Spatt and Zhang (2002a).4The market is effectively complete in the sense of Ross (1987) that a martingale pricing measure

can be defined, even though dividend and capital gains are taxed differently and the same asset receivesdifferent tax treatment if placed in taxable or tax-deferred accounts.

5To be more precise, we derive the replication strategy for “state-independent” tax-deferred strategiesas defined in Definition 1, and show that the optimal tax-deferred strategy is indeed state-independent.The details are provided in Appendix A. We focus on the main intuition for now.

2

The replication costs depend primarily on the portfolio strategy followed in tax-deferred

accounts, and is independent of investor preferences or their overall portfolio allocation

decisions. Second, investing in tax-deferred accounts is equivalent to receiving an “effec-

tive tax subsidy” from the government, with the size of the subsidy fully determined by

the replication cost of the tax-deferred strategy. Following different tax-deferred trading

strategies have only a “wealth effect” on overall portfolio decisions due to the different tax

subsidies generated. The overall risk exposure is not affected since investors can offset

the risk of any tax-deferred strategy by shorting the replicating portfolio in their tax-

able accounts. Finally, the optimal tax-deferred portfolio is chosen only to maximize the

corresponding effective tax subsidy generated, and hence, is independent of the overall

portfolio allocation decision.

One direct implication of the separability result is that there is no interaction between

the location and allocation decisions, and the earlier proposed intuition about how one’s

view on stock returns affect the location choice is in general not correct. Intuitively,

one might think that when investors become more optimistic about stock returns they

might weight their portfolio more towards stocks, and in addition, place more stocks in

their tax-deferred accounts to minimize the anticipated capital gains taxes. However, our

results indicate that the only decision to be changed is the allocation decision of how

much wealth to invest in stocks relative to bonds. The location decision of whether to

place stocks or bonds in tax-deferred accounts remains the same, since the effective tax

subsidy depends only on the replication cost of the assets placed in tax-deferred accounts

and is not affected by the expected return of those assets.

More generally, our second result states that the location decision is determined purely

by tax arbitrage. Investors maximize the effective tax subsidy received by placing only

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the assets that generate the highest effective tax subsidy in tax-deferred accounts. The

only relevant factors for the location decision are tax rates on assets, investment horizon,

and asset characteristics like dividend yield and asset volatility. For example, if dividends

are taxed more heavily than capital gains, assets with higher dividend yields and/or

lower volatility are preferred in tax-deferred accounts since they generate higher effective

tax subsidy if placed in tax-deferred accounts. While other factors, like expected asset

returns, subjective views of market conditions, individual endowments and preferences, are

important for overall allocation decisions, they do not influence the effective tax subsidy

received or the corresponding asset location decision.

The separability between location and allocation decisions and the simple structure of

the location decision lead naturally to our third result: the optimal portfolio allocation

decision can be derived by mapping the two-account-problem to a taxable-account-only

problem. Specifically, having followed the optimal location decision of placing only the

assets that generate the highest effective tax subsidy in tax-deferred accounts, investors

can replace the existence of tax-deferred accounts with a wealth increase equal to the

tax subsidy. The optimal taxable portfolio with two accounts can be constructed by

first solving the optimal portfolio for the taxable-account-only problem with the adjusted

wealth level, and then subtracting the taxable replicating portfolio for assets held in

tax-deferred accounts.

It is worth pointing out that borrowing and short-selling in taxable accounts are nec-

essary to ensure the separability between location and allocation decisions. For example,

assume bonds (relative to stocks) provide higher effective tax subsidy if placed in tax-

deferred accounts.6 The optimal location decision described above calls for holding only

6The effective tax subsidy may not be well-defined without borrowing or short-selling in taxableaccounts. We use the notion loosely here to illustrate the main problem under constraints.

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bonds in tax-deferred accounts, while borrowing in taxable accounts to lever up the stock

exposure if necessary. If borrowing is not allowed, one might think that it is sufficient to

adjust the location decision by following a simple “pecking order rule” that places assets

with higher effective tax subsidy (bonds) in tax-deferred accounts first, followed by lower

tax subsidy assets (stocks). However, the tax subsidy generated for placing a specific as-

set in tax-deferred accounts is determined by the asset returns in different realizations of

states and the corresponding future tax subsidies, which in general depend on the portfo-

lio allocation decision in the future.7 Hence, the ranking of assets by their corresponding

tax subsidy is affected by the distribution of future tax subsidies, and the current-period

location decision is affected by the portfolio allocation decision in the future.

Tepper and Affleck (1974), Black (1980), and Tepper (1981) are the first to address

the problem of wealth allocation when corporations have the choice between two accounts

characterized by different tax treatments. By introducing the concept of tax-arbitrage in

a static setting, they show that, a corporation should fully fund its pension account and

hold only bonds in it, while using tax-deductible borrowing to lever up direct holdings of

stocks in its investment account. Our approach can be viewed as a dynamic version of

the tax-arbitrage argument applied to the individual investor setting.

The dynamic replication argument employed in the paper, on the other hand, owes

much to the elegant approach of Constantinides (1983, 1984) in the context of optimal

capital gains realization with only a taxable account. By combining the two literatures,

we are able to derive the optimal taxable and tax-deferred portfolio strategy in a dynamic

setting. Our main contribution is to identify the separability between location and allo-

cation decisions by exploring the implications of dynamic replication, and to quantify the

7Allowing borrowing and short-selling in taxable accounts effectively ensures that the mixture of assetholding in tax-deferred accounts is independent of overall portfolio holding in all periods.

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impact of tax-deferred investing on overall portfolio decisions.

There is also an extensive literature addressing the optimal life-cycle portfolio prob-

lem of an investor with access to only taxable accounts. Starting with the pioneer-

ing work of Merton (1969, 1971), significant efforts have been devoted to relaxing dif-

ferent dimensions of the original setup, for example, considering the impact of pre-

dictability of asset returns (e.g., Kandel and Stanbaugh (1996)); introducing labor in-

come and liquidity needs over the life-cycle (e.g., Bodie, Merton and Samuelson (1992),

Viceira (2001)); imposing borrowing, short-selling constraints and other forms of incom-

pleteness (e.g., He and Pearson (1991)); and accounting for transaction costs (e.g., Con-

stantinides (1986), Davis and Norman (1990), Dumas and Luciano (1994)) and taxes (e.g.,

Constantinides (1983, 1984), Dybvig-Koo (1996), DeMiguel-Uppal (2002), and Dammon,

Spatt and Zhang (2001, 2002b)).

Our results are complementary to this literature. Instead of addressing the impact of

different factors on the optimal portfolio allocation when there is only a taxable account,

we take as a given the optimal strategy identified in the above literature, and provide a

mapping between the optimal strategy for the taxable-account-only problem and that for

the two-account problem. For application purposes, investors can use the mapping as an

add-on to their existing taxable-account strategies. The optimality of our two-account

solution is only relative to the given single-account strategy, regardless of whether the

single-account strategy is optimal or not. Moreover, the resulting mapping is remarkably

simple: we do not need any individual specific information except for their tax bracket,

investment horizon, and the fact that they are optimizing.

There is also a small literature on the optimal portfolio decision with both taxable and

tax-deferred accounts. Dammon, Spatt and Zhang (2002a) incorporate realistic features

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(such as borrowing and short-selling constraints, labor income, and mortality risks) into

a life-cycle portfolio model, and derive numerically the optimal location and allocation

decisions. They confirm our conjecture that under constraints, instead of placing only

the highest taxed assets (i.e., bonds) in tax-deferred accounts, investors follow a “pecking

order rule” by first placing higher taxed assets (i.e., bonds) in tax-deferred accounts,

followed by lower tax taxed assets (i.e., stocks). Shoven and Sialm (2002) focus on the

impact of municipal bonds on the location decision by solving a single period model with

stocks, bond and municipal bonds. They show that, when stock mutual funds distribute a

high proportion of their total returns as capital gains, investors should hold mutual funds

in the tax-deferred account and municipal bonds in the taxable. On the other hand,

when they distribute a low proportion of total returns, investors should hold bonds in

the tax-deferred account and the mutual fund in the taxable account. Their result is also

consistent with the prediction of our model: municipal bonds and taxable bonds can be

viewed as substitutes for each other and the effective tax rate on the taxable bonds (which

determines the size of tax subsidies and hence the location decision) is the minimum of the

implied tax rate on municipal bonds8 and the tax rate on the taxable bond. The location

decision depends on the relationship between the effective tax rate on bonds and that

on the mutual fund which increases with the proportion of return distributed as capital

gains.

The paper is structured as follows. Section 1 introduces the main intuition of the paper

through a simple numerical example. Section 2 describes the general model setup and the

specifics of the tax environment. Section 3 presents the structure of the optimal solution,

and Section 4 describes the properties of the optimal solution. Section 5 concludes.

8The implied tax rate on the municipal bond is defined as the tax rate that makes the taxable investorindifferent between taxable and municipal bonds.

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Appendix A contains all the proofs, and Appendix B presents the tables and figures.

1. A Numerical Example

Before presenting the general model, we solve a simple numerical example to illustrate

the main intuition of the results. First assume that there are only taxable accounts

in the economy and two assets are traded: a risk-free bond with an expected return of

rf−1 = 5%, taxed at the regular income rate, τB = 40%; and a risky stock with an expected

return9 of E[rs−1] = 50%, taxed at the capital gains rate, τs = 20%. For simplicity, we

further assume that both the gains and losses on the stock are taxed upon accrual each

period, and investors receive full tax credits on losses immediately. An investor with 50

year investment horizon currently has $100 in his taxable accounts. His current-period

optimal portfolio is evenly split between the two assets, i.e., $50 each in stocks and bonds.

Now suppose the investor is allowed to open a new tax-deferred account (Roth IRA) with

his existing taxable wealth and the maximum allowed contribution is $1 (i.e., the investor

can invest $1 in his tax-deferred account and he will have $99 left in his taxable account).10

By the end of 50 years, the investor withdraws money from both accounts and consumes

the total wealth. The two problems facing the investor are (i) how to invest that dollar

in the tax-deferred account, and (ii) how to adjust his taxable portfolio for the additional

opportunity to invest in the tax-deferred account.

9The 50% return can either be the rational expected return on a high beta asset, or the irrationalbelief of an individual investor.

10Since the investment to the Roth account comes from his current taxable wealth, which is alreadyafter-tax, it is fair to assume there is no more tax for the wealth invested in the tax-deferred account overthe 50 year period.

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Tax-deferred Portfolio Strategy

Since the expected tax payment is (50%)×(20%) = 10% for the stock and (5%)×(40%) =

2% for the bond, conventional wisdom suggests investing that tax-deferred dollar in stocks

to minimize expected tax payments. However, the following replication argument shows

that the optimal strategy is to always place only bonds in the tax-deferred account.

Over the last period, for each dollar invested in asset i (i = stock, or bond) in the

tax-deferred account, the investor receives an after-tax return ri (= rf , or rs). If a dollar

is invested in the taxable account, the investor receives after-tax Ri = 1+(1− τi)(ri− 1).

Therefore, we can rewrite the return in the tax-deferred account as

ri =1

1− τi

Ri − τi

1− τi

1

Rf

Rf ,

which is identical to the after-tax return of the following taxable portfolio: 11−τi

in asset

i and − τi

1−τi

1Rf

in bonds. We therefore call the above taxable portfolio the replicating

portfolio, with a replication cost of 11−τi

− τi

1−τi

1Rf

= 1 + τi

1−τi

Rf−1

Rf. Moreover, τi

1−τi

Rf−1

Rf

can be viewed as the “effective tax subsidy” received for investing in asset i over the last

period. Given the after-tax risk free rate Rf = 1 + (1 − .4)(1.05 − 1) = 1.03, this tax

subsidy depends only on the tax rate τi of asset i placed in the tax-deferred account. It

does not depend on the expected return of asset i at all. With a tax rate of τB = 40%,

each tax-deferred dollar invested in bonds yields a tax subsidy of $(

.4

.6

) (0.031.03

)= $0.0194 if

placed in tax-deferred accounts. Similarly, at τs = 20%, each tax-deferred dollar invested

in stocks yields a tax subsidy of only $(

.2

.8

) (0.031.03

)= $0.0073. Clearly, the investor should

hold only bonds in the tax-deferred account over the last period, and they effectively

receive a tax subsidy of $.0194 for each dollar held in the tax-deferred account.11

11So why is minimizing the expected tax payment not the optimal strategy here? Consider the following

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Using backward induction, we can extend the replication argument to all periods and

show that investors optimally hold only bonds in tax-deferred accounts. They effectively

receive a tax subsidy of $(1.0194)n − 1 for each dollar held in the tax-deferred account n

periods before the terminal date.

Taxable Portfolio Strategy

Having derived the optimal tax-deferred strategy of placing only bonds in tax-deferred

accounts, we can now determine the optimal taxable portfolio. With 50 periods to go,

the effective tax subsidy is (1.0194)50 − 1 = $1.62. Therefore, the investor can equate

the opportunity to invest $1 in the Roth account to receiving a tax subsidy of $1.62 and

reduce his optimization problem to that with only a taxable account and an effective

taxable wealth of 100+1.62 = 101.62. By assumption, we know that the optimal portfolio

for the investor is to evenly split between the stock and the bond if he has only a taxable

account. Hence, the investor optimally chooses a risk exposure of $50.81 to each asset.

Moreover, he optimally holds only bonds in tax-deferred accounts. Therefore, the only

exposure to stocks comes from the taxable portfolio holding. The investor optimally

holds $50.81 of his taxable wealth in stocks, and the remaining 99−50.81 = $48.19 of

taxable wealth is invested in bonds. The $1 in tax-deferred account is invested in bonds.

Although it appears that the investor holds only 48.19+1 = $49.19 in bonds, his true risk

two strategies: strategy one invests the one dollar in stocks in the tax-deferred account; and strategytwo invests that tax-deferred dollar in bonds, while matching the risk of the first strategy by selling$1.25 in bonds (short-sell if necessary) to purchase stocks in the taxable account.12 With an extra $1.25in taxable stocks relative to bonds, the second strategy actually has higher expected tax payments of1.25× [(.5)(.2)−(.05)(.4)] = 10%. However, the government is not better off with this higher expected taxpayment since it should be considered a co-investor in the taxable investments. In the second strategy,where the investor holds more stocks in the taxable account and more bonds in the tax-deferred, thegovernment has a riskier co-investment in the taxable account in the form of future tax revenue, and the10% extra tax payment is simply the compensation for the higher risk. Equivalently, investors are notworse off simply for paying higher expected tax payments.

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exposure to bonds is 48.19+2.62 = $50.81, since each tax-deferred dollar placed in bonds

is equivalent to 2.62 dollars of bonds, according to the replication argument above.13

Equilibrium Price Implication

The above example also indicates that, when the government introduces tax-deferred ac-

counts, it is not necessarily true that the price of the more heavily taxed assets should

increase. Specifically, although bonds are more heavily taxed, the nominal demand for

bonds actually decreases from $50 to $49.19, while the nominal demand for stocks in-

creases from $50 to $50.81, when investors are given the opportunity to contribute one

dollar to tax-deferred accounts. This is because each tax-deferred dollar invested in an

asset in general represents higher effective risk exposure to that asset.

2. The Model

In this section, we set up a discrete-time finite-horizon partial equilibrium model to de-

termine the optimal portfolio decisions when investors hold assets in both taxable and

tax-deferred accounts. Investors are risk averse, take prices as given, and solve for the

optimal portfolio by maximizing the utility from intertemporal consumptions, as well as

terminal bequeaths. We describe below the asset return processes, tax environments,

trading constraints, and finally the individual optimization problem.

2.1. Assets and Investors

There exist one risk-free bond (indexed i = 1) and N−1 risky assets (indexed i = 2, · · · , N)

traded in the economy. Let Pi,t and Di,t be the asset price and the dividend payment

13The taxable replication portfolio for one dollar of tax-deferred bonds with 50 periods to the terminaldate is 1

1−τB(1.0194)49 in bonds and − τB

1−τB

1Rf

(1.0194)49 in bonds, or a total of (1.0194)50 = $2.62 ofbonds.

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for asset i at time t, then di,t ≡ Di,t

Pi,t−1denotes the dividend yield, gi,t ≡ Pi,t

Pi,t−1− 1 is the

capital gains rate, and ri,t = 1 + di,t + gi,t is the total return for asset i. The risk-free

bond has constant interest rate over time, paid out in full each period, or, d1,t = rf−1,

and g1,t = 0 at any time t. The total return ri,t for risky asset i is binomial and i.i.d. over

time. Without much loss of generality, we further assume that di and gi evolve according

to the same underlying binomial process:

di =

{dH

i

dLi

, 1 + gi =

{ui with prob. pi

u−1

i with prob. 1− pi

. (1)

The following vector notations are used later to set up the optimization problem:

Pt = [P1,t, · · · , PN,t]′ denotes the vector of asset prices, Dt = [D1,t, · · · , DN,t]

′ is the

dividend payment, and rt = [r1,t, r2,t, · · · , rN,t]′ is the total return process at time t.

2.2. Tax Environments

The U.S. tax codes are very complicated and ever-changing. As much as we would like to,

taking all relevant tax regulations into account would be a futile exercise. We make several

simplifying assumptions to capture the main features of the tax codes while keeping the

model tractable. Following are descriptions of the simplified tax environments in both

taxable and tax-deferred (retirement) accounts.

Taxable Accounts

In their taxable accounts, investors purchase financial assets using their after-tax labor

income or other sources of income. The following two assumptions specify the tax treat-

ments for investment returns over time.

Assumption 1 In taxable accounts, (i) dividends and interest payments are paid out and

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taxed annually at a rate τd,14 and (ii) both long-term and short-term capital gains/losses

are taxed upon realization at the same rate τg≤τd.

This assumption ignores some details of the tax code for tractability: for example, short

term capital gains are taxed at τd while long term gains at τg; investors are only allowed

to realize tax credits on capital losses through offsetting their realized capital gains, div-

idends, or up to $3000 of ordinary incomes, with any extra losses carried over to future

years; and there is a “Wash Sale” rule specifying that investors cannot realize capital

losses on an asset if they repurchase the same asset within 30 days before or after the sale

(i.e., a 60-day window).

Assumption 2 Investors have fixed investment horizons, and on the terminal date T ,

they are always allowed to realize tax losses. There are two possible tax treatments regard-

ing unrealized capital gains in the taxable accounts: (i) escape capital gains taxes through

“step-up of tax basis”, or (ii) be forced to realize all capital gains.

The step-up of tax basis corresponds to a special provision in the current tax code that

allows investors to escape capital gain tax upon their death,15 which makes stocks more

attractive in taxable accounts for older investors. The forced realization of capital gains

approximates the tax treatment for investors who sell assets to meet their liquidity needs.

The finite horizon assumption enables us to study explicitly the horizon effect. More

importantly, the solution to the optimization problem is not well defined when the invest-

ment horizon approaches infinity since tax-deferred accounts are infinitely more valuable

14Different tax treatments for dividends and interest payments as suggested in the recent Bush taxplan can be easily incorporated into the model.

15Specifically, they can pass the taxable assets with embedded capital gains to their estates and theirbeneficiaries can avoid paying capital gains taxes by keeping the assets and recording the current marketprices as their tax basis for the assets.

13

than taxable accounts.16 Although it is more realistic to consider uncertain investment

horizons induced by changing life expectancy or unforeseeable liquidity events, the hori-

zon risk presents an additional source of risk and complicates the problem significantly.17

For simplicity, we assume fixed horizon.

Tax-deferred Accounts

There are several types of tax-deferred accounts in practice. The most popular type is

401(k)/403(b) or Keogh plans, where both ordinary income taxes on initial contributions

and the taxes on interests and capital gains are deferred and taxed as ordinary income

only upon withdrawal. Another popular type is the Roth IRA, in which investors pay

ordinary income taxes on initial contributions but there are no more taxes on future

withdrawal. It turns out that as long as the future tax rates are deterministic, the two

types of tax-deferred accounts are equivalent within a constant factor.

The following example illustrates the point. Let τ0, τT be the ordinary income tax

rates on the initial contribution date and terminal withdrawal date, respectively, and let

rT be the cumulative before-tax return for any portfolio over the T periods. Then the

cumulative after-tax return on $1 investment in 401(k) plans is $1× rT × (1−τT ), whereas

16For example, let Rf = rf − τd(rf − 1) be the after-tax return on risk-free bonds. If investors holdonly risk-free bonds in both accounts for T periods, one dollar grows to rf

T in tax-deferred accounts and

and RfT in taxable accounts. Hence, one dollar in tax-deferred is equivalent to

(rf

Rf

)T

dollars in taxableaccounts.

17Specifically, the market with only the underlying asset and the risk-free bond is no longer complete(see, for example, Constantinides (1983).) We could explicitly assume the existence of insurance contracton the event to complete the market. But this approach complicates the solution without offering muchnew insights. Instead of explicitly modelling the impact of uncertain horizons, we can heuristically addressthis issue using fixed-horizon solutions with different terminal tax treatments. For example, if an investorhas a 5% chance of encountering a large liquidity needs in 10 years upon which he is forced to liquidateall assets, and a 95% chance of living for another 50 years and bequeath his remaining assets to hisbeneficiaries, the optimal portfolio can be approximated by a weighted average of the solutions to thefollowing two problems: (i) a 10-year horizon problem with forced realization of capital gains, and (ii) a50-year horizon problem with step-up of tax basis on the terminal date.

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the return in Roth accounts is $1× (1− τ0)× rT . Clearly, each dollar invested in 401(k)

plans is equivalent to $1−τT

1−τ0in Roth accounts, independent of the realized portfolio return

rT , as long as investors follow the same strategy in both accounts.18 Given the equivalence,

it suffices to consider only the Roth type of tax-deferred accounts for the model.19

Assumption 3 In tax-deferred accounts, investors (i) never pay taxes on either dividend

payments or capital gains/losses, (ii) can contribute a maximum of $M t to their tax-

deferred accounts at time t, and (iii) are allowed to withdraw from tax-deferred accounts

with penalty ηt at time t and are forced to withdraw all tax-deferred wealth on terminal

date T .

At time t, a negative $M t implies that investors are forced to withdraw certain amounts

from their tax-deferred accounts, and investors receive (1−ηt) dollars in taxable accounts

for each dollar withdrawn. In practice, there are different rules about contribution limits

(M t) and withdrawal penalty (ηt) for different types of tax-deferred accounts. For exam-

ple, the current contribution limit is $3, 000 for Roth IRA and $11, 000 for 401(k) plans.

For 401(k) plans, there is a 10% penalty if investors withdraw from tax-deferred accounts

before age 5912

and investors are required to take annual minimum distributions from

tax-deferred accounts that are proportional to their life expectancy after age 70. On the

18Intuitively, the income taxes are government’s claim on investors’ asset positions. In the case of Rothaccounts, the government takes its share before an investor starts investing and decides its own portfolio,while in the case of 401(k) plans, the government leaves its share in the investor’s account to followexactly the same investment strategy as the investor and takes its share out only when the investor startswithdrawing from his accounts. Clearly, whether the government invests on its own or leaves it with theinvestor should have no impact on the return of the investor’s portfolio or his investment decisions.

19Investors need to attach a factor of (1− τT ) to their 401(k) wealth to derive the effective Roth IRAwealth before they can apply the results derived in this paper if they have in practice a 401(k) accountrather than a Roth IRA. Furthermore, contributing $1 to a Roth account is accomplished by simplymoving a cash position of $1 from the taxable to the Roth account, however, to contribute $1 to a 401(k)account, investors increase cash by $1 in the 401(k), but only decrease the cash position in the taxableby 1/(1− τ0) since they do not pay income tax on their contribution.

15

other hand, Roth IRA has no required distribution after age 70 and no penalty for some

qualified early withdrawal. Assumption 3 is flexible enough to capture these features.

2.3. Trading Constraints

Investors are allowed to borrow and short-sell in their taxable accounts as described in

the following assumption.

Assumption 4 In taxable accounts, investors are allowed (i) to borrow at the risk-free

rate with full tax deduction (at rate τd) for interest payments, and (ii) to short-sell risky

assets by paying cash dividends when declared with full tax deduction, and are required to

post 100% of the market value (“mark-to-market”) of the asset as cash collateral and earn

a “rebate interest rate” (rft−1), which is also taxed at rate τd. The rebate rate rft <rf is

specified in Appendix A to rule out arbitrage.

Full tax deduction on interest payments is not too strong an assumption considering the

ability of investors deduct mortgage interest payments. The restrictions on short-selling

is rather complicated and typically institution specific. In general, investors are required

to “mark-to-market” in the sense that the required amount of cash collateral moves with

market prices. Institutional investors need to post 102% of the market value of the

borrowed share upon transaction date as cash collateral,20 and earn a “rebate interest

rate” on the collateral determined by market conditions, like the volatility of the asset,

the difficulty to locate the shares to be shorted, and so on. On the other hand, retail

investors may be required to post as much as 150% of the market value as cash collateral

and they typically do not receive interest payments on their cash collateral since their

shares are held in street name. In summary, the rebate interest rate is lower than the

20Duffie, Garleanu, and Pedersen (2002) provide a nice survey of the institutional details.

16

risk free rate (rft <rf) and is exogenously determined based on investor characteristics and

market conditions. In this paper, for simplicity, we assume rft is defined endogenously to

rule out arbitrage opportunities.21

Assumption 4 is the main assumption of the paper that effectively completes the

market. Although not very realistic, the assumption drastically simplifies the problem

and enables us to derive analytic solutions regarding optimal portfolio decisions. Using

the simple structure of the optimal portfolio as a benchmark, Garlappi and Huang(2002)

derive approximate portfolio rules when investors are not allowed to borrow or short-sell.

Assumption 5 In tax-deferred accounts, investors are not allowed to borrow or short-sell

any asset.

Aside from being consistent with the current regulations on tax-deferred accounts, As-

sumption 5 is also necessary to ensure a properly functioning market, as illustrated in

Section 3.1.3. Finally, for simplicity, we assume there are no further transaction costs in

forms of bid-ask spreads, brokage fees, and so on.

Assumption 6 There is no transaction cost in either taxable or tax-deferred accounts.

21The following example illustrates the necessity to have rft <rf on the short sell proceeds in order torule out arbitrage opportunities (See Constantinides (1983) for a detailed discussion.) If rft =rf , Investorsare guaranteed positive return on zero initial investment if they form a portfolio by shorting one share ofan asset and simultaneously purchasing a share of the same asset using the proceeds. If the asset pricegoes up, investors can close the short position to get tax credits on the losses, re-short one share at thecurrent price (new short proceeds exactly cover the cost to close the old short position), and finally investthe amount (the tax credits on losses) at the risk-free rate. Similarly, if the price goes down, they closethe long position to realize tax credits on losses, re-purchase the share (using the proceeds from closingthe long position), and invest the rest (tax credit on losses) in the risk-free bond. In the end, the longand short positions offset each other and the total bond position is always higher than the capital gainstax due, because the tax credits on losses earn interest over time and the corresponding tax liability oncapital gains are postponed without interest.

17

2.4. Dynamic Optimization Problem

We start describing the optimization problem by identifying the state and choice variables.

Since investors are allowed to postpone realizing capital gains on all shares, the purchase

price of each share in any asset i matters for the overall optimization. Let hi,s,t−1 be

the number of shares of asset i purchased at time s that investors hold in their taxable

accounts before they make their trading decisions at time t, and let yi,s,t ≡ Pi,s/Pi,t be the

corresponding tax cost-basis for the share, where Pi,s is the price of the share at time s

when it was original purchased, and Pi,t is the current market price.22 Then the matrices

Ht−1 = [H1,t−1, · · · , HT ,t−1], where Hs,t−1 = [h1,s,t−1, · · · , hN,s,t−1]′, and Y t = [Y1,t, · · · , YT ,t],

where Ys,t ≡ [y1,s,t, · · · , yN,s,t]′, denote the share holding of all assets before-trading at

time t and the corresponding tax cost-basis,23 and clearly are state variables. Moreover,

since investors are restricted in moving money between the taxable and tax-deferred

accounts, the before-trading wealth in their tax-deferred accounts (after incorporating

the current period prices), W Dt , is also a state variable. The detailed portfolio holding in

tax-deferred accounts, however, is not a state variable, since investors can trade to any

position in tax-deferred accounts given their current position without occurring any cost

by Assumption 6. Finally, the current period stock price Pt, is also a state variable, since

it affects the total wealth in taxable accounts. In summary, the state variables at time t

are {Ht−1,Y t, WDt ,Pt}.

Given the current set of state variables, investors maximize overall utility by optimizing

over the following choice variables at time s = t, · · · , T , (i) the dollar consumption, Cs,

(ii) the dollar contribution to (or withdrawal from) tax-deferred accounts, Ms, (iii) the

22yi,s,t > 1 indicates there are capital losses on the position and < 1 corresponds to capital gains.23We define hi,s,t =yi,s,t =0 for s>t.

18

portfolio holding in taxable accounts after trading,24 Hs, and (iv) the portfolio holding

in tax-deferred accounts, Θs = [θ1,s, · · · , θN,s]′, where θi,s is the percentage of tax-deferred

wealth invested in asset i at time s.

Let Vt(Ht−1, Y t,WDt ,Pt) be the indirect utility function for investors at time t given

the state variables, then the investors’ optimization problem can be stated as follows:

Vt(Ht−1, Y t,WD

t ,Pt) = max{Cs,Ms,Hs,Θs}T

t

{Et

[T−1∑

s=t

[δs−tu(cs)

]+ δT−tΓ(CT )

]}(2)

subject to the following constraints for all s = t, t + 1, . . . , T ,

Cs = diag((Hs−1 −Hs) [(1− τg) + τgY s]

′)′ Ps

+ (1− τd)1′THs−1

′Ds − (1− ηs1{Ms<0})Ms (3)

W D

s+1 = (W D

s + Ms)Θ′srs+1 (4)

−W D

s ≤ Ms ≤ M s, ∅N ≤ Θs ≤ 1N (5)

MT = W D

T , HT = 0 (6)

Equation (2) defines the value function, where δ is the time discount, u(·) is the utility

over consumption, and Γ(·) is the utility over terminal bequeath, which may include

the terminal consumption as well. Equations (3) and (4) are the budget constraints in

taxable and tax-deferred accounts, respectively, where 1T = [1, · · · , 1]′ is a vector of ones

with T elements, diag(A) = [a11, · · · , att]′ is the vector of diagonal elements for matrix

A = [aij], 1{Ms<0} is an indicator function that equals one if and only if Ms < 0, and ηs

is the potential withdrawal penalty for moving money from the tax-deferred to taxable

accounts before the terminal date. Equation (5) specifies the limit (M t) to the amount

24Since the after-trading share holding at time s is identical to the before-trading share holding at times+1, to economize on the notations used, we use the vector Hs to denote both.

19

an investor can contribute to tax-deferred accounts, and that investors cannot withdraw

more than the wealth in tax-deferred accounts, or there are borrowing and short-selling

constraints in tax-deferred accounts, where ∅N = [0, · · · , 0]′ is a vector of zeros with N

elements. Finally, Equation (6) describes the terminal condition when investors liquidate

all wealth (taxable or tax-deferred) for consumption and bequeath motive.

3. Optimal Portfolio Decisions

Solving the individual optimization problem specified in the model section is a formidable

task, considering the high-dimensionality problem induced by the history dependence of

taxable positions.25 Fortunately, it is not necessary to solve simultaneously the taxable

and tax-deferred portfolios in order to characterize the impact of tax-deferred accounts on

overall portfolio decisions. In this section, we introduce a replication argument to study

the relationship between taxable and tax-deferred portfolio decisions, taking as a given

the optimal portfolio with only taxable accounts.

3.1. Replication Arguments

The replication argument states that, tax-deferred assets can be uniquely replicated by a

dynamic portfolio of taxable assets as long as investors are optimizing. As a result, the

additional investment opportunity set introduced by tax-deferred asset payoffs is spanned

by asset payoffs in taxable accounts. The cost of replication, in general, depends both

on the trading strategy followed in tax-deferred accounts and on the replicating strategy

in taxable accounts. We demonstrate the uniqueness of the replication strategy in three

25The optimization problem is hard to solve with only a taxable account. The few papers that dealwith this history dependence problem rely on numerical approach and can only solve for a small numberof time periods, for example, Dybvig-Koo (1996) and DeMiguel-Uppal (2002).

20

steps: First, the optimal trading strategy on any asset positions in taxable accounts is to

postpone gains and to realize losses immediately, independent of investor preference or

overall portfolio decision. Second, if investors follow the above trading strategy, then any

taxable position with embedded capital gains can be uniquely replicated using taxable

assets with no embedded capital gains. Finally, we show that tax-deferred positions can

be uniquely replicated using dynamic portfolios of taxable assets. The first two steps

are similar to the results in Constantinides (1983). We reproduce the results using our

notation for completeness, and to establish the framework for our main results regarding

replication of tax-deferred assets.

3.1.1. Optimal Trading Strategy in the Taxable Accounts

The following lemma shows that the optimal trading strategy on any given asset position in

taxable accounts is uniquely determined by the current price level and the initial purchase

price (the cost basis) of the assets. Using the notation specified in the model section, the

optimal trading strategy can be stated as follows.

Lemma 1 The optimal trading strategy on any existing position of asset i in taxable

accounts is to realize losses immediately and to postpone gains until the terminal date.

Formally, let hi,s,t−1 and yi,s,t be the holding of asset i (purchased at time s) and the

before-trading tax cost basis at time t < T , then the optimal holding of asset i after

trading is

hi,s,t =

{hi,s,t−1, if {yi,s,t < 1 and hi,s,t−1 > 0} or {yi,s,t > 1 and hi,s,t−1 < 0}0, otherwise

. (7)

Proof. See Appendix A.

21

The first case in equation (7) states that the optimal trading strategy is to keep the

current asset position if there are unrealized capital gains, that is, if the current market

price is higher than the purchase prices for long positions (described by conditions in the

first bracket), or if the current price is lower than the purchase price for short positions

(the second bracket). The second case states that if there are capital losses, the optimal

strategy is to liquidate the entire position immediately. The optimality of postponing gains

and realizing losses relies on the fact that investors can always introduce new holdings in

the asset (hi,t,t) to achieve any desired risk profile. For example, if there are losses on a

long position and the optimal portfolio requires retaining the long position, investors do

strictly better by realizing the losses immediately to get tax credits and then re-purchasing

the same asset, because they can earn interests on the tax credits received. Similarly, if

there are capital gains on the long position and investors need to reduce total risk exposure

to the asset, instead of directly liquidating their long position, investors can improve their

utility by retaining the share while simultaneously shorting the same asset. Again, the

benefit is the interest payments saved on the capital gains.

Lemma 1 shows that the current tax basis yi,s,t of any asset position is the only

determinant for the optimal trading strategy in taxable accounts. The fact that the

optimal trading strategy is independent of individual preference or their overall portfolio

decisions ensures that the following replication argument is well-defined.

3.1.2. Replicating Taxable Assets with Embedded Capital Gains

We now show that any taxable position with an embedded capital gain is equivalent to

a cash position in taxable accounts, as long as investors follow the above defined optimal

trading strategies in their taxable accounts. To simplify notation, we remove the subscript

22

{i, s, t} for the tax basis whenever there is no risk of confusion.

Lemma 2 Assume investors follow the optimal trading strategy described in Lemma 1 for

any asset position in their taxable accounts, let y be the tax cost basis for a long position

in asset i at time t, then there exists a unique relative value zi,t(y) such that investors are

indifferent between having one dollar face value in a long position with tax basis y and

having zi,t(y) dollars of cash in taxable accounts. Similarly there exists a unique relative

value si,t(y) for any short position in asset i with cost basis y.


Following is the outline of the proof. Let di, gi be the before-tax dividend and capital

gains rates defined in equation (1), then a long position with face value $1 and tax basis y

evolves into a portfolio of (1− τd)di after-tax cash dividends and a long position in asset

i with $(1+ gi) face value and tax cost basis y/(1+ gi). If investors follow the optimal

trading strategy from time t + 1 on, they are indifferent between holding a long position

in asset i with basis y at time t and receiving the following cash payments at time t + 1,

Ri,t(y) ≡ (1− τd)di + (1+gi)zi,t+1 (y/(1+gi)) . (8)

Let RHi,t(y) and RL

i,t(y) be the realizations of the effective one-period return Ri,t(y) when

the return on asset i is high and low respectively, then we can define a “risk neutral”

measure using a position in asset i with no embedded capital gains and the risk-free bond

as the base assets,

qi,t ≡RH

i,t(1)−Rf

RHi,t(1)−RL

i,t(1). (9)

23

Any other position in asset i can be viewed as a derivative asset whose value can be

calculated by discounting its future cash flows using the above risk neutral measure:

zi,t(y) =1

Rf

[(1− qi,t)R

H

i,t(y) + qi,tRL

i,t(y)]. (10)

It is important to note that, although the effective one-period return and the correspond-

ing replication costs of any taxable positions are defined recursively, they are only func-

tions of the original purchase prices and future price evolutions of asset. Neither investor

preference nor their overall portfolio strategies matter for the calculation of replication

costs.

3.1.3. Replicating Tax-deferred Assets

We now extend the replication argument to assets in tax-deferred accounts. Investors

are defined as following a future trading strategy, Θt:T = [Θt, · · · , ΘT ], on a tax-deferred

dollar if they follow portfolio strategy Θt ≡ [θ1,t, · · · , θN,t]′ on the dollar at time t (i.e.,

place θi,t of that dollar into asset i at time t for i = 1, · · · , n.), and strategy Θs at any

time s > t for all wealth evolved from that original dollar.26 Following is a definition of

“state-independent” future trading strategies.

Definition 1 A tax-deferred future trading strategy Θt:T = [Θt, · · · , ΘT ] is defined as

state-independent if Θs is well-specified at time t and is independent of realizations of

state variables at time s, for any s = t, · · · , T .

The following lemma illustrates the relationship between taxable and tax-deferred

wealth if investors follow pre-specified trading strategies in tax-deferred accounts.

26This is a slight abuse of notations, since in the model setup we used Θt to denote the entire tax-deferred portfolio.

24

Lemma 3 If investors follow the optimal trading strategy described in Lemma 1 for assets

in their taxable accounts, there exists a relative value zDt:T such that investors are indifferent

between having one dollar in their tax-deferred accounts following a state-independent

trading strategy Θt:T and having zDt:T dollars of cash in their taxable accounts.


We sketch below the main steps of the proof which rely on backward induction. Start-

ing with a special case, we consider a state-independent future trading strategy that

holds only asset i in tax-deferred accounts at time t, i.e., Θti:T = [eiN ,Θt+1:T ], where

eiN = [0, · · · , 1, · · · , 0]′ is an index vector of size N with the i-th element equals one and

zero otherwise, and Θt+1:T is any state-independent strategy from time t+1 on. Depend-

ing on the realization of states, one dollar in tax-deferred accounts following strategy Θti:T

evolves into rHi or rL

i dollars in tax-deferred accounts at time t + 1, when each dollar in

tax-deferred accounts is equivalent to zDt+1:T in taxable accounts by induction assumption.

Therefore, having one dollar in tax-deferred accounts at time t following strategy Θti:T is

equivalent to receiving the following cash payments at time t + 1 in taxable accounts:

RDH

i,t ≡ rH

i zD

t+1:T , RDL

i,t ≡ rL

i zD

t+1:T . (11)

Using taxable asset i and the risk-free bond as the base assets, we can calculate the

replication cost zDti:T

of that tax-deferred dollar by discounting its effective future cash

flows RDHi,t and RDL

i,t under the “risk-neutral” measure qi,t (defined in equation (9)):

zD

ti:T=

1

Rf

[(1− qi,t)R

DH

i,t + qi,tRDL

i,t

]. (12)

25

The extension to a general state-independent tax-deferred strategy, Θt:T = [Θt,Θt+1:T ],

is straightforward. Assume Θt = [θ1,t, · · · , θN,t]′, then the replication cost can be expressed

as

zD

t:T = Θ′t[z

D

t0:T , · · · , zD

tN :T ]′.

As long as Θt is state-independent, the replicating cost zDt:T is identical across states.

Moreover, the cost is fully determined by the trading strategy followed in tax-deferred

accounts and is independent of investor preference or their overall portfolio composition

as long as investors are optimizing in their taxable accounts.27

3.1.4. Effective Tax Subsidy

To isolate the impact of tax-deferred accounts, we introduce the concept of effective

tax subsidy, which measures the wealth transfer that would make investors indifferent

between whether or not to have the privilege to defer taxes in their retirement accounts.

For simplicity, we ignore future contributions to tax-deferred accounts until Section 3.5.

Following is the definition of effective tax subsidy.

Definition 2 Assume investors are not allowed to contribute to tax-deferred accounts in

the future, or M = 0 in Equation (5). Let Vt(Ht−1, Y t,WDt |Θt:T ,Pt) and Vt(Ht−1, Y t, 0,Pt)

be the value functions of investors following a state-independent tax-deferred trading strat-

egy Θt:T , or having no access to tax-deferred accounts, respectively. The effective tax

27An implication of Lemma 3 is that the assumption of no borrowing or short-selling in tax-deferredaccounts is necessary to ensure a properly functioning market. It is clearly possible to have differentreplication costs for different tax-deferred trading strategies, to fix ideas, assume the strategy ΘI (or ΘJ)of holding only asset i (or j) in tax-deferred accounts has a replication cost of $3 (or $2), respectively. Ifinvestors are allowed to borrow and short-sell in tax-deferred accounts, they can always short asset j tolong asset i in tax-deferred accounts, while at the same time long the replicating portfolio for strategyΘJ and short the replicating portfolio for strategy ΘI in the taxable accounts. The overall risk exposureis zero by the definition of replication portfolio, and investors pocket the $1 difference in replication costs,generating an arbitrage opportunity.

26

subsidy is defined as the Φt:T that solves the following equation

Vt(Ht−1,Y t,WD

t |Θt:T , Pt) = Vt(Ht−1 + Φt:TW D

t e1,tNT ,Y t, 0, Pt) (13)

where e1,tNT is an index matrix of size N ×T with the (1,t)-th element equals one and zero

otherwise.

The quantity Φt:T in the above definition measures the value of a dollar invested in tax-

deferred accounts. Although in general, the value Φt:T incorporates information regarding

overall portfolio optimization in the future and can only be solved dynamically, Lemma 3

indicates that it can be determined separately from the overall optimization through

tax-arbitrage if investors restrict themselves to state-independent tax-deferred strategies.

Specifically,

Corollary 1 Let zDt:T be the relative value of a dollar in tax-deferred accounts following

future trading strategy Θt:T derived in Lemma 3, then the effective tax subsidy for investing

in tax-deferred accounts in Definition 2 can be expressed as Φt:T = zDt:T−1.

Investors are indifferent between having the opportunity to invest a dollar in tax-deferred

accounts following a state-independent strategy Θt:T or having no access to tax-deferred

accounts while receiving a lump-sum subsidy of the size zDt:T − 1 in their taxable accounts.

The size of the effective tax subsidy received depends only on the future trading strategy

followed in tax-deferred accounts. The independence of the tax subsidy on individual

preference or overall portfolio decisions ensures the separability between the taxable and

tax-deferred portfolio decisions, to which we now turn our attention.

27

3.2. Separation between Taxable and Tax-deferred Portfolios

Since our dynamic replication argument applies only to state-independent tax-deferred

trading strategies, we need to show both that the optimal tax-deferred strategy is state-

independent and that the replication of tax-deferred strategies ensures the separability

between taxable and tax-deferred portfolio decisions.

Proposition 1 (Separability) Let H∗t be the optimal taxable portfolio at time t and

Θ∗t:T be the optimal trading strategy followed in tax-deferred accounts from time t to T ,

then Θ∗t:T (i) is state-independent, (ii) is chosen only to maximize the corresponding

effective tax subsidy Φ∗t:T , and (iii) is independent of the optimal taxable portfolio H∗

t .


All three results are proved simultaneously using backward induction. Assume the op-

timal tax-deferred strategy (Θ∗t+1:T ) is state-independent from time t+1 on. For any (not

necessarily optimal) tax-deferred strategy Θt in the current period, the overall tax-deferred

strategy [Θt,Θ∗t+1:T ] is state-independent and hence is replicable by taxable assets. There-

fore, the portfolio strategies in tax-deferred accounts do not affect the overall risk exposure

of investors, since they can always undo any position taken in tax-deferred accounts by

including an offsetting replicating portfolios in their taxable accounts. Following different

current-period tax-deferred strategies (Θt) has only a “wealth effect” on overall portfolio

due to the different effective tax subsidy generated. Therefore, it is clear that the optimal

current-period tax-deferred strategy (Θ∗t ) is chosen only to maximize the corresponding

effective tax subsidy. Moreover, without any trading friction in tax-deferred accounts, any

strategy that yields the highest subsidy is also feasible in any realization of states. Hence,

the optimal tax-deferred strategy has to be identical across all states, confirming the in-

28

duction assumption.28 Finally, the tax-deferred portfolio is fully specified by the motive

to maximize tax subsidy and is clearly independent of the taxable portfolio strategy.

The proposition implies that investors can solve their optimization problem sequen-

tially by first determining the future tax-deferred strategy (Θ∗t:T ) to maximize the effective

tax subsidy received, and then choosing the optimal taxable portfolio (H∗t ) to achieve op-

timal risk exposure.

The significance of this separability result can be illustrated in the following example.

Assume there are only two assets in the economy, a stock and a bond. There are two

investors who differ drastically in their desired exposure to each asset due either to their

different risk aversion, endowment, or to their different private information regarding the

payoff of assets. In particular, assume investor 1 prefers to hold 100% of his portfolio

in stocks and investor 2 prefers to hold 100% in bonds. If, however, investor 1 and

2 agree that the strategy of holding only bonds in tax-deferred accounts for all future

periods maximizes the effective tax subsidy received, they will hold the same tax-deferred

portfolio, namely, 100% bonds in their tax-deferred accounts. The individual preference

is reflected only in the corresponding taxable portfolio, e.g., investor 1 holds a huge short

position of bonds to offset the undesired exposure to bonds in his tax-deferred account,

while investor 2 holds only bonds to reflect his strong preference for it.

Two questions remain from the above example: First, will investors 1 and 2, given their

different preferences, agree on the tax-deferred strategy to follow in order to maximize

the effective tax subsidy received? More generally, what determines the optimal tax-

deferred strategy? Second, how should investors adjust their taxable portfolios in order

28If there are multiple tax-deferred strategies yielding the highest tax subsidy, the optimal tax-deferredstrategy is still state-independent, since investors are indifferent among any linear combinations of thesestrategies.

29

to account for their desired portfolio holding in tax-deferred accounts? We address these

two questions in the next two sections.

3.3. The Optimal Location Decision

Proposition 1 shows that the optimal tax-deferred strategy is chosen to maximize the

effective tax subsidy. In general, the strategy is rather complicated, since the effective

tax subsidy is an endogenous variable that depends on investor preference and their

trading strategies in future periods. We show that under dynamic replication, however,

the optimal tax-deferred strategy is preference-free and can be determined myopically.

Moreover, the properties of the optimal tax-deferred strategy allow us to decompose the

optimal portfolio decisions into “location” and “allocation” decisions, where the location

decision refers to the preference for holding assets in taxable vs. tax-deferred accounts,

and the allocation decision describes the desired risk exposure to each assets.

We first introduce a measure of the current-period tax subsidy received for investing

in asset i from time t to t + 1, assuming that investors follow the optimal tax-deferred

strategy from time t + 1 on.

Definition 3 At any time t, the current-period effective tax subsidy is defined as

Φit ≡ 1

Rf

[(1− qi,t)rH

i + qi,trL

i ]− 1. (14)

for any asset i, where qi,t is the risk-neutral probability defined in equation (9).

The current-period effective tax subsidy turns out to be a sufficient statistic for the optimal

tax-deferred portfolio decision at time t.

Proposition 2 (Location) The optimal tax-deferred trading strategy Θ∗t:T = [Θ∗

t , · · · , Θ∗T ]

can be determined myopically, that is, the tax-deferred portfolio Θ∗t at any time t can be

30

expressed as

Θ∗t =

∑

i∈I

ωieiN , where

∑

i∈I

ωi = 1, I ≡ {i | Φit = Φ∗t ≡ max

{j=1,···,N}Φjt}, (15)

where Φit and Φjt are the current-period tax subsidies in Definition 3.


Proposition 2 states that investors hold only those assets that generate the highest

current-period effective tax subsidies in tax-deferred accounts. If there are several assets

sharing the highest subsidy Φ∗t , the optimal tax-deferred portfolio is any linear combina-

tion of these assets. We can recursively specify the size of the effective tax subsidy and

the corresponding taxable replication portfolio for the optimal tax-deferred strategy.

Corollary 2 If investors follow the optimal tax-deferred strategy Θ∗t:T = [Θ∗

t , · · · , Θ∗T ]

derived in Proposition 2, they receive at time t an effective tax subsidy Φ∗t:T for each

tax-deferred dollar, where

Φ∗t:T =

T∏

s=t

(1 + Φ∗s)− 1, (16)

and each tax-deferred dollar can be replicated with the following taxable portfolio

∆∗t:T =

∑

i∈I

ωiB∗ite

1N +

∑

i∈I

ωi∆∗ite

iN (17)

where ∆∗it ≡

(rHi − rL

i )(1 + Φ∗t+1:T )

RHi,t(1)−RL

i,t(1), B∗

it = 1 + Φ∗t:T −∆∗

it, (18)

and RHi,t(1), RL

i,t(1) are defined in Equation (8).

Investors determine their tax-deferred portfolio each period by first ranking all assets

by their current-period effective tax subsidy, and then holding only those assets that

provide the highest tax subsidy in tax-deferred accounts. To keep a balanced portfolio,

31

they generally do not hold (or even need to short-sell) those assets with the highest tax

subsidy in taxable accounts. Hence, those assets can be viewed as having a preferred

location in tax-deferred accounts, while all other assets are held only in taxable accounts

and hence have a preferred location in taxable accounts. In this sense, the tax-deferred

portfolio decision fully specifies the asset location decision, and Proposition 1 also implies

the separability between location and allocation decisions.

3.4. The Optimal Allocation Decision

To address the question of how investors should adjust their overall portfolios for their

asset holdings in tax-deferred accounts, we solve for the optimal taxable portfolio, taking

as a given the previously derived optimal tax-deferred strategy. Without loss of generality,

we further simplify the optimization problem by assuming that investors do not derive

utility from intertemporal consumption. We return to contribution and consumption

decisions in Section 3.5.

The following proposition derives the optimal taxable portfolio by transforming the

optimization problem with both taxable and tax-deferred accounts to that with only

taxable accounts.

Proposition 3 (Allocation) Assume M = 0 and u(·) = 0 in the optimization problem

specified by Equations (2)-(6). Let Θ∗t:T , Φ∗

t:T , and ∆∗t:T be the optimal tax-deferred trading

strategy, the effective tax subsidy, and the corresponding taxable replication portfolio in

Proposition 2 and Corollary 2, respectively. Let Ht(Ht−1,Y t, Pt) ≡ Ht(Ht−1, Y t, 0,Pt)

be the optimal portfolio when investors have no access to tax-deferred accounts, then the

32

optimal taxable portfolio for the two-account problem is

Ht(Ht−1,Y t, WD

t ,Pt) = Ht(Ht−1, Y t,Pt)−W D

t ∆∗t:Tet′

T (19)

where Ht−1 = Ht−1 + (1 + Φ∗t:T )W D

t e1,t−1NT is the before-trading taxable portfolio augmented

by the effective tax subsidy, and etT (or e1,t−1

NT ) is an index vector (matrix) that equals to

one for the t-th (or (1, t−1)-th) elements and zero otherwise.


Proposition 3 is a straightforward application of Proposition 1 and 2. Having one dollar

in tax-deferred accounts following strategy Θ∗t:T is equivalent to receiving an effective tax

subsidy Φ∗t:T , hence, the original two-account problem can be mapped to a taxable-account-

only problem by adjusting for the “wealth effect” through the term Ht(Ht−1,Y t,Pt).

Moreover, a dollar following the optimal future trading strategy in tax-deferred accounts

is equivalent to a risk exposure ∆∗t:T in taxable accounts. Therefore, investors need to

subtract the “hedging term”, W Dt ∆∗

t:Tet′T , in order to derive their optimal taxable portfolio.

It is possible that some elements of Ht is less than zero, indicating that investors borrow

or short-sell in their taxable accounts to achieve optimal portfolio allocation.

3.5. Consumption and Contribution Decisions

Without consumption and contribution decisions, we have shown that the only impact

of tax-deferred accounts on the overall portfolio decision is the wealth effect due to the

effective tax subsidy received. With the ability to borrow in taxable accounts, it is clear

that the optimal contribution decision is to always maximize contribution in order to max-

imize the total tax subsidy received. Moreover, the only impact of tax-deferred accounts

on consumption decision is a wealth effect, similar to the impact on optimal portfolio

33

decisions. Formally,

Corollary 3 Consider the optimization problem specified by Equations (2)-(6), let Φ∗t:T

be the effective tax subsidy received for following optimal tax-deferred strategy in Proposi-

tion 2, and Ct(Ht−1,Y t,Pt) ≡ Ct(Ht−1,Y t, 0,Pt)|M=0 be the optimal consumption decision

when investors have no access to tax-deferred accounts, then (i) investors always contribute

the maximum allowed to their tax-deferred accounts, or Mt = M t at any time t, (ii) the

present value of the effective tax subsidy received for all future contributions is

ΦM

t:T =T∑

s=t

1

Rfs−t M sΦ

∗s:T , (20)

and (iii) the optimal consumption decision for the two-account problem is

Ct(Ht−1, Y t,WD

t ,Pt) = Ct(HM

t−1, Y t,Pt) (21)

where HM

t−1 = Ht−1 + [(1 + Φ∗t:T )W D

t + ΦMt:T ] e1,t−1

NT is the before-trading taxable portfolio

augmented by the effective tax subsidies received both for current tax-deferred wealth and

for future contributions to tax-deferred accounts.


The corollary states that the existence of tax-deferred accounts and the opportunity

to make future contributions to tax-deferred accounts are equivalent to increasing the

endowment of safe assets in taxable accounts. Moreover, the optimal tax-deferred portfolio

is not affected at all by the contribution or consumption decision, and the optimal taxable

portfolio is similar to that derived in Proposition 3 except that investors need to account

for future contributions with an additional wealth adjustment ΦMt:T , or effectively replacing

Ht−1 with HM

t−1 in Equation (19).

34

4. Properties of the Optimal Portfolio

The optimal portfolio decisions are centered around the notion of effective tax subsidies.

Specifically, investors identify in each period the assets that provide the highest current-

period effective tax subsidy and hold only those assets in tax-deferred accounts. After

adjusting their overall wealth level by the total tax subsidy received, investors reduce the

two-account problem to that with only a taxable account. For application purposes, in

this section we derive the properties of effective tax subsidies based on characteristics of

both underlying assets and individual investors.

We start with a special case when investors are not allowed to postpone capital gains

in their taxable accounts and derive the closed-form expression for effective tax subsidies.

Later, we rely on numerical results to identify other determinants of effective tax subsidies

when investors have the option to postpone capital gains. In both cases, the current-period

subsidy (Φit) is fully specified by Equations (8), (9), and (14), and the total subsidy (Φ∗t:T )

is specified by Equation (16), as long as investors follow the optimal trading strategy to

postpone gains and to realize losses immediately for assets in their taxable accounts.29

Without Timing Option to Postpone Capital Gains

When investors are not allowed to postpone capital gains, the current-period effective tax

subsidy depends only on an “effective tax rate”, τi,t, which is a weighted average of the

tax rates on dividends and capital gains.

Result 1 Assume assets are taxed according to Assumption 1, except that capital gains/losses

are taxed upon accrual at a rate τg, then the current-period effective tax subsidy of asset

29If, however, investors do not behave optimally, then the effective tax subsidy depends on the overallportfolio strategy and is generally lower. This is most likely when investors face borrowing and short-selling constraints that they need to liquidate positions with capital gains early.

35

i can be expressed as

Φit =1

Rf

[1 +

Rf − 1

1− τi,t

]−1 (22)

τi,t = ω(δRN

i )τd + [1− ω(δRN

i )]τg, δRN

i ≡ (1− qi)dH

i + qidL

i , (23)

where dHi , dL

i are dividend payments, qi and ω(δRNi ) are defined in Appendix A.

We can think of δRNi as the “effective dividend yield”, which measures the risk-adjusted

dividend payments, and ω(δRNi ) as the “effective dividend ratio”, or the percentage of total

return paid out as dividends. Since dividends are taxed more heavily than capital gains,

assets with higher dividends are taxed more heavily and they generate higher effective

tax subsidy if placed in tax-deferred accounts. The optimal location decision is simply to

place the assets with the highest effective dividend yield in tax-deferred accounts.

It is important, however, to distinguish the effective dividend yield and the expected

dividend yield. Consider a simple case when dividend payments are linear functions of

asset returns,30 i.e., di = β0(rf − 1) + βr(ri − 1), we can show that the effective dividend

ratio ω(δRNi ) can be reduced to

ω(δRN

i ) =β0 + βr − εβr

1− ε(1− β0), where ε =

τd − τg

1− τg

,

which roughly equals to β0 + βr. For example, assume the tax rates on dividends and

capital gains are 40% and 20% respectively. Consider a risk-free bond with a 5% return

30The example of linear dividend yields is general enough to cover most asset classes. For example, thecase βr → 0 approximates dividend paying stocks, which usually keep relatively stable dividends over time(or slowly adjusting the dividends level) irrespective of their stock performance, while the case β0 → 0approximates actively managed stock mutual funds that realize most of their capital gains quickly andpass them to investors as dividends, with βr representing the percentage of total returns paid out eachyear. Of course, there are some details left out by this approximation, mostly noticeably the asymmetryin dividend payments for mutual funds in the high and low states of asset return, since the funds cannotpass along capital losses to investors. Moreover, the case of short-term investors who realize short-termcapital gains which are taxed at the same rate as dividends falls in this category as well, even though wedo not explicitly consider the difference between short- and long-term capital gains in the model.

36

that pays out all returns as dividends, and a risky asset with a 20% expected return that

pays out half of the total return as dividends and half as capital gains. Although the bond

has a lower expected dividend yield, it has a higher effective dividend ratio, and hence is

preferred in tax-deferred accounts.31

The main contribution of Result 1 is to show that, when investors are not allowed

to postpone capital gains, the current-period effective tax subsidy depends only on the

effective dividend yields (δRNi ) and the tax brackets of investors (τd, τg). Neither expected

returns of assets, nor individual characteristics (like risk preferences, private information

regarding asset returns, investment horizons, or their overall portfolio strategies) affect

the tax subsidy received, or the optimal location decision of which asset to prefer in

tax-deferred accounts.

With Timing Option to Postpone Capital Gains

When investors are allowed to postpone capital gains, most of the previous intuition still

carries through. The only complication is that the definition of effective tax subsidy

is more complicated since the risk-neutral probability qi,t in Equation (9) can only be

determined recursively and it now depends on individual characteristics like their remain-

ing investment horizon and terminal tax treatments. Instead of solving the effective tax

subsidies in closed-form, we rely on numerical approximation to illustrate its properties.

Table 1 summarizes the base set of parameters used in the numerical analysis. The

risk-free interest rate is rf = 6%, and the risky asset follows a binomial process with mean

µ = 10% and volatility σ = 20%. The dividend process on the risky asset is assumed

31The expected dividend yield is 5% for the bond, and (.5)(20%) = 10% for the stock. On the otherhand, the bond has β0 = 1, βr = 0, and an effective dividend ratio of ω = 1, while the stock has β0 = 0,βr = .5 and an effective dividend ratio of only ω= .5.

37

to be proportional to the total return on the asset, d ≡ δr, with δ = 3%. The tax

rate is τg = 20% for capital gains and τd = 40% for interests and dividend payments.

The maximum investment horizon is 80 years and there are two different terminal tax

treatments regarding unrealized capital gains in taxable accounts: (i) “step-up of tax

basis”, when investors are allowed to escape capital gains taxes, or (ii) “forced realization

of capital gains”, when investors are forced to pay capital gains taxes on the terminal

date.

Figure 1 reports the current-period effective tax subsidy as a function of the remaining

investment horizon (T − t) and terminal tax treatments. Panel A and B represent the

cases when asset dividend yields are δ = 0% and 3%, respectively. The solid lines in

both panels represent the case when investors escape terminal capital gains taxes through

step-up of tax basis, and the dashed lines represent the case when investors are forced to

realize capital gains.

The solid line in Panel A indicates that, when investors escape terminal capital gains

taxes, the current-period effective tax subsidy is always negative, increases with the in-

vestment horizon, and approaches zero. The current-period effective tax subsidy measures

both the benefit (of escaping capital gains taxes) and the cost (of foregoing tax credits on

capital losses) of investing in tax-deferred accounts relative to in taxable accounts. With

the step-up provision, when the asset pays no dividend, investors can escape all capital

gains taxes in taxable accounts (by always postponing gains until the terminal date and

then escape taxes through step-up of basis), and they are allowed to realize tax credits on

capital losses. Hence, investing in taxable accounts dominates investing in tax-deferred

accounts, and the corresponding effective tax subsidy is always negative.

It is a bit surprising that the current-period effective tax subsidy increases with horizon

38

and approaches zero, since the benefit of realizing losses in taxable accounts should be

higher for longer investment horizon and certainly does not approach zero. However, the

current-period effective tax subsidy Φit measures the marginal benefit of investing in tax-

deferred accounts for an extra period from time t to t+1, taken as a given the benefit of

investing in tax-deferred accounts from time t+1 on. When the horizon is long enough, an

extra investment period has little impact on the total capital losses realized since investors

need to cancel out previous capital gains before they can realize a loss.

The dashed line in Panel A indicates that, when investors are forced to realize capital

gains on the terminal date, investing in tax-deferred accounts dominates investing in

taxable accounts. On the terminal date, with all capital gains/losses realized and taxed

at τg =20%, the effective tax subsidy can be calculated using Result 1. When the horizon

is long enough, the effective tax subsidy also converges to zero, the same as that when

investors escape terminal taxes. This is because the terminal tax treatments are not

important for the current-period effective tax rates when the remaining investment horizon

is long enough.

The lines in Panel B are similar in shape to those in Panel A, indicating that dividends

always increase the level of current-period tax subsidy and the impact is not sensitive to

investment horizons. Intuitively, we can decompose the total return into the dividend and

capital gains components. The effective tax subsidy generated by the dividend component

can be approximated using Result 1 and is constant over time, and the effective tax subsidy

for the capital gains component is similar to the subsidy in Panel A which changes with

remaining horizon. The total subsidy can be viewed as a weighted average of the two

components.

We see from Figure 1 that the horizon effect is monotone for each given terminal tax

39

treatment, and that the current-period effective tax subsidy does not change much after

the investment horizon reaches 30 years. Therefore, we present the impacts of dividend

yields and asset volatilities only for the cases of short- and long-horizon (5 and 30 years,

respectively).

Figure 2 reports the current-period effective tax subsidy as a function of dividend

yields when the asset volatility is set at σ=20%. The tax subsidy increases significantly

with dividend yields, indicating that assets with higher dividend yields in general provides

higher effective tax subsidy and should be preferred in tax-deferred accounts. Moreover,

as long as the investment horizon is long or investors are forced to realize capital gains

taxes on the terminal date, the effective tax subsidy is almost always positive, indicating

that it is generally beneficial to be able to invest in tax-deferred accounts. If, however,

investors can escape capital gains taxes in taxable accounts and the investment horizon is

short (less than 5 years), the effective tax subsidy can be negative for low enough dividend

yields (e.g. ≤ 1.5% in the figure), indicating that there are significant benefits to realize

losses in taxable accounts.

Figure 3 reports the current-period effective tax subsidy as a function of asset volatil-

ities when the dividend yield is set at δ = 3%. The effective tax subsidy decreases with

asset volatilities, since the option value of postponing gains and realizing losses in taxable

accounts increases, reducing the benefit of investing in tax-deferred accounts. However,

the impact of asset volatility is usually small. The only exception is for short-horizon

investors who are allowed to escape terminal capital gains taxes, for example, it is no

longer beneficial to place an asset with 45% volatility in tax-deferred accounts if investors

have only a 5-year horizon.

To summarize, the current-period effective tax subsidy is determined mainly by the

40

return distributions of assets and the risk-free rate. The only relevant investor charac-

teristics are their investment horizon and tax status, including both the tax rates on

capital gains and dividends, and the terminal tax treatments. Other things equal, in-

vestors prefer assets with higher effective dividend yields and/or lower asset volatility in

tax-deferred accounts. For long-term investors, terminal tax treatment is not important,

and the dividend yield is the main determinant for asset location decision. On the other

hand, terminal tax treatment is very important for short term investors. If they are forced

to realize terminal capital gains taxes,32 tax-deferred investing is generally beneficial, and

the dividend yield is still the most important factor for location decisions. If however,

investors are allowed to escape terminal capital gains taxes,33 investing in tax-deferred

accounts may not be beneficial, and placing lower volatility assets in tax-deferred accounts

may be the first priority in their location decisions.

Cost of Suboptimal Location Decisions

We calculate the cost of some suboptimal location decisions by comparing the effective tax

subsidy of following those strategy with that of the optimal strategy. Figure 4 plots the

total effective tax subsidy Φt:T received as a function of the remaining investment horizon

when investors follow different tax-deferred trading strategies. Panel A and B refer to the

two different terminal tax-treatments. Clearly, for long-horizon investors, the terminal

tax treatments do not affect the total tax subsidy received. The dotted lines represent

the case when investors place only bonds in tax-deferred accounts. The dash-dotted lines

correspond to holding only assets with δ = 6% dividend yield and σ = 20% volatility

for the remaining horizons. The dashed and solid lines correspond to holding δ = 3%

32For example, investors with future liquidity needs or a short-term trading strategy.33For example, older investors with step-up of tax-basis upon death.

41

and δ = 0 respectively. The effective tax subsidies change dramatically with both the

investment horizon and the dividend yields on assets. For example, if investors keep a

dollar in tax-deferred accounts for 80 years, they receive an effective tax subsidy of about

$5.1 if they invest in the risk-free bond, while the subsidy drops to about $1.5 if they

invest in a stock index fund with 3% dividend yields and 20% volatility. If the risky asset

does not pay dividends, the benefit of holding that asset in tax-deferred accounts is very

small, if not negative.

The effective tax subsidy is also important in understanding the true risk expo-

sure of tax-deferred holdings. Consider an asset i with δ = 3% dividend yield and

σ = 20% volatility, the replication portfolio in Equation (18) consists of roughly ∆∗t:T =

$(

11−0.3

)(1+Φ∗

t+1:T ) of asset i.34 If an investor plans to always hold asset i in tax-deferred

accounts for the next 80 years, then Φ∗t+1:T ≈ 1.5 and each dollar of asset i held in tax-

deferred accounts for the current period is equivalent to ∆∗t:T ≈ 1+1.5

1−.3= $3.6 of taxable

asset i. On the other hand, if the investor plans to hold asset i in tax-deferred accounts

only for the current period and to switch to the optimal strategy of holding only risk-free

bonds for all future periods, then Φ∗t+1:T ≈ 5.2, and the effective taxable holding of asset

i increases to 1+5.11−.3

= 8.9 dollars for each dollar of asset i held in tax-deferred accounts.

Saving Decisions

An interesting public policy issue concerning retirement accounts is whether and to what

extent they stimulate savings. Corollary 3 sheds some light on this issue.

It is reasonable to assume that investors have utility function such that they optimally

increase their consumption with a higher wealth level (that is, Ct increases with the first

34Applying Result 1, the asset has an effective dividend ratio ω = .5 and an effective tax rate about30%.

42

element of HM

t−1). Since the privilege to invest in tax-deferred accounts is equivalent to

receive an ex-ante tax subsidy, investors feel wealthier and optimally increase the level of

their total consumption after the introduction of tax-deferred accounts. Even though in-

vestors are effectively wealthier due to the tax subsidy received, the nominal wealth level is

not changed after introducing tax-deferred accounts since investors fund their tax-deferred

accounts with taxable wealth. Investors only save what they do not consume. The overall

amount of saving always decrease after introducing tax-deferred accounts, contradicting

the popular belief that tax-deferred accounts provide better investing opportunities and

hence encourage savings from investors.

The following numerical example illustrates the point. Assume an investor has $100

total wealth and consumes $2 for the current period. His saving is $98. Now assume he is

allowed to place $30 in a Roth-type tax-deferred account, and he will have $70 left in his

taxable accounts. He feels wealthier since he effectively receives a tax subsidy (say, $50)

for being able to invest in tax-deferred accounts. Although his effective wealth is $150,

nominally, he still has only $100 between the two accounts. The investor will increase

his consumption (say to $3, proportional to his effective wealth level). Therefore, his

observed saving, which is calculated on a nominal basis, reduces to $97.

What is missing in our model is the trade off between current consumption and contri-

bution to tax-deferred accounts. With the ability to borrow in taxable accounts, investors

always fully fund their tax-deferred accounts to receive the maximum tax subsidy. The

only impact of tax-deferred accounts on saving decisions is the above-described “wealth

effect”. However, borrowing is limited in general, and tax-deferred accounts have an

additional “substitution effect” that makes saving more attractive, since investors trade

off current consumption with contributions to tax-deferred accounts which generate ef-

43

fective tax subsidy and provide higher future consumption. This substitution effect is

most pronounced for less wealthy investors who do not have enough wealth to max out

their contributions. It is for this class of investors that the marginal saving increases

the effective tax subsidy received and the existence of tax-deferred accounts may serve to

encourage overall saving. The optimal saving decision can only be solved by balancing the

wealth and substitution effects. In general, for wealthier investors who always contribute

up to the maximum allowed to tax-deferred accounts, the wealth effect dominates and

total savings always decrease in the presence of tax-deferred accounts. On the other hand,

the substitution effect could dominate for poorer investors and they may increase their

total saving after the introduction of tax-deferred accounts. Sorting out these two effects

requires solving the general model with borrowing and short-selling constraints, which we

defer to future research.

Equilibrium Asset Prices

Our model also provides a good framework to think about the impact of tax-deferred

accounts on equilibrium asset prices. Consider an economy with only a stock and a

bond where the bond is taxed more heavily than the stock and there are no tax-deferred

investment opportunities. How would the equilibrium prices change if the government

introduces tax-deferred accounts where neither the stock nor the bond is taxed? Investors

are assumed to fund tax-deferred accounts with their existing taxable wealth and there is

a preset limit on the amount they are allowed to invest in tax-deferred accounts.

If investors are allowed to borrow and short-sell in taxable accounts, Proposition 2

and 3 suggest that investors hold only bonds in tax-deferred accounts, and the pres-

ence of tax-deferred investing opportunities has only a “wealth effect” on asset demands.

44

Specifically, investors feel wealthier due to effective tax subsidy received. As long as the

optimal dollar amount of stock holding increases with individual wealth level, the total

dollar demand for the stock increases, and the dollar demand for the bond has to de-

crease, since the nominal wealth in the economy is not changed after the introduction of

tax-deferred accounts.35 As a result, the bond price decreases relative to the stock price

after the introduction of tax-deferred accounts.

The result seems counter-intuitive. The introduction of tax-deferred accounts allows

investors to hold the heavily-taxed bonds in tax-deferred accounts, making the bond more

attractive. Therefore, the bond price should increase relative to the stock price. However,

this argument fails to recognize that holding one dollar of bond in tax-deferred accounts is

equivalent to holding 1+Φ∗t:T dollars of bonds in taxable accounts. Although the effective

holding (accounting for the subsidy effect) of bonds is higher, the nominal dollar demand

for bonds turns out to always decrease after the introduction of tax-deferred accounts.

Of course, the result is partly driven by the possibility to borrow in taxable accounts.

When there are limits to borrow or short-sell in taxable accounts, and the previous in-

tuition of bonds becoming more attractive does enter into equilibrium asset prices. For

example, in an extreme case when there are no upper limits on the size of tax-deferred

accounts and no borrowing is allowed in taxable accounts, investors would simply move

all their assets to tax-deferred accounts.36 Or equivalently, we shift into a new economy

with no taxes on either the stock or the bond. The overall demand for the bond has to be

higher in this new economy relative to the old, and the bond price has to increase relative

to the stock price. We term this change in opportunity set the substitution effect.

In general, the impact of introducing tax-deferred accounts on equilibrium asset prices

35We ignore the potential increase in personal savings after introducing tax-deferred accounts.36Assume capital gains are taxed upon accrual for the stock, for example.

45

depends on the maximum contribution allowed to tax-deferred accounts. For small con-

tribution limits, the wealth effect dominates, and the bond price decreases, whereas for

large enough contribution limits, the substitution effect may dominates and the bond price

increases. To properly account for the price impact of tax-deferred accounts, we need to

first solve for optimal portfolio decisions under borrowing and short-selling constraints.

We defer the detailed analysis to future research.37

5. Conclusion

When investors are allowed to borrow and short-sell in their taxable accounts, we show

that the optimal taxable and tax-deferred portfolio decisions are separable. The rights to

invest in tax-deferred accounts is equivalent to receiving a fixed effective tax subsidy from

the government, which depends only on the characteristics of assets placed in tax-deferred

accounts, and not on individual preference or their overall portfolio holding. The optimal

tax-deferred portfolio can be determined myopically by placing only those assets that yield

the highest current-period effective tax subsidy in tax-deferred accounts. Therefore, the

tax-deferred portfolio fully specifies investors’ preference for asset location. The overall

allocation is then determined by mapping the two-account problem to a taxable-account-

only problem with overall wealth level augmented by the tax subsidy.

37Garlappi-Huang (2002) provides an approximate solution to optimization problem when borrowingand short-selling are not allowed.

46

A Appendix: Proofs

Proofs of Lemmas 1 and 2.

We prove simultaneously the following three induction assumptions using backward in-

duction on the remaining investment horizon T − t.

1. the optimal trading strategy is to postpone gains and realize losses immediately

(Lemma 1),

2. a long position with basis y is equivalent to zi,t(y) of cash, and a short position is

equivalent to si,t(y) of cash (Lemma 2),

3. it is possible to define a short rebate rate rft such that zi,t(1) = −si,t(1) = 1.

Note that part three of the induction assumption is necessary to rule out arbitrage. We

need to define rft recursively at each period.

On the terminal date T , Lemma 1 holds trivially since investors always realize losses

and they either postpone or realize gains depending on the terminal tax treatment. For

Lemma 2, depending on the terminal tax treatment, any long taxable position in asset i

is equivalent to the following cash positions

zi,T (y) =

1− τg(1− y), if forced to realize capital gains

1− 1{y>1}τg(1− y), if escape tax through step-up of basis(24)

and any short position is equivalent to

si,T (y) =

−1 + τg(1− y), if forced to realize capital gains

−1 + 1{y<1}τg(1− y), if escape tax through step-up of basis(25)

where 1{y>1} is an indicator function that equals to 1 when y > 1, and 0 otherwise, and

similarly for 1{y<1}. Part three of the induction assumption is clearly true for any short

rebate rate on the terminal date.

47

We now assume all three parts hold at time t + 1 and prove the results for time t.

First we define the effective one-period return from time t to t+1 for any long or short

taxable positions assuming investors retain the position until time t+1 and follow optimal

trading strategy (defined in Lemma 1) from time t + 1 on. Let di, gi be the before-tax

dividend and capital gains rates defined in equation (1), then a long position with face

value $1 and tax basis y evolves into a portfolio of (1− τd)di after-tax cash dividends and

a long position in asset i with $(1+gi) face value and tax cost basis y/(1+gi). By induction

assumption, the new long position is equivalent to a cash holding at time t+ 1, therefore,

holding the long position in asset i at time t is equivalent to receiving the following cash

payments at time t + 1

Ri,t(y) ≡ (1− τd)di + (1+gi)zi,t+1(y/(1+gi)),

We define the above Ri,t(y) as the effective one-period return for the long position from

time t to t + 1, which follows a binomial distribution with realizations

RH

i,t(y) ≡ (1− τd)dH

i + uizi,t+1(u−1

i y), RL

i,t(y) ≡ (1− τd)dL

i + u−1

i zi,t+1(uiy).

The effective one-period return on short positions can be derived similarly, except that we

need to account for the difference between the short rebate rate on the cash collateral and

the risk-free rate. According to Assumption 4, investors are required to post $1 as cash

collateral38 on short-sell proceeds and they earn a rebate interest rate rft−1 at time t. Let

Rft−1 = (1−τd)(rft−1) be the after-tax rebate interest rate, then (Rf−Rft) measures the

38Even though the relative value of a short position with face value $1 is in general 6= −1 due tounrealized capital gains/losses, only $1 is required as cash collateral.

48

cost of short selling and should be included in the effective one-period return, specifically,

RH

−i,t(y) ≡ −(1− τd)dH

i + uisi,t+1(u−1

i y)− (Rf − Rft)

RL

−i,t(y) ≡ −(1− τd)dL

i + u−1

i si,t+1(uiy)− (Rf − Rft).

The short rebate rate rft is defined to rule out tax arbitrage by setting the return on

the following zero-cost portfolio to zero: short $1 of asset i at market price and earn

rft−1 on the short proceeds, borrow $1 to buy asset i at market price, and follow optimal

trading strategy on both the long and short positions in all future periods. From induction

assumption, the long position in asset i is equivalent to RHi,t(1) (or RL

i,t(1)) of cash next

period, whereas the short position, along with the difference on rebate interest rate and

the risk free rate on the borrowed amount, is equivalent to a cash position of RH−i,t(1) (or

RL−i,t(1)). If RH

i,t(1) + RH−i,t(1) ≥ RL

i,t(1) + RL−i,t(1), then the return on the above portfolio

can be replicated by a portfolio of long position in asset i and bonds, or effectively, we

can define a “risk-neutral” probability measure qi,t

qi,t ≡RH

i,t(1)−Rf

RHi,t(1)−RL

i,t(1)

and the present value of the portfolio returns can be expressed as

(1− qi,t)(RH

i,t(1) + RH

−i,t(1)) + qi,t(RL

i,t(1) + RL

−i,t(1)).

If on the other hand, RHi,t(1) + RH

−i,t(1) ≤ RLi,t(1) + RL

−i,t(1), then the return on the above

portfolio can be replicated by a portfolio of short positions in asset i and bonds, and we

can define a “risk-neutral” probability measure

q−i,t ≡RH−i,t(1) + Rf

RH−i,t(1)−RL

−i,t(1),

49

The present value of the portfolio returns can be expressed as

(1− q−i,t)(RH

i,t(1) + RH

−i,t(1)) + q−i,t(RL

i,t(1) + RL

−i,t(1)).

It turns out qi,t = q−i,t under the rebated rate rft defined below. Therefore, it does not

matter which replication we use.

Since the net cost of the above portfolio is zero, to rule out arbitrage, the above present

value has to be zero as well. Effectively, under the “risk-neutral measure”, the present

value of shorting one share (including the cost from lower rebate rate) is the same as minus

the present value of of buying one share at current market price, or si,t(1) = −zi,t(1) = −1.

We can solve for the rebate rate from the above equation,

Rft ≡ Rf − [(1− qi,t)ui(zi,t+1(u−1

i ) + si,t+1(u−1

i )) + qi,tu−1

i (zi,t+1(ui) + si,t+1(ui))] .

since Rft depends only on zi,t+1 and si,t+1, it is well defined using induction assumptions.

Note that the after-tax rebate rate Rft ≤ Rf since zi,t+1(u−1

i )+si,t+1(u−1

i ) ≥ 0 and zi,t+1(ui)+

si,t+1(ui) ≥ 0. Intuitively, being able to postpone gains and realize losses immediately on

both the long and short positions gives investors an arbitrage opportunity if they can

make full use of the short sell proceeds. Equivalently, the before-tax short rebate rate

rft < rf is necessary to rule out that arbitrage. This proves the third part of the induction

assumptions.

To prove the first part, we need to consider four possible cases: long or short positions

with gains or losses, respectively. We prove only the two cases for long positions. The

short positions can be proved similarly. First, if there are losses on the long position, or

y > 1, we need to show that it is optimal to realize losses immediately. Assume the result

is not true and investors retain the position and eventually sell it at time s > t (s could be

the terminal date). Without loss of generality, we ignore dividend payments. Let Pt, Ps

50

be the current and future stock prices, then for each share of long position with current

tax basis y, investors would obtain an after-tax proceeds of (1− τg)Ps + τgPty if they are

required to pay capital gains taxes upon liquidation. On the other hand, if the investor

liquidate the position today, he obtains an after-tax proceeds of Pt + τgPt(y − 1) on the

share. He can purchase a new share using Pt of the proceeds and invest the remaining

proceeds τgPt(y − 1) in risk-free bonds. The value of this portfolio at time s will be

(1− τg)Ps + τgPt + τgPt(y − 1)Rfs−t = (1− τg)Ps + τgPty + τgPt(y − 1)(Rf

s−t − 1),

higher than the after-tax proceeds if the investor retains the long position, with the

difference τgPt(y − 1)(Rfs−t − 1) equals to the interest earned on the tax credits over the

period. Similarly, if the investor escapes terminal taxes on date s, then the total proceeds

is Ps if he retains the share, and Ps + τgPt(y − 1)Rfs−t if he liquidates today, purchases

a new share and invests the rest in bonds. The different is the total future value of tax

credits. No matter what happens to the share in the future, investors do better if they

realize the losses immediately.

To show that it is optimal to postpone gains on long positions, consider an investor

who liquidates a long position of an asset with cost basis y to follow certain strategy on

the proceeds that yields r over the next period. Then the total proceeds of the share

at time t + 1 is [Pt − τgPt(1 − y)]r. We show that the investor can do strictly better

if he retains the long position while short-sells one share of the same asset at market

price. The proof relies on the definition of the short rebate rate where the rebate interest

rate plus the option value to postpone gains in the future generates an effective return of

the risk-free rate on the short proceeds. Equivalently, we can simplify the calculation by

setting the short rebate rates to the risk-free rate while assuming investors liquidate both

the long and the short positions on the asset next period to forego the potential option

51

value. Specifically, the investor retains the original long position, short a new share with

proceeds Pt and places [Pt − τgPt(1− y)] of the proceeds in the same strategy that yields

r while investing the rest τgPt(1− y) in risk-free bonds. Then the total proceeds at time

t + 1 on the portfolio is

[(1− τg)Pt+1 + τgPty]− [(1− τg)Pt+1 + τgPt] + [Pt − τgPt(1− y)]r + τgPt(1− y)Rf

= [Pt − τgPt(1− y)]r + τgPt(1− y)(Rf − 1)

higher than that on the original strategy of liquidating immediately, with the difference

τgPt(1− y)(Rf −1) equals to the interest saved on the capital gains taxes. This concludes

the proof for the second part of the induction assumption.

Finally, we show that the relative values of the long position at time t is uniquely

defined. The result holds trivially for the case of y > 1, since we have just shown that

investors optimally liquidate the asset position to realize losses immediately, and the long

position is equivalent to zi,t(y) = 1 − τg(1 − y) of cash. If y ≤ 1, the optimal trading

strategy is to retain the long position, and we have just derived the corresponding effec-

tive one-period return RHi,t(y) and RL

i,t(y). The present value under risk neutral measure

qi,t provides the replication cost for the above return using taxable asset i and bonds

purchased in the spot market.

zi,t(y) =1

Rf

[(1− qi,t)R

H

i,t(y) + qi,tRL

i,t(y)]

(26)

The proof for short positions is identical since the definition of short rebate rates

ensures that, at current market price, a new short position is equivalent to minus a new

long position. If y ≤ 1, investors realize losses on the short positions immediately, and

the relative value is si,t(y) = −1 + τg(1− y). If y > 1, investors retain the short position

52

and the relative value si,t(y) can also be defined as the present value of future cash flows

under the “risk neutral” measure q−i,t,

si,t(y) =1

Rf

[(1− q−i,t)R

H

−i,t(y) + q−i,tRL

−i,t(y)].

where short positions in asset i with basis y = 1 and the risk free bonds are used as the

base assets.

Proof of Lemma 3.

The proof also uses backward induction on the remaining investment horizon T − t. The

induction assumptions are (i) the relative value zDt:T is uniquely defined for any state-

independent tax-deferred strategy Θt:T from time t on, such that having a dollar in tax-

deferred accounts following a state-independent trading strategy Θt:T is equivalent to

having zDt:T dollars of cash in taxable accounts, and (ii) zD

t+1:T is also state-independent.

On the terminal date T , by Assumption 3, investors are forced to liquidate all tax-

deferred assets and move the proceeds to taxable accounts for terminal consumption. As a

result, one dollar in tax-deferred accounts is equivalent to zDT :T = 1− ηT dollars in taxable

accounts, where ηT ≥ 0 is the penalty for withdrawal on the terminal date. Clearly, the

relative value zDT :T is identical across states. We now assume both results hold at time

t + 1 and prove the same for time t.

The induction step is given in Section 3.1.3. Clearly, as long as Θt is state-independent,

the replicating cost zDt:T is identical across states.

Proof of Proposition 1.

Let Xt ≡ [Ht−1,Y t,WDt ,Pt] be the realization of state variables at time t, Vt(Xt) be

the corresponding value function, and Ht∗(Xt), Θ∗

t (Xt) be the optimal taxable and tax-

53

deferred portfolio strategies at time t, respectively. The proof again relies on backward

induction on remaining investment horizon T − t.

There are three induction assumptions at time t,

1. Θ∗s is state-independent for any time s ≥ t, Hence, Θ∗

t:T ≡ [Θ∗t , · · · , Θ∗

T ] is uniquely

defined over all realizations of states at time t.

2. Θ∗t:T maximizes the corresponding replication cost z∗t:T defined in Lemma 3.

3. Θ∗t is independent of Ht

∗,

On the terminal date T , by Assumption 3, investors are forced to liquidate and with-

draw all tax-deferred assets for consumption. The tax-deferred trading strategy is clearly

state-independent, maximizes replication cost (among the only feasible solution), and is

independent of the taxable strategy.

Assume all three induction assumptions are true from time t+1 on. We need to show

the same for time t. Let Θ∗t+1:T be the optimal tax-deferred strategy from time t + 1 on

and z∗t+1:T be the corresponding replication cost. By induction assumption, Θ∗t+1:T is state-

independent. Hence, any optimal tax-deferred strategy Θ∗t:T (Xt) can always be written

as [Θt(Xt),Θ∗t+1:T ], since the optimality of Θ∗

t+1:T implies that investors can improve their

utility by switching to Θ∗t+1:T from time t + 1 on had they not done so already. The

optimal strategy Θt(Xt) at time t, however, may depend on realizations of state variables

Xt at time t.

Given that Θ∗t+1:T and its corresponding replication costs z∗t+1:T are state-independent,

we can follow the same steps as in the proof of Lemma 3 while replacing zDt+1:T with z∗t+1:T

in Equation (11) and the following derivations, and define z∗ti:T as the replication cost for

strategy [eiN ,Θ∗

t+1:T ] where investors hold only asset i in tax-deferred for current period

and following the optimal strategy Θ∗t+1:T for future periods. The replication cost z∗ti:T is

well-defined and also state-independent at time t.

54

Let Θt be any feasible current-period tax-deferred strategy, then a tax-deferred dollar

following any pre-specified strategy [Θt,Θ∗t+1:T ] can be replicated by a taxable portfolio

with costs Θ′t[z

∗to:T , · · · , z∗tn:T ]′ (following the derivation of Equation (??)). Define Θ∗

t to be

the strategy that maximizes this replication cost39, or

Θ∗t = argmaxΘt

Θ′t[z

∗to:T , · · · , z∗tn:T ]′.

We now prove, by contradiction, that the optimal tax-deferred trading strategy at

time t is Θ∗t across all states. Assume not, then there exists at least one state Xt such

that the optimal tax-deferred strategy is different from Θ∗t , or ΘXt ≡ Θt(Xt) 6= Θ∗

t .

Assume Ht(Xt) is the corresponding optimal taxable portfolio in that state. Let ∆∗t:T ,

∆D

t:T (Xt) be the replicating portfolios and z∗t:T , zDt:T (Xt) be the replication costs (similar

to those defined in Equations (17) and( ??)) for tax-deferred strategies [Θ∗t ,Θ

∗t+1:T ] and

[ΘXt ,Θ∗t+1:T ] respectively. We show that investors are better off if they switch to strategy

Θ∗t in tax-deferred accounts, while at the same time shorting the replicating portfolio

for strategy [Θ∗t ,Θ

∗t+1:T ], longing the replicating portfolio for strategy [ΘXt ,Θ

∗t+1:T ] and

finally holding the difference between the replication costs in bonds. Formally, instead

of holding Ht(Xt) in taxable and ΘXt in tax-deferred accounts, investors hold Θ∗t in tax-

deferred accounts while change their taxable holding to

Ht(Xt) = Ht(Xt)−W D

t

[∆∗

t:T −∆D

t:T (Xt)− (z∗t:T − zD

t:T (Xt))e1N

]et′

T ,

where etT = [0, · · · , 1, · · · , 0]′ is an index vector of size T that equals to one for the t-th ele-

ment and zero otherwise. By the definition of replicating portfolios, the total return from

following Θ∗t in tax-deferred accounts and Ht(Xt) −W D

t [∆∗t:T −∆D

t:T (Xt)] et′T in taxable

39Θ∗t may not be unique, in which case the optimal strategy is any linear combination of these Θ∗t , andinvestors are indifferent between these strategies. Any one of these strategy can be defined as the optimaland it will still be state-independent. Without loss of generality, we assume Θ∗t is unique.

55

accounts is equivalent to that of following the original strategy of ΘXt and Ht(Xt). The

definition of Θ∗t ensures that z∗t:T ≥ zD

t:T (Xt), and the new strategy of following Θ∗t in tax-

deferred accounts dominates the original strategy by recovering the original risk profile,

while increasing the taxable wealth by the difference in the replication costs. Hence, the

optimal tax-deferred strategy is state-independent and maximizes the replication costs,

or Θ∗t:T = [Θ∗

t ,Θ∗t+1:T ]. It is also clear that Θ∗

t:T is independent of Ht.

Proof of Proposition 2.

The proof follows directly from those of Lemma 3 and Proposition 1. After replacing

zDt+1:T in Equation (11) by z∗t+1:T , it is easy to see that the optimal tax-deferred strategy

can be determined myopically:

z∗ti:T =1

Rf

[(1− qi,t)rH

i + qi,trL

i ]× z∗t+1:T = zD

i,tz∗t+1:T . (27)

and

z∗t:T = Θ∗′i [z∗to:T , · · · , z∗tn:T ] = Θ∗′

i [zD

0,t, · · · , zD

N,t]z∗t+1:T (28)

Given the linear structure, z∗t:T is maximized by holding only assets that provide the

highest current-period effective tax subsidy zDi,t.

Proofs of Proposition 3 and Corollaries 1, 3.

First, we show that investors always contribute the maximum allowed to their tax-deferred

accounts, or Mt = M t at any time t. If not, they can increase their contribution by

ε, following the exogenously specified optimal tax-deferred strategy, while shorting the

replicating portfolio in taxable accounts. Clearly, they match the risk of the original

profile while increase their taxable wealth by ε(z∗t:T − 1) ≥ 0 as long as z∗t:T > 1 (or as long

56

as there exists an asset with positive tax rate).

Second, We map the two-account problem to that with only a taxable account. Let

Vt(Ht−1,Y t,WDt ,Pt) be the indirect utility function for investors at time t, defined in

Equation (2), we show that there exists an HM

t−1 such that

Vt(Ht−1,Y t, WD

t ,Pt) = Vt(HM

t−1,Y t, 0,Pt).

Since the optimal tax-deferred strategy is well-defined in Proposition 2, the replication

argument in Lemma 3 indicates that investors are indifferent between having W Dt in

tax-deferred accounts or having W Dt z∗t:T dollars of cash in taxable accounts, while each

dollar holding in tax-deferred accounts is equivalent to a holding of taxable replicating

portfolio ∆D

t:T . Therefore, the total current W Dt and future contribution are equivalent to

ZDt:TW D

t + ΦMt:T in taxable accounts, where ΦM

t:T =∑

T

s=t1

Rfs−t M sZ

Ds:T is the present value of

all future contributions.

The optimal consumption can simply be determined using the adjusted wealth level,

since investors can always borrow to finance their consumption.

The overall risk exposure can be determined using the taxable-account-only solution

with the adjusted wealth level. Since the tax-deferred holding is equivalent to a replicating

portfolio in taxable, the optimal taxable portfolio is then equal to the total risk exposure

minus the replicating portfolio for tax-deferred holdings.

Proof of Result 1.

Without the ability to postpone gains, the after-tax returns in Equation (8) can be written

as Rf = rf−τd(rf−1), Rh = 1+(1−τd)dHi +(1−τg)g

Hi , and Rl = 1+(1−τd)d

Li +(1−τg)g

Li .

57

The “risk-neutral” probability in Equation 9 reduces to qi =Rh−Rf

Rh−Rl. Define

ω ≡ (1− τg)δRNi

(1− τd)(rf − 1− δRNi ) + (1− τg)δRN

i

,

the current-period subsidy in Equation (14) reduces to the form in Result 1

58

B Appendix: Tables and Figures

Table 1: Base Set of Parameter Values.

Parameter ValueRiskless rate (rf) 6%Stock risk premium (µ− rf) 4%

Dividend yield (δ ≡ d/r) 3%Stock volatility (σ) 20%Tax rate on capital gains (τg) 20%Tax rate on dividends and interests (τd) 40%

59

Panel A: Dividend Yield δ = 0

0 10 20 30 40 50 60 70 80

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015Step−upRealize Gains

Remaining Investment Horizon (T−t)

Panel B: Dividend Yield δ = 2.5%

0 10 20 30 40 50 60 70 80

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Step−upRealize Gains


Figure 1: Current-Period effective tax subsidy Φit as a function of investment horizon.

Panel A and B represent the cases when dividend yields are di = 0 and 3%, respectively. The solid

lines are for the case when investors are allowed to escape capital gains tax through step-up of tax

basis in their taxable accounts on the terminal date, and the dashed lines are for the case when

investors are forced to realize all capital gains taxes. The other parameters are given in Table 1.

60

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

−0.005

0

0.005

0.01

0.015

0.02

0.025

horizon = 5 Realize Gainshorizon = 30 Realize Gainshorizon = 30 Step−uphorizon = 5 Step−up

Dividend Yield (δ)

Figure 2: Current-Period effective tax subsidy Φit as a function of dividend yields. The

solid and the dashed line report the cases when investors are forced to realize capital gains taxes in

their taxable accounts on the terminal date and the remaining investment horizons are T−t=5 and

30 years respectively. The dotted and dash-dotted lines are for the cases when investors are allowed

to escape capital gains tax through step-up of tax basis. The other parameters are given in Table

1.

61

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6−0.005

0

0.005

0.01

0.015

0.02

0.025horizon = 5 Realize Gainshorizon = 30 Realize Gainshorizon = 30 Step−uphorizon = 5 Step−up

Asset Volatility (σ)

Figure 3: Current-Period effective tax subsidy Φit as a function of asset volatility. Panel

A and B The solid and the dashed line report the cases when investors are forced to realize capital

gains taxes in their taxable accounts on the terminal date and the remaining investment horizons

are T −t = 5 and 30 years respectively. The dotted and dash-dotted lines are for the cases when

investors are allowed to escape capital gains tax through step-up of tax basis. The other parameters

are given in Table 1.

62

Panel A: Escape Taxes through Step-up of Basis

0 10 20 30 40 50 60 70 80

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5δ=0δ=3%δ=6%Bond


Panel B: Forced to Realize Capital Gains Taxes

0 10 20 30 40 50 60 70 80

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5δ=0δ=3%δ=6%Bond


Figure 4: Total effective tax subsidy Φt:T as a function of investment horizon. Panel A

and B represent the two terminal tax treatments when investors are forced to realize capital gains

taxes and are allowed to escape capital gains tax through step-up of tax basis respectively. The

dotted reports the subsidy for holding only bonds in tax-deferred accounts, and the other three lines

correspond respectively to holding only assets with dividend yield δ = 6%, 3% and 0 in tax-deferred

accounts. The other parameters are given in Table 1.

63

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