A Model for Tax Advantages of Portfolios with Many Assets

A Model for Tax Advantages of Portfolios with Many Assets John R. Birge The University of Chicago Graduate School of Business, Chicago, Illinois 6637, USA john.birge@chicagogsb.edu Song Yang Northwestern University, IEMS Department, Evanston, Illinois 628, USA yangsong@northwestern.edu August 29, 25 Abstract Taxable portfolios present challenges for optimization models with even a limited number of assets. Holding many assets, however, has a distinct tax advantage over holding few assets. In this paper, we develop a model that takes an extreme view of a portfolio as a continuum of assets to gain the broadest possible advantage from holding many assets. We find the optimal strategy for trading in this portfolio in the absence of transaction costs and develop bounding approximations on the optimal value. We compare the results in a simulation study to a portfolio consisting only of a market index and show that the multi-asset portfolio s tax advantage can lead either to significant consumption or bequest increases. This work was supported in part by the National Science Foundation under Grants DMI-2429 and 422937 and by The University of Chicago Graduate School of Business. We gratefully acknowledge comments from Antje Berndt, George Constantinides, Victor DeMiguel, Xin Guo, Retsef Levi, David Luenberger, James Primbs, Stan Uryasev, and other participants at the Financial Engineering Seminar at Cornell University and the Conference on Risk Management and Quantitative Finance at the University of Florida. 1

2 Key Words: Taxes, portfolio optimization 1 Introduction Taxes present complications for dynamic portfolio optimization due to the explosion in potential portfolio states that accompanies assets with many cost bases and the non-convexity imposed by high short-term and lower long-term tax rates. Finding optimal strategies with even two assets (see, e.g., Dybvig and Koo (1996) and DeMiguel and Uppal (24)) leads to difficult optimization problems. While limiting the number of assets (or treating groups of assets as a single asset category) may have little effect on optimal non-taxable portfolios, such aggregations for taxable portfolios cannot capture a major individual investor benefit of holding many individual assets to use losses to offset gains and to take advantage of tax basis re-setting at death. This paper aims to quantify this advantages for holding many assets over holding a single asset (e.g., a market index fund) in each asset class. Other relevant papers on the effect of taxes on portfolios include the seminal papers by Constantinides and Scholes (198) and Constantinides (1983, 1984). Constantinides and Scholes show that capital gains taxes can be effectively deferred until death (and thus avoided completely) with the use of derivatives assuming zero transaction costs. Constantinides (1983) proves that an optimal policy (with no transaction costs and equal long-term and short-term rates) is to realize losses as soon as they occur and to defer capital gains until a forced liquidation. Constantinides (1984) considers differences in short-term and long-term rates and provides a critical ratio for the decision to realize a long-term gain (and, thereby, re-set the tax basis for a potential short-term loss). This paper also includes a computational study of different policies including transaction costs that supports a policy of annually realizing short-term losses and long-term gains with sufficiently high short to long-term rate ratios.

3 In more recent studies, Dybvig and Kou (1996) and DeMiguel and Uppal (24) find optimal policies for small portfolios using the exact tax basis. Studies by Dammon, Spatt, and Zhang (21, 24), Garlappi, Naik and Slive (21), and Gallmeyer, Kaniel, and Tompaidis (21) use the average tax basis. In addition, these studies assume full use of capital losses (in contrast to legal limitations) and, in many cases, allow wash sales that are also prohibited under current U.S. guidelines. In contrast, Gallmeyer and Srivastava (23) consider a model with limitations on tax loss uses and without wash sales. Their results generally imply lower equity positions than would be held with unlimited tax loss rebates and allowable wash sales. Other results from this set of papers include the observations in DeMiguel and Uppal (24) that the exact tax basis and average tax basis give similar results, in Garlappi, et al. (21) that multiple assets create advantages over a single periodically re-balanced fund, and in Gallmeyer, et al. (21) that short sales costs can significantly alter optimal portfolio choice. Dammon, et al. (24) also show how tax-deferral options can influence optimal portfolio choice, while Huang (23) shows that location of assets in either a taxable or tax-deferred account can be separated from the allocation of funds to those assets. Other tax considerations include studies of asset pricing in the context of taxes, such as Ross (1987), Dermody and Rockafellar (1991, 1995), and Wang and Poon (2). Our approach in this paper takes a different view of portfolio composition and the implications of taxes by explicitly considering the value of holding multiple assets. This approach follows Garlappi, et al. (21), to some extent, but, instead of considering only a small number of assets that are amenable to exact optimization, we will follow simple policies that can consider any number of assets. The intuitive motivation is that a portfolio with a large number of assets often has loss positions that can be used to offset gains. Our goal is to describe the potential of such portfolios to offset taxes purely through diversification without incurring costs due to short sales or derivative

4 purchases. Our numerical results show that this potential can be considerable, equivalent, in a typical example of 5% annual portfolio liquidation, to an average annual consumption increase over 1 years of 5% over a single-fund portfolio when bequest amounts are held constant or to a 7% increase in bequest amount after 25 years when consumption is equal for the index and multi-asset portfolio. This result does not include the tax consequences of rebalances within the single fund or the use of (a limited amount of) capital losses to offset income taxed at a higher rate, suggesting that investors in multiple individual assets may gain significant tax advantages over those holding indexes, including exchange-traded index funds with limited embedded capital gains. In the next section, we describe the market model and portfolio representation and provide some comparison results among different basic portfolios. Section 3 presents a stochastic dominance argument for the optimal trading strategy in the presented model. Section 4 provides a bounding approximation using a periodic discretization of the asset price distribution. Simulation results appear in Section 5 with conclusions in Section 6. 2 Model 2.1 Market Model Our basic model takes a departure from other portfolio models by our assuming an infinite number of assets to test the limits of diversification in multiple assets. In this way, the portfolio will be characterized by a measure over share prices instead of a number of shares or value of an individual asset. We assume that portfolio rebalances occur only at fixed time intervals (e.g., years) indexed by t. We then say that the portfolio consists of a continuum of assets such that each asset, indexed by θ Θ, has a price S t (θ) with distribution such that, given S t 1 (θ), ln(s t (θ)/s t 1 (θ)) = µ + σ W + σw θ, (1)

5 where W and W θ are independent standard normal random variables and W θ1 is independent of W θ2 for any θ 1 θ 2. In this way, each asset price has a component that depends on the market and an independent component. Our basic assumption is that this continuum of assets ensures full diversification of the idiosyncratic risk of each asset. The result of full diversification is that this portfolio does not require rebalancing to maintain diversification and allow for the gains from volatility pumping (as in, e.g., Luenberger (1998)). Defining such a portfolio precisely requires care due to the non-measurability in general of processes with continuous parameters (see, e.g., examples in Doob (1953, p. 67) and Judd (1985)). Khan and Sung (1997, 21, 23) resolve this dilemma for arbitrage pricing theory (Ross (1976)) using hyperfinite processes (see also Sun (1998)) and the Loeb (1975) product measure to obtain a consistent characterization of full diversification. Our interests in this paper focus on the extremes of portfolio diversification and the limiting case of convergence of the portfolio prices in distribution to a fixed (and, in our analysis, stationary) distribution. We assume this limiting case to bound results for discrete-asset portfolios. To keep the state space manageable, we assume the tax basis for each asset is normalized to 1. The total initial portfolio value is also 1, and the initial carryover loss is. The asset prices in the portfolio at t = 1, conditioning on the return of the market µ 1 = µ + σ W (1), then follow a lognormal distribution where the mean of the log-return is µ 1 and the standard deviation is σ. The total portfolio value, given the market return, is then: e µ 1+y f(y)dy = e µ 1+σ 2 /2, where f(y) is the density of a normal random variable with mean and standard deviation σ. As noted earlier, we wish to restrict ourselves to simple trading strategies that we can analyze for this case of infinitely many assets. For our analysis here, we also assume no transaction costs. Since we have an infinite choice of assets, we do not face a wash sales restriction (i.e., we can always

6 choose an identically distributed but different asset from that sold). We assume that all capital gains are long-term and that capital losses can only be used to offset capital gains. As shown in Constantinides (1983), in this context without transaction costs and with a single long-term tax rate, it is optimal to realize losses as they occur and to defer capital gains until a forced liquidation. We assume here that consumption is only possible through asset sales (i.e., no borrowing) and that, therefore, the investor may need to take capital gains. Our strategy in that case is to sell assets with the least capital gain first. The result is the following trading strategy. Trading Strategy: Realize all capital losses in each period, regardless of consumption; When necessary to realize a capital gain in order to meet the consumption need, sell assets with the lowest embedded gain first; Do not realize capital gains when not necessary for consumption. We suppose that consumption in each period t is C t (generally a fraction of wealth as in Samuelson (1969) and Merton (1969) models). All sale proceeds at t minus C t and taxes are reinvested in equivalent assets and again have a tax basis of 1. In this way, the tax basis of all assets remains at 1. Calculations for the strategy going forward should follow each remaining portfolio asset s price (but this would create a very complex model). To simplify the analysis, we will assume that the prices of all remaining assets will be clustered at a finite number of points representing distinct intervals of assets prices. This approximation effectively replaces the integral (sum) of independent lognormal random variables with different starting prices by a single lognormal distribution. We will show how these points can obtain both lower and upper bounds on the portfolio value.

7 2.2 Portfolio Model To track the portfolio over times, our assumed state will consist of a measure ν t (consisting of continuous and singular components) on the prices of assets in the portfolio and the carryover loss L t. The initial portfolio measure ν consists of an atom at 1 with mass 1 and carryover loss L =. Given that the log-return in year 1 is µ(ω), the portfolio at year 1 before trading has a measure ν 1 corresponding to the lognormal density with mean e µ(ω)+σ2 /2 as described earlier. 2.3 Optimal Problem Formulation Based on the above model, the inter-temporal investment-consumption problem can be described as an optimization problem. We state the problem formally but will develop a simple optimal policy. We assume a data process (determining here the market return and time of death), ω := {ω t : t =, 1, 2,...} in a (canonical) probability space (Ω, Σ, P ). Associated with the data process is a filtration IF := {Σ t } T t=, where Σ t := σ(ω t ) is the σ field of the history process ω t := {ω,..., ω t } and the Σ t satisfy {, Ω} Σ Σ. In this case, our states are random measures ν t (ω) on the current prices of assets in the portfolio and a random variable L t (ω) representing the carryover losses at t. The controls or actions in each state are represented by a measure µ t (ω). In addition, we define an outgoing portfolio (after sales) by νt, C t as consumption at time t; R t as the amount sold and reinvested at t; G t as the taxable gain at time t; and V t as the total wealth (and bequest at T ). If we define this decision process as µ t (ω) M, where M is the space of finite measures on R +, we can let x t = (µ t, ν t, νt, L t, C t, R t, G t, W t ) be the decision process in the product space, X, of the products of M M M R 5 for each t =, 1,.... By defining a σ-field X on X (using, for example, a metric on measures), we can formalize that x t is Σ-measurable. To obtain adaptive control, we also require x t to be Σ t -measurable. A characterization of this nonanticipative property is that x t (ω) = E[x t (ω) Σ t ] a.s., t =, 1, 2,...,, where E[ Σ t ] is conditional expectation with respect to

8 the σ field Σ t. Using the projection operator Π t : z π t z := E[z Σ t ], t =, 1, 2,...,, this is equivalent to (I Π t )x t =, t =, 1, 2,...,. We then let N denote this closed linear subspace of nonanticipative processes defined for X, so that the optimization is over x N. Other notation in the following formulation includes T, horizon length (or time of death assumed here for one life and where T (ω) is necessarily Σ T -measurable); B, the σ-algebra of Lebesque measurable sets of R; u, the utility (assumed concave and increasing) of intermediate consumption ; U, the utility (assumed concave and increasing) of a bequest at T ; ρ, a subjective, constant one-period discount rate. With this formal setup, we can then form the optimization problem as follows. T (ω) 1 1 1 Max x N F (x) = E[ ( 1 + ρ )t u(c t (ω)) + ( 1 + ρ )T (ω) U(V T (ω) (ω))] (2) t=1 s.t. : ν (B) = 1 1 B; B B, (3) ν t (ω, B) = ν t(ω, B) µ t (ω, B) + 1 1 B (R t (ω)), a.s., B B, ν t (ω, B) = ( t = 1,..., T (ω) 1; (4) 1 xe µ+σ W t(ω)+σy B f(y)dy)ν t 1 (ω, dx), B B, G t (ω) L t (ω) 1 C t (ω) + R t (ω) = 1 (x 1)µ t (ω, dx) t = 1,..., T (ω); (5) (1 x)µ t (ω, dx) L t 1 (ω), t = 1,..., T (ω) 1; (6) xµ t (ω, dx) τg t (ω), t = 1,..., T (ω) 1; (7) G t (ω), L t (ω), C t (ω), R t (ω), a.s., t = 1,..., T (ω) 1; (8) V t (ω) = ν t (ω, dx), t = 1,..., T (ω). (9) The objective is the discounted cumulative expected utility for consumption and bequest. Con-

9 straint (3) gives the initial endowment; Constraints (4) ensure no short positions at any price asset; Constraints (5) give the dynamics of determining the distribution in the next period; Constraints (6-8), together with the assumption of concave, increasing utilities, ensures that capital gains taxes and re-investment are subtracted from sales before consumption and that losses, if any remain, are carried over to the next period. Constraints (9) keeps the full wealth process. 3 Optimal Trading Strategy As shown in Dammon, Spatt, and Zhang (21), optimal consumption will vary according to relative values of the utility functions for consumption and bequest and will also be influenced by the embedded capital gain in the portfolio. Our goal here is not to consider the relative effects of the consumption decisions but rather to determine optimal liquidations to finance given consumption or bequest levels. We first demonstrate a general result on the value function. In contrast to other results that hold for any sample path of prices, the optimal policy of selling assets with the lowest price first does not hold on all sample paths in the general case with multiple random assets. 1 Instead of using a sample path argument, we will prove the result using stochastic 2 dominance and the assumptions on the utility functions. The form of stochastic dominance here is, for two measures A and B on R, that A dominates B (A B) if z A(dy) z B(dy) z R. We suppose that Problem (2-9) 1 As an example, suppose that an investor has one share each of two assets A and B where each has a cost basis of 1 while the current prices are S A = 2 and S B = 3 respectively. Suppose in the next period, that S A 1 = 1 and S B 1 = 6. If we follow the proposed strategy in the theorem, we sell.5 share of A now and then.5 Share of A and 1/12 share of B in the next period to finance consumption. The result is a capital gain of.5(2 1) =.5 now and (1/12)(6 1) = 5/12 at time 1, for a total capital gain of 11/12. If, however, we sell B now, then we can sell 1/3 share of B now followed by 1 share of A at time 1. In that case, the total capital gain is 2/3 < 11/12; thus, on this sample path, the strategy is not optimal. 2 We use stochastic dominance although the measures µ are not probability measures.

1 can be interpreted as a dynamic program with a Σ t -measurable state, (ν t (ω), L t(ω)) at time t and value function, F t (ν t (ω), L t(ω)), as an optimal continuation from that state for periods t + 1 to T (ω). Lemma 3.1 If V 1 t (ω) = ν 1 t (ω) ν 2 t yνt 1 (ω), a.s.,, then F t (ν 1 t (ω, dy) yν 2 t (ω, dy) = Vt 2 (ω), a.s., L 1 t (ω) L 2 t (ω), and (ω), L 1 t (ω)) F t (ν 2 t (ω), L 2 t (ω)), a.s., for all t 1. Proof. We proceed by backward induction on t and note that, for t = T (ω), the result holds since U is an increasing function. We assume that the result holds for all t + 1 t T (ω) for all ω and wish to show the result holds for t. In this case, from (5), we must have ν 1 t+1 (ω) νt+1 2 (ω). Suppose (µ2 t+1 (ω), C2 t+1 (ω), R2 t+1 (ω), G2 t+1 (ω), L2 t+1 (ω)) is part of an optimal strategy given the state, (νt 2 (ω), L 2 t (ω)). We will construct µ 1 t+1 (ω) that yields a better state at t + 1. For this construction, there exists some c(ω) such that c(ω) xν 1 t+1 (ω, dx) = µ 2 t+1 (ω, dx) or ν1 t+1 (ω) has a singular value at c(ω) with c(ω) xνt+1 1 (ω, dx) < µ 2 t+1 (ω, dx) and c(ω) xνt+1 1 (ω, dx) > µ 2 t+1 (ω, dx). In either case, this is possible because V 1 t (ω) Vt 2 (ω). We then let µ 1 t+1 (ω, B) = νt+1 1 (ω, B) for all B (, c(ω)) and partition any singular part at c(ω) so that xµ 1 t+1 (ω, dx) = xµ 2 t+1 (ω, dx) and by construction µ1 t+1 µ2 t+1, or z µ1 t+1 (ω, dx) z µ2 t+1 (ω, dx) for any z and, in particular, for z =. The left-hand side of Constraint (6) for t + 1 and ω is then no greater with µ 1 t+1 and L1 t than with µ 2 t+1 and L2 t ; therefore, we can find G 1 t+1 (ω) G2 t+1 (ω) and L 1 t+1 (ω) L2 t+1 (ω) for any ω a.s. Given this relation, we can then choose C1 t+1 (ω) C2 t+1 (ω) and Rt+1 1 (ω) R2 t+1 (ω) and satisfy Constraints (7) and (8). With these values, we then let ν1 t+1 be defined by ν 1 t+1, µ1 t+1, and R1 t+1 according to (4). The overall result is that ν1 t+1 ν2 t+1, V t+1 1 (ω) V 2 t+1 (ω), L1 t+1 (ω) L2 t+1 (ω), and u(c1 t+1 (ω)) u(c2 t+1 (ω)), which implies F t(ν 1 (ω), L 1 t (ω)) F t (νt 2 (ω), L 2 t (ω)), a.s. to complete the induction.

11 The next theorem states that our proposed trading strategy is optimal for Problem (2 9). Theorem 3.2 (Optimal Trading Strategy): An optimal trading strategy for (2 9) includes the following policy: 1. realize all losses in each period (i.e., µ t (ω, B) = ν t (ω, B) B [, 1)), regardless of consumption; 2. do not realize capital gains except when necessary to finance consumption (i.e., µ t (ω, B) = B [1, ) if C t (ω) 1 xµ t(ω, dx)); 3. if necessary to realize capital gains, sell those with the lowest price first (i.e., c xµ t(ω, dx) = c xν t(ω, dx) for some c 1). Proof. The first two items follow from Theorem 1 in Constantinides (1983), although we can also use the result in Lemma 3.1. Item 3 is an immediate result of the lemma. If this policy is not followed for some µ 2 t (ω) resulting in νt 2 (ω), then following the policy in Item 3 would yield some ν 1 t such that ν 1 t ν 2 t with V 1 t (ω) V 2 t (ω) and L 1 t (ω) L 2 t (ω) for any ω. In that case, Lemma 3.1 implies that µ 2 t is dominated by a strategy following Item 3. 4 Bounding the optimal solution with a multi-point discrete approximation While Theorem 3.2 provides an optimal policy for asset sales, computations involving the convolution in Constraint (5) can be complicated for lognormal distributions since this class of distributions is not closed under addition (and, here, integration). To obtain computable results, at each period t, we replace νt with a discrete approximation in such a way that the resulting value is either a lower or upper bound on the optimal value of (2 9). We assume two possible alternatives:

12 Definition 4.1 Measures ν L t and ν U t defined from ν t are called lower and upper bounding measures respectively if they satisfy the following conditions: Lower bound: ν L t Upper bound: ν U t such that xν L t (ω, dx) = xν t (ω, dx) and νl t (ω) ν t such that xν U t (ω, dx) = xν t (ω, dx) and νu t (ω) ν t (ω) for all ω; (ω) for all ω. Corollary 4.1 For any lower and upper bounding measures, ν L t and ν U t, as in Definition 4.1, F t (ν L t (ω), L t (ω)) F t (ν t (ω), L t(ω)) F t (ν U t (ω), L t (ω)), a.s., for all t 1. Proof. This follows immediately from the definition and Lemma 3.1. In our computational results, we re-define νt L and νt U at each t on a given sample path (corresponding to a realization of ω) and obtain overall bounds on the optimal value in (2 9). We use a discrete approximation that depends on a partition (that may depend on t and ω) of R + into intervals I 1, I 2,..., I K, where I j = [a j, b j ). With this partition, we then let ν U t have atoms at a j with mass m U j so that m U j a j = I j xν t (ω, dx). Similarly, νl t has atoms at b j with mass m L j so that m L j b j = I j xν t (ω, dx) for j < K. To maintain a finite lower bound for the last interval, j = K, we replace ν t over I K with an atom at I K xν t (ω, dx)/ I k ν t (ω, dx) and only allow transitions in Constraint (5) from I K(t) at period t to I K(t+1) at period t + 1 for all t. This ensures that no losses result from assets with prices in interval I K and maintains the lower bounding property. 5 Algorithms and Computation Results For the computational results, we suppose that all assets begin with the same basis (normalized to 1). This situation can model the case in which an investor elects a lump-sum distribution of a qualified plan at retirement (and, hence, in the United States, would pay tax on post-1974

13 contributions as ordinary income). The investor then purchases assets with the after-tax amount. We assume that this investment then finances all consumption until death. 3 In comparing results, we will not generally specify the utility forms u and U to keep the results general. Our main comparisons will be with a portfolio that holds only a market index and a hypothetical portfolio that pays no capital gains tax. In our comparison test, we calculated both upper and lower bounding results as well as an average approximation of the portfolio holding a continuum of assets according to the process in (1). The upper and lower bounds were sufficiently close (see Table 1) that our summaries only include the intermediate approximation. Our comparisons focus on either the consumption or bequest amounts separately. In the first set of comparisons (equal-bequest case), we assume that the index portfolio and the continuum portfolio both attain the same bequest amount. In the second set of comparisons (equal-consumption case), we assume the same consumption in the index portfolio and the continuum portfolio. In each case, we observe the difference in the quantity (either consumption or bequest) that varies across the portfolios. In the first set of examples, to maintain the bequest amount, we assume that the different portfolios spend the same total on consumption plus taxes as a fixed percentage of the portfolio value in each period. In the second set of examples, we maintain that consumption alone is a fixed percentage value in each period. (For a specific utility, this percentage may change depending on t and the portfolio s unrealized capital gain. We keep the percentage fixed to make the comparisons across portfolios more direct.) Our results also use a fixed time horizon T, although we will show results for varying T. We 3 Our purpose in choosing the lump-sum distribution is to set the tax basis to 1 as in our model and not to endorse this practice over maintaining a tax-deferred amount or making another distribution election. Our goals, as stated in the introduction, are to measure the potential benefit of broadly diversified portfolios from a tax-savings perspective and not to make specific recommendations on other forms of asset allocation.

14 assume that the market overall has an average return of 12%. As observed in Campbell, et al. (21), our base case assumes that the idiosyncratic volatility (σ) is twice the market volatility (σ ). We also consider cases where the volatility ratio (σ/σ ) is 1 and 3. We also explore a range of consumption fractions (equivalent to selling 5%, 7%, and 1% of the portfolio in each period) to model varying potential investor utilities. In our simulations, we ran cases with no taxes, the continuum of assets (including upper bound, lower bound, and approximation), and the market index with equal-bequest and equal-consumption. The no-tax case is straightforward. The next subsections provide more details on the other cases. 5.1 Continuum Assets Case In the continuum case, we use a dynamically allocated partition to discretize the portfolio and to maintain the upper and lower bounds as in Corollary 4.1. We generate the partition {I 1, I 2,...I K }, such that the value of the portfolio with prices between any two adjacent points is fixed, i.e, for a given constant D, which is.1 in our simulation, and any partition interval, I j = [a j, b j ), (where b j = a j+1 for j < K), we have: b j a j xν t (ω, dx) = D; the value in interval I K is the remainder of the portfolio that is then less than D. In the lower bound case, the distribution places all weight at b j for j < K as noted earlier. For interval I K, we place the remaining assets into an account that is not liquidated until the bequest calculation. In this way, these funds cannot generate losses, maintaining the lower bound. The upper bound proceeds as discussed in Section 4 with the distributional weight of each interval I j placed at a j. In the approximation case, besides generating a partition, we place the weight in each I j at the conditional mean of the interval (i.e., we let m app j = I j ν t (ω, dx) be the mass at point x app j = ( I j xν t (ω, dx))/m app j, where ν t (ω, dx) is the distribution found in the simulation at time t

15 under scenario ω). In this way, the approximation ensures that the share measure and the total value of assets in each interval match the true continuum case (given the previous period s asset distribution). 5.2 Market Index Cases with Equal Bequest and Equal Consumption In the market index case, the portfolio is just a single asset that follows our specified market movement and trading strategy. In the equal-bequest case, we close out the specified percentage or all of the portfolio if the price is lower than the basis. In the equal-consumption case, we match the consumption in each year to the consumption in the lower bound case, which gives a favorable bias to the index case relative to the continuum portfolio. 5.3 Simulation Results The results below assume that µ =.2, µ + (σ 2 + σ2 )/2 =.12, and the tax rate τ =.15. For the volatility, we vary σ/σ = {1, 2, 3} while maintaining σ 2 + σ2 =.2. In the 1 : 2 volatilityratio case, we then have σ =.4 and σ =.2. We also considered varying liquidation fractions, c =.5,.7,.1, which denote the portion of the portfolio that is used for consumption and tax payments. For each combination, we ran 2 simulations up to T = 5. Table 1 gives the results from each combination for ratio of bequest and consumption of the continuum portfolio approximation to the relevant index portfolio (equal-consumption case for bequest comparisons and equal-bequest case for consumption comparisons). The bequest amounts consider the ratio at T = 1 and T = 25 years. The consumption values give the average ratio of consumption with the continuum portfolio approximation to that of the index portfolio for the first 1 years (when the advantage of the continuum portfolio is greatest). We also include the ratio over the first 1 years of the certainty equivalent consumption value for discounted utility using

16 β = 1 1+ρ =.1 and power utility, u(c) = Cγ /γ with γ =.5. Table 1 also includes the average ratios over T {1,..., 1} and T {1,..., 5} between the lower bound and upper bound on consumption. Since these errors are generally small, we focus on the ratios for the intermediate approximation. The table also includes the standard error of the simulations on the average bequest ratio between the continuum approximation and the equalconsumption index case at T = 25. (Standard errors had similar relative orders for other ratios.) The average number of years that the continuum approximation portfolio had no tax obligation is also given. Indiv. to Market Fraction Sold Avg. Ratio of Continuum Approx. to Index Volatility Ratio Each Period Bequest T=1 Bequest T=25 Avg. Cons n. Disc d. Cons n. T=1-1 Cert. Eq. T=1-1 3.1 1.65 1.299 1.35 1.36 3.7 1.44 1.16 1.49 1.47 3.5 1.33 1.69 1.57 1.54 2.1 1.56 1.259 1.29 1.3 2.7 1.41 1.121 1.4 1.39 2.5 1.3 1.71 1.47 1.45 1.1 1.4 1.194 1.18 1.21 1.7 1.28 1.84 1.24 1.27 1.5 1.2 1.44 1.28 1.31 Avg. Upper/Lower Avg. Upper/Lower Average No-Tax Standard Error Bound Bound Years Bequest Ratio (T=1..1) (T=1..5) (T=1..5) T=25 3.1 1.4 1.2 7.9.4 3.7 1.13 1.1 9.6.2 3.5 1.8 1.11 11.5.1 2.1 1.3 1.2 9.2.6 2.7 1.2 1.2 11.1.3 2.5 1.3 1.5 13.1.2 1.1 1.2 1.1 14.7.6 1.7 1.3 1.3 16..3 1.5 1.7 1.7 17.9.2 Table 1: Summary results for simulations with varying volatilities and consumption fractions. From Table 1, we observe, as expected, that higher relative individual asset volatility leads to greater gains relative to the index portfolio. The relative gain also increases for bequest amounts

17 as the portfolio fraction sold for consumption in each period increases (and consumption amounts for the index portfolio equal that of the continuum approximation), but the relative gain in consumption decreases as the portfolio fraction sold increases when bequest amounts are held constant. Intuitively, the advantage of the continuum portfolio is highest relative to the equal-bequest index portfolio when the fraction sold is small and can be covered by losses in the continuum portfolio. Conversely, the advantage of the continuum portfolio relative to the equal-consumption index portfolio increases with the fraction that the equal-consumption index must sacrifice for taxes in each period. That tax fraction in turn increases with the liquidation fraction. The average number of years without taxes in Table 1 contrasts with the index portfolios, which had averages of 4.35 years for the equal-consumption index portfolio and 4.34 years for the equalliquidation-fraction portfolio. The average number of years without tax obligations decreases as the consumption fraction increases in each case as expected. The average number of no-tax periods also decreases with increasing individual asset to market volatility ratio, which seems somewhat counterintuitive. This outcome appears to occur because periods with overall market losses allow lower-idiosyncratic-volatility portfolios to realize more loss positions than can higher-volatility portfolios and may extend no-tax periods in those cases. Figures 1 6 present the results for the average ratio of consumption and bequest in the continuum approximation portfolio to the relevant index portfolio for the initial years in the simulation for the 3 volatility cases. The consumption figures (4 6) also show the ratio of consumption in the no-tax case to the index portfolio for the 5% liquidation case (which is similar to those at 7% and 1% liquidation). The bequest figures (1 3) display the increase in bequest advantage as time increases. The consumption figures (4 6) show that the relative advantage of the continuum portfolio over the index portfolio increases to a maximum (over 8% for 5% annual liquidation with σ = 3σ ) at 5 to 8 years and then declines. The continuum portfolio advantage also closely tracks

18 Figure 1: Ratio of average bequest from the continuum approximation portfolio to the index portfolio with σ = 3σ for years 1 to 25 for 1%, 7%, and 5% annual liquidation. that of the no-tax portfolio until the relative advantage over the index portfolio starts to decline. Overall, the results indicate that a widely diversified portfolio can lead to significant consumption or bequest increases over an index portfolio. Consumption levels can average as high as 5 or 6% over those from an index portfolio over the first 1 years. Bequest levels at 1 years have similar advantages over an equal-consumption index portfolio and have much greater advantages over longer horizons. The impact of diversification on capital gains taxes in this model decreases for later years in the equal-bequest comparisons, but this model does not include intermediate rebalancing that may also take advantage of diversification within an asset class and allow the fully diversified portfolio s relative advantage to continue over a longer horizon than shown here. 6 Conclusions and Areas for Further Study The results in this model suggest that holding many assets in a portfolio can give investors a significant tax advantage over holding a single index portfolio. To some extent, this advantage

19 Figure 2: Ratio of average bequest from the continuum approximation portfolio to the index portfolio with σ = 2σ for years 1 to 25 for 1%, 7%, and 5% annual liquidation. Figure 3: Ratio of average bequest from the continuum approximation portfolio to the index portfolio with σ = σ for years 1 to 25 for 1%, 7%, and 5% annual liquidation.

2 Figure 4: Ratio of average consumption from the continuum approximation portfolio to the index portfolio with σ = 3σ for years 1 to 2 for 1%, 7%, and 5% annual liquidation and for the no-tax portfolio with 5% liquidation. Figure 5: Ratio of average consumption from the continuum approximation portfolio to the index portfolio with σ = 2σ for years 1 to 2 for 1%, 7%, and 5% annual liquidation and for the no-tax portfolio with 5% liquidation.

21 Figure 6: Ratio of average consumption from the continuum approximation portfolio to the index portfolio with σ = σ for years 1 to 2 for 1%, 7%, and 5% annual liquidation and for the no-tax portfolio with 5% liquidation. could be larger in practice since we did not include capital gains of an index fund that rebalances to maintain specified asset weights. We also did not include possibilities for realizing losses more frequently than annually and the ability to use capital gain losses to offset some portion of other income with a higher tax rate. While this model provides some indication of the potential for portfolios with many assets, the model includes many simplifications that may not hold in practice and should be considered for further research. The accuracy of the continuum model as a surrogate for a portfolio of many assets is an open question. An additional study could consider the difference between the continuum portfolio here and a portfolio with a large discrete number of assets and the number of assets necessary to provide a reasonable approximation of the continuum results. Our results have also ignored heterogeneous asset volatilities, multiple return factors, additional asset categories (including borrowing and derivatives), and transaction costs. Additional studies

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