B Online Appendix: Extensions In this online appendix we demonstrate that many important variations of the exact cost-basis LUL framework remain tractable. In particular, dual problem instances corresponding to the optimization problem inside the expectation on the right-hand-side of (31) remain tractable so that tight upper bounds on the optimal value function can be easily computed given a sufficiently good dual feasible penalty. We begin with the problem where no wash-sales constraints are imposed. B.1 No Wash-Sales Constraints An important restriction imposed by 28 the U.S. tax code is the so-called no wash-sales restriction. Under this constraint any security that has been sold at a loss may not be repurchased within 30 days without foregoing the benefit of realizing that loss. More specifically, if such a security is repurchased within 30 days then the realized loss is ignored and the basis and holding period for the newly purchased stock will be adjusted to include the basis and holding period of the stock that was sold. A standard way to model the no wash-sales restriction is to simply forbid the sale and purchase of a stock in the same period. We will follow that convention here through the use of binary decision variables at the cost (in general) of a considerable increase in computational time. For this reason many papers in the literature do not account for the no wash-sales restriction but it can be important 29 to do so in practice. We will argue here that the restriction is easily handled in our dual problem formulations. First let z t,k {0, 1} be a binary decision variable and consider the constraints t 1 (n j,t 1,k n j,t,k ) M 1 (1 z t,k ) and n t,t,k M 2 z t,k k, 1 t T 1 (73) j=0 where M 1 and M 2 are suitably large constants. Note that we do not need to consider the case t = T since the portfolio is liquidated at that point. Once we note that t 1 j=0 (n j,t 1,k n j,t,k ) is the total sales of the k th security at time t it is clear that constraints (73) forbid the simultaneous sale and purchase of security k at time t. Therefore, by adding the new binary variables z k,t and the constraints (73) to the problem formulation in Section 2 we can easily model the no wash-sales restriction. Consider now the dual inner problems corresponding to the optimization problem inside the expectation on the right-hand-side of (31). Imposing the no wash-sales restriction will add O(KT ) binary decision variables 28 Other tax jurisdictions also have similar rules. See for example bed-and-breakfasting in the United Kingdom. 29 These constraints are deemed particularly important in tax-aware index-tracking problems. 1
to these optimization problems. Nonetheless it is well known (see, e.g. [7]) that the sequential QP approach we have adopted for solving the dual problem instances can easily handle these binary variables for the problem sizes (K = 25 and T = 80) we have considered in this paper. Moreover, since simulating the feasible strategies has been the computational bottleneck, solving dual problem instances with these integer variables will only result in a modest overall increase in computational time. With regards to the primal strategies, we note that the tax-blind and tax-aware heuristic strategies can easily accommodate the no wash-sales restriction. The RBH strategy, however, would require an additional O(K) binary decision variables (and corresponding constraints) at each time t to accommodate them. If simulating the RBH strategy turns out as a result to be too time consuming, however, then it would be easy to impose the no wash-sales constraints instead via alternative simple heuristic rules. B.2 Random Time of Death Thus far we have assumed that the time horizon T is fixed. However, for long-term investors, T typically represents the time of death, and is therefore random. Indeed the step-up provision of the U.S. tax code that we discussed in Section 5.4 only applies at the investor s (random) time of death. We argue here that we can easily handle random time horizons in our problem formulations. First, we assume that the probability distribution of T is exogenous in the sense that the realized market returns do not 30 influence T. More specifically, the distribution of T given F t depends only on t. With this assumption it should be clear that calculating the no-tax optimal solution can be performed at relatively little additional cost. For example, one could bound T at some large and very unlikely value and then solve the resulting finite horizon no-tax problem, taking into account at each intermediate time t that death was possible in the next period. Given this approximately optimal no-tax solution it is easy to construct and simulate the tax-blind and tax-aware heuristics. It is also straightforward to simulate the RBH strategy until the random time T and also have the strategy itself account for the uncertainty in T. This can be done by having each of the I simulated paths in (22) to (25) have a random horizon T with the appropriate conditional (on time t) distribution. Handling a random horizon is also very straightforward in the dual problem instances. Following Brown and Haugh [1] it is easy to see that strong duality still holds and that dual problem instances can be obtained by first simulating T from its exogenous distribution and then simulating market returns (and state variables if necessary) until the realized value of T. The values of T will therefore vary across each dual problem 30 We consider this to be a reasonable assumption, notwithstanding the many reports of failed investors jumping to their deaths in the aftermath of the stock market crash of 1929. 2
instance but the dual problems will otherwise have the same form and be similarly easy to solve. B.3 Inter-Temporal Consumption We have assumed the investor is only concerned with the value of terminal wealth. Clearly, it is also possible that she derives utility from inter-temporal consumption q t in addition to terminal wealth. To model this we can follow the usual financial economics approach and assume time-additive preferences. In particular, if we assume a concave utility function u( ) then our new objective function has the form b 1 γ T 1 T 1 γ + u(q t ) t=0 and the budget constraints (9) and (10) are replaced by b 0 + p 0n 0,0 + q 0 = w 0 t t 1 b t + p tn j,t + τ s gt s + τ l gt l + q t = b t 1 r 0 + p tn j,t 1 t 1 j=0 j=0 with the additional non-negativity constraint q t 0 for all t. Our first observation is that the dual inner problems remain tractable since u( ) is concave. Moreover the no-tax solution for the problem with intertemporal consumption can still be computed numerically. This means that we can easily define corresponding tax-blind and tax-aware heuristic policies. We can also use the no-tax solution to construct a (hopefully) good dual feasible penalty for the dual problem. The RBH problem is not so easy to modify, however, as our approach at time t was to only simulate terminal security prices, p T, and not intermediate values. Clearly, an alternative solution or heuristic would need to be developed. One of the well-known weaknesses of time-separable utility functions is that risk aversion and elasticity of inter-temporal substitution are determined by the same parameter, and therefore, cannot be specified independently. Recursive utility preferences of Epstein and Zin [3] overcome this difficulty. Moreover, following 31 Kogan and Mitra [6], the information relaxation approach can also be applied with these preferences to obtain valid lower and upper bounds. We give a brief outline here and note that the same methods can also be applied to our tax-aware setting. These methods are also related to the dynamic zero-sum game work of Haugh and Wang [5]. 31 Kogan and Mitra [6] used the information relaxation approach to verify the accuracy of numerical solutions to general equilibrium models. 3
The Bellman equation for the consumption / asset-allocation problem with Epstein-Zin preferences takes the form V t (w t ) = max q t (0,w t) W (q t, E 0 [V t+1 ]) 1 = max q t (0,w t) γ [ q ρ t + β (γe 0 [V t+1 (w t+1 )]) ρ γ ] γ ρ (74) where w t and q t are time t wealth and consumption, respectively, V t is the time t value function and W is the so-called time aggregator. The elasticity of inter-temporal substitution and coefficient of relative risk aversion are then given by 1/(1 ρ) and 1 γ, respectively. The non time-separability is clear from (74) and this makes the application of the information relaxation approach difficult. However, Geoffard [4] and Dumas et al [2] showed that (74) could be formulated equivalently as a time-separable zero-sum game V 0 (w 0 ) = max min E 0 q t (0,w t) ν t F t [ T ] (1 ν t ) t F (q t, ν t ) t=0 (75) where F is the Legendre transform of the time-aggregator W and ν t can be interpreted as a stochastic discount rate process. An upper bound for the optimal value function can now be found in the usual manner by simply setting ν t to be any feasible 32 discount process ˆν t (q t ) and then maximizing over the consumption process. This latter problem cannot be solved exactly in general but we can use the information relaxation approach to obtain an upper bound on this (upper) bound. It is also interesting to note that valid lower bounds with Epstein-Zin utility cannot be obtained by simply simulating a feasible policy. This is clear from (74) where the non time-separability hinders such an estimation. However, the formulation in (75) again comes to the rescue. We simply first fix a consumption process, ˆq t, and then minimize over the discount process ν t. Again, this latter problem cannot be solved exactly in general but we can use the information relaxation approach to obtain a lower bound on this (lower) bound. See Kogan and Mitra [6] for further details as well as Haugh and Wang [5] for further applications of information relaxations and zero-sum games. B.4 Allowing Excess Long-Term Losses to Offset Excess Short-Term Gains One can extend the formulation in Section 2 to allow for excess long-term losses to offset any remaining short-term gains as follows. Recall that in Section 2 we use ˆl s t to denote the excess short-term losses after 32 The ideal selection would be the (presumably) unknown optimal discount process, νt (qt). Similarly, when constructing a lower bound the best choice of ˆq t would be the unknown optimal consumption process qt. 4
offsetting short-term gains and ĝt l to denote the long-term gains after offsetting the long-term proceeds c l t using only long-term losses. Now let ĝt s denote the short-term gains after offsetting the short-term proceeds c s t using only short-term losses and let ˆl t l denote the excess long-term losses after offsetting long-term gains. The U.S. tax code requires the agent to use carried losses to offset gains of the same type first. We therefore have ĝ s t = max{c s t + l s t 1, 0}, ˆls t = min{c s t + l s t 1, 0} (76) ĝ l t = max{c l t + l l t 1, 0}, ˆll t = min{c l t + l l t 1, 0}. (77) The agent is then allowed to use the available excess short-term (resp. long-term) losses to offset the remaining long-term (resp. short-term) gains. We can model this via the constraints g s t + l l t = ĝ s t + ˆl l t, 0 g s t ĝ s t, ˆll t l l t 0 (78) g l t + l s t = ĝ l t + ˆl s t, 0 g l t ĝ l t, ˆls t l s t 0. (79) The agent will pay short-term (resp. long-term) capital gains taxes τ s g s t (resp. τ l g l t), and carry the remaining unused losses l s t and l l t forward. As in Section 2, the fact that τ s > τ l allows us to linearize (76) according to ĝ s t + ˆl s t = c s t + l s t 1, ĝ s t 0, ˆll t 0. (80) However, in contrast to Section 2, we can no longer linearize (77). This is because there is now an incentive to let the long-term gains be large, and to simultaneously have large long-term losses that can be used to offset excess short-term gains. However, these constraints can be linearized using binary variables. Thus, the portfolio selection problem reduces to a binary program where non-convexity only arises from the binary variables. Again the dual inner problems for this formulation are relatively easy to solve using the SQP solution approach as only O(T ) binary variables are required. References [1] Brown, D.B. and M.B.Haugh. 2014. Information Relaxations Bounds for Infinite Horizon Markov Decision Processes. Working paper, Columbia University. 5
[2] Dumas, B., R. Uppal and T. Wang. 2000. Efficient intertemporal allocations with recursive utility. Journal of Economic Theory, Vol. 93, 240 259. [3] Epstein, L.G., and S.E. Zin. 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica, Vol. 57, 937 969. [4] Geoffard, P. 1996. Discounting and optimizing: Capital accumulation problems as variational minmax problems. Journal of Economic Theory, Vol. 69(1), 53 70. [5] Haugh, M.B. and C. Wang. 2014. Information relaxations and dynamic zero-sum games. Working paper, Columbia University. [6] Kogan, L. and I. Mitra. 2014. Accuracy Verifcation for Numerical Solutions of Equilibrium Models. Working paper, MIT. [7] Mittelmann, H. Mixed Integer Q(C)P Benchmark. 6