INCORPORATING TAXES

When stocks in a portfolio appreciate or depreciate in value, capital gains (respectively, losses) accumulate. When stocks are sold, investors pay taxes on the realized net capital gains. The taxes are computed as a percentage of the difference between the current market value of the stocks and their tax basis, where the tax basis is the price at which the stocks were bought originally.¹⁵ The percentage is less for long-term capital gains (when stocks have been held for more than a year) than it is for short-term capital gains (when stocks have been held for less than a year).¹⁶ Since shares of the same stock could have been bought at different points in time (in different lots), selling one lot of the stock as opposed to another could incur a different amount of tax. In addition to capital gains taxes, investors who are not exempt from taxes owe taxes on the dividends paid on stocks in their portfolios. Those dividends are historically taxed at a higher rate than capital gains, and after 2010 will be taxed as income, i.e., at the investor's personal tax rate. The tax liability of a particular portfolio therefore depends on the timing of the execution of trades, on the tax basis of the portfolio, on the accumulated short-term and long-term capital gains, and on the tax bracket of the investor.

Over two-thirds of marketable portfolio assets in the United States are held by individuals, insurance and holding companies who pay taxes on their returns. (Exceptions are, for example, pension funds, which do not pay taxes year-to-year.) Studies have indicated that taxes are the greatest expense investors face—greater than commissions and investment management fees. To gain some intuition about the effect of taxes on the income of an investor over the investor's lifetime, consider a portfolio that has a capital appreciation of 6.00% per year. After 30 years, $1,000 invested in that portfolio will turn into $1,000 · (1 + 0.06)³⁰ = $5,743.49. Now suppose that the capital gains are realized each year, and a tax of 35% is paid on the gains (the remainder is reinvested). After 30 years, $1,000 invested in the portfolio will turn into $1,000 · (1+(1 − 0.35) · 0.06)³⁰ = $3,151.13, about half of the amount without taxes even when the tax is about one third of the capital gains. In fact, in order to provide the same return as the portfolio with no taxes, the portfolio with annual realized capital gains would need to generate a capital appreciation of 9.23% per year! One can imagine that the same logic would make benchmark tracking and performance measurement very difficult on an after-tax basis.

As investors have become more aware of the dramatic impact of taxes on their returns, there is increasing pressure on portfolio managers to include tax considerations in their portfolio rebalancing decisions and to report after-tax performance. Consequently, the demand for computationally efficient and quantitatively rigorous methods for taking taxes into consideration in portfolio allocation decisions has grown in recent years. The complexity of the problem of incorporating taxes, however, is considerable, both from a theoretical and practical perspective:

1. The presence of tax liabilities changes the interpretation of even fundamental portfolio performance summary measures such as market value and risk. Thus, well-established methods for evaluating portfolio performance on a pretax basis do not work well in the case of tax-aware portfolio optimization. For example, in traditional portfolio management a loss is associated with risk, and is therefore minimized whenever possible. However, in the presence of taxes, losses may be less damaging, because they can be used to offset capital gains and reduce the tax burden of portfolio rebalancing strategies. Benchmarking is also not obvious in the presence of taxes: two portfolios that have exactly the same current holdings are not equivalent if the holdings have a different tax basis.¹⁷
2. Tax considerations are too complex to implement in a nonautomated fashion; at the same time, their automatic inclusion in portfolio rebalancing algorithms requires the ability to solve very difficult, large-scale optimization problems.
3. The best approach for portfolio management with tax considerations is optimization problem formulations that look at return forecasts over several time periods (such as, until the end of the year) before recommending new portfolio weights. However, the latter multiperiod view of the portfolio optimization problem is very difficult to handle computationally—the dimension of the optimization problem; that is, the number of variables and constraints, increases exponentially with the number of time periods under considerations.

We need to emphasize that while many of the techniques described in the previous sections of this chapter are widely known, there are no standard practices for tax-aware portfolio management that appear to be established. Different asset management firms interpret tax-aware portfolio allocation and approach the problem differently. To some firms, minimizing turnover,¹⁸ such as, by investing in index funds, or selecting strategies that minimize the portfolio dividend yield¹⁹ qualify as tax-aware portfolio strategies. Other asset management firms employ complex optimization algorithms that incorporate tax considerations directly in portfolio rebalancing decisions, so that they can keep up with the considerable burden of keeping track of thousands of managed accounts and their tax preferences. The fact is, even using simple rules of thumb, such as always selling stocks from the oldest lots after rebalancing the portfolio with classical portfolio optimization routines, can have a positive effect on after-tax portfolio returns. The latter strategy minimizes the likelihood that short-term gains will be incurred, which in turn reduces taxes, because short-term capital gains are taxed at a higher rate than long-term capital gains.

Apelfeld, Fowler, and Gordon suggest a tax-aware portfolio rebalancing framework that incorporates taxes directly into the portfolio optimization process.²⁰ The main idea of the approach is to treat different lots of the same stock as different securities and then penalize for taxes as if they were different transaction costs associated with the sale of each lot. (This means, for example, that Microsoft stock bought on Date 1 is treated as a different security from Microsoft stock bought on Date 2.) Many tax-aware quantitative investment strategies employ versions of this approach, but there are a few issues to beware when using it in practice:

• The first one is a general problem for all tax-aware approaches when they are used in the context of active portfolio management. For a portfolio manager who handles thousands of different accounts with different tax exposures, it is virtually impossible to pay attention to the tax cost incurred by each individual investor. While the tax-aware method described above minimizes the overall tax burden by reducing the amount of realized short-term sales, it has no provisions for differentiating between investors in different tax brackets because it is difficult to think of each trade as divided between all investors, and adjusted for each individual investor's tax circumstances. This issue is so intractable, that in practice it is not really brought under consideration.
• The dimension of the problem can become unmanageable very quickly. For example, a portfolio of 1,000 securities, each of which has 10 different lots, is equivalent to a portfolio of 10,000 securities when each lot is treated as a different security. Every time a new purchase is realized, a new security is added to the portfolio, since a new lot is created. One needs to exercise care and “clean up” lots that have been sold and therefore have holdings of zero each time the portfolio is rebalanced.
• Practitioners typically use factor models for forecasting returns and estimating risk. One of the assumptions when measuring portfolio risk through factor models is that the specific risk of a particular security is uncorrelated with the specific risk of other securities. (The only risk they share is the risk expressed through the factors in the factor model.) This assumption clearly does not hold when different “securities” are in fact different lots of the same stock.

DiBartolomeo describes a modification to the model used by Northfield Information Service's portfolio management software that eliminates the last two problems.²¹ Instead of treating each lot as a separate security, the software imposes a piecewise linear transaction costs (see Exhibit 18.1) where the break points on the horizontal axis correspond to the current size of different lots of the same security. The portfolio rebalancing algorithm goes through several iterations for the portfolio weights, and at each iteration, only the shares in the highest cost basis tax lot can be traded. Other shares of the same stock can be traded in subsequent iterations of the algorithm, with their appropriate tax costs attached.

The approaches we described so far take into consideration the short-term or long-term nature of capital gains but do not incorporate the ability the offset capital gains and losses accumulated over the year. This is an inherent limitation of single-period portfolio rebalancing approaches and is a strong argument in favor of adopting more realistic multiperiod portfolio optimization approaches. The rebalancing of the portfolio at each point in time should be made not only by considering the immediate consequences for the market value of the portfolio, but also the opportunity to correct for tax liabilities by realizing other capital gains or losses by the end of the taxable year. The scarce theoretical literature on multiperiod tax-aware portfolio optimization contains some characterizations of optimal portfolio strategies under numerous simplifying assumptions.²² However, even under such simplifying assumptions, the dimension of the problem grows exponentially with the number of stocks in a portfolio, and it is difficult to come up with computationally viable algorithms for portfolios of realistic size.