Embedded Tax Liabilities & Portfolio Choice

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1 Embedded Tax Liabilities & Portfolio Choice Phillip Turvey, Anup Basu and Peter Verhoeven This study presents an improved method of dealing with embedded tax liabilities in portfolio choice. We argue that using a risk-free discount rate is appropriate for calculating the present value of future tax liabilities. Supportive of recent research, our results found a taxation-induced preference of holding equities over bonds, and a location preference of holding equities in the taxable account and bonds in retirement accounts. These important findings contrast with traditional investment advice which suggests a greater capacity for risk in retirement accounts. While taxes are only one factor in portfolio selection and evaluation, it is an important factor. Recent research deals with the issues of tax efficiency of different asset classes as well as their location preferences between available savings vehicles due to differential tax treatments. However, with multiple tax environments and deferred capital gains, the effect can be complex so how investment portfolio decisions should incorporate taxes is not always obvious. Nevertheless, practitioners need to identify and address these complexities in order to devise an optimal investment program for their clients. While there are numerous papers in the area of after-tax investing i, one area where the literature has not made much headway is how to deal with embedded capital gains and implied deferred tax liabilities. ii One approach treats embedded capital gain taxes as a transaction cost in the portfolio optimisation process (Arnott, Berkin, & Ye, 2000). While this approach shows that investors should avoid turning over assets with embedded capital gains, it does little for evaluating the after-tax value or after-tax return of the asset. Phillip Turvey is a PhD candidate in finance, Anup Basu is a senior lecturer and Peter Verhoeven is an associate professor. All authors are from School of Economics and Finance at Queensland University of Technology, Australia. 1 Electronic copy available at:

2 Reichenstein (2006, 2007a, 2007b, 2008a, 2008b) shows that $0.70 in a Tax Deferred Account (TDA) is equivalent to $1 in a Roth account and recommends adjusting effective asset allocations to reflect this fact. He also suggests adjusting the taxable account for the value of any embedded tax liability as if it were sold today, but this overlooks the time value of money the asset may not need to be sold today, but at some uncertain time in the future. Because a dollar tomorrow is worth less than a dollar today, the principal values need to be adjusted for the present value of the implied capital gains tax. Indeed, Horan and Al Zaman (2008) [H&A hereafter] use this present value approach within a mean-variance optimisation framework. They argue that when a capital gains tax is delayed, more wealth can remain invested in the asset, and conclude that the after-tax return on the asset is the correct discount rate to use. Other authors like Berkin and Luck (2010) make similar arguments. This line of reasoning has intuitive appeal but, as we show in his paper, it does not properly consider the increase in risk to an investor s after-tax wealth. In this article, we propose a number of improvements to the H&A model most importantly the way in which embedded capital gains are valued. Embedded capital gains represent an implicit future tax liability and can be considered an interest-free loan from the government. While the H&A model attempts to value this future tax liability using an equity rate of return, we show that using an after-tax risk-free discount rate is more appropriate as it accounts for the change in risk to the investor s after-tax wealth. This suggests that embedded tax liabilities have a greater impact on asset allocation than assumed by the current literature and they dramatically increase preference for holding equities in taxable accounts and bonds in retirement accounts. These results are contrary to traditional investment planning advice, which generally views retirement accounts as more suited to taking on risk due to their long investment horizon. The most likely explanation for these results is the preferential taxation treatment of equities compared to bonds. iii 2 Electronic copy available at:

3 Current Treatment of Taxes in the Literature Adjusting returns for immediate taxation and where tax credits are given on losses is straightforward. After-tax investment returns ( ) are calculated as the pre-tax return ( ) multiplied by one less the applicable tax rate ( ): 1 (1) The adjustment to an asset s risk (or standard deviation) for taxes is less obvious than the method for adjusting returns. Say that the pre-tax return of an investment is 20% in a good year and 0% in a bad year (i.e. 10% ± 10%). Also, assume a tax rate of 30% and that tax credits are given for any losses generated. Substituting these numbers into equation (1), a good year will produce an after-tax return of 14%, while a bad year will produce an after-tax return of 0%. The expected tax-adjusted return is therefore 7% ± 7%. Effectively, the government shares both risk and return with the investor such that the risk to the investor drops proportionally with after-tax returns. That is, if the after-tax return drops by 30% compared to the pre-tax returns, the standard deviation will also reduce by 30%. As the government is risk-sharing in gains and losses, the after-tax standard deviation ( ) of an asset s return also reduces by (1 ): 1 (2) However, what adds complexity to the issue is that taxes on capital gains are only payable when they are realised. This leaves us with two problems: (i) how do we value the future tax payable on an unrealised capital gain?; And (ii) how does this affect effective annual returns for assets that generate capital gains? The H&A paper presents a tax-adjusted optimisation model in the presence of multiple taxation environments based on the work of Wilcox, Horvitz and dibartolomeo (2006). In their world, there are three tax environments (i) the normal taxable account iv with various marginal tax rates; (ii) the aptly named Tax Deferred Account (TDA) where 3 Electronic copy available at:

4 withdrawals are taxed at a flat rate of 30%; and (iii) the tax-free Roth account. v Proportions invested in each location are fixed and cannot be reallocated. There are two assets equities and bonds and investments can flow into them through any of the three accounts. For the taxable account, returns from investments can be broken up into ordinary income, dividend income, immediate capital gains, and deferred capital gains each attracting their own tax rates. When these assets are invested through the different tax environments, they are treated as distinct investment opportunities because the after-tax cash flows associated with each asset are different. H&A offer two separate approaches to dealing with deferred tax liabilities one using a pre-tax principal value and the other using a post-tax principal value. Both approaches first calculate what the value of the taxable account will grow to by the end of the investment period after all taxes, including final capital gains tax, are considered. The pre-tax principal method then calculates an effective compounding rate for the annual return to achieve the same terminal value. In contrast, the post-tax principal value determines what the after-tax return should be and discounts the expected future value using this discount rate to determine the asset s post-tax principal value. Due to the problems associated with their pre-tax principal method mainly due to the fact the tax adjustments can yield negative asset returns we focus on their post-tax principal method. The investor s expected terminal wealth value of the taxable account ( ) is based on the present value of taxable account ( ), expected returns less immediate taxes ( 1 ) compounded for the investment time horizon ( ), and adjusted for the tax on final capital gains: (3) where = initial cost basis on the investment 4

5 = tax rate on deferred capital gains = effective annualised tax rate for deferred capital gains The adjustment is important because the components of return that are taxed along the way and reinvested are not taxed again as a capital gain. It is calculated as: where 1 (4) = proportion of the returns that represent deferred capital gains = effective tax rate on gross income that accounts for all immediate sources of taxation. H&A specifies as a function of the proportion of income that are ordinary income ( ), dividend income ( ), immediate capital gains ( ), and their respective tax rates (,, ): (5) Because an unpaid tax liability can remain invested, the authors conclude that the appropriate discount rate should reflect the after-tax cost of equity for that asset. They approximate this by making adjustments to the standard CAPM as follows vi : where 1 (6) = market risk of the asset = expected return on the market portfolio = risk-free rate Using this discount rate, along with the expected future value already calculated in equation (3), they calculate the after-tax present value of an asset in the taxable account ( ) as follows: 5

6 1 P (7) From the above it is clear that they effectively adjust the principal value for all current and future expected capital gains taxes. For the TDA, the investor must pay a 30% withdrawal tax on the gross amount and is essentially a caretaker of this money, with the government taking on all the risk associated with this 30%. As such, the principal value of the TDA needs only be lowered by a flat 30% to make it effectively tax-free. The Roth account being untaxed requires no adjustments. The investor s total post-tax wealth is simply the sum of these aftertax values, while the post-tax weights are measured as the account s post-tax values divided by the investor s total post tax wealth. As it is assumed that the government equally shares in both gains and losses, taxes have no impact on the correlations between the assets. vii Nevertheless, the covariance terms do become smaller since the average deviation from the mean will be less as a result of this risk-sharing. H&A generate formulas for the after-tax covariances between the different asset classes and locations. Their adjusted equations for standard deviation and covariance are as follows: and where cases. viii 1, (8), 1, 1, (9), = effective tax rates on gross returns for immediate income for assets i. Note that for the Roth account and TDA are zero, so these terms drop out in some 6

7 Improved Method for Considering Taxes When looking back at equations (6) and (7) above, one question that may crop up is why combine a pre-tax risk-free rate with an after-tax risk premium? H&A (2008, pp ) argue that the government is a silent risk-sharing partner (we agree) and conclude that this only relates to the risk premium (we disagree). They argue that this risk-sharing does not extend to returns on the risk-free rate, because it by definition has no risk and the government cannot share in something that does not exist. However, it is not difficult to recognize that the government shares in risk-free returns regardless of the lack of risk. For example, the government still manages to tax interest on government bonds despite its essentially riskless nature. This renders the H&A discount rate inherently flawed (Reichenstein, 2007c). ix Simply correcting the above flaw, however, would not account for the change in risk to the investor s equity. To illustrate this changing nature of the risk, consider the following example. An investor has a $100,000 asset with an embedded tax liability that has a present value of $10,000 leaving the investor with true wealth of $90,000. Using H&A s method, we are lead to believe that his after-tax allocation is $90,000 and that $100,000 with embedded capital gains is equivalent to owning a $90,000 asset without embedded gains. However, the investor still has $100,000 exposure to shares so that if the underlying asset increases by 10%, the investor s asset will grow by $10,000, not $9,000. This means that 111% of the investor s true wealth is invested in shares and -11% is invested in cash as a $10,000 loan from the government which is effectively a leveraged investment. In order to maintain the investor s desired asset allocation of say 100% equity, he would need to reallocate 11% of his wealth (or $10,000) to the risk-free asset to meet the future tax liability when it falls due. The amount invested would grow at the risk-free rate less tax on ordinary income, compounding yearly (n). That is, it would grow by 1 1. This method 7

8 of investing the present value of the embedded tax liability into the risk-free asset maintains a constant risk level for the investor. Hence, using the after-tax risk-free rate is the most appropriate rate for calculating the present value of tax liabilities. The present value of the tax liability is therefore calculated as: where = deferred capital gain = tax on those gains 1 1 (10) There are other ways to calculate the present value of the tax liability including: (i) an amortisation approach where the liability is paid gradually; and (ii) a reducing balance approach, e.g. 20% of the embedded liability is triggered each year. Alternatively, there could be a combination of these methods. The method chosen may depend on the type of investment. For example, a company stock directly owned by the investor is likely to only have a capital gains tax (CGT) event at sale, while an indirectly owned managed fund may regularly generate CGT events due to normal fund turnover. However, the after-tax risk-free rate remains the appropriate discount rate to use. Equation (10) leads us to the new equation for the after-tax present value of the taxable account ( ): (11) Although equation (11) calculates the after-tax value of the taxable account, it does not consider the implicit leveraging effect within the taxable account. Hence, further examination is required. The H&A approach deducts the tax liability from equity and bonds on a pro-rated basis for all accounts. Instead of adjusting the bond and equity assets equally for the tax liability, we propose deducting the present value of the tax liability from the risk-free 8

9 allocation to reflect the effective interest free loan from the government. In our proposed method, the post-tax principal value of the equity allocation is unaffected by the future tax liability and,,. x The present value of the future tax liability is instead deducted from the allocation to the risk-free asset, such that:,. (12) These proposed changes will correctly adjust for the change in risk to the investor s wealth and are able to incorporate the implicit leveraging effect. If preferred, the present value of the tax liability could be deducted from the bond allocation, but in this case the present value calculation should use the return on those bonds as well. The investor should use whichever method best reflects their investment opportunities. Importantly, while the H&A method incorporates the existing and future capital gains into their present value calculations, our method adjusts present values only for existing capital gains. To achieve this, our model incorporates the present value of any new (or reduced) capital gains tax generated in any period as part of that period s return. Intuitively this makes more sense since the increase or decrease in the embedded tax liability is in fact a component of that asset s investment returns. To accomplish this, we rewrite equation (1) incorporating the present value of any new embedded liabilities created or lost during the period. The formula becomes: 1 t (13) 1 1 where = proportion of the pre-tax returns that are deferred capital gains That is, the after-tax return is equal to pre-tax returns ( ) multiplied by 1 less the tax rate on gross returns for all immediate sources of income ( ), less the present value of the 9

10 future tax liability. Here, n refers to the investment horizon, (n-1) reflects how long the capital gains tax is deferred compared to taxes on dividends, and is the tax on deferred capital gains. Taxes on dividends are assumed to occur at the end of the year in which they are received. With respect to capital gains tax, if this same logic is followed, the tax would occur at the end of the n-th year i.e. (n -1) years after the tax on dividends is due. xi The adjustment to after-tax standard deviation is proportional to after-tax returns, which yields: 1 t. (14) 1 1 For covariance calculations with embedded taxes all that is required is to use the aftertax standard deviation of each asset and substitute these into the equation for calculating the covariances. The covariance between assets i and j ( ) equals the correlation of assets and ( ) multiplied by their respective after-tax standard deviations ( and ): (15) The use of standard deviation and covariance as above is in contrast to the H&A method where the beta measure is used. We choose to avoid using beta as the mean-variance optimisation model should be based on total risk, not market risk. We explain the impact of this decision further in the next section. Empirical Results In this section we apply our modifications to the H&A method by using after-tax input parameters as outlined above. We use the same initial inputs as the H&A paper to facilitate comparison of our results to theirs and these are presented in Panel A of Table 1. Panels B and C show the tax-adjusted inputs using the H&A method and our method respectively. [Table 1 here] 10

11 Most of the differences between the two methods lie in the taxable account. As mentioned in the previous section, we do avoid using a beta term in our method, unlike H&A. When dealing with mean-variance optimisation matrices, the covariance matrices pick up the co-variation between the idiosyncratic returns, choosing the highest return for any given level of risk. By using beta in their model, H&A inherently consider only market risk, instead of total risk, creating a fundament flaw in their model. This shortcoming is apparent when looking at the tax-adjusted inputs for bonds in the taxable account in Panel B. The taxadjusted standard deviation in the taxable account drops from 6% pre-tax to 2.1% after-tax a huge 65% drop, while the tax-adjusted return is reduced by a mere 9%, even although bond returns are taxed at a rate of 30%. Instead, our method adjusts the actual standard deviation (the total risk, not just the market risk) of the asset, yielding a tax-adjusted standard deviation of 4.2% (Panel C). With respect to the tax-adjusted bond returns, the H&A method assumes that the government only shares in the market risk premium and not the base return for the risk-free rate. As a relatively small portion of bond returns is reward for market risk, this assumption means that bonds are largely tax-free! The reality, however, is that the government shares in the entire return. Therefore, our method imposes taxes on both the market premium as well as the risk-free component resulting in a tax-adjusted return of 2.8% for the taxable account s bonds, a 30% drop from pre-tax levels commensurate with a 30% tax rate. For the taxable account s equity return their assumption is not an issue because the market risk of the equities is also its total risk. There are marked differences between the methods for the taxable account for the present value of the embedded tax liability and the equity rate of return. While this is partly attributable to the non-taxation of the risk-free component of return, it is also a result of the way in which deferred capital gains are dealt with. The H&A method deducts the future 11

12 deferred capital gains taxes from the initial principal value rather than the annual return. Our method takes a more intuitive approach, acknowledging that future capital gains taxes generated are part of that period s return and not an existing liability. This results in our method reaching a relatively small present value for the tax liability at 0.66% of pre-tax wealth, compared to 7.5% for H&A. Also, our tax-adjusted return is 6.47% compared to H&A s 7.43%, the latter ignoring tax on deferred gains and tax on the risk-free component. Finally, for the taxable account we deduct the present value of the tax liability from the cash allocation rather than from the equity allocation, which more correctly reflects the loan from the government. With our assumption of 10% embedded capital gains, this creates a slight leveraging effect within the taxable account. By contrast, in adjusting the taxable account for all existing and future deferred capital gains, the H&A method overadjusts for taxes in the taxable account compared to the TDA and Roth accounts. This is evident in the difference between post-tax allocations of wealth. Following the H&A model would result in 40% of post-tax wealth attributable to the taxable account compared to a 45% post-tax wealth allocation with our model. Table 2 presents the efficient frontiers resulting from the two methods, along with the resulting asset allocations for each point along the frontier. The optimisation routine varies the pre-tax weights in each asset, subject to borrowing and lending restrictions that ensure that no short selling occurs and that maximum investment in each category does not exceed its initial endowment. Because the Roth account and TDA have identical after-tax returns after the TDA s principal value has been adjusted, we hold the ratio of equity to bonds constant between these two accounts. xii Table 2 presents the asset allocations based on pre-tax principal weights, although the post-tax asset weights (not reported here) are used to calculate the portfolio s return and risk. [Table 2 here] 12

13 Both methods produce a consistent reduction in the overall allocation of equity across the accounts, as expected. The H&A method in Panel A shows that in order to reduce risk, investors should start reducing allocation to risky assets in their retirement accounts. However, this trend is not monotonic about half way down the risk spectrum, investors should switch the entire portion of equity to bonds in the taxable account for the lower half of the risk spectrum. In their paper, they also comment on a preference to replace equity in the TDA rather than in the Roth account (p. 68), but this can be attributed to the optimisation routine, because both accounts have the same tax-adjusted inputs. xiii Our results in Panel B create an even stronger, and more monotonic, preference for holding equities within the taxable account, suggesting that investors switch to bonds in the retirement accounts first. The only exception is at the bottom end of the spectrum, where 0% equity is held in the taxable account, and a small portion of the retirement account is held in equities. This leads us to conclude that all but the most risk averse investors should maintain a 100% allocation to equities in the taxable account and adjust their risk by switching to bonds in their retirement accounts. This preference for holding equities in the taxable account is most likely due to the relatively favourable taxation of equity returns compared to bond returns, which leads to an increased after-tax market risk premium. These findings bring the H&A results in line with other works in the literature. The first body of work is Leibowitz (2003), and Leibowitz and Bova (2009) who succinctly illustrate the increased after-tax market risk premium. And second, the work of Dammon, Spatt and Zhang (2004) who find a preference for investing the taxable account balance into equities, and the tax-deferred account balance into bonds. In the latter study, the authors examine the case in an intertemporal context, and use numerical methods to calculate the optimal allocation and locations. The confirmation of their multi period findings with our single period model 13

14 further suggests that our model is picking up some extra timing effects that the H&A method was missing. [Figure 1 here] So how do these efficient frontiers compare? Figure 1 plots the mean-variance efficient frontiers using the results from Table 2 together with the pre-tax efficient frontier. Our model creates a noticeable downward shift in the efficient frontier when compared to the H&A paper. In other words, the after-tax efficient frontier is less efficient than what was suggested by H&A. This shift is attributable to two reasons. First, H&A do not adjust the risk-free component of returns and only tax the beta component of returns. Second, they adjust for change in deferred capital gains in the post-tax principal value while we adjust for this in the expected returns. [Figure 2 here] Assuming that our method reveals the true post-tax mean-variance optimisation, how inefficient are the allocations suggested by the H&A model, and a naive tax unaware model that calculates a single risky weight, and applies this to all tax environments? Figure 2 plots these results, and surprisingly, the H&A method is less efficient than the naive allocation generated in the pre-tax method! xiv The underperformance is most noticeable in the middle of the risk spectrum, and at a standard deviation of 8.7% the H&A method underperforms by 15.8 basis points while the pre-tax method underperforms by just 6.7 basis points. Over a 20 year investment period, the investor could expect to have up to 3.2% more wealth at retirement using our improved method, instead of H&A s method. We expect that our method would consistently outperform the alternative methods, even in the presence of estimation error. [Figure 3 here] 14

15 As a final comparison, we examine how each model values deferred capital gains. For the H&A method, they implicitly discount the deferred capital gains using an after-tax discount rate of 7.43% for equity. We, on the other hand, use the after-tax return of the riskfree rate to calculate present values. The capital gains tax multiplier for both methods are displayed in Figure 3. By using a higher discount rate, the H&A model finds the present value of deferred capital gains tax to be a mere 5.7% of its future value after 40 years. The corresponding estimate using our method is 43.5%. Our approach shows that considering the impact of deferred capital gains on wealth, be they existing embedded capital gains or future capital gains yet to be accrued, is important even at longer time horizons. Conclusion This study advanced the literature on dealing with taxation in a mean-variance optimisation model in a number of important ways. Firstly, it acknowledged the leveraging effect caused by embedded capital gains, as a result of the implicit interest-free loan from the government. Using this insight, we explained how the investor s risk can be returned to a non-leveraged level using risk-free investments, and concluded that this rate is therefore the most appropriate to use when valuing these implicit future tax liabilities. This approach correctly incorporates the time value of money and the change in the nature of the risk to the investor. We adapted the model used by H&A using this new method to value embedded capital gains, while making a number of other minor adjustments such as ensuring that total investment risk is adjusted for taxes, and not just for market risk. Using our approach, the effective post-tax asset allocations are adjusted for embedded capital gains. Subsequently, expected returns and standard deviations are adjusted for the present value of any new or reduced deferred capital gains generated during that period. We find that the presence of embedded taxes increases the investor s equity risk and must be considered when calculating their current asset allocation. Our findings also suggest that 15

16 investors should prefer holding their risky assets (like stocks) in the tax environment with the highest tax rate and prefer to hold fixed income assets (like bonds) in their retirement accounts. This most important finding contradicts the traditional investment heuristic that retirement accounts are more suited to taking on equity risk due to their long investment horizon, and increased tolerance for risk. This confirms and strengthens the findings of Horan and Al Zaman (2008). Furthermore, these findings support the results of Dammon, Spatt and Zhang (2004) who use a numerical approach, and also Leibowitz and Bova (2009) who show that the presence of taxes increases the equity risk premium by taxing bond returns at higher average rate than equity returns. 16

17 Table 1. Pre-Tax and Tax-Adjusted Mean-Variance Inputs Notes: This table displays the pre-tax and post-tax mean-variance inputs for our improved method and that of Horan and Al Zaman (2008). The risk-free rate is assumed to be 3%. *The -0.76% allocation to bonds is actually the allocation to the risk-free rate. 17

18 Table 2. Tax-Adjusted Mean-Variance Efficient Frontiers and Asset Allocations Notes: This table compares the mean-variance efficient asset allocations computed using our improved method and that of Horan and Al Zaman (2008). The mean-variance inputs are from Table 1. While the expected returns and standard deviations are based on the post-tax weights, the displayed asset allocations are pre-tax. 18

19 Figure 1. Pre- and Post-tax Mean-Variance Efficient Frontiers Notes: This graph compares the post-tax efficient frontiers of our improved method with that of Horan and Al Zaman (2008). 19

20 Figure 2. Mean-Variance Efficient Frontiers Notes: This graph shows the relative inefficiency of the naïve (pre-tax) and the Horan and Al Zaman optimal weights compared to the improved method. 20

21 Figure 3. Present Value of Tax Liabilities with Different Investment Horizons Notes: This graph compares the reduction in the present value of the capital gains tax over time for our improved method with that of Horan and Al Zaman (2008). 21

22 References Arnott, R. D., Berkin, A. L., & Ye, J. (2000). How Well Have Taxable Investors Been Served in the 1980s and 1990s? Journal of Portfolio Management, 26(4), Berkin, A. L., & Luck, C. G. (2010). Having Your Cake and Eating It Too: The Before- and After-Tax Efficiencies of an Extended Equity Mandate. Financial Analysts Journal, 66(4), Dammon, R. M., Spatt, C. S., & Zhang, H. H. (2004). Optimal Asset Location and Allocation with Taxable and Tax-Deferred Investing. Journal of Finance, 59(3), Horan, S. M. (2007). Applying After-Tax Asset Allocation. The Journal of Wealth Management, 10(2), Horan, S. M., & Al Zaman, A. (2008). Tax-Adjusted Portfolio Optimization and Asset Location: Extensions and Synthesis. Journal of Wealth Management, 11(3), Leibowitz, M. L. (2003). The Higher Equity Risk Premium Created by Taxation. [Article]. Financial Analysts Journal, 59(5), Leibowitz, M. L., & Bova, A. (2009). Return-Risk Ratios Under Taxation. Journal of Portfolio Management, 35(4), Reichenstein, W. (2006). After-Tax Asset Allocation. Financial Analysts Journal, 62(4), Reichenstein, W. (2007a). Calculating After-Tax Asset Allocation Is Key To Determining Risk, Returns and Asset Location. Journal of Financial Planning, 20(7), Reichenstein, W. (2007b). Implications of Principal, Risk, and Returns Sharing Across Savings Vehicles. Financial Services Review, 16(1), Reichenstein, W. (2007c). Note on "Applying After-Tax Asset Allocation". Journal of Wealth Management, 10(2), Reichenstein, W. (2008a). How to Calculate an After-Tax Asset Allocation. Journal of Financial Planning, 21(8), Reichenstein, W. (2008b). One Concept and Some of Its Applications. Journal of Wealth Management, 11(3), Wilcox, J., Horvitz, J. E., & dibartolomeo, D. (2006). Investment Management for Taxable Private Investors: Research Foundation of CFA Institute. 22

23 Notes i See, for example, Wilcox, Horvitz & dibartolomeo (2006). ii Embedded Tax Liabilities are only incurred by mutual fund investors and not by individuals who invest via separate accounts or hedge funds. They are also only incurred by taxable accounts. iii Leibowitz (2003) discusses the implication of preferential taxation, finding that although the total return reduces, the risk premium associated with market risk actually increases. (See also Leibowitz & Bova, 2009). iv Taxable accounts are the normal tax accounts as opposed to the concessionally-taxed retirement accounts. v Note that a TDA can be considered tax-free if you notionally adjust the principal value down by 30%, because adjusting the final wealth value for tax at the end is equivalent to adjusting it at the beginning ( ). Using this knowledge, the number of effective tax environments can be reduced to two. vi The derivation of this approximation is shown in the appendix of Horan (2007). vii Note that in a more realistic setting, where taxes are asymmetrical and the government only shares in positive returns it is likely that correlations would change. viii There is an oddity in these adjustments. This oddity lies in the use of beta in the calculations, which in effect is calculating the investments systematic risk, when optimisation models specifically call for the total risk of the asset. This is an obvious error when one sets the tax rate to 0%. ix Reichenstein addresses this in an article discussing what he describes as their philosophical differences. Reichenstein illustrates that under Horan and Al Zaman s equation in (7), when the investment horizon is set to 1, with a risk-free rate of 5% and a tax rate of 30% present value becomes: $1 $1. $ Surely $1 held in a bank account. today is worth $1 today! x A new risk-free asset must be introduced to the model if one does not already exist. Alternatively, this can be deducted from the bond allocation, although it will lead to a slight discrepancy between the investor s true risk and the risk portrayed by the model. xi For practitioners it is generally fairly easy to manipulate the timing of capital gains, for example to occur at the beginning of the n+1 period instead of at the end of the n-th period, so this discounting formula could just as easily be n instead of n-1. In order to maintain consistency, if n is used in equation (13), then n+1 should be used in formulas (10) and (11). The difference between the two approaches is marginal. xii Horan and Al Zaman (2008) do not hold this ratio constant. However, it is held constant for both our method and the H&A model. xiii This optimisation problem is not an issue here because the split between equity and bonds between the TDA and Roth accounts is the same. xiv A strategy that prefers to hold equities in the superannuation account, and cash in the taxable account would have similar inefficiencies to the H&A model in the lower part of the frontier, but worse in the upper part. 23

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