*By Victor Haghani and James White*

^{1}
For investors who feel the expected return of equities isn’t as attractive today as it was 5 to 10 years ago when PEs were lower, it’s natural to be thinking about reducing exposure to the market. For taxable investors, reducing your allocation will also mean paying capital gains tax.^{2} As we’ve done in past notes on coin flipping and optimal trade-sizing, we’ll use a few basic assumptions about individual risk aversion to suggest a simple, intuitive rule-of-thumb for weighing up the trade-off between saving on tax versus achieving your target equity allocation.

Our general finding is that if you’ve already held your equities long enough to qualify for long-term capital gains treatment, then the virtues of rebalancing to your target allocation far outweigh the efficiency from deferring a tax payment. In this case, it’s optimal to rebalance your portfolio most of the way to your target – and simply rebalancing all the way to target is almost as good. *However*, if waiting to rebalance will deliver the relatively large benefit of converting short-term gains to long-term gains, or you expect a significant cut in capital gains tax rates in the next few years, then you will probably be willing to keep an allocation to equities well above the optimal level you’d desire if you were investing from scratch.

Let’s illustrate the question with an example. Say 6 years ago you had a view that the long-term pre-tax excess return of US equities over cash was 5%,^{3} and based on that you invested 50% of your taxable savings in US stocks, and the other 50% you kept in cash. Since that time, your equities have doubled in value, excluding dividends, and they’d have grown to represent 67% of your portfolio.^{4} But you now find equities less attractive to own than you did back then, as higher current PEs suggest lower future returns. Let’s say you think equities’ expected excess return is now down from 5% to 3%, and if you were investing your portfolio from scratch you’d want to have an allocation to equities of just 30%.

If you reduced your exposure down to 30% from 67%, you’d have to make a hefty capital gains tax payment of roughly $12.50 per $100 of equities you own, or 4.6% of the value of your portfolio, assuming a long-term capital gains rate of 25%.^{5}

…So what to do?

You’d think this is a pretty basic question that would have some well-distilled answers. Well, we asked Google ‘how to decide whether to realize a capital gain’ and we found a lot of answers, more than 15 million in fact. But we didn’t see any in the top 100 that went beyond ‘it depends’ and ‘consult your tax advisor.’^{6} We are most definitely not qualified tax advisors, and this is not tax advice, but let’s see if we can come up with a more useful analysis than ‘it depends.’

First, we need to quantify how much tax you’d save by not selling your appreciated equities today. We’ll start off by assuming the long-term tax rate is constant at 25% for your remaining investment horizon – later we’ll discuss results with modified assumptions. In this case, your tax savings arises entirely from deferring the realization of the capital gain – in doing so, you are in effect getting the use of the funds you’d have paid in taxes to invest in your portfolio. If the expected return on your portfolio of equities and T-bills is 1.9% – the blend of 30% in equities with a nominal expected return of 4% and 70% in T-Bills earning 1% – then for every $100 of equities you don’t sell, you get to earn 1.9% on the $12.5 of tax you’re not paying right now. This equates to a benefit of 0.24% per annum.^{7}

Great. We’re halfway there. Now we just have to figure out what it costs us to hold more equities than our original target of 30%. Here’s where we have to draw on a bit of our past discussions about optimal trade-sizing (you can find a brief refresher, illustrated with ketchup and fries, here).

Recall that, if you were investing the portfolio from scratch, we assumed that you want to have 30% of your savings in equities based on your view that equities had an expected excess return of 3%. A basic model of personal risk-taking, known as the Merton Rule,^{8} says that your optimal holding of a risky asset should increase in direct proportion to its expected return. That’s as simple a financial formula as you’ll find, and it seems pretty reasonable too. It’s simply saying that if you want to have 30% of your savings in equities when you expect equities to deliver a 3% excess return, then, all else equal, if you feel that the expected return of equities is 4%, you should want to have 40% in equities; more generally, for every extra 1% of return, you’d want to have an extra 10% allocated to equities.

We now put the two halves of the analysis together to get our answer. Every extra dollar of appreciated equities you don’t sell has an extra return of 0.24% a year. If you want to hold an extra 10% of equities for every 1% of extra expected return, then it follows that for an extra 0.24% of return, you’d optimally like to own an extra 2.4% of equities. So, your tax-adjusted optimal allocation to equities is 32.4%, very close to your optimal allocation ignoring taxes of 30%.^{9} This suggests you should move about 95% of the way to your 30% target allocation from your current allocation of 67%.^{10}

Working through this example suggests this very simple rule-of-thumb:

Extra equities to own above your optimal allocation ignoring taxes = where

*B = annual tax benefit on appreciated equities not sold*

*Q = how much equities you’d want to own ignoring taxes*

*r = your expected excess return on equities*

The bigger the unrealized gain, the higher the tax rate, the higher the return on equities or T-bills or the less risk-averse you are, the further you should be willing to diverge from your optimal allocation ignoring taxes. In our example, the benefit from holding a 32% allocation versus 30% is relatively modest, but the benefit in rebalancing down from 67% to 30-32% is high. At 67% allocation, the portfolio earns a risk-adjusted rate of -0.1%, suggesting you’d be better off owning nothing than maintaining your allocation at that level.

Let’s put this rule-of-thumb to work on a few other salient examples representing commonly encountered tax situations. All cases keep to our Base Case assumptions regarding expected excess return for equities, level of personal risk aversion and amount of capital gain:^{11}

Description |
Current Tax Rate |
Long-term Tax Rate |
Horizon (Years) |
Extra Allocation vs. Optimal from “Scratch” Allocation |

Base Case: deferring long-term cap gains | ||||

Hold to death for basis step-up, or donate | ||||

Move from high-tax to low-tax state (e.g. NY to WY) | ||||

Big Long-Term CG tax cut | ||||

Hold on for Short-term CG to turn into Long-term CG | ||||

Tax rate increase that exactly offsets value of deferring |

**What if I Expect Higher Tax Rates in the Future?**

The last entry in the above table shows how much tax rates would need to rise over the next year to exactly balance the value of deferral: just 0.35% with the assumptions we’ve used. In practice, the problem is more complex than this, as, for one thing, you would want to consider the analysis to multiple horizons. If the expected increase in tax rates outweighs the value of deferral to the relevant horizon, then this simplified analysis suggests you should realize capital gains entirely, pay the resulting capital gains tax and then reinvest in equities to your desired allocation from scratch.

**Conclusions**

Our simple rule-of-thumb suggests that in most scenarios involving a realization of long-term capital gains, when you do not expect a reduction in capital gains rates in the next few years and you’re pretty far from where you’d like to be, your optimal move is fairly close, but not all the way, to your desired allocation ignoring taxes completely. However, when it comes to bearing extra risk in order for a short-term capital gain to convert into a long-term capital gain, you should be willing to take quite a lot of extra risk to enjoy the benefit of that lower tax rate.

At Elm Partners, in rebalancing portfolios for both our Fund and SMA clients we are willing to tolerate quite high deviations from target allocations to avoid short-term capital gains, but we are much less tolerant for avoiding long-term gains. Thus, our approach to tax-efficient investing is broadly consistent with the rule-of-thumb we have presented here.

**Caveat Emptor!**

The purpose of this note was to provide a simple rule-of-thumb to help illustrate some, but not all, of the factors investors should consider in trying to take account of tax effects in their investment decisions. The tax code is much more complex than the simple representation we have used in our examples. For example, we did not give any treatment to the fact that there is an asymmetry in capital gains taxation, in that we pay tax on gains, but we may get very limited refunds from the government on capital losses. Or more succinctly, as some unfortunate soul once said, “Man cannot live on capital loss carry-forwards alone.” We likewise did not treat whether investors form their target allocation views based on pre-tax or post-tax returns, or whether investors view portfolio risk inclusive or exclusive of tax liabilities. Also, state taxation varies from state to state and adds another layer of complexity. And adding insult to injury, tax policy changes over time in hard-to-predict ways. In the words of Albert Einstein, *“The hardest thing in the world to understand is the income tax.”*

**A few references:**

Balcer, Y., and Judd, K., 1987, “Effects of Capital Gains Taxation on Life-Cycle Investment and Portfolio Management,” Journal of Finance, 42, 743-761.

Dammon, R., C. Spatt, and H. Zhang, 2001a, “Optimal Consumption and Investment with Capital Gains Taxes,” Review of Financial Studies, 14, 583-616

Dybvig, P., and Koo, H. K., 1996, “Investment with Taxes,” unpublished working paper, Washington University in St. Louis.

Merton, R., 1971, “Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” Journal of Economic Theory, 3, 373-413.

Odean, T., 1998, “Are Investors Reluctant to Realize Their Losses?,” Journal of Finance, 53, 1775-1798.

^{[1]}Victor is the Founder and CIO of Elm Partners, and James is Elm’s CEO.**Past returns are not indicative of future performance.**This not is not an offer or solicitation to invest.A big thank you to Larry Hilibrand, Ayman Hindy and WK, as well as our colleagues at Elm – Jeff, Arjun, Bruce, Gregg and Tara – for your help with this note.

^{[2]}Assuming the investments are not held in tax-advantaged retirement accounts. Also, we’re assuming the investor is not able to achieve total portfolio level asset allocation changes either through making exaggerated changes in non-taxable retirement accounts or through adding to investment accounts with further savings from earnings each year. Both of these can be tax efficient ways of moving one’s asset allocation without incurring capital gains taxes.^{[3]}To be precise, that 5% is the expected arithmetic return in excess of T-bills, a standard proxy for cash. Also, for the purposes of this example, we’re assuming that the T-bill rate = 1%. We are using US equities for this example for simplicity, but as you know, at Elm Partners we believe in global diversification.^{[4]}Assuming you spent your investment income. Also, more significantly, we’re ignoring the fact that you have a tax liability (think of it as negative cash) of $12.5 (25% of 50% of $100 of appreciated equities), which, if you think of it as negative cash, you may wish to subtract from your $50 of cash, leaving you with $41.7 of cash. Your allocation to equities taking this tax liability into account is actually 71%, higher than the more conventionally calculated allocation of 67% we’re using here.^{[5]}Federal + Obamacare tax + a gross-up for the phase-out of itemized deductions, but assuming you live in a state with no capital gains tax like Wyoming, Florida or Texas. See https://taxfoundation.org/how-high-are-capital-gains-tax-rates-your-state/. As per the previous footnote, we are not taking the embedded capital gains tax liability into account in doing this calculation of how much you have invested in equities. You’d need to sell even more equities to get to 30% in equities taking the capital gains tax liability into account.^{[6]}We did find several thorough treatments of this question in the academic literature, which we listed in the reference section below. For an example of the more typical treatment of this question, see this recent NY Times article here. Perhaps an explanation for why this problem hasn’t gotten the attention it deserves is because we so often delegate to active managers who don’t have an incentive to defer the realization of gains. However, long-term investing in broad index funds, as typified by the type of investing we do at Elm Partners, will force us to confront the question posed in this note more frequently.^{[7]}While we have considered plausible arguments that would lead us to value deferral at either the equity expected return or the risk-free rate, it should be noted that your choice of rate will not alter the general conclusion that the value of deferral is not significant enough to move you far from your optimal allocation ignoring taxes, unless your basis is very low. This is the case wherein you defer the capital gains indefinitely and your post-tax portfolio return converges to the pre-tax return, as shown by this formula:After tax rate of return = ((1-tau) * (1+r)

^{T}+ tau)^{(1/T)}– 1where r is the pre-tax return on your portfolio, tau is the tax rate and T is your horizon. You can also see this by realizing that in the very, very long-term, if you never sold your equities, the initial basis would converge to a tiny fraction relative to the value of your holding. So, at that point, you’d be earning the pre-tax return on your entire tax liability which would result in an after-tax return equal to the pre-tax return.

^{[8]}Merton rule: for a portfolio with cash earning rate r and a risky investment with expected return µ and volatility σ, the optimal fraction of wealth to invest in the risky asset is (µ-r) / (γσ²), where γ reflects the investor’s idiosyncratic level of risk-aversion. For an investor who has an optimal holding of equities of 30% for a 3% expected excess return, and assuming equity market annual volatility of 18%, we get a coefficient of risk aversion of 3, signifying the investor is 3 times as risk averse as a Kelly bettor.^{[9]}This is a rule-of-thumb, so we’ve left out the effect of your blended rate at the adjusted allocation being different than the blended rate at the optimal allocation ignoring taxes. This is a relatively small effect, and seems worth putting to the side for a rule-of-thumb.^{[10]}If, even after the 100% equity rally, the investor still felt that the expected excess return of the stock market was unchanged at 5%, then 50% would remain her optimal allocation ignoring taxes, and the optimal rebalancing factoring in taxes would be to reduce the allocation to equities from 67% to 54%.^{[11]}Calculations for tax benefit in each case are as follows, where g=gain, r = blended return on portfolio at desired allocation ignoring taxes, T= horizon in years, tau_{0 }= tax rate at start, tau_{T }= tax rate at horizon:- Base Case: g
_{ *}tau_{0 * }r - Hold-until-you-die: Base Case + g * tau
_{0 }/ (1- tau_{0}) / T - Tax cut or move to lower tax jurisdiction: g
_{ *}tau_{ o * }r + g * (tau_{0}-tau_{T}) / (1 – tau_{0}) / T - Converting ST to LT: g
_{*}tau_{ T * }r + g * (tau_{0}-tau_{T}) / (1 – tau_{T}) / T

In case of Converting ST to LT, the maximum deviation is the starting allocation minus desired allocation.

- Base Case: g