Introduction
In the context of US tax law, does it ever make sense to harvest long term capital gains (LTCG) when one's income puts one into the $15\%$ LTCG tax bracket? Perhaps there is value to be gained by paying LTCG taxes now and increasing the basis so that one pays lower LTCG taxes in the future. Is that opportunity cost worth it?
This post investigates the conditions for LTCG harvesting to yield a net benefit in after-tax spendable wealth under a simplified model. It is found that LTCG harvesting has a clear long-term advantage when done in the $0\%$ LTCG tax bracket, whereas harvesting in a positive LTCG tax bracket can provide an advantage only over a finite investment horizon assuming future LTCG tax rates are sufficiently higher. The longer the post-harvest investing duration, the greater the erosion in the advantage. Eventually, once the post-harvest investing duration crosses a break-even horizon, any advantage due to LTCG harvesting (done in a positive LTCG tax bracket) is completely lost.
Terminology
| Term | Description |
|---|---|
| $P_0$ | Initial principal that will be harvested |
| $B_0$ | Cost basis of $P_0$ |
| $m$ | Duration of analysis (years) |
| $r_c$ | Capital appreciation rate $(r_c \ge 0)$ |
| $r_d$ | Dividend yield $(r_d \ge 0)$ |
| $T_{c_0}$ | Effective capital gains tax rate at the start $(0 \le T_{c_0} < 1)$ |
| $T_{c_m}$ | Effective capital gains tax rate at the end of the analysis period $(0 \le T_{c_m} < 1)$ |
| $T_d$ | Effective dividend tax rate $(0 \le T_d < 1)$ |
Assumptions
- $B_0$ has been invested long enough to qualify for LTCG
- $B_0 < P_0$ so that LTCG taxes will be due if liquidated
- LTCG taxes are paid from the harvesting proceeds, and the remainder is reinvested
- All dividends are taxable immediately at a constant effective $T_d$
- Dividends, net taxes, are reinvested
- $r_c$ and $r_d$ are annual returns measured relative to the beginning-of-year portfolio value
- $r_c$, $r_d$, and $T_d$ stay constant over the duration of the analysis ($m$ years)
- At the end of year $m$, the strategies under comparison are fully liquidated and the resulting after-tax spendable amount is compared
- US tax laws from 2026 apply
Preliminaries
Refer to the calculations in Redirecting Taxable investments to Roth 401(k) - An Improved Analysis to motivate the definitions below:
$\begin{align}G_\text{div} &= r_d (1 - T_d)\\G_\text{tax} &= 1 + r_c + G_\text{div}\\G_\text{basis}(m) &= G_\text{div}\left( \frac{G_\text{tax}^m - 1}{G_\text{tax} - 1} \right) &(\text{for } G_\text{tax} \ne 1)\\&= m G_\text{div} &(\text{for } G_\text{tax} = 1)\end{align}$
For passive long-term investing it's reasonable to take $G_\text{tax} > 1$ so that $r_c + G_\text{div} > 0$.
Default strategy
"Do nothing."
In $m$ years, $P_0$ grows to
$$P_m = P_0 G_\text{tax}^m$$
Dividend reinvestment increases the basis (see detailed workthrough here) by
$$\Delta B = P_0 G_\text{basis}(m)$$
so that new basis is
$$B_m = B_0 + \Delta B$$
After-tax spendable amount in the default strategy is
$\begin{equation}P_\text{default, after-tax} = P_m (1 - T_{c_m}) + B_m T_{c_m}\end{equation}$
LTCG harvesting strategy
"Harvest LTCG at the start, pay taxes from the proceeds, reinvest the rest, and let the account grow."
The strategy is deliberately defined this way in order to have the same starting assets in both strategies.
Post-harvest account value
$$P'_0 = P_0 (1 - T_{c_0}) + B_0 T_{c_0}$$
Post-harvest basis
$$B'_0 = P'_0$$
Note that $P'_0 < P_0$ as $B_0 < P_0$ by assumption. Conversely, $B'_0 > B_0$.
In $m$ years, $P'_0$ grows to
$$P'_m = P'_0 G_\text{tax}^m$$
Note that $P'_m < P_m$ as $P'_0 < P_0$.
As before dividend reinvestment increases the basis by
$$\Delta B' = P'_0 G_\text{basis}(m)$$
so that new basis is
$$B'_m = B'_0 + \Delta B'$$
Note that $\Delta B' < \Delta B$ as $P'_0 < P_0$.
After-tax spendable amount in the LTCG harvesting strategy is
$\begin{equation}P_\text{harvesting, after-tax} = P'_m(1-T_{c_m}) + B'_m T_{c_m}\end{equation}$
Analysis
The advantage of the default strategy over the LTCG harvesting strategy is
$$\begin{align*}\Delta &= P_\text{default, after-tax} - P_\text{harvesting, after-tax}\\&=(P_m - P'_m)(1 - T_{c_m}) + (B_m - B'_m)T_{c_m}\\&=(P_0 - P'_0)G_\text{tax}^m(1 - T_{c_m}) + (B_0 - B'_0 + \Delta B - \Delta B')T_{c_m}\end{align*}$$
Now,
$$P_0 - P'_0 = (P_0 - B_0) T_{c_0}$$
$$B_0 - B'_0 = - (P_0 - B_0)(1 - T_{c_0})$$
$$\Delta B - \Delta B' = (P_0 - P'_0)G_\text{basis}(m) = (P_0 - B_0) T_{c_0}G_\text{basis}(m)$$
Plugging the above into the equation for $\Delta$
$$\begin{align*}\Delta &= (P_0 - B_0) G_\text{tax}^mT_{c_0}(1 - T_{c_m})\\&\quad\ - (P_0 - B_0)(1 - T_{c_0})T_{c_m}\\&\quad\ + (P_0 - B_0) G_\text{basis}(m)T_{c_0}T_{c_m}\end{align*}$$
Since, by assumption, $P_0 - B_0 > 0$, the sign of $\Delta$ is the same as that of $\Delta/(P_0 - B_0)$.
In the above expression:
- $G_\text{tax}^mT_{c_0}(1 - T_{c_m})$ represents the (normalised) future cost of having paid the initial tax and therefore having less money invested. This favours the default strategy.
- $- (1 - T_{c_0})T_{c_m}$ represents the (normalised) benefit of the basis reset and favours the LTCG harvesting strategy.
- $G_\text{basis}(m)T_{c_0}T_{c_m}$ is a (normalised) second-order effect caused by the fact that the default strategy has a larger portfolio and therefore reinvests more after-tax dividends, generating more future basis. This favours the default strategy.
A special case is $T_{c_0} = T_{c_m} = T_c > 0$ (i.e., LTCG tax rates are the same at the start and end). Then,
$$\frac{\Delta}{P_0 - B_0} = (G_\text{tax}^m - 1)(1 - T_c)T_c + G_\text{basis}(m)T_c^2$$
If $G_\text{tax} > 1$, then the right-hand side is always positive and the default strategy has the better outcome. This is why the vast majority of investors do not routinely reset their basis by LTCG harvesting as they are very likely going to stay in the same LTCG tax bracket (usually, $15\%$) for their foreseeable future.
Going back to equation (\ref{eqn:delta:normalised}), if $T_{c_0} > 0$, then $G_\text{tax}^mT_{c_0}(1 - T_{c_m})$ and $G_\text{basis}(m)T_{c_0}T_{c_m}$ are both positive. Further, if $G_\text{tax} > 1$, then they both grow monotonically with $m$. On the other hand, the negative term $-(1 - T_{c_0})T_{c_m}$ is a constant. Thus, the right-hand side of the above equation can be made positive by a large enough choice of $m$. This is independent of the values of $T_{c_0}$ and $T_{c_m}$!
On the other hand, when $T_{c_0} = 0$, the right-hand side becomes $-T_{c_m}$ and is negative if $T_{c_m} > 0$.
This is usually the scenario for a retired individual with a taxable income low enough to provide ample space in the $0\%$ LTCG tax bracket for LTCG harvesting and who may be at risk of being pushed into a higher income bracket due to Required Minimum Distributions from their tax-deferred accounts.
The above discussion raises the question: Is there a window of time during which the LTCG harvesting strategy has the edge whenever both $T_{c_0}, T_{c_m} > 0$?
Setting $\Delta < 0$ (which is the condition for LTCG harvesting strategy to be beneficial) and solving for $T_{c_m}$,
provided the denominator on the right-hand side is positive (proof left as an exercise for the reader; hint: use Appendix A). This constrains the values that $m$ can take for fixed values for $T_{c_0}$ and $T_{c_m}$.
It can be shown that the expression on the right-hand side increases with $m$ (see Appendix B). The expression has its smallest value $T_{c_0}$ at $m = 0$ and, for $G_\text{tax} > 1$, becomes $1 + G_\text{div}/r_c > 1$ as $m \rightarrow \infty$. But, as $T_{c_m}$ cannot exceed $1$, the maximum value it can have is $1$. Thus, for a given $T_{c_m}$, there is a critical value of $m$ above which the inequality of relation (\ref{eqn:Tcm}) ceases to hold.
See Appendix C, equation (\ref{eqn:mBE}), which gives an analytical expression for $m_\text{BE}$.
As an illustration, for $r_c = 7.177\%/\text{year}$, $r_d = 2.75\%/\text{year}$, $T_d = 19\%$ (assuming mix of ordinary and qualified dividends), and $T_{c_0} = 15\%$ here's how the right-hand side (RHS) of relation (\ref{eqn:Tcm}) varies with $m$
| $m$ (yr) | RHS (%) |
|---|---|
| $0$ | $15$ |
| $1$ | $16.31$ |
| $2$ | $17.73$ |
| $3$ | $19.24$ |
| $4$ | $20.86$ |
| $5$ | $22.59$ |
If $T_{c_m} = 20\%$ (the highest LTCG tax rate in the US), the LTCG harvesting strategy has an advantage only for the first $3$ years. By the $4^\text{th}$ year, it's already starting to fall behind. And by the $5^\text{th}$ year the default strategy is comfortably in the lead. Thus, while there is a window during which the LTCG harvesting strategy holds a net benefit over the default strategy, that advantage quickly erodes particularly for those with a long-term investment focus.
The value of $m_\text{BE}$ for $T_{c_m} = 20\%$ calculated using equation (\ref{eqn:mBE}) is $3.7$ years.
Conclusion
Given the step-up in the cost basis of taxable accounts upon inheritance, the LTCG harvesting strategy yields absolutely no benefit for an inheritance/legacy goal. Hence the after-tax spendable amount is the only relevant consideration. This is what the post above explored.
LTCG harvesting at a $0\%$ tax rate is fundamentally different from harvesting when the current LTCG tax rate is positive. At a $0\%$ rate there is no upfront tax cost to resetting the basis, so harvesting can be advantageous when future capital gains are expected to be taxed at a higher rate.
When the current LTCG tax rate is positive, however, harvesting requires paying tax immediately and consequently reduces the amount of capital that remains invested. The resulting loss of future compounding increasingly offsets the benefit of the higher basis. If the current and future LTCG tax rates are the same, harvesting is strictly worse for any positive investing duration.
A sufficiently higher future LTCG tax rate can nevertheless make harvesting advantageous over a finite investment horizon. The required future tax rate increases monotonically with the length of that horizon, however, and eventually exceeds $100\%$. Consequently, for any positive starting LTCG tax rate, there is always a sufficiently long holding period after which the default strategy overtakes the harvesting strategy regardless of the future LTCG tax rate.
Thus, LTCG harvesting in a positive tax bracket should not generally be viewed as a long-term basis-reset strategy. It can make sense when there is a credible expectation of a substantially higher future effective capital-gains tax rate and the gains are likely to be realized within the relatively short window in which that higher rate justifies paying the tax today. Otherwise, allowing the gains to compound and paying the tax later is preferable under this analysis.
Appendix A: $G_\text{basis}(m) < G_\text{tax}^m$
By definition,
$$G_\text{basis}(m) = r_d(1-T_d)\frac{G_\text{tax}^m-1}{G_\text{tax}-1}$$
If $G_\text{tax} > 1$, then $r_c + r_d(1-T_d) > 0$, and $r_c$ and $r_d$ cannot both be $0$ simultaneously. Then,
$$0 \le \frac{r_d(1 - T_d)}{G_\text{tax}-1} = \frac{r_d(1 - T_d)}{r_c + r_d(1-T_d)} \le 1$$
because (a) the denominator is larger by the amount $r_c$, (b) $r_c$, $r_d$, $T_d \ge 0$, and (c) $T_d < 1$. Ergo,
$\begin{equation}G_\text{basis}(m) = \frac{r_d(1 - T_d)}{G_\text{tax}-1}(G_\text{tax}^m-1) \le G_\text{tax}^m - 1 < G_\text{tax}^m\end{equation}$
QED.
Appendix B: Right-hand side of relation (\ref{eqn:Tcm}) versus $m$
Let
$\begin{equation}f(m) = \frac{G_\text{tax}^m T_{c_0}}{G_\text{tax}^m T_{c_0} + (1-T_{c_0}) - G_\text{basis}(m)T_{c_0}}\end{equation}$
For simplicity, write $G_\text{tax} = G$, $G_\text{div} = D$, and $T_{c_0} = T_c$. Then, for $G > 1$ (a reasonable assumption for passive long-term investing), and using $G = 1 + r_c + D$ (where, by implication, $r_c + D > 0$),
where, given $0 \le T_c < 1$, $r_c \ge 0$, $r_c + D > 0$
$\begin{gather}\alpha = T_c,\quad 0 \le \alpha < 1\\\beta = \alpha(\frac{r_c}{r_c + D}),\quad 0 \le \beta < \alpha < 1\\\gamma = 1-\beta,\quad 0 < \gamma \le 1\end{gather}$
Then,
$\begin{align}\frac{\mathrm{d}f(m)}{\mathrm{d}G^m} &= \frac{(G^m \beta + \gamma)\alpha - G^m \alpha (\beta)}{(G^m \beta + \gamma)^2}\nonumber\\&=\frac{\alpha \gamma}{(G^m \beta + \gamma)^2} > 0\end{align}$
Thus, $f(m)$ is an increasing function of $G^m$ where $G^m$ itself is an increasing function of $m$. It follows that $f(m)$ is a monotonically increasing function of $m$.
Further,
$\begin{equation}f(0) = T_c\ (= T_{c_0})\end{equation}$
and, provided $r_c \ne 0$ (i.e., $\beta \ne 0$),
$\begin{equation}\lim_{m\rightarrow\infty} f(m) = \frac{\alpha}{\beta} = 1 + \frac{D}{r_c}\ (= 1 + \frac{G_\text{div}}{r_c})\end{equation}$
Appendix C: Break-even $m$ for relation (\ref{eqn:Tcm})
Using the definitions of $\alpha$, $\beta$, and $\gamma$ from Appendix B, relation (\ref{eqn:Tcm}) can be written as
$\begin{equation}T_{c_m} \ge \frac{\alpha G_\text{tax}^m}{\beta G_\text{tax}^m + \gamma}\end{equation}$
which after some algebra can be recast as
Solving for $m$ and using the equal sign for the break-even value $m_\text{BE}$
$$m_\text{BE} = \ln(\frac{\gamma T_{c_m}}{\alpha - \beta T_{c_m}}) / \ln(G_\text{tax})$$
Substituting all values and simplifying gives
The numerator of equation (\ref{eqn:mBE}) has an interesting symmetry. Let the argument of $\ln$ in the numerator be $R(T_{c_0}, T_{c_m})$:
Define
$\begin{equation}h(T) = \frac{T}{(1-T)r_c + G_\text{div}}\end{equation}$
Then
$$R(T_{c_0}, T_{c_m}) = \frac{h(T_{c_m})}{h(T_{c_0})}$$
and
$\begin{equation}m_\text{BE}(T_{c_0},T_{c_m}) = \frac{\ln[h(T_{c_m})] - \ln[h(T_{c_0})]}{\ln(G_\text{tax})}\end{equation}$
Now, $h(T)$ is an increasing function of $T$ (proof left as an exercise for the reader: find $\mathrm{d}h(T)/\mathrm{d}T$ and determine its sign). Hence,
$$T_{c_m} > T_{c_0} \iff h(T_{c_m}) > h(T_{c_0}) \iff R > 1 \iff m_\text{BE} > 0$$
Thus, a positive break-even horizon exists if, and only if, the future LTCG tax rate exceeds the current rate.
Swapping $T_{c_m} \leftrightarrow T_{c_0}$ shows that $m_\text{BE}$ is anti-symmetric in its arguments.
$$m_\text{BE}(T_{c_m},T_{c_0}) = -m_\text{BE}(T_{c_0},T_{c_m})$$
It follows that when the future LTCG tax rate is lower than the current rate (i.e., $T_{c_m} < T_{c_0}$), the break-even horizon is in the past. Thus, in this case, there is no positive holding period over which the LTCG harvesting strategy has an advantage.
The anti-symmetry of $m_\text{BE}$ further implies that
$$m_\text{BE}(T_c, T_c) = 0$$
To wit, right from the get go, the LTCG harvesting strategy is at a disadvantage if the expected future capital gains tax rate is the same as the current one.
All of the above results may be obvious. It is, nevertheless, nice to see them clearly in light of the symmetry of $R$.
For future: Why is equation (\ref{eqn:mBE}) symmetric in that way? What is the significance of $h(T)$?
No comments:
Post a Comment