Harvesting as a Stochastic Dividend | Research

Working paper · SSRN 7507839 · Posted 23 Sep 2026 · 30 pages

If tax-loss harvesting pays a dividend that grows with volatility, how much more risk should a taxable mean-variance investor hold?

Anirban Majumdar · InnovationStrat Consulting, LLC · ORCID

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Key results

At the base calibration the optimal equity weight moves from 0.65 to 0.92, about 41%, and the after-tax reward-to-risk slope from 0.31 to 0.43.
The dividend is convex on the downside, so a power-utility investor tilts by 48 to 49% instead of 42%.
Fitted to index data from 1993 to 2026, the dividend is 91 basis points in calm markets and 319 in turbulent ones.
In one simulated account the dividend falls from 160 basis points in year one to 21 by year five.

Summary

A cash flow that grows with volatility

When a stock in a taxable account falls below its cost basis, its owner can sell it, buy a close substitute and bank a capital loss without leaving the market. That loss shelters gains realized elsewhere, and the tax it saves is cash. How much an account collects this way depends on how far and how often its holdings fall, so the benefit rises with volatility, which is an odd property for anything a mean-variance investor receives. Volatility is what that investor pays to avoid.

Papers on portfolio choice price variance as a cost, and papers on harvesting price volatility as a source of extra return. We wanted both in one objective. So the paper treats the yearly harvest benefit as a dividend: roughly proportional to volatility, and larger in the years the portfolio does badly. We add it to a one-period mean-variance problem and solve for the best share in the risky asset again.

The base case

The dividend changes the solution in two ways. Its expected level adds return, and because the dividend pays more when markets fall, it also offsets part of the variance the investor carries. Both push the optimal weight up.

We calibrated the dividend with a fixed-seed simulation of a 60-name sleeve that harvests whenever a name trades 5% below its basis. Take portfolio volatility of 16%, an equity premium of 5%, a blended tax rate of 0.30 and a usage factor of 0.35, meaning roughly a third of the idealized paper losses become usable tax savings. Those inputs give a dividend of 1.6% a year. For a moderately risk-averse investor the optimal equity weight moves from 0.65 to 0.92, about 41% higher, and the after-tax reward-to-risk slope goes from 0.31 to 0.43.

The percentage increase does not depend on risk aversion. Table 1 shows 41.5% at every level, while the move measured in weight shrinks from 0.41 for an aggressive investor to 0.14 for a conservative one. We had expected the hedging channel to count for more. It adds about 7%; the level of the dividend supplies the rest.

There is also an asymmetry between the two sides of the problem. Losses are harvested name by name, so the dividend follows the volatility of individual stocks, while the variance penalty falls on the diversified portfolio. In the calibration that raises the scaling against portfolio volatility by a factor of 1.79.

Where the straight line holds

Figure 1 plots the simulated harvest yield against name volatility. Between about 0.16 and 0.40 the curve is nearly straight. Below that band few names fall 5% and harvesting is starved; above it the curve bends over as capacity binds. The usage factor is the softest input we have, and moving it from 0.20 to 1.0 takes the weight increase from 23% to 134%.

Figure 1. Simulated annual harvest yield against name-level volatility. The relationship is near-linear through the practical band and then saturates.

Removing the simplifications

Version 2 of the paper drops, one at a time, the linear dividend, the single period and the assumption that the inputs are known. Start with shape. On 3,000 fresh one-year paths the dividend is convex on the downside: in the worst tenth of years, when the sleeve loses 19%, it averages 284 basis points where the straight line predicts 252, and in no year does it go below zero, although the line says it can. A power-utility investor likes that floor. She tilts by 48 to 49% instead of 42%.

Noise? An uncertain usage factor? Neither moves the optimum by as much as a tenth of a percent, because both enter the variance penalty in units of the dividend squared, and the dividend is small next to volatility. A cap at a weight of one binds for investors whose risk aversion is below 2.76. It cuts the tilt short and never reverses it. At the base case the tilt is worth 26 basis points a year in certainty-equivalent terms.

The dividend changes with the market and with age

Does the dividend stay put? No. We fitted a two-regime hidden Markov model to daily returns on the S&P 500 trust from February 1993 to August 2026, 8,445 observations in all, and pushed each regime's volatility through the simulated yield curve: 91 basis points of dividend in calm spells, 319 in turbulent ones. Calendar years spread wider still, from 41 basis points in 2017 to 440 in 2008. In the six years the index fell the figure averaged 282, against 151 in the years it rose, and its correlation with the year's return is -0.54.

Account age matters as much. Over five simulated years one account's dividend runs 160 basis points, then 55, and reaches 21 by year five, about an eighth of where it began, since every recovery embeds a gain and fewer names end up below cost. What forecasts next year's harvest best is the account's own basis position, which custodians already report; market variables add little. Out of sample, R² goes from 0.16 using the year alone to 0.35 once last year's yield and the two basis measures are in. In the model, then, the tilt is largest in an account's first year and fades after it.

A recurrent network, finally, read each simulated year's harvest off its daily path with an R² of 0.99, against 0.58 for a rule on realized volatility. That helps in measuring a dividend after the fact. By then, of course, the loss is already on the tax return.

Limits

The yield curve comes from simulated prices with a fixed correlation, and the yearly figures push real index volatility through that curve, so they know nothing about the actual stocks in a given year. Repurchasing at the same price, as the simulation does, is forbidden by the 30-day wash-sale rule; the usage factor stands in for that gap. This paper is educational; it describes a model and no real account.

Who this is for: Researchers in after-tax portfolio choice, CPAs who work with harvested losses, and journalists covering direct indexing.

Figures

Figure 2. The countercyclical loading, the slope of harvest yield on the year's market return, against name-level volatility. It stays negative throughout, so the dividend pays more when markets fall.

Figure 3. Expected after-tax return against after-tax portfolio volatility. Both allocation lines run through the risk-free rate; the tax-aware line is the steeper, and the marked points are the optimum on each.

Figure 4. The optimal risky weight against risk aversion. The tax-aware weight sits above the tax-indifferent one at every level, with the mean-only curve between them.

Figure 5. The realized harvest dividend against the sleeve's pre-tax return, one dot per simulated year at the base calibration. The straight line is the linear projection Section 3.2 assumes; the marked curve joins the ten decile means. The dividend is countercyclical throughout and convex on the… — Figure 5. The realized harvest dividend against the sleeve's pre-tax return, one dot per simulated year at the base calibration. The straight line is the linear projection Section 3.2 assumes; the marked curve joins the ten decile means. The dividend is countercyclical throughout and convex on the downside, and it never turns negative where the line does.

Figure 6. After-tax one-year return at γ=3 on the same three thousand paths: the tax-indifferent investor at w0*=0.65, the tax-aware weight wD*=0.92 without the dividend, and the same weight with it. The dotted verticals mark the fifth percentile of the last two; the dividend moves it from -14.5%… — Figure 6. After-tax one-year return at γ=3 on the same three thousand paths: the tax-indifferent investor at w0*=0.65, the tax-aware weight wD*=0.92 without the dividend, and the same weight with it. The dotted verticals mark the fifth percentile of the last two; the dividend moves it from -14.5% to -11.9%.

Figure 7. The model's dependency graph. Blue nodes carry the return side, orange the harvest side, purple the investor's choice and its outcome. Name volatility feeds the harvest yield directly and the portfolio volatility only through the diversification factor, and the single market shock drives… — Figure 7. The model's dependency graph. Blue nodes carry the return side, orange the harvest side, purple the investor's choice and its outcome. Name volatility feeds the harvest yield directly and the portfolio volatility only through the diversification factor, and the single market shock drives both the pre-tax return and, countercyclically, the yield.

Figure 8. Top: 63-day realized volatility of the S&P 500 trust, February 1993 to August 2026, with the two regime volatilities of the fitted hidden Markov model as dashed lines and the turbulent-regime days shaded. Bottom: the harvest dividend implied by the rolling volatility through the simulated… — Figure 8. Top: 63-day realized volatility of the S&P 500 trust, February 1993 to August 2026, with the two regime volatilities of the fitted hidden Markov model as dashed lines and the turbulent-regime days shaded. Bottom: the harvest dividend implied by the rolling volatility through the simulated yield curve, against the base-case 162 basis points.

Figure 9. The implied harvest dividend by calendar year, realized S&P 500 trust volatility mapped through the simulated yield curve at τ=0.30, φ=0.35. Orange bars are years the index fell. The dotted line is the base case of Section 4.

Figure 10. Left: the harvest dividend by year of the account's life, with and without an 8% drift. Right: the myopic tax-aware weight that follows it at γ=3, against the tax-indifferent weight. The first-year tilt of Section 5 is the top of a curve, not a level.

Figure 11. Predicted against realized harvest yield on the 600 held-out simulated years for three of the readers in Table 10. The affine rule scatters widely, the ridge on path features tightens, and the recurrent network sits on the diagonal.

Figure 12. The proportional increase in the optimal risky weight over the blended tax rate on harvested losses and the usage factor, with contours at 10, 25, 50, 75, 100 and 125%. The base case of Section 4 and the three households of Table 11 are marked. The two inputs enter only through their… — Figure 12. The proportional increase in the optimal risky weight over the blended tax rate on harvested losses and the usage factor, with contours at 10, 25, 50, 75, 100 and 125%. The base case of Section 4 and the three households of Table 11 are marked. The two inputs enter only through their product, so the contours are hyperbolas.

Tables

Table 1. Optimal risky weight, tax-indifferent vs. tax-aware (σ=0.16, π=0.05, D0=0.0162, β=0.033).
γ	w0*	wD* (mean only)	wD* (full)	wD* (brute force)	Δ w	% increase
2	0.977	1.293	1.382	1.390	0.405	41.5
3	0.651	0.862	0.921	0.925	0.270	41.5
4	0.488	0.647	0.691	0.695	0.203	41.5
6	0.326	0.431	0.461	0.465	0.135	41.5

Table 2. Weight shift by blended tax rate (γ=3, σ=0.16).
τ	dividend D	w0*	wD*	Δ w
0.000	0.0000	0.651	0.651	0.000
0.150	0.0081	0.651	0.782	0.131
0.238	0.0129	0.651	0.863	0.212
0.300	0.0162	0.651	0.921	0.270
0.408	0.0221	0.651	1.027	0.376

Table 3. Weight shift by usage factor φ (γ=3, σ=0.16, τ=0.30).
φ	dividend D	wD*	% increase
0.20	0.0093	0.801	23.1
0.35	0.0162	0.921	41.5
0.50	0.0232	1.048	61.0
0.70	0.0324	1.228	88.7
1.00	0.0463	1.525	134.3

Table 4. The optimum on the full simulated paths, by criterion (σ=0.16, π=0.05, 3,000 paths).
γ	w0*	closed form, Section 3	closed form at the paths' D, β	mean-variance on paths	CRRA, no dividend	CRRA, with dividend	CRRA % increase
2	0.977	1.382	1.390	1.390	1.110	1.640	47.7
3	0.651	0.921	0.927	0.925	0.740	1.105	49.3
4	0.488	0.691	0.695	0.695	0.555	0.825	48.6
6	0.326	0.461	0.463	0.465	0.370	0.550	48.6

Table 5. Refinements to the optimum at γ=3 (w0*=0.651).
variant	w*	% increase
closed form, Section 3	0.921	41.5
plus the dividend's residual noise (sε = 0.0037)	0.921	41.4
plus an uncertain φ, uniform on [0.20, 0.50] (sD = 0.0040)	0.921	41.4
both	0.920	41.3
worst case, φ = 0.20	0.801	23.1
CRRA on the simulated paths	1.105	49.3
capped at w ≤ 1 (cap slack at γ=3)	0.921	41.5

Table 6. The cap w ≤ 1 and the certainty-equivalent value of the tilt, in basis points per year.
γ	w0*	wD*	capped	cap binds	value of tilt, uncapped	value that survives the cap	cost of the cap
1.5	1.302	1.843	1.000	yes	52.5	0.0	127.6
2.0	0.977	1.382	1.000	yes	39.4	4.4	34.9
2.5	0.781	1.106	1.000	yes	31.5	28.2	3.3
3.0	0.651	0.921	0.921	no	26.2	26.2	0.0
4.0	0.488	0.691	0.691	no	19.7	19.7	0.0
6.0	0.326	0.461	0.461	no	13.1	13.1	0.0

Table 7. The dividend and the optimal weight by volatility regime (γ=3, π=0.05).
regime	index vol	name vol	Y	D (bp)	β	w0*	wD*	% increase	time share	mean stay (days)
calm	0.108	0.19	0.087	91	0.034	1.439	1.821	26.6	0.69	71
turbulent	0.292	0.52	0.304	319	0.028	0.196	0.339	73.3	0.31	32
mixture, regime unknown						0.486	0.680	39.8

Table 8. The implied harvest dividend and the optimal weights by calendar year (γ=3, π=0.05, τ=0.30, φ=0.35). Index volatility and return are the S&P 500 trust's; name volatility is index volatility over k=0.56; the yield is read off Figure 1.
year	index vol	index return (%)	name vol	D (bp)	w0*	wD*	capped
1994	0.105	+0.4	0.19	89	1.50	1.89	1.00
1995	0.085	+38.0	0.15	65	2.29	2.77	1.00
1996	0.131	+22.5	0.23	121	0.98	1.30	1.00
1997	0.201	+33.5	0.36	217	0.41	0.63	0.63
1998	0.223	+28.7	0.40	251	0.34	0.54	0.54
1999	0.181	+20.4	0.32	184	0.51	0.74	0.74
2000	0.239	-9.7	0.43	267	0.29	0.48	0.48
2001	0.221	-11.8	0.40	249	0.34	0.54	0.54
2002	0.264	-21.6	0.47	291	0.24	0.40	0.40
2003	0.165	+28.2	0.30	167	0.61	0.87	0.87
2004	0.112	+10.7	0.20	95	1.34	1.71	1.00
2005	0.103	+4.8	0.18	86	1.56	1.96	1.00
2006	0.100	+15.8	0.18	83	1.66	2.08	1.00
2007	0.159	+5.1	0.29	162	0.66	0.93	0.93
2008	0.412	-36.8	0.74	440	0.10	0.19	0.19
2009	0.266	+26.4	0.48	293	0.24	0.40	0.40
2010	0.179	+15.1	0.32	181	0.52	0.75	0.75
2011	0.231	+1.9	0.41	259	0.31	0.51	0.51
2012	0.127	+16.0	0.23	117	1.03	1.36	1.00
2013	0.111	+32.3	0.20	94	1.36	1.73	1.00
2014	0.113	+13.5	0.20	97	1.32	1.68	1.00
2015	0.155	+1.2	0.28	156	0.70	0.98	0.98
2016	0.131	+12.0	0.23	122	0.97	1.30	1.00
2017	0.067	+21.7	0.12	41	3.66	4.21	1.00
2018	0.171	-4.6	0.31	173	0.57	0.82	0.82
2019	0.125	+31.2	0.22	114	1.06	1.40	1.00
2020	0.336	+18.3	0.60	369	0.15	0.27	0.27
2021	0.130	+28.7	0.23	121	0.99	1.31	1.00
2022	0.243	-18.2	0.43	270	0.28	0.46	0.46
2023	0.131	+26.2	0.23	122	0.98	1.30	1.00
2024	0.126	+24.9	0.23	115	1.05	1.39	1.00
2025	0.193	+17.7	0.35	204	0.45	0.67	0.67

Table 9. The dividend, the loading and the myopic optimum by year of the account's life (1,000 five-year paths, base calibration, γ=3, w0*=0.651).
year	Y	D (bp), 8% drift	β	wD*	% increase	value of tilt (bp)	names below basis	price / basis	D (bp), 0% drift
1	0.152	160	0.032	0.917	40.9	25.4	6.9%	1.27	192
2	0.052	55	0.017	0.747	14.8	3.4	4.4%	1.45	80
3	0.033	35	0.010	0.711	9.2	1.3	3.5%	1.64	56
4	0.025	26	0.008	0.697	7.0	0.8	2.8%	1.81	47
5	0.021	21	0.007	0.688	5.7	0.5	2.5%	2.00	40

Table 10. Reading the year's harvest yield off its price path, out of sample (600 held-out paths; the yield's standard deviation is 0.114).
reader	inputs	RMSE	R²	RMSE, no volatility shift	RMSE, with shift	implied error in w* at γ=3
affine in realized volatility	1 number	0.074	0.58	0.073	0.075	0.108
ridge on path features	7 numbers	0.037	0.90	0.035	0.039	0.054
ridge plus timing features	10 numbers	0.037	0.90	0.035	0.039	0.053
LSTM on the daily sequence	252 × 3	0.009	0.99	0.008	0.011	0.014
LSTM, squared channels appended	252 × 6	0.010	0.99	0.008	0.012	0.015

Table 11. Three stylized households (σ=0.16, π=0.05, Y0=0.154; the weight is capped at one).
	A: professional	B: retiree	C: early-career
taxable wealth	$2,000,000	$800,000	$250,000
rate on harvested losses τ	0.408	0.238	0.300
usage factor φ	0.50	0.35	0.20
risk aversion γ	3	4	2
dividend D (bp / yr)	315	129	93
loading β	0.063	0.026	0.019
w0*	0.651	0.488	0.977
wD*, uncapped	1.210	0.647	1.202
wD*, capped	1.000	0.647	1.000
% increase, uncapped	85.8	32.5	23.1
equity dollars, tax-indifferent	$1,302,083	$390,625	$244,141
equity dollars, tax-aware	$2,000,000	$517,551	$250,000
extra equity dollars	$697,917	$126,926	$5,859
dividend, dollars per year	$63,018	$6,659	$2,317
value of the tilt (bp / yr)	90.3	12.2	2.5
value of the tilt, dollars per year	$18,063	$978	$62

Abstract

Systematic tax-loss harvesting produces a benefit that grows with volatility, which is what a mean-variance investor pays to avoid. This paper writes the harvest benefit into the objective as a volatility-linked, stochastic cash flow that pays more when the portfolio does worse. We model the annual harvest benefit as a yield approximately proportional to volatility, add it to a single-period mean-variance problem, and re-derive the optimal risky weight and the after-tax capital allocation line: w* = (pi + D) / (gamma * sigma^2 * (1-beta)^2), with D the dividend level and beta its countercyclical loading. For a diversified equity sleeve calibrated to a fixed-seed harvesting simulation (base dividend 1.6% per year, tax rate 0.30, usage factor 0.35), the optimal allocation to the volatile asset rises about 41% relative to the tax-indifferent case and the after-tax reward-to-risk slope steepens about 37%. What is new is the frontier geometry: the slope of the after-tax frontier changes, not a single point, and the dividend rides name-level volatility while the variance penalty falls on the diversified portfolio.

Version 2 removes the linearization, the single period and the fixed parameters. On the full simulated paths the dividend is convex on the downside and a power-utility investor tilts by 48 to 49% rather than 42%; the dividend's noise and an uncertain usage factor barely move the optimum; a cap on the finished allocation truncates the tilt below a risk aversion of 2.8. A two-regime model fitted to 33 years of index data puts the dividend at 91 basis points in calm markets and 319 in turbulent ones, and year by year from 41 basis points in 2017 to 440 in 2008. The dividend decays to an eighth of its first-year level by year five of an account's life, and the account's embedded gain, not realized volatility, predicts next year's yield. A recurrent network reads the year's harvest off the daily path almost exactly; we say what that buys and what it does not.

Keywords: tax-loss harvesting, after-tax portfolio choice, efficient frontier, mean-variance optimization, direct indexing, stochastic dividend, volatility, tax-aware investing, certainty equivalent, risk-taking, regime switching, sequence models

How to cite

Majumdar, A. (2026). Harvesting as a Stochastic Dividend: The After-Tax Efficient Frontier Revisited. SSRN Working Paper No. 7507839. https://ssrn.com/abstract=7507839

@techreport{majumdar_harvesting_stochastic_dividend,
  author={Majumdar, Anirban},
  title={Harvesting as a Stochastic Dividend: The After-Tax Efficient Frontier Revisited},
  institution={SSRN},
  number={7507839},
  year={2026},
  url={https://ssrn.com/abstract=7507839}}

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Harvesting as a Stochastic Dividend: The After-Tax Efficient Frontier Revisited