After-tax Return, Effective Long-Term Capital Gains Tax Rate, Future Value of the Portfolio, and Accrual Equivalent Return

Investment Portfolio = V₀ = $750,000
Average Annual Portfolio Pre-tax Return = r = 8.0%
Investment Time Horizon = n = 5 (years)
Cost Basis as a % of current market value = 100% = 1

Portfolio Expected Return Distribution

Source	First Year	Annual Proportion (p)		Tax Rate (t)
Tax Interest	$ 18,000	30.00 %		25 %
Dividends	$ 5,000	8.33 %		15 %
Short-Term Capital Gains	$12,000	20.00 %		25 %
Long-Term Capital Gains	$ 25,000	41.67 %		20 %
Total	$ 60,000	100.00 %

---

[1] After-tax Return

r^* = r (1 - p_i t_i - p_d t_d - p_sg t_sg)

r^* = 8.00% * (1 - 0.3000 * 25% - 0.0833 * 15% - 0.2000 * 25%) = 6.90%


[2] Effective Long-Term Capital Gains Tax Rate

r^* T^* = r (p_lg t_lg)T^* = (r/r^*) (p_lg t_lg)

T^* = (8.00%/6.90%) * (41.67% * 20%)
= 9.66260869565217% = 9.66%

Also,
(r/r^*) = 1/(1 - p_i t_i - p_d t_d - p_sg t_sg)
# from the result of [1]
T^* = (p_lg t_lg)/(1 - p_i t_i - p_d t_d - p_sg t_sg)


[3] Future Value of the Portfolio

V_n = V₀ [1 + {(1 + r^*)ⁿ - 1} (1 - T^*)  -  (1 - B) t_lc]
= V₀ [{(1 + r^*)ⁿ (1 - T^*) + T^*}  -  (1 - B) t_lc]

V_n = $750,000 [{(1 + 6.90%)⁵ (1 - 9.66%) + 9.66%}  -  (1 - 1) 20%]
= 750000 * (((1 + 6.90%)^5 ^* (1 - 9.66%) + 9.66%)  -  (1 - 1) * 20%)
= 1,018,316.56847811 = 1,018,316.57


[4] Accrual Equivalent Return

V₀ (1 + R_AE)ⁿ = V_n
R_AE = (V_n/V₀)^1/n - 1

R_AE = (1,018,316.57 / 750,000)^1/5 - 1
= (1018316.57 / 750000)^0.20 - 1
= 0.0630759897687938 = 6.31%