Capital Gains and Dividends
motivated by a discussion on bogleheads.

Okay, here's the problem:

• Each year we get dividends.
• The portfolio grows at a certain annual rate ... and the dividend amounts increase at a certain annual rate.
• If the dividends are taxed each year at some dividend-rate ... and the portfolio is taxed after n years at a capital-gains rate, what's our portfolio worth at year n, after taxes?

We asume:
A0 = initial portfolio.
g = annual growth factor for the assets in the portfolio (so \$1 grows to \$g in 1 year).
d = annual dividend.
t = dividend tax rate.
D = d (1-t) = the first after-tax dividend (re-invested).
x = annual growth factor for the dividends (so a \$1 dividend in one year would be \$x the following year).
T = capital gains tax rate.

Consider only the growth of the original portfolio.

At the end of k years it'd be worth:
[0]     gk A0 ... since it grows by a factor g each year.

>Where's the dividends?
Patience. We'll get to them soon enuff.

All of the gain after n years, namely [gn A0 - A0] = [gn - 1] A0, is taxable at the capital gains rate T, leaving:
[1]     P1 = gnA0 - T [gn - 1] A0 = { (1-T) gn + T } A0   after-tax balance.

Now consider the dividends.
The first dividend, D, has a final worth of gn-1D and a \$gain of (gn-1 - 1) D   ... being invested for n-1 years.
The second dividend, xD, has a final worth of gn-2xD and a \$gain of (gn-2 - 1) xD   ... being invested for n-2 years.
The third dividend, x2D, has a final worth of gn-3x2D and a \$gain of (gn-3 - 1) x2D   ... being invested for n-3 years.
...
The next-to-last dividend, xn-2 D, has a final worth of g xn-2D and a \$gain of (g - 1) xn-2 D   ... being invested for 1 year.
The last dividend, xn-1 D, has a final worth of xn-1D and a \$gain of 0   ... being invested for 0 years.

>You're considering the dividends as being in a separate investment account?
Uh ... yes. For purposes of the calculations.

The total dividend gain is then the sum of the above \$gains, namely:
D { (gn-1-1) + (gn-2-1) x + (gn-3 - 1) x2 + ... + (g-1) xn-2} = D {gn-1 + gn-2x + gn-3x2 + ... + gxn-2 + xn-1 - (1 + x + x2 + ... + xn-2 + xn-1)}.

Or, summing the series we get the total \$gain in our "dividend account as:
D {(gn - xn) / (g - x) - (xn - 1) / (x - 1) }

The tax on this total \$gain (at the capital-gains rate T) is:
T D {(gn - xn) / (g - x) - (xn - 1) / (x - 1) }.

Similarly, the final value of the dividend portfolio is the sum of all the final dividend values (after umpteen years of investing), namely:
D {(gn - xn) / (g - x) }

>Huh?
Just stare intently at what we did above. The final \$value of each dividend is given there. We just add 'em all up.

Subtracting, from the final portfolio generated by the dividends, the capital-gains tax (on the dividend gains) leaves:
[2]     P2 = (1 - T) D (gn - xn) / (g - x) + T (xn - 1) / (x - 1).

Finally (!) the after-tax portfolio is then P1 + P2, namely:   ta-DUM !
 [A] Final after-tax Portfolio = { (1-T) gn + T } A0 + (1 - T) D (gn - xn) / (g - x) + T D (xn - 1) / (x - 1)

>I think you're in trouble if x = 1 and ...
Meaning the dividends don't increase. Yes, but in that case:
If the dividends are constant, then x = 1 and the above reduces to:
(1 - T) [gn - 1] A0 + (1 - T) D (gn - 1) / (g - 1) + n R D
If the dividend increases equal the return on the assets, that is x = g, then the above reduces to:
(1 - T) [gn - 1] A0 + (1 - T) D n gn-1 + R D (gn - 1) / (g - 1)

>That's it?
Well, yes ... except, maybe, for a chart:

>So all the capital gains are taxed at the end of year n. What if ... ?
Some of these gains are taxed each year? Okay, let's condsider that.
>And what if there are additional investments each year?
Okay, let's do that, too.

 Capital Gains and Dividends ... again

Here we'll add an additional investment \$B each year and capital-gains-tax a fraction f of the portfolio each year.

In the above stuff, the portfolio balance at the end of year n is the sum of two balances: the original portfolio P1 and the dividend account P2.
Let's consider each, at the end of year k.

Consider only the growth of the original portfolio.

[3a]     P1(k) = g P1(k-1)     ... applying the annual gain factor to the previous year's balance.
Note that: P1(0) = A0.

Now apply a capital gains tax T to a fraction f of the annual gain.
In other words, we'll apply the tax R to the fraction: f {P1(k) - P1(k-1)} = f (g - 1) P1(k-1) .
That'd leave the after-tax balance as:

[3b]     P1(k) = g P1(k-1) - T f (g - 1) P1(k-1) = {g - T f (g - 1)}P1(k-1)     ... applying the annual gain then subtracting the capital gains tax on a fraction of the \$gain.

Note that this is similar to equation [0] above, but with g replaced by:
G = g - T f (g - 1).

>So it's like ... uh, we reduce the annual gain factor to incorporate capital gains, right?
Right, but just to a fraction f of the capital \$gains. If f = 0, we're back to G = g again.

Now we add our after-tax dividend and the additional (constant) investment of \$B and get:
[3c]     P1(k) = G P1(k-1) + xk-1D + B.

>That after-tax dividend is after the dividend tax?
Yes, and at year k it's increased by a factor xk-1 from the initial dividend D, received at the end of year 1.

After n years our "partially-taxed" balance would be:
[3d]     P1(n) = Gn A0 + {(Gn - xn) / (G - x) } D + {(Gn - 1)/(G - 1)}B         ... where G = g - T f (g - 1).
The first two terms are what we got before, but with the tax-reduced gain factor G (rather than g).

>What about all the taxes that wasn't paid?
Patience.
After subtracting the unpaid taxes, our portfolio will look like: P1(n) - T(untaxed gains) ... so now we calculate those (untaxed gains).

Note that, each year, we've subtracted a percentage f of the capital gains tax.
The resultant portfolios then had that tax-reduced gain G factor.
After n years of applying that tax-reduced gain G factor:
>>>> the initial portfolio A0 becomes Gn A0 with a \$gain of (Gn - 1) A0.
After n years:
>>>> the annually increased dividends D, xD, x2D ... xn-1D have a \$gain of D{(Gn - xn) / (G - x) - (xn - 1) / (x - 1)} ... as we got earlier, but using the tax-reduced gain factor G.
After n years:
>>>> the annual investments of \$B have a \$gain of {(Gn-1 - 1)B + (Gn-2 - 1)B + ... + (G - 1)B + (1-1)B} = {(Gn - 1) / (G - 1) - n}B,
Note that the last investment has no \$gain at all ... but we left it in 'cause it's easier to sum the series.

Adding the three \$gains and applying the capital gains tax T and subtracting from our portfolio from [3d] we get:   ta-DUM !

>Is ta-DUM really necessary?

Final after-tax Portfolio = [ Gn A0 + {(Gn - xn) / (G - x) } D + {(Gn - 1)/(G - 1)}B] - T[(Gn - 1) A0 + D{(Gn - xn) / (G - x) - (xn - 1) / (x - 1)} + {(Gn - 1) / (G - 1) - n}B]

Or, (perhaps) more simply:
 [B] Final after-tax Portfolio = {(1 - T) Gn + T} A0 + (1 - T) D (Gn - xn) / (G - x) + T D (xn - 1) / (x - 1) + {(1 - T)(Gn - 1) / (G - 1) + nT}B where: A0 = initial portfolio g = annual growth factor for the assets in the portfolio (so \$1 grows to \$g in 1 year) d = annual dividend t = dividend tax rate D = d (1-t) = the first after-tax dividend (re-invested) x = annual growth factor for the dividends (so a \$1 dividend in one year would be \$x the following year) T = capital gains tax rate f = fraction of annual portfolio that is capital-gain-taxed B = additional annual investment G = g - T f (g - 1) = a tax -reduced annual gain factor

>Mamma mia! Do I have to memorize that for the final exam?
Note that, if f = 0 (so nothing is capital-gains-taxed until year n), then G = g.
Further, if B = 0 (so no new investments are made), then [B] reduces to [A].

>And you still have that problem with x = 1 or x = G.
Uh ... yes.
We used a magic formula for summing a series: Gn-1 + Gn-2x + Gn-3x2 + ... + G xn-2 + xn-1 = (Gn - xn) / (G - x).
However, if x = G then we'd get 0 / 0 ... but then the series is easy to sum: Gn-1 + Gn-2G + Gn-3G2 + ... + G gn-2 + Gn-1 = n Gn-1.