Can someone explain the concept of effective capital gains tax rate t_ecg? I’ll use example 8 from the CFAI reading 70 to explain what confuses me: We have a 100’000 EURO portfolio, with an average return of 8% and we invest for 5 years. -5% of the return is ordinary income (I guess interest) , taxed at 35% -25% of the return is dividends, taxed at 15% -45% are realized capital gains, taxed at 15% So now we have an annual return after realized taxes of r* = 0.08 (1 - 0.05 * 0.35 - 0.25 * 0.15 - 0.45 * 0.15) = 0.0702, or 7.02%. So far so good - for every EURO we earn, 7.02 cents are taken away at the end of the year. After 5 years, you have 100’000 (1+r*)^5 = 140386.30 EUROs. Now the portfolio is sold. And here’s what I don’t get: the extra 40386.30 EUROs are a capital gain, and if I were the government, I’d tax them at the CG tax rate of 15%. But that’s not what happens. Instead you’re taxing them at the _effective_ CG tax rate of T* = 0.15 (1 - 0.05 - 0.25 - 0.45) / (1 - 0.05 * 0.35 - 0.25 * 0.15 - 0.45 * 0.15) = 4.27%. Since the cost basis B=1 the value of the portfolio after taxes is 100’000 (1+r*)^5 (1-T*) + T* Basically, I’m okay with the formula, I just don’t understand why we’re using T* instead of T_CG. After all, we’re taxing a return of 40386.30 EUROs as capital gains… Any tax experts?
Correction: forget the sentence “for every EURO we earn, 7.02 cents are taken away at the end of the year.”
I got it now. The point is, we’re not taxing the whole return of 40386.30 EUROs as capital gain - only the part that hasn’t been taxed yet. If you do the math, you’ll find that a total of 11506 EUROs hasn’t been taxed yet (unrealized capital gains). At a tax rate of 15%, this makes 1725.91 EURO of tax. Compare that to the 40386.3 EUROs, and you’ll find the effective capital gains tax rate of 4.27%, as they give it in the example in the book. In summary, it’s really just what they say: The effective tax rate that takes into account previously paid taxes. So you can apply it to the 40386.3 EUROs even though taxes on dividends and previously realized capital gains have already been paid. Was it just me or did anybody else have problems with that?
I actually had a similar problem understanding this. Would you mind sharing how you did “the math” to finally understand this? I understand the concept, I just don’t understand how they derive the formula for the ECG tax… appreciate your help!
Will do. I wrote it down, got it at home. I’ll post it tonight.
where did you get this problem from?
OK, so again, we’re talking about example 8 from the CFAI reading 70. First, I made an Excel sheet where for every year I listed the return, divided up into dividends, realized capital gains and unrealized capital gains. I compute the taxes based on those. At the end of the 5 years, I look at how much unrealized capital gain there is, and how much tax you have to pay for it now. Then you compare that to the total return and voila, you get the effective tax rate of t_ECG = 4.27%. This is just the tax rate that you apply to the total return in order to figure out the taxes on the unrealized part (that, for me, was the sticking point). The Excel sheet won’t look very good if posted here, so I put it up here: http://fb.esnips.com/web/underthebridgesStuff Note: It’s converted from an openoffice document, and I haven’t verified that it’s all ok. Let me know if it’s not. I also figured out how you get the formula for t_ECG. You can find that in the pdf file (tax_ecg.pdf) in the same web folder. Below is the pure text version, just doesn’t look that great: Let B_t be the value of a portfolio at time t. r is the rate of return, p_R is the realized part of r, taxed at t_R, p_U=1 - p_R is the unrealized part of r$, taxed at t_U. B_1 = (1+r) * B_0 - r * p_R * t_R * B_0 The tax paid on the return r * B_0 is equal to r * p_R * t_R * B_0. The unrealized part of the gain is equal to B_0 * r * (1-p_R), the hypothetical tax paid on it would be B_0 * r * (1-p_R) * t_U. We are looking for a t_ecg such that B_0 r (1-p_R) t_U = (B_1 - B_0) t_ecg = ( (1+r) B_0 - rp_Rt_RB_0 - B_0 ) t_ecg B_0 r (1-p_R) t_U = (B_1 - B_0) t_ecg = B_0 (1 + r - r p_R t_R - 1) t_ecg = B_0 t_ecg r (1-p_Rt) t_ecg = (1 - p_R)/(1 - p_R * t_R) * t_U This is basically the normal tax rate t_U, adjusted by 1/(1-p_Rt_R) to account for the fact that B_1 - B_0 is after tax, and only applied to the part that is not realized yet (1-p_R). So this is the rate that can be applied against all of B_1 - B_0 and takes into account that taxes on the realized part have already been paid.
This is very helpful, thanks!
For the FVIF of all the taxable amounts, can you use an equation similar to 70-3B in the CFAI book instead of 70-3A? In the examples they give B=1 so I haven’t been able to test it out
bump. Naze Duck is on fire
The denominator is not 40386 or the whole return!
Could someone please please please explain this!