tax formulas

CFA Level III Portfolio Management
#1

Need help with deriving the following two formulas, do not understand them or should I just memorize them?

effective capital gains tax rate

T_ECG=T_CG(1-P_I-P_D-P_CG)/(1-P_I*T_I-P_D*T_D-P_CG*TCG)

future value interest factor after all taxes:

FVIF_T=(1+R_ART)^N(1-T_ECG)+T_ECG-(1-B)T_CG

#2

Next time when you ask a question, please give the volume and page numbers in the CFAI curriculum.

The first T_cg is just the capital gains rate.

1 - P_i - P_d - P_cg is the portion of your return that is subject to deferred capital gains tax, since the 3 P_x entries are for the return that you’ve already paid tax on and will not have to pay again.

This return was earned on (1 - P_i * T_i - P_d * T_d - P_cg * T_cg) of your principal (not on the complete principal). So your effective tax rate on $1 of principal is T_cg * effective return = T_cg * (1 - P_i - P_d - P_cg) / (1 - P_i * T_i - P_d * T_d - P_cg * T_cg). OK this is a bit fuzzy but hopefully not too much.

After than the FVIF_t formula is easy to make sense. r* is the return, T* is the tax rate, you pay tax at the end (not every year), so FVIF_t has the (1 + r*)^n (1 - T*) + T* portion (Look at vol 2 p 226 FVIF_cg for where the last T* comes from. You’re being taxed not on (1+r)^n but on (1+r^n)-1.) And (1-B)T_cg is the tax you should have paid at the beginning but didn’t and deferred till now (your basis was B and the investment was worth 1, so tax = (1-B)T_cg.)

HTH.

#3

heres a good way to understand the first formula.

easiest way to gain intuition is too make up some numbers…

You have a 100,000 portfolio today.

this year you make 15%

of that 15%,

  • 5% is interest, taxed at 30%

  • 3% is dividends, taxed at 25%

  • 3% is realized capital gains, taxed at 20%

  • 4% is unrealized capital gains, taxed at 20%

So at the end of the year your account is worth before realized taxes = 100,000 * 1.15 = 115,000

and you have a current tax liability of:

100,000 * 5% * 30% = 1,500

100,000 * 3% * 25% = 750

100,000 * 3% * 20% = 600

= 2,850

Therefore your account after paying realized taxes is worth 115,000 - 2,850 = 112,150 for an after realized tax return of 12.15%

Calculating the realized yearly tax liability is easy as we just did above, but the complicating matter is how to adjust for the deferred tax on the unrealized capital gains.

Heres how… you know that your account will grow by 12,150 this year after you pay realized taxes. you also know that you deferred 100,000 * 4% * 20% = $800 until you sell your investment… so the effective deferred capital gains tax rate is simply 800 / 12,150 = 6.58%. That is to say that for every 12.15% your account grows, if you generate returns in the same proportion as above, you will effectively still owe 6.58% of this amount (800/12,150) once you sell your investment. Therefore in the future value formula for your ending account balance, this is the effective tax rate you should use against the 12.15% growth rate.

Using the terrible textbook formula…

proportions

  • interest = 5000/15000 = 33.33%

  • dividends = 3000/15000 = 20%

  • realized cap = 3000/15000 = 20%

= 20% * (1 - 33.33% - 20% - 20%) / ( 1 - 33.33% * 30% - 20% * 25% - 20% * 20%) = 6.58%

cheers!

#4

Thanks for both, now it is getting much more clear

#5

Thanks 1recho , you summarized even better than the CFAI book . Helps to think of the effective tax rate in the order you gave

#6

For me the calcutions are obvious in the chapter 10 on taxes and PWM and I don’t use the formulas.

But should we have to be able to reproduce them on the exam?!

#7