I don’t think I can think clearly now. Can’t even understand what this question is asking?

Anyone can translate into plain english for me please?

You compute that an investment with a current value and basis equal to $20,000 will have an annual return after realized taxes equal to 10% for the next 12 years until it is sold. The effective capital gains rate will be 15%. What will be the accrual equivalent after-tax return?

The balance in the account after payment of all taxes in 12 years uses the future value interest factor after all taxes: Future value = $20,000 × [((1.1)12) × (1 - 0.15) + 0.15] = $56,353 The accrual equivalent after-tax return is then ($56,353 / $20,000)1/12 - 1 = 9.02%. But, WHY??

The question asks the future value of $20k invested over 12 years at 10%, and then taxed at a capital gains rate of 15%. This is where the future value equation comes into play. You add back the .15 beecause your original basis ($20k) is not taxed.

Next, the question asks for the accural equivalent after-tax return. In my mind, I think of this as your effective after-tax return. It is the same equation as effective interest rate (i think). So you take (your after tax return and divided it by your initial investestment) - this equals your cumulative return then you take it to the power of 1/number of years invested - this gets you an annualized figure and finally - 1 to get your effective after tax return.

The accrual equivalent after tax return gives you the geometric return of your investment after all taxes are paid (accrual taxes). Its just another return term, but it net of taxes, and it is a geometric mean.

The accrual equivalent tax rate is the similar idea - it is the actual tax rate you are realizing on the investment. Cap gains may be 20%, and Div 15% and interest 10% (just random numbers chosen). There will also be unrealized gains, which pay no gains until sold. So the accrual equivalent tax rate is the actual tax rate you are realizing if there were accrual taxes paid.

Sorry just to check, for this realised capital gain formula being used: $20,000 × [((1.1)^12) × (1 - 0.15) + 0.15] = $56,353 from (FVIF_{AT }= (1 + r*)^{n}(1-t_{cg}) + t_{cg)}

Isnt the ‘r’ a pre-tax return, but the question states “annual return _ after _ realized taxes equal to 10%”