Reading 11 Question 6 - Partially deferred Capital Gains Tax

CFA Level II Economics
#1

Essence of the question is: you have 250k, you’ll earn 7.5% half of which will be realized and taxed at 10% the other half will be unrealized and taxed at 10% at the end of 15 years. What is the ending wealth?

My first problem is that I cannot understand the logic behind the formula that CFAI gives solve this so if anyone can explain the answer intuitively I’d appreciate it.

However, even when I put into a spreadsheet and and run the calc’s for each of the 15 years I still come up with an ending value of 675,525. Correct answer is 678,158.

#2

I gave this a quick shot in Excel: I get an ending value of 678,157.5755

I split the 7.5% interest stream into two pieces: one which gets taxed at the end of the year at 10%, the other which rolls up with no taxes until year 15. In the first year, the interest payments are $8,437.50 (250,000 * 0.5 *0.075 * (1-0.9)) and $9,375 (250,000 * 0.075 * 0.5) for the anually taxed and deferred portions, respectively. The new principal at the start of the 2nd year is $267,812.50 and the interest amounts are $9,038.67 and $10,042.97 for the 2nd year. At the end of the 15th year, sum up the interest payments for the deferred portion and multiply by 10%: your tax bill should be $23,786.50.

I’m trying to do this algebraically too, but I’m tired and am coming down with a cold.

Hope this helps!

#3

effective return = 7.5*(1-0.1*0.5) = 7.125% [50% is taxed at 10%]

Effective CG Tax = 10*(1-0.5) / (1-0.1*0.5) (you deferred tax on 95% of the balance, and paid 50% of the tax.]

= 5.26%

now apply the standard formula = (1.07125^15)(1-0.0526) + 0.0526

and arrive at 2.7xxx factor

(1+r*)^n(1-T*)+T*

#4

Yeah, what cpk123 said… yes

#5

I got $678,158.

I don’t have the curriculum, and I don’t have the SchweserNotes with me (I’m out of town teaching), so I don’t know what the formula is, but that doesn’t matter: I wouldn’t bother trying to remember another stupid formula. I simply think through the steps: what happens to the money? I’ll go through them.

We’re earning 7.5% per year, half of which is taxed at 10%. That’s an effective tax rate of ½ × 10% = 5%, so, net of taxes, we’re earning 7.5% × (100% - 5%) = 7.5% × 95% = 7.125% per year, compounded.

Thus, our ending account balance (before we close the account and pay the rest of the taxes) is:

$250,000 × (1.07125)^15 = $701,944.

That gives us a gain of $701,944 – $250,000 = $451,944.

Now comes the tricky bit: 45 parts of that gain represent the (previously) realized (and taxed) portion, and 50 parts represent the (previously) unrealized (and untaxed) portion. The way I got 45 and 50 parts is this:

Realized (and taxed) return: ½ × 90% = 45% (of the pre-tax return). (We got to keep 90% of the half that got taxed)

Unrealized (and untaxed) return: ½ × 100% = 50% (of the pre-tax return). (We got to keep 100% of the half that didn’t get taxed.)

Total return: 45% + 50% = 95% (of the pre-tax return).

So 45/95 has already been taxed, 50/95 hasn’t.

50/95 × $451,944 = $237,865.

The tax on $237,865 (at 10%) is $23,787.

Finally, after paying that tax, we’re left with $701,944 – $23,787 = $678,158 (except for rounding).

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Let’s look at another example of tax splits, so you can see how to handle them; this will help make clear the 45, 50, and 95 parts above.

Suppose that ¾ was realized and ¼ unrealized, and the tax on the realized portion was 20%; the tax on the unrealized portion will be 30%. Your annual, after-tax return would be (0.75 × 80% + 0.25) × 7.5% = 0.85 × 7.5% = 6.375%

After 15 years you’d have $250,000 × (1.06375)^15 = $631,733.

The total gain is $631,733 – $250,000 = $381,733.

Taxed portion: 0.75 × 80% = 60% (of the pre-tax return).

Untaxed portion: 0.25 × 100% = 25% (of the pre-tax return).

Total return = 60% + 25% = 85% (of the pretax return).

Untaxed gain = 25/85 × $381,733 = $112,274.

Tax = 30% × $112,274 = $33,682.

Final balance = $631,733 – 33,682 = $598,051.

I hope that this helps.

#6

I just applied the formula and I got $678,158

R* = 0.1( 1-0.05) = 0.07125

T* = tcg( 1-pcg)/(1-pcg*tcg) = 0.1 *0.5 / 0.95 = 0.05263158

n=15 ,T*=0.05263158,r*=0.07125

FVIF = (1+r*)^n(1-T*)+T*

So at the end of 15 years = $250,000 * ( ( 1+0.07125)^15 ) ( 1 - 0.05263158 ) + 0.05263158 )) ~$678,158

#7

Thanks to all of those who replied and especially to S2000. That explaination was very intuitive and I’m sure saved me another couple of hours trying to reason it out myself.

#8

My pleasure.

I have only a tiny bit of room in my brain, not nearly enough to try to remember all of the tax formulae that CFA Institutes gives you guys. I’d much rather reason through the problem step-by-step; then I don’t have to worry that either I’ve used the wrong formula, or that I’ve used the correct one, but remembered it incorrectly.

#9

A quick follow up to this thread…

In the CFAI text Book 2 pg. 243 I am able to calculate $138,662 amount with no problem using the logic above. My confusion comes in on how to calculate the Effective capital gains tax rate of 4.27% by using the intuition above instead of the formula in the book. It seems like it should be straight forward but I can’t seem to think it through.

Thanks,

Dan

#10

take this example

so you paid 5$ tax on 45$ you got a return of. 5/45 * 10% = 5.26% – same as above

10*(1-0.5) / (1-0.1*0.5)