Can someone explain the formula for deferred capital gains taxation
FVIF at = (1+r)^n (1-tcg) + tcg B
Why cant we just compound the value of the asset to r for n periods and deduct the tax on the last period by multiplying that future value by the tax rate?
When I first saw this formula I was also puzzled by the last term, TcgB and why we add it.
FVIF is a factor, a multiplier, but TcgB is a dollar amount…
Tcg x B means already taxed basis value, thus it’s been added back. Otherwise it would be taxed twice.
Think of it this way…the way the equation 3b on page 237 is initially written is that 100% of the asset is taxed at the capital gains tax rate. We know that this incorrect. Capital gains is the difference between the original cost basis (B) and the newer market value. So the B represents the cost basis. The equation corrects for the 100% tax of the asset by adding back the cost basis times the capital gains tax rate.
EASY METHOD
Return 6% of 10 years
MV = 100
Cost (B) = 80
Cap Gain = 100-80=20
CGtax 30% thus 20 x .3 = 6 in tax
Portfolio value a/t = 100 - 6 = 94
TEXT BOOK APPROACH
(1.06)^10(1-0.30) + (0.3)(0.8) = 1.2536 + 0.24 = 1.4936
or (1.06)^10(1-0.3) + (.3)(1.0) = 1.5536
Difference between 1.5536 - 1.4936 = 6 same as above
Vol. 2, Study Session 4, Reading 9, Section 3.1.3 bottom of page 327 using equation 3b and that’s the equation you want to use not 3a.
I hope this helps,
Marc A. LeFebvre, CFA
Founder & President - LevelUp, LLC and LevelUp BootCamps
If you want to, you can do the algebra:
FVIF = Future value (of 1) before taxes − taxes
= 1 × (1 + r)ⁿ − tcg × [1 × (1 + r)ⁿ − B]
= (1 + r)ⁿ − tcg(1 + r)ⁿ + tcgB
= (1 + r)ⁿ(1 – tcg) +tcgB
There’s your formula. And, as I tell my candidates, do your level best to blot it from your mind! Do not memorize this formula; it’s a waste of precious neurons! (I don’t know about you, but I have only about two neurons left; I cannot afford to waste them on anything.) What you should do is understand the formula: it’s the first line of the algebra: future value before taxes less taxes. On the exam, if you need to calculate it, you should be able to calculate it easily without memorizing a formula.
You have $1 million growing for 20 years at 5%. The tax basis is $200,000, and the capital gains tax rate is 25%. What’s the final value?
Future value before taxes = $1,000,000(1.05)²º = $2,653,298.
Capital gain = $2,653,298 − $200,000 = $2,453,298.
Capital gains tax = $2,453,298 × 25% = $613,324.
Future value after taxes = $2,653,298 − $613,324 = $2,039,974.
Excellent! I knew this has to do with algebra but didn’t delve into it given plenty of time to the exam. Thank you both.
Marc, did you know using the browser version of Vitalsource Bookshelf you can actually hyperlink any section of the curriculum with pinpoint accuracy. How cool is that!?
For example, here’s that formula.
(Login required)
Yeah…that is pretty cool. I did not know that. I went to the link and saw a whole different view than what I’m used to so I’ll use that source of the book in the videos. I just answered another question earlier and simply cut and pasted but a hyper link would be far better. Thanks for helping me! Awesome!
I see you guys really devoted, committed, engaged to CFA 3 wisdoms.
Can you send me some motivations?
Thank you.
^ Celebrate a little victory every day, solve a question 
Haha nice one, I feel refreshed
What happens when the temperature of water is 211 degrees? Nothing. What happens when water is at 212 degrees? It boils, it powers steam engines, it supplies energy to cities, it powers locomotives, it is powerful and can accomplish a whole lot with only 1 degree of difference. Think about the Grand Prix in Monaco, Olympic 100 meter freestyle or the Green Jacket at Augusta, what is the difference between winning and second place, one degree of difference. So between now and that first Saturday in June make a concerted effort to make one more degree of difference in your reading, note taking, problem solving and concentration. If you add one more degree of difference, you’ll pass that final exam, earn that CFA designation and enhance the value of our profession. Now git’to work!
Marc A. LeFebvre, CFA
Founder & President, LevelUp, LLC and LevelUp BootCamps
Thank you Marc, valuable words and a really nice advice. First time in my whole path in the CFA studies I feel a bit tired about continue studying. I hope the time lost won’t become critical.
Thank you again!
Harrogath
Fatigue is a temporary state of being. Achievement lasts a lifetime. You got this!
Your more than welcome, ask for motivation anytime, I got all kinds of it I can dish out 
-Marc
Thank you gents for the explanations. Made life easier.
Magic, Schweser deduct the investment $1,000 from FV(at) to get the gain
They did not deduct $200,000 like you did here. They deducted the $1 million. Any thoughts?
Shweser page 48 deduct the investment $1,000 instead of deducting the basis $800 to find the gain. Any thoughts?
What does $1 invested for N years at r% return become?
It becomes (1 + r)ⁿ
What is your gain?
Gain = (1 + r)ⁿ - 1
What is your Tax on the Gain?
Tax on Gain = [(1 + r)ⁿ - 1] * Tcg
So what do you have Left?
End Balance = (1 + r)ⁿ - [(1 + r)ⁿ - 1] * Tcg
= (1 + r)ⁿ (1-Tcg) + Tcg [some simplification using Alge]
==> Same as if you but B=1 in the other formula. …
You always deduct the tax basis to get the gain. Check the Schweser problem again to verify what the tax basis is. They’re not likely to make a simple mistake such as that.
They did actually deduct the investment 1000 instead of the basis 800 or 1200 
Oops!
E-mail Schweser and tell them that they face-planted.