Reading 10- Pg 242 equation 5

[Reallyfrustrated]

I dont understand what equation 5 represents. It’s the after tax accumulatin of each unit of currency in a taxable portfolio. Can someone explain? Also, what does the effective capital gains tax rate mean? I don’t get how the book says that it adjusts capital gains taxes to reflect prior taxes on dividends/int, etc.

 

[Galli]

Effective capital gains tax rate takes into account the varying tax % differences in taxable investments. For example, your capital gains might be taxed at a different rate than the dividends you received. Therefore your effective tax rate on capital gains would be lower or higher depending on the specific tax % on each investment.

 

[AndrewWheeler87]

I’m stuck on this one too man. I just cannot conceptualizer what this rate actually represents.

 

[lawrence90sg]

i not sure whether i am right or wrong on this 1, i just want to help out. I am quite weak in math but the way i conceptualise is that this is like a deferred captial tax rate on the investment that is untaxed.

 

[cpk123]

r* = r [1-(pi*ti+pd*td+pcg*tcg)]
This part is easy to understand. It is the effective rate after deducting taxes on the proportion of the particular component i=interest, d=dividend, cg=capital gains in a particular year
The equation which is a little more difficult to comprehend
t* = tcg * [1-(pi+pd+pcg)] / [1-(pi*ti+pd*td+pcg*tcg)]

• tcg = (Capital tax gains rate)
• [1-(pi+pd+pcg)]= [Proportion of portfolio not yet taxed)
• [1-(pi*ti+pd*td+pcg*tcg)] = (Proportion already taxed)

Then you have the standard formula seen before (for the deferred tax case)
FVIFTaxable = (1 + r *)^n (1 − T *) + T* −(1 − B)t cg
only here r becomes r*, tcg becomes t*
and B as before is the basis component.
I hope this was the equation being talked about, and hope this helps provide some context.
And I have slighlty modified the equation in the book by using brackets and combining like terms e.g.
when the book says
1-pi*ti - pd*td - pcg * tcg -> does not provide context
but
1-(pi*ti+pd*td+pcg*tcg) -> which is the same expression - does provide context as to what is happening.

 

[AndrewWheeler87]

CPK - few questions on the above:
- When you reference the denominator as portion already taxed, isn’t actually the portion not yet taxed? [1-(pi*ti….)] is what i’m referring to. Doesn’t this represent the % of the return left after tax assessment? So, that times the pre-tax return would yield r*, right?
- The forumula assumes that the proportions of total gain each year are constant throughout the period, yes? Meaning if I have $1,000 gain at the end of year 1 and that includes $500 of dividends that were reinvested, each year dividend income will continue to represent 50% of the total gain in value, right?
-Lastly, is the overall point of this effective CG number is yield a rate that can be applied to the FVIC (entire) and give me the same ending value that I would get if I simply took the ending value, subtracted the basis, applied the normal CG rate to gains, and subtracted? I thought this was the case but when I try to work out a simplified version using say 2 periods, I wind up getting ending values that differ slighty - here i’m talking about long hand givind me something like $1,043 ending value vs FVIC and T* giving me something like $1,050.
Thanks for your help

 

[cpk123]

Andrew - you are right.

• tcg = (Capital tax gains rate)
• [1-(pi+pd+pcg)]= [Proportion of portfolio not yet taxed)
• [1-(pi*ti+pd*td+pcg*tcg)] = (Proportion NOT already taxed)

Proportion of the individual components in the return and the tax rates for each of those components are constant in this formula.
I am not sure about working the whole thing out by hand…. and getting the same answer - esp. with a blended rate as above.

 

[AndrewWheeler87]

Thank you - after I posted that I thought about it some more and when I was working it out by hand I was deducting the starting value and apply CG to that however, I was not accounting for the fact that the basis grows as well…The long hand answer worked out to be much closer once I remembered this. Thanks!