Before tax real returns

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blackomen

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I’m a little confused how to calculate this. I’ll give some numbers and ask which is the correct method. This is a practice problem with the numbers changed.
Givens:
Required after tax real return: 10%
Expected inflation: 10%
Tax rate: 25%
Method 1:
Convert to before tax nominal return then take out inflation
Before tax nominal return =( 1.1 *1.1 -1 )/(1-0.25) = 28%
Take out inflation: 1.28/1.1-1= 16.36%
Method 2 (actual solution from Finquiz):
Directly take out the effect of taxes from after tax real rate:
0.1/(1-0.25) = 13.33%
Which of these is correct? What’s wrong with method 1?
 
before tax nominal - inflation = before tax real.
before tax real * (1-t) = After tax real.
so before tax real = 10 / 0.75 = 13.33%
and before tax nominal = 13.33 + 10 = 23.33% (additive)
before tax nominal = 1.1333 * 1.1 - 1 = 24.66 (multiplicative)
 
Are all returns (required real after-tax return and inflation) taxed, or is inflation untaxed?
 
cpk123 wrote:before tax nominal - inflation = before tax real.
before tax real * (1-t) = After tax real.
so before tax real = 10 / 0.75 = 13.33%
and before tax nominal = 13.33 + 10 = 23.33% (additive)
before tax nominal = 1.1333 * 1.1 - 1 = 24.66 (multiplicative)
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I’ve never understood the reason for the multiplicative answer.
 
single period = additive, multiple period = multiplicative (for the compounding effect)
- is my understanding.
S2000 - please correct me if I am wrong.
 
cpk123 wrote:single period = additive, multiple period = multiplicative (for the compounding effect)
- is my understanding.
S2000 - please correct me if I am wrong.
Click to expand...
That is exactly how it is taught, but if you think about how these returns are used, you’ll see that that makes no sense.
Ignoring taxes (which only complicates the discussion), imagine that you have a $1,000,000 portfolio, that you need $50,000 on which to live, and that inflation will be 3%. Your required nominal return is 8%, not 8.15% (= 1.05 × 1.03 – 1). What happens this year is:
  • The 5% return ($50,000 = 5% × $1,000,000) is removed from the portfolio and spent
  • The 3% return ($30,000 = 3% × $1,000,000)) grows the portfolio value to $1,030,000 for next year
What happens next year is:
  • The 5% return ($51,500 = 5% × $1,030,000), 3% more than last year) is removed from the portfolio and spent
  • The 3% return ($30,090 = 3% × $1,030,000) grows the portfolio value to $1,060,090 for next year
and so on.
The compunding occurs because this year’s inflation return is retained; you don’t need to compound the returns again.
 
in your example you withdraw living expenses once a year . If expenses are withdrawn in continuous time and inflation is also given in continuous time then the required rate would be 8.33%. if you withdraw it once a year then the required rate is only 8%. so the geometric compounding rate is in between at 8.15% , putting the withdrawal rate somewhere between continuous and annually
 
janakisri wrote: in your example you withdraw living expenses once a year . If expenses are withdrawn in continuous time and inflation is also given in continuous time then the required rate would be 8.33%. if you withdraw it once a year then the required rate is only 8%. so the geometric compounding rate is in between at 8.15% , putting the withdrawal rate somewhere between continuous and annually
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You may very well be correct. If so, that’s a silly reason to compound 5% and 3%; it’s a good reason to compute the required return correctly.
 
I think the compounding ( maybe this is not the term which should be used , but it looks like compoundng in the calculation ) makes sense because :
1. Withdrawal rate may be stated as an average . It could fluctuate . Also withdrawals might be taken at other ( even irregular ) intervals than annual
2. Inflation affects both principal ( through diminished purchasing power and therefore future value in real terms ) and spend .
3. The two , inflation and spend , are sure to interact given 1 & 2 above.
Given 1,2 & 3 all taken together the simple average rate of required return ( inflation + spend ) would definitely be lowest possible , and naive at worst. particularly because of # 3 above
inflation + spend + inflation*spend would be at least more accurate ( my opinion , I might be wrong )
 
Assume inflation is taxed (nowhere do I see any mention of being able to deduct inflation in my tax returns, at least in the US and I assume this is the case in most other countries.)
I purposely picked high but realistic values for inflation and required returns to magnify the difference between using 2 different methods rather than saying either one’s OK since the answers are close enough.
In method 2, by going from after tax real returns to before tax real returns, you’re ignoring the tax on inflation.. correct me if i’m wrong.
 
not sure what you were doing in method 1 anyways. It looks incorrect to me
 
cpk123 wrote:
not sure what you were doing in method 1 anyways. It looks incorrect to me
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Before Tax Nominal Returns are something we all know a priori, at least if you can read some brokerage statements. This is what’s being referred to if someone mentions to you “My portfolio returned X% last year”.
Taxes are a function of Before Tax Nominal Returns, assuming you started day 1 with 100% cash and established your positions and liquidated them all on the last day.
You need to remove Taxes AND inflation to get to after tax real returns. The order you remove them matters. If you remove inflation first then remove taxes, then you’re assuming taxes are not paid on the return due to inflation. So you remove taxes first and see what part of the return you have left. Then calculate how much of your return is due to inflation is remove that as well and you get the after-tax real return.
To go from after-tax real return to a before-tax nominal return, just reverse the steps in the above paragraph.
Let R = before tax nominal return
Let r = after tax nominal return
Let i = inflation rate
Let T = tax rate
(1+R*(1-T)) / (1 + i) = 1 + r
1 + R*(1 - T) = (1 + r)*(1 + i)
R*(1 - T) = (1 + r)*(1 + i) - 1
R = [(1 + r)*(1 + i) - 1] / (1 - T)
Let r = 0.1, i = 0.1, T = 0.25 (from the original post), crunch the numbers, R = 0.28 = 28%
Let R2 = before tax real return:
1+ R2 = (1 + R) / (1 + i)
R2 = (1 + R / (1 + i) - 1 = 1.28 / 1.1 - 1 = 16.36%
And to get before-tax real returns, take a before-tax nominal return and remove the effects of inflation.
 
blackomen wrote:Assume inflation is taxed (nowhere do I see any mention of being able to deduct inflation in my tax returns, at least in the US and I assume this is the case in most other countries.)
I purposely picked high but realistic values for inflation and required returns to magnify the difference between using 2 different methods rather than saying either one’s OK since the answers are close enough.
In method 2, by going from after tax real returns to before tax real returns, you’re ignoring the tax on inflation.. correct me if i’m wrong.
Click to expand...
If you have your portfolio in a nontaxable account, then you’re taxed only on the amount you take out. You’ll take out what you need to spend this period, but leave the additional (inflation) in the account, untaxed, to grow for the next period.
 
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