Check my math

[benjamen]

With the idea based on this thread:
http://www.pmbug.com/forum/f4/blowing-up-stock-examples-amusement-2369/

I am trying to solve for the interest rate needed to just maintain purchasing power after accounting for a given time, tax rate, and inflation rate.

Amount invested: x
Years: n
Required return: i
Inflation rate: j
Tax rate: t

Equation I came up with to solve for i:
x = {[x*(1+i)^n-x]*(1-t)+x}*(1-j)^n

1: Base equation
x = {[x*(1+i)^n-x]*(1-t)+x}*(1-j)^n
2: move over the inflation term
x*(1-j)^-n = [x*(1+i)^n-x]*(1-t) + x
3: distribute the tax term
x*(1-j)^-n = x*(1-t)*(1+i)^n - x*(1-t) + x
4: factor out the x
(1-j)^-n = (1-t)*(1+i)^n - (1-t) + 1
5: move over the 1
(1-j)^(-n) -1 = (1-t)*(1+i)^n - (1-t)
6: factor out the tax term and move it over
[(1-j)^(-n)-1]/(1-t) = (1+i)^n - 1
7: move over the 1
[(1-j)^(-n)-1]/(1-t) + 1 = (1+i)^n
8: distribute the 1 that just got moved over by turning it into a (1-t)/(1-t) term
[(1-j)^(-n)-t]/(1-t) = (1+i)^n
9: move over the n exponent
{[(1-j)^(-n)-t]/(1-t)}^(1/n) = 1 + i
10: final
i = {[(1-j)^(-n)-t]/(1-t)}^(1/n) - 1

Plugging in an example of
Years: 10
Inflation rate: 3%
Tax rate: 30%
I got a required return to just break even at 4.198%

I believe this is correct. If anyone who is math inclined wants to check this out. Are there any errors? Can the result be reduced further?

 

[benjamen]

Going further with this idea, what about pre tax 401k type plans where your employer matches you dollar for dollar? What return do you need to at least maintain the purchasing power of the pre tax dollars you put in the account? Where would you have been better off just paying tax on that income and taking it as income?

Equation I came up with to solve for i:
x*(1-t) = {[2x*(1+i)^n]*(1-t)}*(1-j)^n

1: base equation
x(1-t) = {[2x*(1+i)^n]*(1-t)}*(1-j)^n
2: move over the inflation term
x*(1-t)/(1-j)^n=[2x*(1+i)^n]*(1-t)
3: move over the tax term
x*(1-t)/[(1-t)*(1-j)^n] = 2x*(1+i)^n
4: Factor out x and bring over the 2
(1-t)/[2*(1-t)*(1-j)^n] = (1+i)^n
5: factor out the taxes
1/[2*(1-j)^n] = (1+i)^n
6: final
i = 1/[(1-j)*2^(1/n)] - 1

With the same example from above you get a negative 3.811% return needed, which is mostly because the match outweighs the other factors. Assuming 30% tax, as long as inflation stay under ~6.7% you need a zero rate of return to break even.

 

[Golsil]

Both seem correct to me. Having said that, I wonder if asking Warren Buffett, the genius of stock and other matters what his opinion would be? He sure has a way of knowing those pertinent facts and has a keen observational mind. That gentleman sure has an exceptional record that would be difficult for any normal person to match. He sure is in a league all by himself. His mastery of the stock market is absolutely remarkable.

 

[DoChenRollingBearing]

...

Warren Buffett also likes sending oil on his trains...!