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?
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?