Formulas II
The problem of interest period:
FV = PV(1 + — )mt, m
where rn...nominal interest interest rate,
m...# of conversions {how many time interest is calculated to given nominal interest rate),
t...gives time over interest periods, mt...^number of inter est periods
if m —> oo —> concept of continues interest :
FV = PVeft,
where c.Euler's number (2,182781...), f ...inter est intensity, t...time {given to period of /).
Effective interest rate:
An effective interest rate (re) is an interest rate that satisfy:
((l + -r = e/ = (l + re) m
Value of capital and taxes:
There are three different situations:
1. IP = TP,
2. IP < TP,
3. Tax is paid only once when the money is withdraw.
1. IP = TP How much is the FV if the tax rate is 15 % and tax is paid when interest is accrued. You save 7.000,00. The bank promised you to pay monthly interest in the amount of 0.3 %. The maturity of your account (when you withraw your money) is 6 years.
Solution:
FVtax = 7000(1 + 0.003 * (1 - 0.15))12*6,
where 0.003...monthlyinterestrate, (1—0.15)...howmuchmoneyleftmeaftertaxpaiment, 12 * 6...# of inter est periods (12m in one year/6 years).
1
Note: We multiply exactly the interest rate.
2. IP < TP How much is the FV if the tax rate is 15 % and tax is paid once a year (after one year). You save 7.000,00. The bank promised you to pay monthly interest in the amount of 0.3 %. The maturity of your account (when you withraw your money) is 6 years.
Solution:
FVtax = 7000(((1 + 0.003)12 - 1) * (1 - 0.15) + l)6,
where we have to calculate interest first and after we have the amount of interest in one year we can pay the tax.
Here 12..of month (as well IPs in one tax period — year), 6...# of tax periods.
Note that the expresion ((1 + 0.003)12 - 1)
is nothing, but the effective interest rate re, so
again you follow the logic
from 1 [interest rate multiply by (1 — tax).
3. T ax paid only once inT  How much is the FV if the tax rate is 15 % and
tax is paid when you withdraw your money. You save 7.000,00. The bank promised you to pay monthly interest in the amount of 0.3 %. The maturity of your account (when you withraw your money) is 6 years.
Solution:
FVtax = 7000(((1 + 0.003)12*6 - 1) * (1 - 0.15) + 1),
the magic here is that you calculate FV using compound interest and then pay 15 % tax.
Crucial moment you pay tax ALWAYS only from INTEREST!!!
2
Tax & inflation
!!!Alway calculate the tax liability from nominal value. After you separate the tax from earned interest first then you can depreciate the money with inflation: