1 Interest and Tax
• Consider that you deposit on your saving account $1,000.00. The interest
rate is 4 % p. a. The interest period is
one year. Every year you have to pay the tax from earned interest (at the
end of the year). The tax rate is 15 %.
What is the value on your bank account after 5 years?
Note! Tax is paid only from received interest, NO FROM THE PRINCI-
PAL!
FV = 1000 ∗ (1 + 0, 04 ∗ 0, 85)5
FV = 1181.95977
FV = PV (1 + r (1 − T))
n
• Take the same example if the interest periodis quarterly.
FV1 = 1000 ∗ ((1 + 0.04
4 )4
− 1000) ∗ 0.85 + 1000 . . . after one year the FV
will be 1034.51341
FV2 = 1034.51341 ∗ ((1 + 0.04
4 )4
− 1000) ∗ 0.85 + 1000 . . . after second year
the FV will be 1070.21799
FV3 = 1034.51341 ∗ ((1 + 0.04
4 )4
− 1000) ∗ 0.85 + 1000 . . . after third year
the FV will be 1107.15486
FV4 = 1034.51341 ∗ ((1 + 0.04
4 )4
− 1000) ∗ 0.85 + 1000 . . . after fourth year
the FV will be 1145.36665
FV5 = 1034.51341 ∗ ((1 + 0.04
4 )4
− 1000) ∗ 0.85 + 1000 . . . after fifth year
the FV will be 1184.89705
or an easier way:
FV 0,5 = 1000 ∗ (((1 + 0.04
4 )4
− 1) ∗ 0.85 + 1)5
. . . is ;)
FV = PV (((1 +
rannual
m
)m
− 1)(1 − T) + 1)n
T . . . Tax rate
m . . . number of interest periods in one year (in one Tax pariod)
n . . . number of years (number of Tax periods)
Note! The effect of the number of interest periods (more IPs generate greater
FV).
2 Effective Interest Rate
In order to have the same impact on the future value of capital at varying
interest periods, we use the EFFECTIVE INTEREST RATE.
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• The annual interest rate is 5 %. Your bank calculates the interest on the
monthly basis. What is the future value of
your 10,000.00 after one year?
FV = 10000 ∗ (1 + 0.05
12 )12
= 10511.619
The same effect on the future value must be reached if we use the effective
interest rate.
The calculation of effective interest rate:
ref = (1 +
rannual
m
)m
− 1)
Proof:
ref = (1 + 0.05
12 )12
− 1) = 0.0511619
FV ref
= 10000 ∗ (1 + 0.0511619) = 10511.619
3 Continuous Interest
If the number of interest periods goes to infinity then we speak about continuous
interest calculation.
The expression is:
Let PV=1, than the FV in one year will be if Limm→∞(1 + r
m )m
−→
Limm→∞ (1 + 1
m
r
)
r
m
r
= er
Through the usage of the effective interest rate we can achieve the same
effect on the FV, as in the discrete
calculation (simple, compound).
FV = PV eft
Note. Instead of interest rate, we use the term interest intensity (f).
The relation between the effective interest rate and the interest intensity is
following:
ref = ef
− 1 then,
f = ln(ref + 1)
In our example:
f = ln(1.0511619) = 0.0489612
FVcontInt = 10000 ∗ e0.0489612
= 10511.619
4 Tax payment by the continuous interest
FVcontInt = PV ∗ ((ef
− 1) ∗ (1 − T) + 1)n
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5 Effect of inflation
The Purchasing Power Parity of our money will be lower. To take the effect of
inflation in account we use it as a discounting factor.
1. Situation when inflation is constant in average
df =
1
(1 + πaveragen)n
2. inflation is different every year
df =
1
(1 + π0,1)(1 + π0,2)(1 + π0,3) ∗ ... ∗ (1 + π0,n)
3. Inflation is constant and we use the concept of continuous calculation.
Inflationintensity . . . fΠ = ln(AnnualAverInflation + 1),then
df =
1
efπ
Our pevious example with 4 Interest Periods in one year, interest rate 4 %
p. a., 5 years to maturity and continuous interest + continuous inflation (if the
approximation of an annual average inflation is 1.7 %) will be:
fΠ = ln(1, 017) = 0.01685712
ref = (1 + 0.04
4 )4
− 1 = 0.04060401
f = ln(1, 04060401) = 0.03980132
FV Π = 1000 ∗ e(0.03980132−0.01685712)∗5
= 1121.56051
6 The effect of Tax and Inflation
Note! Tax is paid from the nominal interes, first then the effect of inflation can
be considered.
FVconT axInfla = 1000 ∗ ((e0.03980132
− 1) ∗ 0.85 + 1)5
∗ e−0.01685712∗5
=
1089.12031
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