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The measurement of interest
• The accumulation and amount functions
1. a(t): The accumulation function which gives the accumulated value at time t ≥ 0 of an original investment
of 1.
(a) a(0) = 1.
(b) a(t) is generally an increasing function.
2. A(t): the amount function which gives the accumulated value at time t ≥ 0 of an original investment of k.
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Then we have
A(t) = k · a(t)
and
A(0) = k
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The effective rate of interest
The effective rate of interest i is the amount of
money that are unit invested at the beginning of a
period will earn during the period, where interest is
paid at the end of the period.
equivalently,
i = a(1) − a(0) ⇐⇒ a(1) = 1 + i
(1 + i) − 1 a(1) − a(0) A(1) − A(0)
I1
=
=
=
=
i
a(0)
A(0)
A(0)
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Let in be the effective rate of interest during the nth
period from the date of investment. Then we have
A(n) − A(n − 1)
In
in =
=
for integral n ≥ 1.
A(n − 1)
A(n − 1)
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Simple interest
a(t) = 1 + it for integral t ≥ 0.
The accruing of interest according to this pattern is
called simple interest. A more rigorous mathematical
approach to the definition of a(t) is
a(t + s) = a(t) + a(s) − 1 for t ≥ 0 and s ≥ 0.
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0
a (t) =
=
=
=
=
a(t + s) − a(t)
lim
s→0
s
[a(t) + a(s) − 1] − a(t)
lim
s→0
s
a(s) − 1
lim
s→0
s
a(s) − a(0)
lim
s→0
s
a0(0), a constant.
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Thus,
Z
r
a0(r)dr =
0
Z
r
a0(0)dr
0
a(t) − a(0) = t · a0(0)
a(t) = 1 + t · a0(0).
Since a(1) = 1 + i, we have
a(1) = 1 + i = 1 + a0(0) =⇒ a0(0) = i.
Thus, a(t) = 1 + it for t ≥ 0.
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Compound interest
The theory of compound interest handles this problem by
assuming that the interest earned is automatically
reinvested. i.e.,
a(t + s) = a(t) · a(s) for t ≥ 0 and s ≥ 0.
a(t + s) − a(t)
a (t) = lim
s→0
s
0
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a(t) · a(s) − a(t)
= lim
s→0
s
a(s) − 1
= a(t)lim
s→0
s
= a(t) · a0(0).
Thus
d
a0(t)
= ln a(t) = a0(0).
a(t)
dt
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t
d
a0(0)dr
ln a(r)dr =
0 dr
0
0
ln a(t) − ln a(0) = t · a (0)
Z
t
Z
ln a(t) = t · a0(0)
since ln a(0) = 0. And a(1) = 1 + i, we have
ln a(1) = ln(1 + i) = a0(0).
Thus,
ln a(t) = t ln(1 + i) = ln(1 + i)t =⇒ a(t) = (1 + i)t
t ≥ 0.
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Present value and the discount function
We have seen that an investment of 1 will accumulate to
1 + i at the end of one period. The term 1 + i is often
called an accumulation factor.
It is often necessary to determine how much a person
must invest initially so that the balance will be 1 at the
end of one period. The answer is (1 + i)−1. We define a
new symbol v, such that
1
v=
1+i
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The term v is often called a discount factor.
How much a person must invest in order to accumulate
an amount of 1 at the end of t periods. The answer is
the reciprocal of the accumulation function a−1(t), since
a−1(t)a(t) = 1.
We will call a−1(t) the discount function.
Thus,
1
• For simple interest: a (t) =
1 + it
−1
13
1
t
=
v
• For compound interest: a (t) =
(1 + i)t
−1
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The effective rate of discount
The effective rate of discount d is the ratio of the
amount of interest (sometimes called the “amount of
discoun” or just “discount”) earned during the period to
the amount invested at the end of the period.
Let dn be the effective rate of discount during the nth
period from the date of investment. Then
In
A(n) − A(n − 1)
=
dn =
A(n)
A(n)
In may be commonly called either the “amount of
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discount” or the “amount of interest”.
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The relation between i, d and v
d
=⇒ i − id = d =⇒ d(1 + i) = i
i =
1−d
i
1
1+i
1
d =
=i
=
−
1+i
1+i 1+i 1+i
= iv
= 1−v
d = iv
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= i(1 − d)
= i − id
i − d = id
However, it is possible to define simple discount in a
manner analogous to the definition of simple interest.
Consider a situation in which the amount of discount
earned during each periods is constant. Then, the
original principal which will produce an accumulated
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value of 1 at the end of t periods is
a−1(t) = 1 − dt for 0 ≤ t < 1/d.
This contrasts with compound discount, in which case
the present value is
a−1(t) = v t = (1 − d)t
for t ≥ 0.