Transcript Document

Sections 1.7, 1.8
Recall:
The effective rate of interest i is the amount of money that one unit (one
dollar) invested at the beginning of a (the first) period will earn during
the period, with interest being paid at the end of the (first) period. That
is,
i = a(1) – a(0) = a(1) – 1 .
We can also say that the effective rate of interest i is the ratio of the
amount of interest earned during the (first) period to the amount invested
A(1) – A(0)
at the beginning of the period. That is,
i = ————— .
A(0)
The effective rate of discount d is the ratio of the amount of interest
(discount) earned during the (first) period to the amount invested at the
A(1) – A(0)
end of the period. That is,
d = ————— .
A(1)
The effective rate of discount during the nth period from the date of
investment is
dn
a(n) – a(n – 1)
= ——————
a(n)
=
A(n) – A(n – 1)
——————
A(n)
(The effective rate of interest can be considered interest paid at the end
of the period on the balance at the beginning of the period, whereas the
effective rate of discount can be considered interest paid at the beginning
of the period on the balance at the end of the period.)
Recall that compound interest implies a constant rate of effective
interest; it will be proven in the exercises that compound interest implies
a constant rate of effective discount which can be called “compound
discount”.
t2
1
3
5
If a(t) = 1 + — , then d1 = — ,
d2 = — ,
and
d3 = — .
25
26
29
34
In = A(n) – A(n – 1) can be called the “amount of interest” or the “amount
of discount” for the nth period.
A(1) – A(0)
i = —————
A(0)
d
A(1) – A(0)
d = —————
A(1)
iA(0) = dA(1) = A(1) – A(0)
A(1) – A(0)
A(0)
A(1)
= ————— = 1 – —— = 1 – ——
A(1)
A(1)
A(0)
A(1) – A(0) + A(0)
1 – ———————–
A(0)
–1
–1
=
1
i
= ——
= 1 – (i + 1)–1 = 1 – ——
1+i
1+i
The amount of interest earned for one year when X is invested is $108.
The amount of discount earned when an investment grows to value X at
the end of one year is $100. Find X, i, and d.
i = 0.08
108
i
iX = 108
—— = 100
—— X = 100
d = 2 / 27
1
+
i
1
+
i
dX = 100
X = $1350
a(t) = amount to which 1 unit accumulates in t periods
[a(t)]–1 = amount which must be invested to accumulate 1 unit in t
periods
The accumulation function for simple interest is one where the amount of
interest earned in t periods is the effective rate of interest times t:
a(t) = 1 + it
The accumulation function for simple discount is defined to be one
where the amount of discount earned in t periods is the effective rate of
discount times t:
[a(t)]–1 = 1 – dt
for 0  t < 1/d
1
a(t) = ——– for 0  t < 1/d
1 – dt
(Note that the condition t < 1/d
is required to insure that a(t) is
defined and positive.)
Are simple interest and simple discount the same?
With a(t) =
1
——– for 0  t < 1/d , the effective rate of discount is
1 – dt
a(1) – a(0)
1/(1 – d) – 1
———— = ————— = d (as expected), and the effective rate
a(1)
1/(1 – d)
a(1) – a(0)
1/(1– d) – 1
d
of interest is i = ———— = ————— = —— .
1–d
a(0)
1
d
1
Are 1 + it = 1 + ——– t and ——– the same for all t?
1–d
1 – dt
No, so simple interest and simple discount are not the same.
The accumulation function for compound interest is one where the
effective rate of interest earned each period is constant:
a(t) = (1 + i)t
The accumulation function for compound discount is defined to be one
where the effective rate of discount earned each period is constant:
1
We see that the effective rate of discount earned
a(t) = ——–
in period n is
(1 – d)t
a(n) – a(n–1)
—————– =
a(n)
1/(1 – d)n – 1/(1 – d)n–1
————————— = d
1/(1 – d)n
which is indeed constant.
(This is the proof needed for Text Exercise #24(a).)
Are compound interest and compound discount the same?
t
d
1
Since (1 + i)t = 1 + ——
= ———t
for all t,
1–d
(1 – d)
compound interest is the same as compound discount.
We say that a rate of interest and a rate of discount are equivalent, if a
given amount of principal invested for the same length of time at each of
the rates produces the same accumulated value (i.e., the accumulation
functions are identical).
We have previously seen that an effective rate of simple interest can
never be equivalent to an effective rate of simple discount.
We have previously seen that an effective rate of compound discount d is
equivalent to an effective rate of compound interest i = d / (1 – d).
Observe that several identities can be derived from this equivalency:
d Interpretation: the
i Interpretation: the
i = ——
d = ——
1 – d ratio of amount of
1 + i ratio of amount of
interest to beginning
interest to ending
principal.
principal.
d=1–v
d = iv
or
v = 1 – d Interpretation: v and 1–d are
Interpretation: discount i
each the present value of 1
from the end of a period to
paid at the end of the period.
the beginning of a period.
Compound interest or discount will always be assumed, unless specified
otherwise.
An interest (discount) rate may be stated in terms of one period, which
we have called an effective rate of interest (discount), or in terms of
some fraction of a period, which we shall call a nominal rate of interest
(discount). When interest is paid (i.e., reinvested) more frequently than
once per period, we say it is payable (convertible, compounded) each
fraction of a period, and this fractional period is called the interest
conversion period.
For a positive integer m, we let i(m) represent a nominal rate of interest
payable m times per period; that is, the rate of interest is i(m)/m for each
mth of a period.
The accumulation function with the nominal rate of interest i(m) is
mt
(m)
i
for t  0
a(t) = 1 + —
m
Find the accumulated value of $3000 to be paid at the end of 8 years with
a rate of compound interest of 5%
(a) per annum.
3000(1 + 0.05)8 = $4432.37
(b) convertible quarterly.
3000(1 + 0.05/4)4•8 = $4464.39
(c) convertible monthly.
3000(1 + 0.05/12)12•8 = $4471.76
We say that two rates of interest are equivalent, if a given amount of
principal invested for the same length of time at each of the rates
produces the same accumulated value.
If two rates of compound interest i and i(m) are equivalent, then
m
mt
(m)
i
t
1 + i = 1 + — , which implies (1 + i) = 1 + — for all t  0 .
m
m
i(m)
We find that with compound interest, the rates that are equivalent do not
depend on the period of time chosen for the comparison, but this may not
necessarily be true for another patterns of interest. Observe that nominal
rates of interest are irrelevant with simple interest.
Find the compound yearly interest rate i which is equivalent to a rate of
compound interest of 8% convertible quarterly.
4
4
0.08
0.08
1 + i = 1 + ——  i = 1 + —— – 1  i = 0.08243216
4
4
Find the compound interest rate i(2) which is equivalent to a rate of
compound yearly interest of 8%.
i(2)
1 + 0.08 = 1 + —
2
2
i(2)
 1 + — = (1.08)1/2  i(2) = 2[(1.08)1/2 – 1] 
2
0.07846
Find the compound interest rate i(4) which is equivalent to a rate of
compound yearly interest of 8%.
4
i(4)
i(4)
1 + 0.08 = 1 + —  1 + — = (1.08)1/4  i(4) = 4[(1.08)1/4 – 1] 
4
4
0.07771
For a positive integer m, we let d(m) represent a nominal rate of discount
payable at the beginning of mths of a period.
The accumulation function with the nominal rate of discount d(m) is
–mt
(m)
d
a(t) = 1 – — for t  0
m
Find the present value of $8000 to be paid at the end of 5 years at a rate
of compound interest of 7% yearly
(a) convertible semiannually.
8000
—————— = $5671.35
(1 + 0.07/2)2•5
(b) payable in advance and convertible semiannually.
8000
—————— = 8000(1 – 0.07/2)2•5 = $5602.26
(1 – 0.07/2)–2•5
We say that two rates of discount are equivalent, if a given amount of
principal invested for the same length of time at each of the rates
produces the same accumulated value.
If two rates of compound discount d and d(m) are equivalent, then
–m
–mt
d(m)
d(m)
for all t  0 .
(1 – d)–1 = 1 – — , which implies (1 – d)–t = 1 – —
m
m
We see that with compound discount, the rates that are equivalent do not
depend on the period of time chosen for the comparison, but this may not
necessarily be true for another patterns of interest. Observe that nominal
rates of discount are irrelevant with simple discount.
Equivalent interest and discount rates can be summarized as follows:
–p
i(m) m
d(p)
1 + — = 1 + i = v–1 = (1 – d)–1 = 1 – —
.
m
p
Find the compound yearly discount rate d which is equivalent to a rate of
compound discount of 8% convertible quarterly.
–4
4
0.08
0.08
 d = 1 – 1 – ——  d = 0.07763
(1 – d)–1 = 1 – ——
4
4
Find the compound discount rate d(12) which is equivalent to a rate of
compound yearly discount of 6%.
–12
(12)
d
1–—
= (1 – 0.06)–1  d(12) = 12[1 – (1 – 0.06)1/12] 
12
d(12) = 0.06172
Find the nominal discount rate convertible semiannually which is
equivalent to a nominal interest rate 5% convertible quarterly.
–2
4
–2
(2)
d
0.05
0.05
1–—
= 1 + ——  d(12) = 2 1 – 1 + ——

2
4
4
d(2) = 0.04908