Sections 4.7, 4.8, 4.9
Consider an annuity-immediate with a term of n periods where the
interest rate is i per period, and where the first payment is 1 with
successive payments each having 1 + k multiplied by the previous
payment where k > 1, i.e., the payments form a geometric progression.
The present value for this annuity is
v + (1 + k)v2 + (1 + k)2v3 + … + (1 + k)n–1vn =
n
1+k
1 – ——
1+i
1
—— —————
1+i
1+k
1 – ——
1+i
n
1+k
1 – ——
1+i
= —————
i–k
nv
if k  i
if k = i
The accumulated value after n periods for this annuity-immediate is
(1 + i)n – (1 + k)n
——————– if k  i and is n(1 + i)n  1 if k = i.
i–k
i–k
Suppose k < i , and let i* = —— . If payments occur at the beginning
1+k
of each period, the present value for this annuity-due is
1 + (1 + k)v + (1 + k)2v2 + (1 + k)3v3 + … + (1 + k)n–1vn–1 =
n
n
..
1
–
(v*)
1
–
(v*)
2
3
n–1
1 + v* + (v*) + (v*) + … + (v*) = ———– = ———– = a –n| i*
1 – v*
d*
We see that the present value of an n-period annuity-due with payments
increasing in geometric progression with common ratio (1 + k) and
effective interest rate i is equal to the present value of a level n-period
annuity-due with interest rate i*.
The accumulated value after n periods for this annuity-due is
..
n
(1 + i) a –n| i*
Consider a perpetuity with a payments that form a geometric progression
where 1 < k < i.
The present value for such a perpetuity with the first payment at the end
1
of the first period is
——
i–k
The present value for such a perpetuity with the first payment at the
beginning of the first period is 1 + i
——
i–k
Observe that the value for these perpetuities cannot exist if 1 + k  1 + i.
The first of 30 payments of an annuity occurs in exactly one year and is
equal to $500. The payments increase so that each payment is 5%
greater than the preceding payment. Find the present value of this
annuity with an annual effective rate of interest of 8%.
100k = 5% is the periodic (yearly) increase in payments.
30
1.05
1 – ——
1.08
500 ————— = $9508.28
0.08 – 0.05
Suppose 100k = 5% is the periodic (yearly) increase in payments.
Then the present value would be
30
0.95
1 – ——
1.08
500 ————— =
0.08 + 0.05
$3764.11
In exactly one year from today, a perpetuity-immediate begins making
annual payments. The first payment is $600. With an effective rate of
interest of 7%, find the present value of this perpetuity if
(a) each future payment is increased by 3% more than the previous
payment,
(b) each future payment up to and including the 15th payment is
increased by 3% more than the previous payment, and payments starting
with the 15th payment remain level thereafter.
600
For part (a), the present value is ————— = $15,000
0.07 – 0.03
For part (b), the present value is the sum of the present value for the
first 15 payments and the present value for the remaining payments.
For part (b), the present value is the sum of the present value for the
first 15 payments and the present value for the remaining payments.
15
1.03
1 – ——
1.07
600 ————— =
0.07 – 0.03
1
——
1.07
15
600(1.03)15
$6529.81
1
—— = $4840.11
0.07
$6529.81 + $4840.11 = $11,369.92
A participant in a pension plan will earn $40,000 salary in the upcoming
year, and this salary will be increased by 4% each year after. For each of
the next 25 years, a contribution of 3% of the salary for the year together
with a matching amount from the employer is made to the pension fund
at the end of the year. If the pension fund earns 8% effective per year,
find the accumulated value of these contributions at the end of the 25year period.
1.0825 – 1.0425
40000(0.06) —————— = $250,958.33
0.08 – 0.04
Consider annuities payable less frequently than interest is convertible.
We let
k = the number of interest conversion periods in one payment period,
n = the term of the annuity measured in interest conversion periods,
i = rate of interest per conversion period.
It follows that
n / k = the number of annuity payments made.
Let A be the present value of a generalized increasing annuity with
payments 1, 2, 3, etc. occurring at the end of each interval of k interest
conversion periods. Then,
A=
vk
+
2v2k
n
n
n–k
+ … + — – 1 v + — vn
k
k
n
n
(1 + i)kA = 1 + 2vk + 3v2k + … + — – 1 vn–2k + — vn–k
k
k
n
n
A = vk + 2v2k + … + — – 1 vn–k + — vn
k
k
(1 +
i)kA
=1+
2vk
+
3v2k
n
n
n–2k
+…+ —–1 v
+ — vn–k
k
k
Subtracting the first equation from the second, we have
A[(1 +
i)k
– 1] = 1 +
vk
+
v2k
+…+
vn–k
n
– — vn
k
a –n|
a –n|
a –k|
n n
v This is the present value of an increasing annuity
a –k| – —
k
with payments 1, 2, 3, etc. occurring at the end
A = ——————
of each interval of k interest conversion periods.
i s k|
–
Similar derivations can be done when payments increase in some other
arithmetic progression.
Page 9 of the Class Notes:
With an effective rate of interest is 6%, find the present value of
payments of $8 at the end of three years, $16 at the end of six years,
$24 at the end of nine years, etc., if
(a) the last payment is at the end of 27 years.
a –n|
a27–|
n n
27 27
v
– —v
a –k| – —
a
–
k
3
3|
(8) —————— = (8) —————— = $128.82
i s k|
(0.06) s –3|
–
Consider an increasing annuity payable more frequently than interest is
convertible with constant payments during each interest conversion
period. Let the annuity be payable mthly with each payment in the first
period equal to 1/m, each payment in the second period equal to 2/m, etc.
Letting i = rate of interest per period, we use i(m) to write a formula for
the present value of the n-period annuity-immediate denoted by (Ia)(m)
–n| .
Making the mthly payments of k/m in period k is the same as making one
payment equal to the accumulated value of the mthly payments at the end
of the period, that is, one payment of
k
k (1 + i(m)/m)m – 1
i
– (m)
= — ———————
— s m|
= k —
(m)
i
/m
m
m
i /m
i(m)
1 1
——
m m
1 1
——
m m
…
0
k–1 k k
—– — —
m m m
…
1
k k
——
m m
…
k–1
n–1 n n
—– — —
m m m
…
k
m n
——
m m
…
n–1
n
Consequently, the increasing annuity payable mthly is the same as an
i , which implies
increasing annuity with P = Q = —
i(m)
..
i
a –n| – nvn
(m)
(Ia) –n| = —————
(Ia) –n| = —
(m)
i
i(m)
i
1
1
We may choose to write (Ia)(m)
––
— + —
| = —
(m)
i
i
i2
i
—(m)
i
1 1
——
m m
0
i
k —
i(m)
1 1
——
m m
…
k–1 k k
—– — —
m m m
…
1
1
1
= —
+ ——
(m)
i
i(m) i
k k
——
m m
…
k–1
i
n —
i(m)
n–1 n n
—– — —
m m m
…
k
m n
——
m m
…
n–1
n
Page 10 of the Class Notes:
With an effective rate of interest is 6%, find the present value of
payments of $8 at the end of every 4 months in the first year, $16 at
the end of every 4 months in the second year, $24 at the end of every 4
months in the third year, etc., if
(a) the last payment is at the end of 27 years.
..
i
a27–| – 27v27
(3)
(m)
(Ia)27–| = 24 ————— =
8m(Ia) –n| = 24 (Ia)27–| = 24 —
(3)
i
i(3)
14.003 – 5.599
= $3428.06
24 ———————
1/3
3[(1.06) – 1]
(b) the payments continue forever in perpetuity.
1
1
1
1
8m(Ia)(m)
––
= 24 ——— + —————— =
+ ——
 | = 24 —
(m)
(m)
0.05884
(0.05884)(0.06)
i
i i
$7206.17
Find the present value of a perpetuity which pays 1 at the end of the third
year, 2 at the end of the sixth year, 3 at the end of the ninth year, etc.
Let i be the effective rate of interest, and treat 3 years as one period.
Then i* = (1 + i)3 – 1 is the rate of interest for three years.
With P = Q = 1, the present value of the perpetuity is
P
Q
1
1
— + —2 = —————
+ —————
=
3
3
2
i*
i*
(1 + i) – 1
[(1 + i) – 1]
v3
(1 + i)3
—————
= ———
3
2
(1 – v3)2
[(1 + i) – 1]
Observe that if the payments are at the end of ever kth year (period),
the general formula is
vk
———
(1 – vk)2
Also, look at Example 4.15 in the textbook (page 139) for an
alternative way to obtain this present value.
Page 9 of the Class Notes:
With an effective rate of interest is 6%, find the present value of
payments of $8 at the end of three years, $16 at the end of six years,
$24 at the end of nine years, etc., if
(a) the last payment is at the end of 27 years.
a –n|
a27–|
n n
27 27
v
– —v
a –k| – —
a
–
k
3
3|
(8) —————— = (8) —————— = $128.82
i s k|
(0.06) s –3|
–
(b) the payments continue forever in perpetuity.
v3
(8) ——— = $261.14
(1 – v3)2
Find the accumulated value at the end of eight years of an annuity in
which payments are made at the beginning of each quarter for four
years. The first payment is $3000, and each of the other payments is
95% of the previous payment. Interest is credited at 6% convertible
semiannually.
3000 (1.03)16 + (0.95)(1.03)15.5 + (0.95)2(1.03)15 +… + (0.95)15(1.03)8.5
= 3000 (1.03)16 1 + (0.95/1.03) + (0.95/1.03)2 +…+ (0.95/1.03)15
= 3000 (1.03)16
1 – (0.95/1.03)16
——————— = $49,134.18
1 – (0.95/1.03)
Consider an increasing annuity payable more frequently than interest is
convertible where the first mthly payment in the first period is 1/m2 with
each mthly payment increasing by 1/m2 throughout and across all of the n
interest conversion periods. Letting i = rate of interest per period, we use
i(m) and the formula for (Ia) –n| to write a formula for present value of the
n-period annuity-immediate denoted by (I (m)a)(m)
–n| .
..
a –n| – nvn
Recall that (Ia) –n| = ————— where the annuity payments are 1,
i
2, 3, … n respectively at the end of n periods with interest rate i per
period. With an interest rate of i(m) / m for each of nm periods and
respective payments of 1/m2 , 2/m2 , 3/m2 , … nm/m2 at the end of each
..
1
period, we have
a –––
(m)
nm | i /m – nm ——————
(1 + i(m)/m)nm
(m)
1
(m)
(I a) –n| = —
=
2
(m)
m
i
—
m
(m)
1
(I (m)a) –n| = —
m2
..
a –––
nm | i(m)/m
1
– nm ——————
(1 + i(m)/m)nm
=
i(m)
—
m
1 1 – 1/(1 + i(m)/m)nm
1
— ————————
– n ——————
(m)
(m)
m (i /m)/(1 + i /m)
(1 + i(m)/m)nm
i(m)
=
(m)
1
(I (m)a) –n| = —
m2
..
a –––
nm | i(m)/m
1
– nm ——————
(1 + i(m)/m)nm
=
i(m)
—
m
1 1 – 1/(1 + i(m)/m)nm
1
— ————————
– n ——————
(m)
(m)
m (i /m)/(1 + i /m)
(1 + i(m)/m)nm
i(m)
1 – 1/[(1 + i(m)/m)m]n
1
————————
– n ——————
(m)
(m)
i /(1 + i /m)
[(1 + i(m)/m)m]n
i(m)
=
=
(m)
1
(I (m)a) –n| = —
m2
..
a –––
nm | i(m)/m
1
– nm ——————
(1 + i(m)/m)nm
=
i(m)
—
m
1 1 – 1/(1 + i(m)/m)nm
1
— ————————
– n ——————
(m)
(m)
m (i /m)/(1 + i /m)
(1 + i(m)/m)nm
i(m)
1 – 1/[(1 + i(m)/m)m]n
1
————————
– n ——————
(m)
(m)
i /(1 + i /m)
[(1 + i(m)/m)m]n
i(m)
1 – 1/(1 + i)n
1
——————
– n ———n
(m)
(m)
i /(1 + i /m)
(1 + i)
i(m)
=
=
=
..(m)
a –n| – nvn
—————
i(m)
Look at each of the following:
In Section 4.9, find the symbol for, and definition of, the present
value for an increasing n-period annuity with payments made
continuously at the rate of t per period at exact moment t.
In Section 4.9, find the formula for a general, continuous varying
annuity, and look at Example 4.18.
In Section 4.10, consider Table 4.1 (page 142) in the textbook.