Transcript Document
Sections 3.1, 3.2, 3.3
A series of payments made at equal intervals of time is called an annuity.
An annuity where payments are guaranteed to occur for a fixed period of
time is called an annuity-certain. Unless otherwise indicated, this is the
type of annuity we will assume, and the “certain” will be dropped from
the name.
An annuity where payments occur only under certain conditions is called
an annuity-contingent.
The interval between annuity payments is called an payment period,
often just called a period.
An annuity under which payments of 1 are made at the end of each of
period for n periods is called an annuity-immediate.
Payments
Periods
0
1
1
1
1
2 … n–1
1
n
The present value of the annuity
The accumulated value of the
– ,
at time 0 is denoted a –n|i , where
annuity at time n is denoted s n|i
the interest rate i is generally
where the interest rate i is
included only if not clear from
generally included only if not
the context.
clear from the context.
a –n| = v + v2 + … + vn =
s n|
– = (1 + i)n–1 + (1 + i)n–2 + … + 1 =
1 – vn 1 – v n
(1 + i)n – 1
(1 + i)n – 1
v —— = ——
———— = ————
1–v
i
(1 + i) – 1
i
– are available from certain calculators & Excel.
Values for a –
n| and s n|
1 – vn
a –n| = ——
i
1 = i a –n| + vn
The right hand side can be interpreted as the sum of
the “present value of the interest payments” and the
“present value of 1 (the original investment)”
(1 + i)n – 1
s n|
– = ———— 1 = (1 + i)n is –
n|
i
The right hand side can be interpreted as the
“accumulated value of 1 (the original investment)”
minus the “accumulated value of the interest payments”
– = a –n| (1 + i)n
Observe that s n|
1
i
i(1 + i)n
i
1
+ i = ————
+ i = ————
= ——n = ——
Also, ——
n
n
s n|
–
a –n|
(1 + i) – 1
(1 + i) – 1
1–v
This identity will be important in a future chapter.
Find the present value of an annuity which pays $200 at the end of each
quarter-year for 12 years if the rate of interest is 6% convertible
quarterly.
200 a ––
48 | 0.015 =
An investment of $5000 is made at 6% convertible semiannually. How
much can be withdrawn each half-year to use up the fund exactly at the
end of 20 years?
To calculate 200 a ––
48 | 0.015
on the TI-84 calculator, do the following:
(Note: On the TI-83 calculator, the | 2nd | | FINANCE | keys should be
used in place of the | APPS | key and Finance option.)
Press the | APPS | key, select the Finance option, and select the
TVM_Solver option. Enter the following values for the variables
displayed:
N = 48
I% = 1.5
PV = 0
PMT = –200
FV = 0
P/Y = 1
C/Y = 1
Select the END option for PMT , press the | APPS | key, and select the
Finance option.
Select the tvm_PV option, and after pressing the | ENTER | key, the
desired result should be displayed.
To calculate 200 a ––
48 | 0.015
in Excel, enter the following formula:
=PV(0.015,48,-200,0,0)
This is the balance
remaining (generally 0)
This implies payments at the end of
each period. A 1 implies payments
at the beginning of each period.
Find the present value of an annuity which pays $200 at the end of each
quarter-year for 12 years if the rate of interest is 6% convertible
quarterly.
200 a ––
48 | 0.015 = $6808.51
An investment of $5000 is made at 6% convertible semiannually. How
much can be withdrawn each half-year to use up the fund exactly at the
end of 20 years?
Let R be the amount withdrawn (i.e., the payments) at each half-year.
The present value at the time the investment begins is $5000, so the
equation of value is
5000 = R a ––
40 | 0.03
R=
5000
To calculate R = ———
a ––
40 | 0.03
on the TI-84 calculator, do the following:
(Note: On the TI-83 calculator, the | 2nd | | FINANCE | keys should be
used in place of the | APPS | key and Finance option.)
Press the | APPS | key, select the Finance option, and select the
TVM_Solver option. Enter the following values for the variables
displayed:
N = 40
I% = 3
PV = –5000
PMT = 0
FV = 0
P/Y = 1
C/Y = 1
Select the END option for PMT , press the | APPS | key, and select the
Finance option.
Select the tvm_Pmt option, and after pressing the | ENTER | key, the
desired result should be displayed.
5000
To calculate R = ———
a ––
40 | 0.03
in Excel, enter the following formula:
=5000/PV(0.03,40,-1,0,0)
Find the present value of an annuity which pays $200 at the end of each
quarter-year for 12 years if the rate of interest is 6% convertible
quarterly.
200 a ––
48 | 0.015 = $6808.51
An investment of $5000 is made at 6% convertible semiannually. How
much can be withdrawn each half-year to use up the fund exactly at the
end of 20 years?
Let R be the amount withdrawn (i.e., the payments) at each half-year.
The present value at the time the investment begins is $5000, so the
equation of value is
5000 = R a ––
40 | 0.03
R = $216.31
Compare the total amount of interest that would be paid on a $3000 loan
over a 6-year period with an effective rate of interest of 7.5% per annum,
under each of the following repayment plans:
(a)
The entire loan plus accumulated interest is paid in one lump sum
at the end of 6 years.
3000(1.075)6 = $4629.90
(b)
Total Interest Paid = $1629.90
Interest is paid each year as accrued, and the principal is repaid at
the end of 6 years.
Each year, the interest on the loan is 3000(0.075) = $225
Total Interest Paid = $1350
(c)
The loan is repaid with level payments at the end of each year
over the 6-year period. Let R be the level payments.
3000 = R a ––
R = $639.13
6 | 0.075
Total Interest Paid = 6(639.13) – 3000 = $834.78
An annuity under which payments of 1 are made at the beginning of each
period for n periods is called an annuity-due.
Payments
1
1
1
Periods
0
1
2 … n–1
1
n
The present value of ..the annuity
The accumulated value of the
..
a
–
at time 0 is denoted n|i , where
annuity at time n is denoted s –n|i ,
the interest rate i is generally
where the interest rate i is
included only if not clear from
generally included only if not
clear from the context.
..the context. 2
..
n–1
a–
– = (1 + i)n + (1 + i)n–1 + … + (1 + i) =
n| = 1 + v + v + … + v = s n|
1 – vn
1 – vn
(1 + i)n – 1
(1 + i)n – 1
—— = ——
(1 + i) ———— = ————
1–v
d
(1 + i) – 1
d
.. = ..
–
a –n| (1 + i)n
Observe that s n|
1
d
d(1 + i)n
d
1
+ d = ————
= ——n = ——
.. + d = ————
..
Also, ——
n
n
(1 + i) – 1
(1 + i) – 1
1–v
s n|
–
a –n|
..
In addition, observe that a –n| = a –n| (1 + i)
..
– (1 + i)
s n|
– = s n|
..
a –n| = 1 + a –––
n–1|
..
––– – 1
s n|
– = s n+1|
These last four formulas demonstrate that annuity-immediate and
annuity-due are really just the same thing at two different points in time,
as is illustrated graphically in Figure 3.3 of the textbook.
An investor wishes to accumulate $3000 at the end of 15 years in a fund
which earns 8% effective. To accomplish this, the investor plans to make
deposits at the end of each year, with the final payment to be made one
year prior to the end of the investment period. How large should each
deposit be?
Let R be the payments each year. The accumulated value of the
investment at the end of the investment period is to be $3000, so the
equation of value is
3000
3000
..
————
= —————— =
3000 = R s ––
R
=
..
14 | 0.08
s ––
s ––
–1
14 | 0.08
15 | 0.08
3000
To calculate R = ———
..
s ––
14 | 0.08
on the TI-84 calculator, do the following:
(Note: On the TI-83 calculator, the | 2nd | | FINANCE | keys should be
used in place of the | APPS | key and Finance option.)
Press the | APPS | key, select the Finance option, and select the
TVM_Solver option. Enter the following values for the variables
displayed:
N = 14
I% = 8
PV = 0
PMT = 0
FV = –3000
P/Y = 1
C/Y = 1
Select the BEGIN option for PMT , press the | APPS | key, and select
the Finance option.
Select the tvm_Pmt option, and after pressing the | ENTER | key, the
desired result should be displayed.
3000
To calculate R = ———
..
s ––
14 | 0.08
in Excel, enter the following formula:
=3000/FV(0.08,14,-1,0,1)
An investor wishes to accumulate $3000 at the end of 15 years in a fund
which earns 8% effective. To accomplish this, the investor plans to make
deposits at the end of each year, with the final payment to be made one
year prior to the end of the investment period. How large should each
deposit be?
Let R be the payments each year. The accumulated value of the
investment at the end of the investment period is to be $3000, so the
equation of value is
3000
3000
..
————
= —————— =
3000 = R s ––
R
=
..
14 | 0.08
s ––
s ––
–1
14 | 0.08
15 | 0.08
3000
——— = $114.71
26.1521