Transcript Document

Sections 4.5, 4.6
An annuity where the frequency of payment becomes infinite (i.e.,
continuous) is of considerable theoretical and analytical significance.
Formulas corresponding to such annuities are useful as approximations
corresponding to annuities payable with great frequency (e.g., daily).
The present value of an annuity payable continuously for n interest
conversion periods so that 1 is the total amount paid during each interest
conversion period is
n
n
n– 1
n– 1
_
t
n
v
v
v
1
–
v
a –n| = vt dt =
—— = —— = —————— = ——
ln(v)
ln(1) – ln(1+i)
ln(v)

0
0
(The formula could also be derived by taking the limit as m goes to 
..(m)
of either a(m)
–n| or a –n| .)
_
i
Observe that a – = — a –n|
n|

The accumulated value of a continuous annuity at the end of n interest
conversion periods so that 1 is the total amount paid during each interest
conversion period is
n
n
n–1
n–1
_
t
(1
+
i)
(1
+
i)
(1
+
i)
s n|
– = (1 + i)t dt = ——— = ———— = ————
ln(1 + i)

ln(1 + i)
0
0
(The formula could could also be derived by taking the limit as m goes
..(m) .)
to  of either s (m)
or
–
s n|
–
n|
_
Note that this first factor is s – .
1|
_
i
–
Observe that s – = — s n|
n|

–n
_
1
–
e
Observe that a – = ———
n|

and
_
en – 1
s n|
– = ———

Find the force of interest at which the accumulated value of a continuous
payment of 1 every year for 8 years will be equal to four times the
accumulated value of a continuous payment of 1 every year for four
_
_
years.
s 8|
– = 4 s 4|
–
e8 – 1
e4 – 1
——— = 4 ———


e8 – 1 = 4e4 – 4
e8 – 4e4 + 3 = 0
(e4 – 3)(e4 – 1) = 0
e4 = 3   = ln(3) / 4  0.275
e4 = 1   = 0 which is not a possibility.
Consider an annuity-immediate with a term of n periods where the
interest rate is i per period, and where the first payment is P (>0) with
successive payments each having Q (possibly negative in certain
situations) added to the previous payment, i.e., the payments form an
arithmetic progression.
Payments
P P+Q P+2Q
P+(n–2)Q P+(n–1)Q
Periods
0
1
2
… n–1
3
n
If A is the present value for this annuity, then
A = Pv + (P + Q)v2 + (P + 2Q)v3 + …
+ [P + (n – 2)Q]vn–1 + [P + (n – 1)Q]vn
(1 + i)A = P + (P + Q)v + (P + 2Q)v2 + …
+ [P + (n – 2)Q]vn–2 + [P + (n – 1)Q]vn –1
A = Pv + (P + Q)v2 + (P + 2Q)v3 + …
+ [P + (n – 2)Q]vn–1 + [P + (n – 1)Q]vn
(1 + i)A = P + (P + Q)v + (P + 2Q)v2 + …
+ [P + (n – 2)Q]vn–2 + [P + (n – 1)Q]vn –1
Subtracting the first equation from the second, we have
iA = P – Pvn + Q(v + v2 + v3 + … + vn–1) – (n – 1)Qvn
iA = P(1 – vn)+ Q(v + v2 + v3 + … + vn–1 + vn) – Qnvn
a –n|
1 – vn
Qnvn
A = P ——– + Q ——— – —— =
i
i
i
a –n| – nvn
P a – + Q —————
n|
i
The accumulated value of this annuity over the n periods is
A(1 +
i)n
=
s n|
– – n
P s – + Q ————
n|
i
Page 7 of the Class Notes:
With an 8% effective annual interest rate, find the present value for
each of the following payment streams:
(a) four yearly payments beginning one year from today, where the
first payment is $250 and each successive payment is increased by
$250
a –n| – nvn
a –4| – 4v4
P a – + Q ————— = 250 a – + 250 ————— =
n|
4|
i
0.08
3.312127 – 4 / 1.084
250(3.312127) + 250 ————————— =
0.08
$1990.55
Page 7 of the Class Notes:
With an 8% effective annual interest rate, find the present value for
each of the following payment streams:
(b) eight yearly payments beginning one year from today, where the
first payment is $250, each of the next four payments increases by
$250 over the previous payment, and the last three payments are
all equal to fifth payment
Using the result from part (a), the present value is
1990.55 + 1250v4 a –4| = 1990.55 + 1250(1/1.08)4(3.312127) =
1990.55 + 3043.14 = $5033.69
If P = Q = 1, then the annuity is called an increasing annuity, and the
present value of such an annuity is
..
n
n
a –n| – nvn
a –n| – nvn
a
–
1
–
v
+
–
nv
n|
(Ia) –n| = a – + ————— = ———————— = —————
n|
i
i
i
The accumulated value of this annuity over the n periods is
..
s n|
– – n
(Is) n|
– = (Ia) –n| (1 + i)n = ————
i
Observe that (Ia) –n| can be interpreted as a sum of level deferred
annuities after realizing that
..
n–1
n–1
n–t
a –n| – nvn
1
–
v
t
 vt a –––

=
v
——– = ————— = (Ia) –
n|
n–t|
i
i
t=0
t=0
..
Writing the last equation as a –n| = i (Ia) –n| + nvn suggests the
following verbal interpretation: The present value of an investment of 1
at the beginning of each period for n periods is equal to the sum of the
interest to be earned and the present value of the return of the principal.
Page 7 of the Class Notes:
With an 8% effective annual interest rate, find the present value for
each of the following payment streams:
(a) four yearly payments beginning one year from today, where the
first payment is $250 and each successive payment is increased by
$250
a –n| – nvn
a –4| – 4v4
P a – + Q ————— = 250 a – + 250 ————— =
n|
4|
i
0.08
3.312127 – 4 / 1.084
250(3.312127) + 250 ————————— =
0.08
We can also get this present value as follows:
..
a –4|0.08 – 4v4
250 (Ia) –4| 0.08 = 250 ——————— =
0.08
3.577097 – 4 / 1.084
250 ————————— = $1990.55
0.08
$1990.55
If P = n and Q = –1, then the annuity is called a decreasing annuity, and
the present value of such an annuity is
n – a – + nvn
a –n| – nvn
n
–
nv
n– a–
n|
n|
(Da) n|
– = n a – – ————— = —————————
=
———
n|
i
i
i
The accumulated value of this annuity over the n periods is
–
n(1 + i)n – s n|
(Ds) n|
– = (Da) n|
– (1 + i)n = ——————
i
– can be interpreted as a sum of level annuities after
Observe that (Da) n|
realizing observing that
n

t=1
n
a t–| =

t=1
1 – vt
n– a–
–
——– = ———n| = (Da) n|
i
i
Note: changing i to d in the denominator of any of the formulas derived
for an annuity-immediate with payments in an arithmetic progression
will give a corresponding formula for an annuity-due.
Page 8 of the Class Notes:
With an 8% effective annual interest rate, find the present value for
each of the following payment streams:
(c) twelve yearly payments beginning one year from today, where the
first payment is $250, each of the next four payments increase by
$250 over the previous payment, and the sixth, seventh, and eighth
payments are all equal to fifth payment, and each of the last four
payments decreases by $250 from the previous payment
Using the results from parts (a) and (b), the present value of
the first eight payments is
1990.55 + 1250v4 a –4| = 1990.55 + 1250(1/1.08)4(3.312127) =
1990.55 + 3043.14 = $5033.69
We see that the last four payments are $1000, $750, $500, $250.
Letting P = 4 and Q = –1, we see that the last four payments are
250P, 250(P + Q), 250(P + 2Q), 250(P + 3Q).
Page 8 of the Class Notes:
With an 8% effective annual interest rate, find the present value for
each of the following payment streams:
(c) twelve yearly payments beginning one year from today, where the
first payment is $250, each of the next four payments increase by
$250 over the previous payment, and the sixth, seventh, and eighth
payments are all equal to fifth payment, and each of the last four
payments decreases by $250 from the previous payment
We see that the last four payments are $1000, $750, $500, $250.
Letting P = 4 and Q = –1, we see that the last four payments are
250P, 250(P + Q), 250(P + 2Q), 250(P + 3Q).
At the time of the eighth payment, the current value of the
last four payments is
4 – a –4 | 0.08
250 (Da) 4–| 0.08 = 250 ————— =
0.08
We see that the last four payments are $1000, $750, $500, $250.
Letting P = 4 and Q = –1, we see that the last four payments are
250P, 250(P + Q), 250(P + 2Q), 250(P + 3Q).
At the time of the eighth payment, the current value of the
last four payments is
4 – a –4 | 0.08
250 (Da) 4–| 0.08 = 250 ————— =
0.08
4 – 3.312127
250 —————— = $2149.60
0.08
Using the results from parts (a) and (b), the present value of
all twelve payments is
1990.55 + 3043.14 + 2149.60/1.088 = $6195.05
Page 9 of the Class Notes:
With an 5% effective annual interest rate, annuity A pays $500 at the
end of this year and at the end of the following nine years, and annuity
B pays X at the end of this year with payments at the end of the
following nine years where each payment decreases by X / 10 from the
previous payment. Find the value of X for which the two annuities
have the same present value.
The present value of annuity A is 500 a10–| 0.05 = $3860.87
– | 0.05 =
The present value of annuity B is (X / 10) (Da) 10
10 – 7.721735
10 – a10–| 0.05
(X / 10) —————— = (X / 10) ——————— = 4.556530X
0.05
0.05
X=
$847.33
Consider a perpetuity with a payments that form an arithmetic
progression (and of course P > 0 and Q > 0). The present value for such
a perpetuity with the first payment at the end of the first period is
lim
n
a –n| – nvn
P a–
n| + Q —————
i
=
lim a – – lim nvn
n| n
n
P lim a –n| + Q —————————
n
i
1
— – 0
1
i
—
————
P
+Q
i
i
=
P
Q
= — + —
i
i2
If P = Q = 1, then the annuity is called an increasing annuity, and the
present value of such an annuity is
1
1
— + —
(Ia) ––
=
|
i
i2
Find the present value of a perpetuity-immediate whose successive
payments are 1, 2, 3, 4, … at an effective rate of 6%.
1
1
—— + ———
= $294.44
2
0.06 (0.06)
With varying annuities, it can be helpful to use the following quantities:
Fn = the present value of a payment of 1 at the end of n periods = vn
Gn = the present value of a level perpetuity of 1 per period with
the first payment at the end of n periods =
n
1
v
vn + vn+1 + vn+2 + … = vn(1 + v + v2 + v3 + …) = vn —— = —
1–v
d
Hn = the present value of an increasing perpetuity of 1, 2, 3, …,
with the first payment at the end of n periods =
vn + 2vn+1 + 3vn+2 + … = vn(1 + 2v + 3v2 + 4v3 + …) =
d
d
1
n
2
3
n
v — (1 + v + v + v + …) = v — ——
dv
dv 1 – v
n
1
v
= vn ———2 = —2
(1 – v)
d
These quantities can be used in an alternative derivation of the formulas
for (Ia) –n| and (Da) n|
– . Appendix 4 (at the end of Chapter 4) in the
textbook illustrates some examples of the use of Fn , Hn , and Gn .
Page 10 of the Class Notes:
With an 8% effective annual interest rate, a steam of twelve yearly
payments begins one year from today, where the first payment is $250,
each of the next four payments increase by $250 over the previous
payment, and the sixth, seventh, and eighth payments are all equal to
fifth payment, and each of the last four payments decreases by $250
from the previous payment. (This is the same steam of payments
described in part (c) on page 8.)
(a) Use Fn , Gn , and Hn to define the present value of this stream of
payments.
H1 1
 H6
 H9
+ H14
1
2
2
3
4
5
5
5
5
4
3
2
1
0
2
3
4
5
6
7
8
9
10
11
12
1
3
4
5
6
7
8
9
10
11
12
1 2 3 4 5 6 7
1 2 3 4
Page 10 of the Class Notes:
With an 8% effective annual interest rate, a steam of twelve yearly
payments begins one year from today, where the first payment is $250,
each of the next four payments increase by $250 over the previous
payment, and the sixth, seventh, and eighth payments are all equal to
fifth payment, and each of the last four payments decreases by $250
from the previous payment. (This is the same steam of payments
described in part (c) on page 8.)
(a) Use Fn , Gn , and Hn to define the present value of this stream of
payments.
250 H1  H6  H9 + H14
(b) Use the formula from part (a) to calculate the present value.
v = 1 / 1.08 = 0.925
v1
v6
v9
v14
250 —2  —2  —2 + —2
d
d
d
d
d = 0.08 / 1.08 = 0.074
= $6195.06
Find the present value of a perpetuity-immediate whose successive
payments are 1, 2, 3, 4, … at an effective rate of 6%.
1
1
—— + ———
= $294.44
2
0.06 (0.06)
H1 =
1 / 1.06
—————2 = $294.44
(0.06 / 1.06)
Find the present value of an annuity-immediate where payments start at
1, increase by 1 each period up to a payment of n, and then decrease by 1
each period up a final payment of 1.
..
a –n| – nvn
a –––
(n
–
1)
–
n–1|
(Ia) –n| + vn (Da) n–1|
––– = ————— + vn ——————— =
i
i
..
..
a –n| – nvn + nvn – vn – vn a –––
a –n| – vn (1 + a ––– )
n–1| = ———————————
n–1|
————————————————
=
i
i
..
..
n
a –n| – vn a –n|
..
1
–
v
——————— = a –n| ——— =
i
i
..
a – a –n|
n|
In one of the homework exercises, you are asked to give a verbal
interpretation of this formula.
Look at Example 4.11 (page 133) in the textbook for an example (and
symbol) for an annuity which increases for a time and then stays level.