Sections 4.5, 4.6.
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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
_
a –
n
vn – 1
—— =
ln(v)
vn – 1
—————— =
ln(1) – ln(1+i)
vt
—— =
ln(v)
1 –vn
——
=
vt dt =
0
0
(The formula could also be derived by taking the limit as m goes to of either or .)
(m)
a –
n
..(m)
a –
n
_
a –
n
i
—
Observe that
=
a –
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
_
s –
n
(1 + i)n– 1
———— =
ln(1 + i)
(1 + i)n– 1
————
(1 + i)t
——— =
ln(1 + i)
=
(1 + i)t dt =
0
0
(The formula could could also be derived by taking the limit as m goes to of either or .)
(m)
s –
n
..(m)
s –
n
_
s –
1
Note that this first factor is .
_
s –
n
i
—
Observe that
=
s –
n
_
a –
n
_
s –
n
1 –e–n
———
en – 1
———
Observe that
=
and
=
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
_
s –
4
= 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 annuityimmediate with a term 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.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
3
…
n– 1
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 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. = 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
1 – vnQnvn– nvn
A = P ——– + Q——— – —— = P + Q—————
iiii
a –
n
a –
n
a –
n
The accumulated value of this annuity over the n periods is
– n
P + Q————
i
s –
n
A(1 + i)n =
s –
n
Page 7 of the Class Notes: 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.
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
– nvn
P + Q————— =
i
– 4v4
250 + 250 ————— =
0.08
a –
n
a –
4
a –
n
a –
4
3.312127 – 4/1.084
250(3.312127) + 250 ————————— =
0.08
$1990.55
Page 7 of the Class Notes: 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.
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 =
1990.55 +
1250(1/1.08)4(3.312127) =
a –
4
1990.55 + 3043.14 =
$5033.69
If 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.P = Q = 1, then the annuity is called an increasing annuity, and the present value of such an annuity is
..
a –
n
– nvn
————— =
i
1 – vn + – nvn
———————— =
i
– nvn
—————
i
(Ia)–
n
a –
n
=
a –
n
+
a –
n
The accumulated value of this annuity over the n periods is
..
s –
n
– n
————
i
(Is) –
n
(Ia)–
n
=
(1 + i)n =
(Ia)–
n
Observe that can be interpreted as a sum of level deferred annuities after realizing that
..
a –
n
n–1
t=0
n–1
t=0
1 – vn–t
——– =
i
– nvn
————— =
i
(Ia)–
n
vt
=
vt
a –––
n–t
..
a –
n
(Ia)–
n
Writing the last equation as 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.
i + nvn
=
Page 7 of the Class Notes: 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.
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
– nvn
P + Q————— =
i
– 4v4
250 + 250 ————— =
0.08
a –
n
a –
4
a –
n
a –
4
3.312127 – 4/1.084
250(3.312127) + 250 ————————— =
0.08
$1990.55
We can also get this present value as follows:
..
a –
40.08
– 4v4
———————
0.08
(Ia)–
4 0.08
250
250
=
=
3.577097 – 4/1.084
250 ————————— =
0.08
$1990.55
If 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.P = n and Q = –1, then the annuity is called a decreasing annuity, and the present value of such an annuity is
a –
n
– nvn
————— =
i
n– nvn – + nvn
————————— =
i
n –
———
i
(Da)–
n
a –
n
= n
a –
n
–
a –
n
The accumulated value of this annuity over the n periods is
s –
n
n(1 + i)n –
——————
i
(Ds) –
n
(Da)–
n
=
(1 + i)n =
(Da)–
n
Observe that can be interpreted as a sum of level annuities after realizing observing that
a –
n
n
t=1
n
t=1
1 – vt
——– =
i
n –
——— =
i
(Da)–
n
=
a–
t
Note: changing i to d in the denominator of any of the formulas derived for an annuityimmediate with payments in an arithmetic progression will give a corresponding formula for an annuitydue.
Page 8 of the Class Notes: 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.
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 =
1990.55 +
1250(1/1.08)4(3.312127) =
a –
4
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: 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.
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
a –
4  0.08
4 –
—————
0.08
(Da)–
4  0.08
250
=
250
=
We see that the last four payments are 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.
$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
a –
4  0.08
4 –
—————
0.08
(Da)–
4  0.08
250
=
250
=
4 – 3.312127
250 —————— =
0.08
$2149.60
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: 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.
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.
$3860.87
500 =
The present value of annuity A is
a –
10 0.05
(Da)– =
10 0.05
(X/10)
The present value of annuity B is
10 – 7.721735
(X/10)——————— =
0.05
10 –
——————
0.05
a –
10 0.05
4.556530X
(X/10)
=
$847.33
X =
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
– nvn
P + Q————— =
i
a –
n
lim
n
a –
n
– nvn
P + Q————————— =
i
lim
n
lim
n
a –
n
lim
n
a –
n
1
—
i
–
0
1
—
i
———— =
i
PQ
— + —
ii2
P + Q
If P = Q = 1, then the annuity is called an increasing annuity, and the present value of such an annuity is
1 1
— + —
ii2
(Ia)––

=
Find the present value of a perpetuityimmediate whose successive payments are 1, 2, 3, 4, … at an effective rate of 6%.
1 1
—— + ——— =
0.06 (0.06)2
$294.44
With varying annuities, it can be helpful to use the following quantities:
Fn =
Gn =
Hn =
the present value of a payment of 1 at the end of n periods =
vn
the present value of a level perpetuity of 1 per period with the first payment at the end of n periods =
1
vn—— =
1 – v
vn
—
d
vn + vn+1 + vn+2 + … =
vn(1 + v + v2 + v3 + …) =
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
vn— (1 + v + v2 + v3 + …) =
dv
d1
vn — —— =
dv1 – v
1
vn ——— =
(1 – v)2
vn
—
d2
These quantities can be used in an alternative derivation of the formulas for and . Appendix 4 (at the end of Chapter 4) in the textbook illustrates some examples of the use of Fn , Hn , and Gn .
(Ia)–
n
(Da)–
n
Page 10 of the Class Notes: following quantities:
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
2
3
4
5
6
7
8
9
10
11
12
H6
1
2
3
4
5
6
7
H9
1
2
3
4
+ H14
1
2
3
4
5
5
5
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
Page 10 of the Class Notes: following quantities:
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
d =
0.08 / 1.08 = 0.074
v1
—
d2
v6
—
d2
v9
—
d2
v14
— =
d2
$6195.06
250
+
Find the present value of a perpetuityimmediate whose successive payments are 1, 2, 3, 4, … at an effective rate of 6%.
1 1
—— + ——— =
0.06 (0.06)2
$294.44
1 / 1.06
————— =
(0.06 / 1.06)2
$294.44
H1 =
Find the present value of an annuityimmediate 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
a –––
n–1
– nvn
—————
i
(n – 1) –
———————
i
(Ia)–
n
(Da)–––
n–1
+ vn
=
+ vn
=
..
a –
n
..
a –
n
a –––
n–1
– nvn+ nvn– vn– vn
———————————————— =
i
a –––
n–1
– vn(1 + )
——————————— =
i
..
a –
n
..
a –
n
– vn
——————— =
i
..
a –
n
1 – vn
——— =
i
..
a –
n
a –
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.