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

_a – n| =

0

n

vt dt = vt

—— =ln(v)

0

nvn – 1—— = ln(v)

vn – 1—————— =ln(1) – ln(1+i)

1 – vn

——

(The formula could also be derived by taking the limit as m goes to of either or .) (m)

a – n|..(m)a – n|

Observe that_a – n| =

i—

a – n|

Sections 4.5, 4.6

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

_s – n| =

0

n

(1 + i)t dt = (1 + i)t

——— =ln(1 + i)

0

n(1 + i)n – 1———— = ln(1 + i)

(1 + i)n – 1————

(The formula could could also be derived by taking the limit as m goes to of either or .) (m)

s – n|..(m)s – n|

Observe that_s – n| =

i—

s – n|

Note that this first factor is ._s – 1|

Observe that_a – n| = 1 – e–n

———

_s – n| =

en – 1———

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| = 4_s – 4|

e8 – 1——— =

e4 – 14 ———

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.

0 1 2 … n – 1 n

Payments

Periods

P P+Q P+(n–2)Q P+(n–1)Q

3

P+2Q

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

(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

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 – vn Qnvn – nvn

A = P ——– + Q ——— – —— = P + Q ————— i i i i

a – n|a – n|

a – n|

The accumulated value of this annuity over the n periods isA(1 + i)n =

– n P + Q ————

is – n|s – n|

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 ————— = ia – n|a – n| – 4v4

250 + 250 ————— = 0.08a – 4|a – 4|

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

Page 7 of the Class Notes:

If P = Q = 1, then the annuity is called an increasing annuity, and the present value of such an annuity is

(Ia) – n|= a – n|

– nvn

————— = i

a – n|+1 – vn + – nvn

———————— =i

a – n| – nvn

————— i

..a – n|

The accumulated value of this annuity over the n periods is

(1 + i)n =(Is) – n|

=(Ia) – n|

– n———— i

..s – n|

Observe that can be interpreted as a sum of level deferred annuities after realizing that(Ia) – n|

n–1t=0

vt a ––– n–t| =n–1t=0

vt1 – vn–t

——– = i

– nvn

————— = i

..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..a – n| = (Ia) – n|

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 ————— = ia – n|a – n| – 4v4

250 + 250 ————— = 0.08a – 4|a – 4|

3.312127 – 4 / 1.084

250(3.312127) + 250 ————————— =0.08

$1990.55

We can also get this present value as follows:

(Ia) – 4| 0.08=250 250

– 4v4

——————— 0.08

..a – 4|0.08 =

3.577097 – 4 / 1.084

250 ————————— = 0.08

$1990.55

Page 7 of the Class Notes:

If P = n and Q = –1, then the annuity is called a decreasing annuity, and the present value of such an annuity is

(Da) – n|= n a – n|

– nvn

————— = i

a – n|–n – nvn – + nvn

————————— =i

a – n| n –——— i

a – n|

The accumulated value of this annuity over the n periods is

(1 + i)n =(Ds) – n|

=(Da) – n|

n(1 + i)n –——————

i

s – n|

Observe that can be interpreted as a sum of level annuities after realizing observing that(Da) – n|

nt=1

a – t | =

nt=1

1 – vt

——– = i

n – ——— = i

a – n|(Da) – n|

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.

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 paymentUsing 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

(Da) – 4 | 0.08250 =

4 –————— 0.08

a – 4 | 0.08250 =

Page 8 of the Class Notes:

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

(Da) – 4 | 0.08250 =

4 –————— 0.08

a – 4 | 0.08250 =

4 – 3.312127250 —————— =

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

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

The present value of annuity B is

500 =a – 10| 0.05 $3860.87

(Da) – = 10| 0.05(X / 10)

10 –—————— 0.05

=(X / 10)a – 10| 0.05 4.556530X

10 – 7.721735(X / 10) ——————— =

0.05

X = $847.33

Page 9 of the Class Notes:

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|a – n|lim

n

– nvn

P + Q ————————— = i

a – n|

a – n|limn

limn

limn

P + Q 1— i ———— =

i

1— i

– 0 P Q— + — i i2

(Ia) –– |=

If P = Q = 1, then the annuity is called an increasing annuity, and the present value of such an annuity is

1 1— + — 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—— + ——— =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 =vn + vn+1 + vn+2 + … = vn(1 + v + v2 + v3 + …) =

1vn —— = 1 – v

vn

— d

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 + …) =

dvn — (1 + v + v2 + v3 + …) = dv

d 1vn — —— = dv 1 – v

vn

— d2

1vn ——— = (1 – v)2

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|

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.

0 1 2 3 4 5 6 7 8 9 10 11 12

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

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 H6 H9 + H14

(b) Use the formula from part (a) to calculate the present value.

v1

— d2

v6

— d2

v9

— d2

v14

— = d2

+

d = 0.08 / 1.08 = 0.074v = 1 / 1.08 = 0.925

250

$6195.06250

Page 10 of the Class Notes:

Find the present value of a perpetuity-immediate whose successive payments are 1, 2, 3, 4, … at an effective rate of 6%.

1 1—— + ——— =0.06 (0.06)2

$294.44

H1 = 1 / 1.06————— =(0.06 / 1.06)2

$294.44

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.

(Ia) – n|+ vn

(Da) ––– n–1|=

– nvn

————— i

..a – n|+ vn

(n – 1) –———————

i

a ––– n–1| =

– nvn + nvn – vn – vn ———————————————— =

i

..a – n| – vn (1 + )——————————— =

i

a ––– n–1|..a – n| a ––– n–1|

– vn

——————— =i

..a – n|..a – n| ..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.