CAP A UNA PEDAGOGIA DEL PATRIMONI ARQUITECTÒNIC

An annuity lasting n interest periods with payments at the beginning of each interest period is called an annuity-due. By a basic annuity-due, we mean an annuity-due

with level payments, each of which is equal to 1. The value at a given time of a level annuity-due with payments of Q is just Q times the value of the basic annuity-due.

Having fixed a choice of accumulation function a.t/, let Ran denote the value at the time of the first payment of the basic annuity-due that lasts n periods, and write Rsⁿ for the value at the end of the n-th payment period of this annuity. Note that time nis one period after the final payment of an annuity-due with duration n periods.

If the chosen accumulation function is the compound interest accumulation function a.t / D .1 C i/^t, the symbol Ra^{n i} is commonly used for the value at the time of the first payment, and the symbol Rs^{n i}indicates the value at time n. The annuity symbol Ra^{n i}is read “a double-dot, angle n, at interest rate i ” or “a due, angle n, at interest rate i .” Likewise, Rs^{n i} is “s double-dot, angle n, at interest rate i ” or “s due, angle n, at interest rate i .”

PAYMENT: 1 1 1 1


1

TIME: 0 1 2 3


n 1 n

VALUE: Raⁿ Rsⁿ

FIGURE (3.3.1) Analogously to (3.2.8) and (3.2.9), we have

(3.3.2) Rs^{n i} D .1 C i/ⁿRa^{n i} and Ra^{n i}D vⁿRs^{n i}; and more generally,

(3.3.3) Rsⁿ D a.n/Raⁿ and Raⁿ D v.n/Rsⁿ:

IMPORTANT FACT (3.3.4)

With an annuity-due, you start receiving payments immediately. The value of a basic annuity-due lasting n periods at the time of the first payment is denoted by Raⁿ. The last payment is made at the beginning of the n-th period, and Rsⁿ gives the annuity’s value one period after this last payment, that is to say at the end of the n-th period.

The annuity symbol Ra^{n i}is given by a geometric series, the sum of which may be computed using (3.2.2). Recalling that 1 v D d (Equation 1.9.7), we find

(3.3.5) Ra^{n i}D 1CvCv²Cv³C


Cv^{n 1}D 1.1 vⁿ/

1 v D 1 vⁿ d : A formula for the annuity symbol

Rs^{n i} D .1 C i/ⁿC .1 C i/^{n 1}C .1 C i/^{n 2}C


C .1 C i/¹

can also be computed using the formula for summing a geometric series (3.2.2), but it is somewhat simpler to use (3.3.2) and (3.3.5). One arrives at

(3.3.6) Rs^{n i} D .1 C i/ⁿC .1 C i/^{n 1}C .1 C i/^{n 2}C


C .1 C i/¹ D ^.1Ci/d^{n 1}:

Note that the formula for the present value Ran igiven in (3.3.5) is the same as the formula for the present value a_{n i} presented in (3.2.4) except that in the formula for the due there is a d in the denominator, while the equation for the annuity-immediate has an i . There is a similar statement for the accumulated values. Perhaps you will find the following memory aid helpful. The word “immediate” begins with the letter “i” and the annuity symbols for a basic annuity-immediate have an “i ” in the denominator. In contrast, “due” begins with “d” and the annuity symbols for a basic annuity-due have a “d ” in the denominator.

The symbols Ra^{n i}and Rs^{n i} are closely related to the symbols an iand sn i. Since Ra^{n i} measures the value of an n-payment basic annuity at the time of the first payment, and an i measures the value of an n-payment basic annuity one period before the first payment,

(3.3.7) Ra^{n i} D .1 C i/a^{n i}:

Similarly,

(3.3.8) Rs^{n i} D .1 C i/s^{n i}:

There is a second type of relationship between the annuity-immediate symbols and the annuity-due symbols. The annuity symbol Ra^{n i} measures the value at time 0 of payments of 1 made at times 0; 1; 2; : : : ; n 1, while the annuity symbol a_{n 1 i} measures the value at time 0 of payments of 1 made at times 1; 2; : : : ; n 1:So, these two symbols give the time 0 value of the same series of payments, except that

Ra^{n i}includes the value of an additional payment of 1 at time 0. Therefore

(3.3.9) Ra^{n i} D an 1 i C 1:

Similarly,

(3.3.10) Rs^{n i}C 1 D s_{nC1 i};

since s_{nC1 i} measures the value at time n C 1 of payments of 1 made at times 1; 2; : : : ; n C 1, while the annuity symbol Rs^{n i} measures the value at time n C 1 of payments of 1 made at times 1; 2; : : : ; n; this interpretation of Rs^{n i} is dependent on accumulation being by compound interest at a constant rate i .

Given the relations (3.3.7)–(3.3.10) between the due and annuity-immediate symbols, it is clear that the annuity-due symbols are not strictly speaking

essential. However, if you continue your actuarial studies, you will likely encounter heavy use of the annuity-due symbols. The following examples will hopefully con-vince you of their convenience and encourage you to master them. Moreover, calcu-lators with annuity buttons are usually designed to handle the annuity-due symbols as well as the annuity-immediate symbols, and this adds to their usefulness.

The BA II Plus calculator handles problems with level annuities-due just like those with level annuities-immediate except now one wishes to have “BGN"

showing on the display before making TVM worksheet computations. The state of having “BGN" displayed is called BGN mode (“begin mode") and if the calcu-lator is in END mode, BGN mode is activated by pushing

2ND BGN 2ND SET 2ND QUIT .

This same sequence of keystrokes will also change the calculator’s status from BGN mode to END mode.

EXAMPLE 3.3.11

Problem:During 1998, Omar had a dividend payment of $100 directly deposited to his savings account on the first day of each month. Find the accumulated value of these payments at the end of the year if his savings account had a nominal discount rate of 4.8% payable monthly.

Solution His account has a monthly discount rate of 4.8%/12 = .4% = .004. This is equivalent to a monthly interest rate of _{1 :004}^:004 D 996⁴ . The accumulated value at the end of the year is therefore $100Rs12₉₉₆⁴ D $100

"

1000 996

¹²

1 :004

#

 $1;231:79.

BA II Plus calculator solution With the calculator in BGN mode and P/Y = 1 and C/Y= 1, depress the following sequence of keys

1 2 N 4  9 9 6 D  1 0 0 D I/Y 0 PV 1 0 0 C= PMT CPT FV .

The calculator display then shows 1,231.791249, and Omar’s accumulated value

is$1,231.79. 

EXAMPLE 3.3.12

Problem: Dr. Hillary Street began making contributions to a new retirement ac-count on her thirtieth birthday. She made a contribution of $4,000 at the beginning of each year through her sixty-fourth birthday. Starting at age sixty-five and continuing through her eightieth birthday, she made a level withdrawal on her birthday. Find the amount of these withdrawals if they completely exhaust the balance in her account, and the annual effective interest rate is 6% until she is sixty-five, then 5% thereafter.

Solution Let W denote the amount of the level withdrawals. A time diagram for the problem is as follows.

PAYMENT: $4,000 $4,000 ... $4,000 W W ... W

TIME: 30 31 ... 64 65 66 ... 80

RATE: „ ƒ‚ …

6% = .06

„ ƒ‚ …

5% = .05

There are thirty-five deposits of $4,000 and the accumulated value of these deposits one period after the last of them, that is to say on her sixty-fifth birth-day, is $4;000Rs35 :06. There are also sixteen withdrawals of W and the value of these at time 65 (her sixty-fifth birthday) is W Ra16 :05. Since the withdrawals exactly exhaust the balance, we must therefore have $4;000Rs35 :06 D W Ra16 :05: Thus, W D^{$4;000 Rs}_{Ra16 :05}^{35 :06}  ^$472;483:4667

11:37965804  $41;520:01:

BA II Plus calculator solution With the calculator in BGN mode and P/Y = 1 and C/Y= 1, depress the following sequence of keys

3 5 N 6 I/Y 0 PV 4 0 0 0 C= PMT CPT FV

to calculate Dr. Street’s accumulated value at the time of her sixty-fifth birthday.

With the accumulated value still displayed on the calculator, push PV 1 6 N 5 I/Y 0 FV CPT PMT .

The calculator now shows 41,520.0057. The retirement plan pays out to Dr.

Street$41,520.01 each year. 

3.4 PERPETUITIES

A perpetuity is an annuity with an infinite term. That is to say, a perpetuity has payments lasting forever. The basic perpetuity-immediate has payments of 1 at the end of each period, and we denote its present value (calculated using an effective rate of i per payment period) by a_{1 i}. The symbol Ra1 i gives the present value (calculated using an effective rate of i per payment period) of the basic perpetuity-due, the perpetuity that has payments of 1 at the beginning of each period. More generally, for an arbitrary accumulation function, the symbol a₁ gives the value one period before the first payment of a perpetuity with payments of 1, and Ra1

gives the value of such a perpetuity at the time of the first payment.

The annuity involved in the definition of a₁ has payments of 1 at times 1, 2, 3, : : : . The annuity involved in the definition of Ra1 has all these payments and an additional payment of 1 at time 0. Therefore,

(3.4.1) Ra1 D 1 C a1:

Suppose one deposits ¹_i in a bank account, which pays out interest at the end of each period at an effective rate of i > 0 per period. Then, the balance at the beginning of each period is ¹_i; and hence the amount of each interest payment is .¹_i/i D 1: Therefore, it takes ¹i remaining on deposit forever (beginning at time 0)

to create a basic level annuity-immediate. So,

(3.4.2) a_{1 i}D 1

i:

It follows from (1.9.4) that 1 C ¹i D ^1Cii D ¹d. Equations (3.4.1) and (3.4.2) therefore combine to yield

(3.4.3) Ra1 i D 1

d:

Those who know calculus should note alternate derivations of (3.4.2) and (3.4.3).

The annuity symbols a_{1 i}and Ra1 iequal the sums of infinite geometric series. More precisely,

(3.4.4) a_{1 i}D v C v²C v³C


C vⁿC


; and

(3.4.5) Ra1 i D 1 C v C v²C v³C


C vⁿC


:

To sum an infinite series, look at the limit of the partial sums. Assuming i > 0, 0 < v < ¹

1Ci, and therefore a_{1 i} D lim

n!1an iD lim

n!1

1 vⁿ

i D 1

i; and

Ra1 i D lim

n!1Ran iD lim

n!1

1 vⁿ

d D 1

d: EXAMPLE 3.4.6

Problem:Gustave Larson has saved $20,000. On January 1, he purchases a perpe-tuity with annual end-of-year payments. The perpeperpe-tuity price is based on an annual effective interest rate of 5%. What are the annual payments for Gustave’s perpetuity?

Solution The value on January 1 of a perpetuity with annual end-of-year payments of Q is Q._:05¹ / D 20Q: Setting this value equal to $20,000, we find Q D $1,000.

 EXAMPLE 3.4.7

Problem:Norman wishes to leave an inheritance to three charities. The total inher-itance is a series of level payments at the beginning of each year forever. He wishes charities A and B (which support medical research) to share the payments equally for ten years, and then all payments revert to charity C (that helps needy children).

If the shares received by all three charities have the same value at the time of the bequest, find the annual effective interest rate i .

Solution Since the shares have the same value at the time of the bequest, they have the same value at the time of the first January 1. Let P denote the amount paid out each January 1. Since charities A and B share the first ten payments equally, each receives a level annuity-due with payments of ^P₂. The value of charity A’s share on the first January 1, the time of the first payment, is therefore ^P₂Ra10 i. Recalling (3.3.5), the value of charity A’s share may also be expressed as ^P₂ 

1 v¹⁰ d



. Charity C receives level payments of P beginning exactly ten years after the bequest. The value of charity C’s share at the time of its first cash inflow is ^P_d, and the value on the first January 1 is^P_dv¹⁰. Equating the values of charity A’s and charity C’s shares on that first January 1, we have ^P₂.^{1 v}_d¹⁰/ D ^Pdv¹⁰. Equivalently, 1 v¹⁰ D 2v¹⁰: Therefore, v¹⁰D ¹3 and i D .1 C i/ 1 D v ¹ 1 D 3¹⁰¹ 1  :116123174: