1

ANNUITIES

Sequences and SeriesConsider the following sequences of numbers:

2, 4, 6, 8, . . . .

2, 4, 8, 16, . . . .

What is the mathematical difference between the two?

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The first is called an arithmetic sequence since it can be described in the following way:

2, 2+2, 2+2+2, ...

The second is a geometric sequence since it can be described as:

2, 2×2, 2×2×2, 2×2×2×2, ...

The geometric sequence is of interest to us.

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In general terms, the first n terms of the sequence can be described by

(In the previous example, a = 2, r = 2).

2 3 1, , , , .. . . ., na ar ar ar ar

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a = 3, r = 2

Example 1

. . . . . 23 ,23 2,3 3, 32

. . . . . 24, 12, 6, ,3

. . . . . , , , , 32 ararara

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Example 2

. . . . . ,, , ,

1.5 1,32 ararara

ra

,.....1(1.5) ,1(1.5) 5),.1(1 1, 32

3.375,.... 2.25, 1.5, 1,

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Example 3

$500 invested at 12% compounded annually.

500……………….$500

12.1 ,500 ra

560$).........12.01(500

20.627.......$)12.01(500 2

46.702........$)12.01(500 376.786........$)12.01(500 4

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An annuity is a sequence of payments made regularly over a given time interval (eg loan repayments).

The time interval is called the term of the annuity. The regular places where repayments are made are called payment periods.

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Payments made at the end of the payment period are called ordinary annuities.

When payments are made at the beginning of the payment period, the process is called an annuity due.

We first consider an ordinary annuity.

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0 1 2 3 4 n − 1 n

R R R R R R

10

r

rRA

n

)1(1

It can be shown that the present value of these payments is given by

This gives the present value A of an annuity of $R per payment period for n periods at the rate r per period.

11

nrAr

R

)1(1

This formulation can be rearranged so that

which gives the periodic payment R of an annuity whose present value is A.

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Example 4

The present value of quarterly payments of $250 for 5 years at 12% compounded quarterly is

201 (1 0.03)250

0.03A

2045

03.0412.0

n

r

37.3719$

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Example 5

$25,000 is borrowed over 8 years. What will be the monthly repayments at 18% compounded monthly?

015.012

18.0r 96128 n

96

25,000 0.015

1 (1 0.015)R

08.493$

Monthly repayments should be $493.08.

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If the loan is taken over 5 years, then

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25,000 0.0151 (1 0.015)

R

84.634$

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Example 6

A person wishes to borrow $5000 now and $4000 two years from now.

Both loans are to be repaid with equal monthly payments made at the end of the month for the next five years.

What is the monthly payment?(Assume 10% compounded monthly.)

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0 1 2 3 4 5

R R R R R

40005000

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Bring everything back to the present value.Loans are presently worth

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12

1.0140005000

64.32775000

64.8277$

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The present value of the repayments is given by

12

1.012

1.011

60

RA

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Equating the PV of the loan and payments we obtain 600.1

1 1128277.64

0.112

R

60

0.18277.64

12 175.880.1

1 112

R

The monthly repayment will be $175.88.

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Note well the discussion on page 148 of the study guide regarding avoiding rounding errors in calculations.

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The future value of an annuity is the value of all payments at the end of the term.

1 2 3 4 5 n − 3 n − 2 n − 1 n

R R R R R _ _ _ _ _ _ R R R R

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It can be shown that the future value of an annuity of n periodic payments of $R with a rate r per period is

r

rRS

n 1)1(

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Manipulating this equation gives a formula for R.

1)1(

nrSr

R

R is the periodic payment that must be made to amount to S at the end of the term.

Investing in this way to meet some future obligation is commonly called a sinking fund.

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Example 7

If you wish an annuity to grow to $17,000 over 5 years so that you can replace your car, what monthly deposit would be required if you could invest at 12% compounded monthly?

60125

01.012

12.0

n

r

60

17,000 0.01

(1 0.01) 1R

16.208$

The monthly payment should be $208.16.

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Example 8

An annuity consists of monthly repayments of $600 made over 20 years.(a) What is the present value of the annuity?(b) How much money is repaid?(c) What is the future value of the payments?(Assume 14% compounded monthly.)

0.1412

20 12 240

r

n

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Question (a)

The present value of the annuity is $48,250.10.

rr

RAn)1(1

2400.141 1

126000.1412

$48,250.10

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Question (b)

The amount repaid is

2012600

$144,000

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Question (c)

r

rRS

n 1)1(

2400.141 1

126000.1412

$780,699.45

The future value of the annuity is $780,699.45.

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An annuity where each payment is due at the beginning of the payment period is called an annuity due.

0 1 2 3 4

(1) R R R R (ordinary)

(2) R R R R (annuity due)

r

rRA

4)1(1 )1(

r

rRR

3)1(1 )2(

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The second case, describing the annuity due, can be thought of as an initial payment followed by an ordinary annuity of shorter duration (one payment period shorter).

OD ArA )1( In a similar way

OD SrS )1(

It can be shown that the present value of an annuity due is equal to

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Example 9

If payments of $100 are received at the beginning of each payment period for 4 years, once a year, at a rate of 15% p.a. compounded annually, what is the present value and the future value?0 1 2 3 4

100 100 100 100

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100 86.96 75.61 65.75 $328.32

Using the formula(1 )D OA r A

41 (1 0.15)(1 0.15) 100

0.15

$328.32

The present value is given by-1 2 3100 100(1.15) 100(1.15) 100(1.15)

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0 1 2 3 4

100 100 100 100

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The future value is given by

4

321

)15.1(100

)15.1(100)15.1(100)15.1(100

90.17409.15225.132115

24.574$

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In summary, treat all annuity problems as ordinary and then make the correction (multiply by 1+r) for annuities due if necessary.

24.574$

15.0

115.110015.1

4

Using the formula

OD SrS )1(

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Example 10

A company wishes to lease temporary office space for a period of 6 months.

The rental fee is $500 a month payable in advance.

Suppose that the company wants to make a lump-sum payment, at the beginning of the rental period, to cover all rental fees due over the 6-month period.

If money is worth 9% compounded monthly, how much should the payment be?

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This is an annuity due. However, treating the problem as an ordinary annuity, the present value is given by

r

rRA

n

)1(1

61 (1.0075)500

0.0075

80.2922

500 ,6 ,0075.01209.0 Rnr

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72.2944$

A lump sum payment of $2944.72 should be made to cover the 6 month rental.

Correcting for an annuity due

80.29220075.1 A

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Example 11

A owes B the sum of $5000 and agrees to pay B the sum of $1000 at the end of each year for 5 years and a final payment at the end of the sixth year.

How much should the final payment be if interest is at 8% compounded annually?

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This is an ordinary annuity.The future value of the annuity is given by

1000 ,5 ,08.0 Rnr

rr

RSn 1)1(

08.0108.1

10005

60.5866$

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The value of this money at the end of the sixth year is

The value of the debt $5000 six years into the future is equal to

0.08) (1 5866.60 93.6335$

6)08.01(5000 37.7934$

The final payment is then

44.1598$93.633537.7934

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0 1 2 3 4 5 6

5000 7934.37

6

Debt:

5000 1 0 08S .

37.7934

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0 1 2 3 4 5 6

1000 1000 1000 1000 1000

5000 7934.37

The value of the repayments after the 5th year is51.08 1

10000.08

S

60.5866

5866.60

44

5866.60

At the end of the next year the money has the value of 15866.60 (1 0.08)S

93.6335

6335.93

0 1 2 3 4 5 6

5000 7934.37

1000 1000 1000 1000 1000

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The equation of value at the end of the sixth year is

37.793493.6335 x

44.1598x

The final payment should be $1598.44.

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Example 12

In 10 years a $40,000 machine will have a salvage value of $4000. A new machine at that time is expected to sell for $52,000. In order to provide funds for the difference between the replacement cost and the salvage value, a sinking fund is set up into which equal payments are placed at the end of each year. If the fund earns 7% compounded annually, how much should each payment be?

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0.07

10

r

n

0 10

40,000 4000

52,000

-- R -- R 48,000

48

r

rRS

n 1)1(

101.07 148,000

0.07R

48,000 13.82R

48,000

13.82R

12.3474Sinking fund repayments should be $3474.12.

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Example 13

A paper company is considering the purchase of a forest that is estimated to yield an annual return of $50,000 for 10 years, after which the forest will have no value. The company wants to earn 8% on its investment and also set up a sinking fund to replace the purchase price. If money is placed in the fund at the end of each year and earns 6% compounded annually, find the price the company should pay for the forest. Give your answer to the nearest hundred dollars.

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50,000

8% return sinking fund repayment

Let the purchase price be x.

50,000 0.08 (*)x R

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To recoup the purchase price, the repayments R must amount to x in 10 years.

101.06 1

0.06x R

Rx 18.13

10

06.0

n

r

52

(**) 18.13

xR

Substitute (**) into (*).

50,000 0.0813.18

xx

The purchase price for the forest should be$320,800 (nearest hundred).

320,784.34x

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Example 14

In order to replace a machine in the future, a company is placing equal payments into a sinking fund at the end of each year so that after 10 years the amount in the fund is $25,000. The fund earns 6% compounded annually. After 6 years, the interest rate increases and the fund pays 7% compounded annually. Because of the higher interest rate, the company decreases the amount of the remaining payments. Find the amount of the new payment. Give your answer to the nearest dollar.

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0 6 100.06 0.07

101.06 125,000

0.06R

70.1896$R

25,000

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0 6 100.06 0.07

R13,230.08

25,000

6

After 6 years the repayments amount to

1.06 11896.70

0.06S

$13,230.08

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0 6 100.06 0.07

R13,230.08

25,000

17,341.94

$17,341.94

413,230.08 (1.07)S

The value of this money after a further 4 years at 7% is

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7658.06This is to be done with 4 payments.

41.07 17658.06

0.07R

81.1724R

The reduced repayment should be $1725 (to the nearest dollar).

Amount to be raised

25,000 17,341.94