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Special

Situations

Chapter 12

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Calculate the…After completing this chapter, you will be able to:

Learning Objectives

Present Value of a perpetuity or

deferred perpetuity

LO 2.LO 2.

LO 1.LO 1.

Present Value and Future Value of an annuity whose payment size

grows at a constant rate

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Calculation of…

Perpetuity or

Deferred Perpetuity

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Ordinary PerpetuitiesOrdinary Perpetuities

A perpetuity

is an annuity whose

payments continue forever.

A perpetuity

is an annuity whose

payments continue forever.

A $100,000 bequest is made to Seneca College to establish a perpetual bursary fund. If the college invests

the funds to earn 6% compounded annually, the

maximum amount that can be paid out on each anniversary

of the bequest is …$100,000 * 0.06 = $6,000

If more than this was to be paid out, a loss of principal would result.

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Ordinary PerpetuitiesOrdinary Perpetuities

Present Value of:

If the payment interval equals

the compounding interval, the perpetuity is an ordinary simple

perpetuity

If the payment interval equals

the compounding interval, the perpetuity is an ordinary simple

perpetuity

it is an ordinary general annuity

it is an ordinary general annuity

Otherwise…PV = PMT / iFormulaFormula

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What endowment is required to establish a perpetuity with an ongoing cost of $6,000 at

the end of each month if interest is 6.0% compounded monthly in perpetuity?

= 6000 / (.06/12)

PV = PMT / iFormulaFormula

= $1,200,000

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What monthly compounded nominal rate of return must an endowment of $1 million

earn to fully fund a perpetuity with an ongoing cost of $4,000 at the end of each

month?PV = PMT/ iFormulaFormulaReorganize

to find iReorganize

to find i

i = 4000 / 1 000 000= 0.004

i = PMT / PV

= 0.4% per monthThe required nominal rate of return is:

12 * 0.4% = 4.8% compounded monthly

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What endowment is required to establish a perpetuity with an ongoing cost of $6,000 at the end of each month if interest

is 6.0% compounded annually in perpetuity?

Since this is a general perpetuity, we need to determine c and i2Since this is a general perpetuity, we need to determine c and i2

C =number of compoundings per year

number of payments per year = .0833= 112

i2 = (1+i)c - 1= (1.06)0.0833-1= 0.00486755= 6000 / 0.00486755

= $ 1,232,652.83= $ 1,232,652.83

PV

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Calculating initial endowment for a

General Perpetuity

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Since this is a general perpetuity, we need to determine c and i2Since this is a general perpetuity, we need to determine c and i2

C =number of compoundings per year

number of payments per year = .1667= 212

i2 = (1+i)c - 1= (1.02)0.1667-1

= 0.00330589= 700 / 0.00486755

= $ 211,743.26

What amount must be placed in a perpetual

fund today if it earns 4.0% compounded semi-annually and monthly payments of $700 in perpetuity are to start 1 month from

now?

PV

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What amount must be placed in a perpetual fund today if it earns 4.0% compounded semi-annually and monthly payments of

$700 in perpetuity are to start 1 YEAR from now?

We have already determined the value at the beginning of

the payments

We have already determined the value at the beginning of

the paymentsPV = $ 211,743.26

PV = FV(1 + i)-nFormula Formula

PV = 211743.26 (1 + 0.00330589)-11

= $204,193.83 This is the value now

This is the value now

This is the value 11 months

from now

This is the value 11 months

from now

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Constant Growth

Annuities

Constant Growth

Annuities

LO 2.LO 2.

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Constant Growth

Annuities

Constant Growth

Annuities

… Annuities in which the payments change by the same percentage

from one payment to another Let g = rate of growth in payment size

between successive paymentsLet g = rate of growth in payment size

between successive payments

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FV = PMT (1+ i)n - (1+g)n[ i - g ]Formula Formula

PV = PMT 1- (1+g)n(1+ i)-n[ i - g ]Formula Formula

Constant Growth

Annuities

Constant Growth

AnnuitiesThe following formulae will be used:

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You intend to make RRSP contributions on Feb.28 of each year. You plan to contribute $2,000 in the

first year and increase the contribution by 4% every year thereafter.

a) How much will you have in your RRSP at the time of your 20th contribution if the plan earns 7.5%

compounded annually?

b) What will be the amount of your last contribution?

Constant Growth

Annuities

Constant Growth

Annuities

Extract necessary data...

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PMT = i = 0.075 n = 20$2000PV = 0 FV = ?g = 4%

You intend to make RRSP contributions

on Feb.28 of each year. You plan to contribute $2000 in the first year and

increase the contribution

by 4% every year thereafter.

You intend to make RRSP contributions

on Feb.28 of each year. You plan to contribute $2000 in the first year and

increase the contribution

by 4% every year thereafter.

Solve …

FV = PMT (1+ i)n - (1+g)n[i - g

]

[FV = 2000 (1.075)20 - (1.04)20

0.075 - 0.04]

Solve …

a) How much will you have in your RRSP at the time of your 20th contribution if the plan

earns 7.5% compounded annually?

…

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1.04

0.035

201.075

2000

20

2.1911 4.24792.0567117,527.31Solve … Amount in the

RRSP at the time of the 20th

contribution

Amount in the RRSP at the time

of the 20th contribution

PMT = i = 0.075 n = 20$2000PV = 0 FV = ?g = 4%

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You intend to make RRSP contributions

on Feb.28 of each year. You plan to contribute $2000 in the first year and

increase the contribution

by 4% every year thereafter.

You intend to make RRSP contributions

on Feb.28 of each year. You plan to contribute $2000 in the first year and

increase the contribution

by 4% every year thereafter.

(b) What will be the amount of your last contribution?

The final payment will be the Future Value of $2000 after 19 compoundings

at 4%

= 2000( 1+ 0.04)19

= $4,213.70

FV = PV(1 + i)nFormula Formula

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Constant Growth

Annuities

Constant Growth

AnnuitiesHow much will it cost to purchase a

25-year ordinary annuity making semiannual payments

that grow at the rate of 3% compounded semiannually?

The first payment is $10,000 and the funds used

to purchase the annuity

earn 5% compounded semiannually. SolutionSolution

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Constant Growth

Annuities

Constant Growth

AnnuitiesThe cost will be the PV of the payments.

Extract necessary data... How much

will it cost to purchase a 25-year ordinary

annuity making semiannual payments that grow at the rate of 3% compounded

semiannually? The first

payment is $10,000 and the

funds used to

purchase the annuity earn 5%

compounded semiannually.

How much will it cost to purchase

a 25-year ordinary annuity making

semiannual payments that grow at the rate of 3% compounded

semiannually? The first

payment is $10,000 and the

funds used to

purchase the annuity earn 5%

compounded semiannually.

PMT = i = 0.05/2 = 0.025

n = 50 $10000

PV = ?g = 3%/2 = 0.015

Solve …

PV = PMT [ 1- (1+ g)n(1+ i)-n

i - g ][PV = 10000 1- (1.015)50(1.025)-50

.025 – .015 ]Solve …

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[PV = 10000 1- (1.015)50(1.025)-50

.025 – .015 ]

0.2909 2.10520.6125-0.3875

1.025

0.01

50

1.015

10000

50

1

387,496.12 Cost of the annuity

Cost of the annuity

How much will it cost to purchase

a 25-year ordinary annuity making

semiannual payments that grow at the rate of 3% compounded

semiannually? The first

payment is $10,000 and the

funds used to

purchase the annuity earn 5%

compounded semiannually.

How much will it cost to purchase

a 25-year ordinary annuity making

semiannual payments that grow at the rate of 3% compounded

semiannually? The first

payment is $10,000 and the

funds used to

purchase the annuity earn 5%

compounded semiannually.

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This completes Chapter 12This completes Chapter 12