Business Mathematics Jerome Chapter 13

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McGraw-Hill Ryerson©

1313 1313Annuities

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Annuities

Due AnnuitiesAnnuities

Chapter 13Chapter 13


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Calculate

… the Future Value and Present Value ofLO 1.LO 1.

After completing this chapter, you will be able to:

…the payment size, number of payments, and interest rate in

Learning ObjectivesLearning

Objectives

LO 2.LO 2.

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Membership dues are usually paid in advance Membership dues are usually paid in advance

Only a small modification to the math of Ordinary Annuities is needed to handle

Only a small modification to the math of Ordinary Annuities is needed to handle

If you lease equipment, a vehicle, or rent property,

the typical lease contract requires payments at the

beginning of each

period of coverage

If you lease equipment, a vehicle, or rent property,

the typical lease contract requires payments at the

beginning of each

period of coverage

1st.1st.

15th.15th.

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Ordinary AnnuityOrdinary Annuity

Payments / Deposits

are made at the end of the period

Payments / Deposits

are made at the end of the period

Payments / Deposits

are made at the beginning of

the period

Payments / Deposits

are made at the beginning of

the period

Classification of Annuities


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Annuity Category

Annuity Category

Is the payment at the end or

at the beginning of each payment

interval?

Is the payment at the end or

at the beginning of each payment

interval?

Compare the payment

interval to the compounding

interval

Compare the payment

interval to the compounding

interval

Ordinary Simple

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Annuities

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End Equal

Ordinary General End

Not Equal

Beginning

Not Equal

Equal

Beginning

Simple Annuity Due

General Annuity Due

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FVdue = PMT (1+ i)n - 1[ i ]Formula Formula * (1+ i)

If the payments form a general annuity due,

use i2 = (1+i)c – 1 in the formula for i

FormulaeFormulaeLO 1.LO 1.

PVdue = PMT 1-(1+ i)-n[ i ]Formula Formula * (1+ i)

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Annuities

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“Payments…in advance”

“First payment … made today”

“Payments at the beginning of each…..”

“Payments ….. starting now”

Clues to help identify annuities due

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How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of $1,000

starting on her 30th birthday?

Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made

on her 60th birthday.

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How much will Elyse

accumulate in her RRSP by age 60 if she makes semi-annual contributions of $1,000 starting on her

30th birthday? Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made


How much will Elyse




Step 1 – Set your calculator to the “BGN mode”

END

Step 2…Step 2…

BGN BGN0

Notice the tiny BGN above the 0

Notice the tiny BGN above the 0

Your calculator is now set for annuity due


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Step 2 – Solve for the FV of the annuity due

How much will Elyse




How much will Elyse




BGN0FV= 247,510.31 Elyse’s RRSP value on her 60th birthday

Elyse’s RRSP value on her 60th birthday

2

1000

0

60

8

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n =i =c = 10000.08/2

1.04

1000

60

1 0.04

PMT =

10.5196 9.5196 247,510.31

601

1.04

How much will Elyse




How much will Elyse




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



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A lottery offers the winner a choice between a $300,000 cash prize, or quarterly payments

of $7,000 beginning immediately and continuing for 20 years. Which alternative should the winner pick if money is worth

8% compounded quarterly?

Need to find the PV today of the payment alternative and compare with the $300,000 cash.

Need to find the PV today of the payment alternative and compare with the $300,000 cash.

What do we have to find?What do we have to find?

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A lottery offers the winner a choice between a $300,000

cash prize, or quarterly payments

of $7,000 beginning

immediately

and continuing for 20 years. Which alternative should the winner pick

if money is worth




of $7,000 beginning

immediately


if money is worth


END

Step 2…Step 2…

BGN BGN0Your calculator

is now set for annuity due



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Step 2 – Solve for the PV of the annuity due

BGN0PV= 283,775.83 Cash Value

of the payment option

Cash Value of the payment option

4

7000

0

80

8



of $7,000 beginning

immediately


if money is worth




of $7,000 beginning

immediately


if money is worth


The $300,000 cash is a better offer …$16,224 more in current day

dollars…($300,000 - $283,775.83)

The $300,000 cash is a better offer …$16,224 more in current day

dollars…($300,000 - $283,775.83)

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1313 1313Annuities

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Annuities

Due A lottery offers the winner a choice between a $300,000 cash prize, or quarterly

payments of $7,000 beginning immediately and continuing for 20 years. Which alternative should

the winner pick if money is worth 8% compounded quarterly?

= $283,775.83

The $300,000 cash is a better offer… $16,224 more in current day dollars…($300,000 - $283,775.83)

The $300,000 cash is a better offer… $16,224 more in current day dollars…($300,000 - $283,775.83)

(1 -(1 + .08/4)-80) .02

*(1.02)= 7000 [ ]

PVdue = PMT 1-(1+ i)-n[ i ]* (1+ i)

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You have accumulated $104,000 in your

RRSP. Your goal is to build it to $250,000

with equal contributions every 6 months for the next 7 years. If you

earn 8.5% compounded semi-annually, and start today, find the size

of your contributions.


BGN

Step 2 – Solve for the PMT of the annuity due

104000

250000

14 8.5

Payment needed is $3286.10


Step 1Step 1 Step 2Step 2BGN0PMT= -3,286.10

2

LO 2.LO 2.

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n =i =c = ?0.085/2 PMT = 141FV = 250 000 PV = 104 000

Extract necessary data...

a.a. Find the FV 7 years from now of the $104,000 already saved.FV = 104 000(1+.085/2)14

= $186,250.84

b.b. Subtract this value from the $250,000 target to get the FV of

the annuity.

= $63,749.16= $63,749.16

You have accumulated $104,000 in your

RRSP. Your goal is to build it to $250,000

with equal contributions every 6 months for the next 7 years. If you

earn 8.5% compounded semi-annually, and start today, find the size

of your contributions.

Formula Formula FVdue = PMT[ (1+ i)n - 1

i]*(1+ i)

$250,000 - $186,250.84

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1313 1313Annuities

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Annuities

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

PV = 0FV = $63,749.16

63749.16 = PMT(19.3997)

PMT = $3286.10PMT = $3286.10

n =i =c = ?0.085/2 PMT = 141

(1.0425)14 – 10.0425

* (1.0425)63749.16 = PMT ][

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1313 1313Annuities

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Annuities

Due A car sells for $27,900. The manufacturer offers an interest rate of 1.8% compounded monthly on a three year lease. If the residual value is $14,500,

find the lease payments assuming a $2,500 down payment.

+

=Purchase Price

Down Payment

Present Value of Lease Payments

Present Value of Residual Value

+=$27,900 $2,500Present Value

of $14,500Present Value of Lease Payments

Present Value of Lease Payments

= + Present Value of $14,500$25,400

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A car sells for $27,900.

The manufacturer

offers an interest rate of 1.8% compounded monthly on a three year lease. If the

residual value is $14,500, find the lease payments

assuming a $2,500 down payment.


BGN

Step 2 – Solve for the PMT of the annuity due

25400

14500

36 1.8



Step 1Step 1 Step 2Step 2BGN0PMT= 332.50

12

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n =i =c = ?0.018/12 PMT = 361Extract necessary data...

A car sells for $27,900.

The manufacturer

offers an interest rate of 1.8% compounded monthly on a three year lease. If the

residual value is $14,500, find the lease payments

assuming a $2,500 down payment.

a.a. Find the PV of the $14,500 residual value

PV = 14 500(1+.018/12)-36

= $13,738.32

b.b. PV annuity =

= $11,661.68= $11,661.68

= $27,900 - $2500 - $13,738.32SP – DP – PV of residual value

See Next

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1313 1313Annuities

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Due Choose appropriate formula and Solve

PV = $11,661.68FV = 0

11661.68 = PMT (35.0722)

PMT = $332.50 Lease PaymentPMT = $332.50 Lease Payment

n =i =c = ?0.018/12 PMT = 361

[ ]

PVdue = PMT 1-(1+ i)-n[ i ]* (1+ i)

11661.68 = 1 - (1.0015)-36

0.0015* (1.0015)PMT

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Due How long will it take to accumulate $1 million in an RRSP if the first quarterly payment of $2000 is made today? Assume the RRSP

earns 6% compounded quarterly.

How long will it take to accumulate $1 million in an RRSP if the first quarterly payment of $2000 is made today? Assume the RRSP


2000

6 0

1 000 000

4

n =?PV = $0 FV = $1 000 000 i = .06/4

35.75 years35.75 years

Step 1Step 1 Step 2Step 2BGN BGN0N = 142.9

143 Quarterly

payments

143 Quarterly

payments

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…using the algebraic method to

determine Number of Payments

Formulae Formulae

PMT

FVdue i *

i1ln

1ln[ ]n

+

PMT i*

i1ln

1ln[ ]n

PVdue

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Annuities

Due How long will it take to accumulate $1 million in an RRSP if the first quarterly payment of $2000 is made today? Assume the RRSP


How long will it take to accumulate $1 million in an RRSP if the first quarterly payment of $2000 is made today? Assume the RRSP


n =?PV = $0 FV = $1000000 i = .06/4

= 142.9 i.e. 143 quarterly payments= 142.9 i.e. 143 quarterly payments

= 2.1269 0.0149

or 35.75 yearsor 35.75 years

][ 1,000,000 * .015

ln (1.015)2000(1.015)

1+n =

ln

PMT

FVdue i *

i1ln

1ln[ ]n =

+ Formula Formula

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1313 1313Annuities

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Due A $100,000 life insurance policy requires an

annual premium of $420 or a monthly premium of $37. In either case, the premium is due at the beginning of the period of coverage. What is the effective rate of

interest charged to those who pay monthly?

1

12BGN modeSet to

37

12 420

0

BGN0

Effective Interest rate on the monthly payment plan = 13.04%

Effective Interest rate on the monthly payment plan = 13.04%

P/Y = 12C/Y = 1I/Y = 13.04

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This completes Chapter 13This completes Chapter 13

Business Mathematics Jerome Chapter 13

Education

current day

interest rate

step 2 solve

hill ryerson

bgnstep 2

lottery offers

quarterly

payment interval