13-2 · annuity due - regular deposits/payments ... assume that the rrsp earns 8% compounded...

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MH Ryerson

Annuities DueAnnuities DueAnnuities Due

Chapter 13Chapter 13Chapter 13

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MH Ryerson

Learning ObjectivesLearning Objectives

After completing this chapter, you will be able to:

> Calculate the future value and presentvalue of annuities due.

> Calculate the payment size, number ofpayments, and interest rate in annuitiesdue.

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Classification of AnnuitiesClassification of Annuities

Ordinary annuity -regular

deposits/paymentsmade at the end of

the period

Annuity due -regular

deposits/paymentsmade at the

beginning of theperiod

Jan. 31 Monthly Jan. 1

Jun. 30 Quarterly Apr. 1

Dec. 31 Semiannually Jul. 1

Dec. 31 Annually Jan. 1

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TABLE 13.1Distinguishing Characteristics of Annuity Categories

Annuity Category

Is the payment at theend or at the beginning

of each payment interval?

Compare thepayment interval to thecompounding interval.

Ordinary simple annuity End EqualOrdinary general annuity End Not equalSimple annuity due Beginning EqualGeneral annuity due Beginning Not equal

Distinguishing Characteristics of Annuity CategoriesDistinguishing Characteristics of Annuity Categories

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PV

Time Diagram for an n-Payment Annuity DueTime Diagram for an n-Payment Annuity Due

PMT PMTPMT PMT PMT PMT PMT PMT

0 1 2 3 4 5 n-2 n-1 n

Interval Number

FV

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Future Value of a Simple Annuity DueFuture Value of a Simple Annuity Due

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

i* ( 1 + i )

= FV * (1 + i)

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How much will Elyse accumulate in herHow much will Elyse accumulate in herRRSP by age 60 if she makes semi-annualRRSP by age 60 if she makes semi-annualcontributions of $1000 starting on her 30thcontributions of $1000 starting on her 30thbirthday? Assume that the RRSP earns 8%birthday? Assume that the RRSP earns 8%compounded semi-annually and that nocompounded semi-annually and that nocontribution is made on her 60th birthday.contribution is made on her 60th birthday.

Set your financial calculator to the BGN mode

8/2 =

n

I/Y

FV

PMT

comp

PV0

30*2=

1000 +/- FV(due) = $66 519.97

Financial Calculator sol�n:

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MH Ryerson

How much will Elyse accumulate in herHow much will Elyse accumulate in herRRSP by age 60 if she makes semi-annualRRSP by age 60 if she makes semi-annualcontributions of $1000 starting on her 30thcontributions of $1000 starting on her 30thbirthday? Assume that the RRSP earns 8%birthday? Assume that the RRSP earns 8%compounded semi-annually and that nocompounded semi-annually and that nocontribution is made on her 60th birthday.contribution is made on her 60th birthday.

Algebraic Solution:

FV (due) =$66 519.97

.08 / 2 = sto1

+ 1 = sto 2

yx 60 =

- 1 =

/ rcl 1 * 500 * rcl 2 =

Now let�s use the formulato answer the same question

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

i* ( 1 + i )

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MH Ryerson

Present Value of an Annuity DuePresent Value of an Annuity Due

= PV * (1 + i)

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

i* ( 1 + i )

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A lottery offers the winner a choice betweenA lottery offers the winner a choice betweena $300 000 cash prize, or quarterlya $300 000 cash prize, or quarterlypayments of $7000 beginning immediatelypayments of $7000 beginning immediatelyand continuing for 20 years. Whichand continuing for 20 years. Whichalternative should the winner pick if moneyalternative should the winner pick if moneyis worth 8% compounded quarterly?is worth 8% compounded quarterly?


8/4 =

n

I/Y

PV

PMT

comp

FV0

20*4=

7000 +/-

PV(due) = $283 776

The cash prize should betaken - it is worth $16224more in current dollars


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MH Ryerson

A lottery offers the winner a choice betweenA lottery offers the winner a choice betweena $300 000 cash prize, or quarterlya $300 000 cash prize, or quarterlypayments of $7000 beginning immediatelypayments of $7000 beginning immediatelyand continuing for 20 years. Whichand continuing for 20 years. Whichalternative should the winner pick if moneyalternative should the winner pick if moneyis worth 8% compounded quarterly?is worth 8% compounded quarterly?

( ) ( )ii

iPMT

iPVPVn

+111

=

)+(1(due)

×

+−

×=−

Now let�s use the formulato answer the same question

.08 / 4 = sto1

+ 1 = sto 2

yx 80 +/- =

- 1 = +/-

/ rcl 1 * 7000 * rcl 2 =

PV(due) = $283 776The cash prize should be taken

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MH Ryerson

Mark has already accumulated $104 000 inMark has already accumulated $104 000 inhis RRSP. His goal is to build it to $250 000his RRSP. His goal is to build it to $250 000with equal contributions every 6 months forwith equal contributions every 6 months forthe next 7 years. If he earns 8.5% cmpd sa,the next 7 years. If he earns 8.5% cmpd sa,and starts today, find the size of hisand starts today, find the size of hiscontributions?contributions?


8.5/2 =

n

I/Y

PMT

FV

comp

PV104000

7*2=

250000

Payment needed = $3286.10


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A car sells for $27 900. The manufacturerA car sells for $27 900. The manufactureroffers an interest rate of 1.8% compoundedoffers an interest rate of 1.8% compoundedmonthly on a three year lease. If themonthly on a three year lease. If theresidual value is $14 500, find the leaseresidual value is $14 500, find the leasepayments assuming $2500 down payment.payments assuming $2500 down payment.

−

valueresidual the

of luePresent va +

payments lease the

of luePresent va =

payment

Down

price

Purchase

−

500 $14 the

of luePresent va +

payments lease the

of luePresent va =

$2500

900 $27

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MH Ryerson

A car sells for $27 900. The manufacturerA car sells for $27 900. The manufactureroffers an interest rate of 1.8% compoundedoffers an interest rate of 1.8% compoundedmonthly on a three year lease. If themonthly on a three year lease. If theresidual value is $14 500, find the leaseresidual value is $14 500, find the leasepayments assuming $2500 down payment.payments assuming $2500 down payment.


1.8/12 =

n

I/Y

PMT

FV

comp

PV25400 +/-

3*12 =

14500 Lease Payment = $332.50

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MH Ryerson

Calculating the Number of PaymentsCalculating the Number of Payments

using the algebraic methodusing the algebraic method

( )( )i

iPMT

FVi

n+

+

×+

1ln

1

(due)1ln

=

( )( )i

iPMT

PVi

n+

+

×−−

1ln

1

(due)1ln

=

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MH Ryerson

How long will it take to accumulate $1 millionHow long will it take to accumulate $1 millionin an RRSP if the first quarterly payment ofin an RRSP if the first quarterly payment of$2000 is made today? Assume the RRSP earns$2000 is made today? Assume the RRSP earns8% cmpd quarterly.8% cmpd quarterly.


8/4 =

n

I/Y

PMT

FV

comp

PV0

2000 +/-

1000000 120.18 payments = 120*3 =360 months = 30 years + �a bit�

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MH Ryerson


( )( )i

iPMT

FVi

n+

+

×+

1ln

1

(due)1ln

= ( )

( )iiPMT

PVi

n+

+

×−−

1ln

1

(due)1ln

=

Which do we use?Does this involve FV or PV?

Algebraic Sol�n

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120.18 payments = 120*3 =360 months = 30 years + �a bit�

( )( )i

iPMT

FVi

n+

+

×+

1ln

1

(due)1ln

=

Algebraic Sol�n

.08 / 4 = sto1

+ 1 = sto2

lnx sto3

rcl1 * 1000000

/ 2000 / rcl2 =

+ 1 = lnx / rcl3 =

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MH Ryerson

A $100 000 life insurance policy requires anA $100 000 life insurance policy requires anannual premium of $420 or a monthlyannual premium of $420 or a monthlypremium of $37. In either case, the premiumpremium of $37. In either case, the premiumis due at the beginning of the period ofis due at the beginning of the period ofcoverage. What is the effective rate ofcoverage. What is the effective rate ofinterest charged to those who pay monthly?interest charged to those who pay monthly?

Set your financial calculator to the BGN mode12

I/Y

n

PMT

FV

comp

PV420

37 +/-

0

1.0269

f = (1+i)m - 1

= (1 + 0.010269)12 - 1

= 0.1304 = 13.04%

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MH Ryerson

THE END

13-2 · annuity due - regular deposits/payments ... assume that the rrsp earns 8% compounded...

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