Defferal Annuities

8/21/2019 Defferal Annuities

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Chapter 11

Ordinary

Annuities

McGraw-Hill Ryerson


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Calculate the

LearningObjectives

After completing this chapter, you will be able to:

number of paymentsin ordinary

and deferred annuities

payment sizein ordinary and

deferred annuities

interest ratein ordinary annuities

LO-1

LO-2

LO-3


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Using your financial calculator

we need to reorganize the formulae to solve

algebraically

solve for payment numberor

size or interest rateusing the

same steps as before


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Finding the Payment Size.

PMT


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Your life partner somehow convinced you that you cantafford the car of your dreams, priced at $28800. You are

advised to Save up for 4 yearsand then buy the car forcash. How much would you have to save each month, if

you could invest with a return of10%compounded monthly?

You need to decide if this situation involves a PVor a FV and then use the appropriate formula...

PMT

As you have to save up the$28,800,i.e. in the future,

FV= $28,800

Assume you have no savings PV= 0

FindingPaymentSizeof an

Ordinary Simple Annuity


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Your life partnersomehow convinced youthat you cant afford the

car of your dreams,priced at $28800. (At leastnot right now).You are

advised toSave up for 4 yearsand

then buy the car for cash.

How much would youhave to save each month, if

you could invest with areturn of 10%

compounded monthly?

FindingPaymentSizeof an


48

12

PMT = - 490.44

10 0

28800

Formula solution


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

]i

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

Which Formula?

Algebraic Method of Solving for PMT

(a) If the paymentsform a Simple Annuity go directly to 2.1.

If the annuitys

PV is known,substitute values

of PV, n,andi

into PVformula.

If the annuitys

FVis known,

substitute values

of FV, n,and i

into FVformula.

3. & 4.

(b) If the paymentsform a General Annuity, find candi2

2.


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Calculate the quantity within the square brackets.

Rearrange the equationto solve for PMT.

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

iPV =PMT 1-(1+ i)

-n[ ]

Which Formula?

Algebraic Method of Solving for PMT

3.

4.

Applying Method


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11-9FindingPaymentSizeof an


Which Formula?Your life partner

somehow convinced youthat you cant afford the



then buy the car for

cash.How much would you

have to save each month,if you could invest with a

return of 10%

compounded monthly?

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

]

As the annuitys FVis known,

therefore, the FVformula is used

2.

Extract necessarydata...

FV =28800 n = 4*12 = 48

i = .10/12 c= 1 PMT = ?PV =0


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12.10

28800

481

1

Your life partnersomehow convinced youthat you cant afford the



then buy the car for

cash.How much would you

have to save each month,if you could invest with a

return of 10%

compounded monthly?

0.00831.00831.48940.48940.008358.7225490.44

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

]FormulaFV =28800 n = 4*12 = 48

i = .10/12 c= 1 PMT = ?

PV =0

another example


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The


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Your parents are discussing the terms of the$100 000 mortgagethat they have offered to

hold in the purchase of your first home.

They are considering aninterest rateof 5%compounded monthly.

If you were to take 20 yearsto repay the mortgage,

find the sizeof the monthly payment.

2405

0

100 000

PMT = -659.9612

n =12*20 = 240PV =$100000 FV =0

Formula solution


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Your parents arediscussing the terms ofthe 100 000 mortgage

that they have offered tohold in the purchase ofyour first home. They

are considering aninterest rate of 5%

compounded monthly.If you were to take 20

yearsto repay themortgage, find the size

of the monthly payment.

i=.05/12

Extract necessary data...

n =12*20 = 240

PV =$100000

FV =0C=1


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Choose appropriate formula and Solve

As the annuitys PVis known, the PVformula is used2.

i=.05/12n =12*20 =240 PV =$100000i

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

12.05 1

100 000

240 1

0.00421.00420.3686-0.6314151.530.0015659.96

Size of monthlymortgagepayment


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Amount

$

How much interest will you pay your

parents over the 20 year period?

Monthly Payment Number of Payments

659.96 240 158,390.40

Amount Borrowed 100,000.00

Total Interest Paid 58,390.40

x


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PMT = -700

As this amountof interest

shocks you,you discuss thepossibility of

making paymentsof $700/month,

to save some timeand interest costs.

Determinethe timeit wil l take you torepay yourmortgage

at this new rate.

700

N = 217.52

Formula solution

218 payments = 18 yrs 2months


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Formula)i

PMT

PV i

+

*

1ln

1ln[ ]n

0.0042

i = .05/12 PMT=$700PV= $100,000 C= 1n 0

12.05 1

700

100 000

1

1.00420.00420.5952-0.4048-0.9045-217.52217.52

218payments= 18 yrs 2months


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1. Base formula+

=

i

-ni)PMTPV

(11

[ ]2. To isolaten, divide bothsides by PMT

PMTPMT

Continue

Developing the Formula

PMT=PV +

i

-ni)(11[ ]

+=

i

-ni)PMTPV

(11 ][

Formula)i

PMT

+

PV i*

1ln

1ln[ ]n


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(a) Multiply both sides by i

3. Continue to isolate n.

PMT=

PV+

i

-ni)(11[ ]

PMT=PV

+ -ni)(11

i[ ]*i *i

(b) Reorganize equation

(c) Now Take the natural logarithm

(ln or lnx) of both sides-n* ln

PMT

*i

=

PV- -n1 +i)(1

-n+i)(1

+i)(1 = ln

(d) Solving forn

divide both sides by ln(1+i)ln(1+i) ln(1+i)

from 2.

=PMT

*iPV1 -

[ ]PMT*iPV1 -

-n* ln +i)(1 = ln[ ]PMT*iPV1 -PMTPV*i)i1ln

1ln

n[ ]


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700.00 217.52 152,264.00

Total Interest Saved 6,126.40

Approximately how much money do you save

in interest charges by paying $700/month,

rather than $659.91/month?

Amount

$

Monthly Payment Number of Paymentsx

158,390.40659.96 240


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If you could

see your way to a

further increase

of $25/month,

(a) how muchfaster would you

pay off the

mortgage, and

(b) approximatelyhow much less

interest would be

involved?

725

PMT = -725N = 205.62

Paying $725206 payments = 17 yrs 2months

Formula solution


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i = .05/12 PMT=$725PV= $100,000 C= 1n 0

0.0042

12.05 1

725

100 000

1

1.00420.00420.5747-0.4253-0.8550-205.52

206payments= 17 yrs 2months

205.52

Formula)i

PMT

+

PV i*

1ln

1ln[ ]n


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(b)Total Interest Saved 3,189.50

700.00 217.52 or 218 152,264.00

Amount

$

Monthly Payment Number of Paymentsx

725.00 205.62 or 206 149,074.50

(a) Payments Saved 12


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York Furniture has a promotion on a bedroom setselling for $2250. Buyers will pay

no money down and no payments for 12 months.

The first of 24 equal monthly paymentsis due

12 monthsf rom the purchase date.What should the monthlypaymentsbeif York Furnitureearns 10%compounded monthly on its accountreceivableduring boththe deferral period and

the repayment period?

Since you want the furni ture now, this involves a PV

PMT

PV= $2250 Once you repay the loan, FV= 0

Payments are deferredfor 11months.

DEFERRAL

FindingPaymentSizein a

Deferred Annuity


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11-26York Furniture has a promotion on a

bedroom set selling for $2250. Buyers will payno money down and no payments for 12 months.

The first of 24 equal monthly paymentsisdue 12 monthsf rom the purchase date.

What should the monthlypaymentsbeif YorkFurniture earns 10%compounded monthly

on its accountreceivableduring boththe deferralperiodand the repayment period?

In effect, York furniture has

given a loan to a buyerof $2,250on the day of the sale!

When the payments begin, the buyer owes $2,250

plus accrued interest!


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d= 11i= 0.10/12

n= 24

$2250 PMT PMT PMT Payments

$2250

PVAnnuity

FV

PVof the payments at

the end of month 11

FVof the $2,250 loan

at the

end of month 11

=

Months0 11 12 13 35 36


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11

12

FV = 2,465.06

10 0

Find the amount owedafter 11months:

2250

$2,465.06is the PVof the annuity

York Furniturehas a promotion

on a bedroom set sellingfor $2250. Buyers will pay

no money down and no

payments for 12 months.The first of 24 equalmonthly paymentsis due

12 monthsfrom thepurchase date. Whatshould the monthlypaymentsbeif York

Furniture earns 10%compounded monthly on

its accountreceivableduring boththe deferral

periodandthe repayment per iod?


Deferred Annuity


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2465.06

24

FV = 2,465.06

Now find the PMT of the annuity

0

24 monthly paymentsof $113.75wil l repay the loan.

PV = - 2,465.06PMT = 113.75









Formula solution


Deferred Annuity


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FV = PV(1 + i)nFormula

FV = 2250(1 + 0.10/12)11= $2,465.06

2465.06

+=

i

-ni)PMTPV

(11[ ]= PMT[1-(1+.10/12)-24]

.10/12

= $113.75PMT









24 monthly paymentsof $113.75

will repay the loan.

Find the amount owedafter 11months:


Deferred Annuity


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i .e....Number Of Payments


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$20,000is invested in a fundearning 8%compounded quarterly.

The first quarterly withdrawal

of $1,000 will be taken from the fundfive years from now.How many withdrawalswill it take to

deplete the fund?N

Payments are deferredfor 19quarters

DEFERRAL

FindingNumber Of Paymentsin a

Deferred Annuity

The FVof $20,000after the deferral, becomes thePV of the annuity...


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Years

0 4.75 5 6 7 8

d= 19

$20,000

PV1

n= ?

This FV1 then becomes the PVof theannuity of $1000/quarter

The $20000

earnsinterest for 4years 9months

Payments of $1000/quarter

FV1

i= 0. 08/4 = .02PMT = $1000


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$20,000isinvestedin a fund

earning 8%compounded

quarterly.The first quarterlywithdrawal

of $1000 will betaken from thefundf ive years

from now.How many

withdrawalswill it take to

deplete the fund?

19

48 0

Find the FVof$20,000in 4.75 years

20000

$29,136.22is the PVof the annuity

FV = 29,136.22


Deferred Annuity


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1000

Now find the PMT of the annuity

044.1 quarterly paymentswill deplete the

fund(44 full paymentsand 1partial)

29136.22

FV = 29,136.22PV = - 29136.22N = 44.1

Formula solution





from now.How many


deplete the fund?


Deferred Annuity


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FV= 20000(1 + 0.08/4)19

= $29,136.22

Find the FVof$20,000in 4.75 years

PMT

PV *i

)i1ln

1ln

n

[ ]

ln(1.02)

=44.1 paymentsor 11 years





from now.How many


deplete the fund?


Deferred Annuity

[ln 1 - ]29136.22 *.021000


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Whennumber of compoundings per year

number of payments per year


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Since you get paid

every second

Thursday you

decide to pay $350

everytwo weeksto make your

budgeting easier.

F ind the new term

of your mortgage if

the interest charges

remain at 5%

compounded

monthly.

12

26350

P/Y = 26

415bi-weekly

payments or15 yrs 11.4 months

C/Y= 12PMT = -350N = 414.74

Formula solution


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Since you get paid

every second

Thursday you

decide to pay $350

everytwo weeksto make your

budgeting easier.

F ind the new term

of your mortgage ifthe interest charges

remain at 5%

compounded

monthly.

Determine cStep 1

C= 12 / 26= .4615

i2= (1+i)c- 1

i2= (1+ .05/12) .4615-1

i2= 0.0019

Use cto determinei2Step 2

C=number of compoundings per yearnumber of payments per year

Step 3


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as the valueforiin the appropriate annuity formula

Step 3 Use this rate i2= 0.0019

Formula)i

PMT

+

PV i*

1ln

1ln[ ]n

1.00190.00190.5428-0.4571-0.7828

1

350100 000

1

0.0019-414.74

415 paymentsor 15 yrs 11.4 months

$100 000 T


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# ofPayments

PaymentAmount

Total CostScenario

1.

2.

$100,000 Twenty-year

Mortgage

Interest 5%per annum

$659.96 $158,390.40

$152,264.00

$149,074.50

$700.00

3.$725.00

$145,250.004. $350.00

Terms

Per

month

Permonth

Per

month

Everytwo

weeks

240

218

206

415 $145,250.004.Every

twoweeks

415

Best Scenario

11 42


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350

You are nowconsidering delaying

the purchaseof your first house to

allow for a largerdown payment. If

you save $350per pay,how longwould it taketo have anadditional

$15000, if you canearn 8%compounded

monthly on yoursavings?

12

26

150000

N = 40.32

8

= FV

New requiredFormula

11 43


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Determine cStep 1

C= 12 / 26= .4615

i2= (1+i)c- 1

i2= (1+ .08/12).4615

-1

i2= 0.0031

Use cto determinei2Step 2

C=number of compoundings per year

number of payments per year

Step 3


the purchase

of your first house toallow for a largerdown payment. Ifyou save $350per

pay, how longwould

it take to haveanadditional$15000,if you can earn 8%compounded monthly

on your savings?

11 44

i[ ]


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the purchase

of your first house toallow for a largerdown payment. Ifyou save $350per

pay, how longwould

it take to haveanadditional$15000,if you can earn 8%compounded monthly

on your savings?

15000

1

1.00310.00310.13161.1316

Formula n)i

PMT

+

FV i*

1ln

1ln[ ]+

0.12370.003140.3

40.3 bi-weekly payments

= approx 1yr 7months

1

350

11 45

i[ ]


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1. Base formula-

=

i

PMTFV+

ni)(1 1

[ ]2. To isolaten, divide bothsides by PMT PMTPMT

continued

Developing the Formula

PMT=FV

Formula)i

PMT

+

FV i*

1ln

1ln[ ]n

+

-=

iPMTFV

+ni)(1 1[ ]

-

i+

ni)(1 1[ ]

11 46


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from2.PMT

=FV

-

i+

ni)(1 1[ ]

(a) Multiply both sides by i3. Continue to isolate n

PMT

FV= -

i+

ni)(1 1[ ]*i*i

[ ]PMTFV

-+ni)(1 1

*i

(b) Reorganize equation =PMT

FV+

ni)(1 1 *i

(c) Now Take the natural logarithm

(ln or lnx) of both sides nln (1+ i) = ln[ ]PMT*iFV

1 +

(d) Solving forn

divide both sides by ln(1+i)ln(1+i) ln(1+i)

nln (1+ i) = ln[ ]PMT*iFV1 +PMT

FV * i

)

i1ln

1 +ln

n[ ]

11 47


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You already have

$10000saved

for your down

payment. If you

save $350per pay,

how longwould it take

to have an additional

$15000? Assumeyou can earn 8%

compounded monthly

on all of your savings.

12

26

Alreadyentered

N = 37.25

350

10000

25000

8

37.5 bi-weekly payments

= approx 1 yr 5months

11 48


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You already have

$10000saved

for your down

payment. If yousave $350per pay,

for the next 2 years,

find the size ofyour

available downpayment. Assume

you can earn 8%

compounded monthly


Alreadyentered

12

26

FV = 31430.12

10000

52

8

Formula solution

350

11 49


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3 Steps

Formula Solution

This is more complicatedto solve when usingalgebraic equations!

1.

2.

3.

Find the FVof the $10 000in 2 years

Find the FVof the $350 per pay

Add totals together

The $10 000continues to earn

interestdur ing the new savings per iod!

You already have

$10000saved

for your down





you can earn 8%

compounded monthly


11 50


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Formula Solution


= 10000(1 + 0.08/12) 24

= $11,728.88

1.

2.

i2= (1+i)c- 1

= (1+ .08/12).4615-1

= 0.0031

3.

$

11,728.88

= 350[(1+.0031)521]

.0031

= $19701.24 19,701.24

31,430.12Total

=

PMTFV

-

i

+ni)(1 1

[ ]

You already have

$10000saved

for your down





you can earn 8%

compounded monthly


11 51


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11 52


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A life insurance company advertises that$50,000will purchase a 20-year annuitypaying $341.13at the end of each month.

What nominal rate of returndoes the annuityinvestment earn?

1

12

341.13240

050000

C/Y = 1I/Y = 5.54

The annuity earns 5.54% pa

11-53


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to solve for i without

a financial calculator

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This completes Chapter 11

Documents

Defferal Annuities