Business Mathematics Jerome Chapter 11

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McGraw-Hill Ryerson©

Chapter 11

OrdinaryAnnuities


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Calculate the…

Learning ObjectivesAfter completing this chapter, you will be able to:

… number of payments in ordinary and deferred annuities

… payment size in ordinary and deferred annuities

… interest rate in 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 number or size or interest rate using 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 can’t afford the car of your dreams, priced at $28800. You are advised to… “Save up for 4 years and 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?

You need to decide if this situation involves… a PV or 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

Finding Payment Size of an

Ordinary Simple Annuity

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Your life partner somehow convinced you that you can’t afford the

car of your dreams, priced at $28800. (At least

not right now). You are advised to… “Save up for 4 years and 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?



48

12PMT = - 490.44

10 028800

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 payments form a Simple Annuity go directly to 2. 1.

If the annuity’s PV is

known, substitute values of PV, n,

and i into PV formula.

If the annuity’s FV is known,

substitute values of FV, n, and i

into FV formula.

3. & 4.

(b) If the payments form a General Annuity, find c and i2

2.

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Calculate the quantity within the square brackets.

Rearrange the equation to 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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Which Formula? Your life partner somehow convinced you that you can’t afford the


not right now). You are advised to… “Save up for 4

years and 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 annuity’s FV is known, therefore, the FV formula is used

2.

Extract necessary data...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 partner somehow convinced you that you can’t afford the


not right now). You are advised to… “Save up for 4

years and 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?

0.00831.0083 1.48940.48940.008358.7225490.44

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

FV = 28800 n = 4*12 = 48i = .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 mortgage that they have offered to

hold in the purchase of your first home. They are considering an interest rate of 5% compounded monthly. If you were to take 20 years to

repay the mortgage, find the size of the

monthly payment.

24050

100 000

PMT = -659.9612

n =12*20 = 240PV = $100000

FV = 0

Formula solution

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Your parents are

discussing the terms of the 100 000 mortgage

that they have offered to hold in the purchase of your first home. They

are considering an interest rate of 5%

compounded monthly. If you were to take 20

years to repay the mortgage, 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 annuity’s PV is known, the PV formula is used2.

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

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

12.05 1

100 000

240 1

0.00421.0042 0.3686-0.6314151.530.0015659.96

Size of monthly mortgage

payment

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Amount$

How much interest will you pay your parents over the 20 year period?

Monthly Payment Number of Payments659.96 240 158,390.40

Amount Borrowed 100,000.00

Total Interest Paid 58,390.40

x

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PMT = -700

As this amount of interest

shocks you, you discuss the

possibility of making payments

of $700/month, to save some time

and interest costs.

Determine the time

it will take you to repay your

mortgage at this new

rate.

700

N = 217.52

Formula solution

218 payments = 18 yrs 2months

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Formula iPMTPV i

1ln

1ln[ ]n

0.0042

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

12.05 1

700

100 000

1

1.0042 0.00420.5952-0.4048-0.9045-217.52 217.52


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1. Base formulai

-ni)PMTPV

(11[ ]2. To isolate n, divide both

sides by PMT PMTPMT

…Continue…

Developing the Formula

PMTPV

i-ni)(11[ ]

i

-ni)PMTPV

(11 ][

Formula iPMT

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[ ]

PMTPV -ni)(11

i[ ] *i *i

(b) Reorganize equation

(c) Now Take the natural logarithm (ln or lnx) of both sides

-n* ln

PMT *iPV -n1 i)(1

-ni)(1

i) (1 ln

(d) Solving for n… 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

1lnn

[ ]

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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 Paymentsx158,390.4

0659.96 240

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If you could see your way to a further increase

of $25/month, (a) how much

faster would you pay off the

mortgage, and (b) approximately

how much less interest would be

involved?

725

PMT = -725N = 205.62

Paying $725 206 payments = 17 yrs

2months

Formula solution

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

0.0042

12.05 1

725

100 000

1

1.0042 0.00420.5747-0.4253-0.8550-205.52


205.52

Formula iPMT

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 set selling for $2250. Buyers will pay “no money down and no payments for 12 months.”

The first of 24 equal monthly payments is due 12 months from the purchase date. What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both the deferral period

and the repayment period?

Since you want the furniture now, this involves a PV

PMT

PV = $2250 Once you repay the loan, FV = 0 Payments are deferred for 11 months.

DEFERRAL

Finding Payment Size in a

Deferred Annuity

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York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.” The first of 24 equal monthly payments is due 12 months from the purchase date.

What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both

the deferral period and the repayment period?In effect, York furniture has given a loan to a buyer of $2,250

on 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

PV of the payments at the end of month 11

FV of 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 owed after 11 months:

2250

$2,465.06 is the PV of the annuity

York Furniture

has a promotion on a bedroom set selling

for $2250. Buyers will pay “no money down and no

payments for 12 months.” The first of 24

equal monthly payments is due 12 months from the

purchase date. What should the monthly payments be if York

Furniture earns 10% compounded monthly on

its account receivable during both the deferral

period and the repayment period?


Deferred Annuity

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2465.0624

FV = 2,465.06

Now find the PMT of the annuity …

0

24 monthly payments of $113.75 will repay the loan.

PV = - 2,465.06PMT = 113.75

York Furniture








period and the repayment period? 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

York Furniture








period and the repayment period?

24 monthly payments of $113.75 will repay the loan.

Find the amount owed after 11 months:


Deferred Annuity

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i.e....Number Of Payments

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$20,000 is invested in a fund earning 8% compounded quarterly. The first quarterly withdrawal of $1,000 will be taken from the fund five years from now. How many

withdrawals will it take to deplete the fund?

N

Payments are deferred for 19 quarters

DEFERRAL

Finding Number Of Payments in a

Deferred Annuity

The FV of $20,000 after the deferral, becomes the PV of the annuity ...

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Years0 4.75 5 6 7 8

d = 19$20,000

PV1

n = ?

This FV1 then becomes the PV of the annuity of $1000/quarter

The $20000 earns

interest for 4 years 9 months

Payments of $1000/quarter

FV1

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

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$20,000 is invested in a fund earning 8% compounded

quarterly. The first

quarterly withdrawal

of $1000 will be taken from the fund five years

from now. How

many withdrawals will it

take to deplete the fund?

19

48 0

Find the FV of $20,000 in 4.75 years

20000

$29,136.22 is the PV of the annuity

FV = 29,136.22


Deferred Annuity

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1000

Now find the PMT of the annuity …

044.1 quarterly payments will deplete the

fund(44 full payments and 1 partial)

29136.22

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

Formula solution





from now. How




Deferred Annuity

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FV = 20000(1 + 0.08/4)19

= $29,136.22

Find the FV of $20,000 in 4.75 years

PMTPV *i

i1ln

1lnn

[ ]

ln(1.02)

= 44.1 payments or 11 years





from now. How




Deferred Annuity

[ln 1 - ]29136.22 *.021000

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When…number 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 every two weeks

to make your budgeting easier.

Find the new term of your mortgage if the interest charges

remain at 5% compounded

monthly.

12

26350

P/Y = 26

415 bi-weekly payments or 15 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

every two weeks to make your

budgeting easier. Find the new term of your mortgage

if the interest charges remain at 5% compounded

monthly.

Determine c Step 1

C= 12 / 26 = .4615

i2 = (1+i)c - 1

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

i2 = 0.0019

Use c to determine i2 Step 2

C =number of compoundings per year


Step 3

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as the value for “i” in the appropriate annuity formula

Step 3 Use this rate i2 = 0.0019

Formula iPMT

PV i*

1ln

1ln[ ]n

1.0019 0.00190.5428-0.4571-0.7828

1

350100 000

1

0.0019-414.74

415 payments or 15 yrs 11.4 months

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# of Payments

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

Per month

Per month

Every two

weeks

240

218

206

415 $145,250.004. Every two

weeks415

Best Scenario

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350

You are now considering delaying the purchase of your first house to

allow for a larger down payment. If

you save $350 per pay, how long would it take to have an additional

$15000, if you can earn 8% compounded

monthly on your savings?

12

26

150000

N = 40.32

8

= FV

New requiredFormula

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Determine c Step 1

C= 12 / 26 = .4615

i2 = (1+i)c - 1

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

i2 = 0.0031

Use c to determine i2 Step 2

C =number of compoundings per year


Step 3


allow for a larger down payment. If you save $350 per

pay, how long would it take to have

an additional $15000, if you can earn 8%

compounded monthly on your

savings?

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allow for a larger down payment. If you save $350 per

pay, how long would it take to have

an additional $15000, if you can earn 8%

compounded monthly on your

savings?

15000

1

1.00310.00310.13161.1316

Formula n iPMT

FV i*

1ln

1ln[ ]+

0.1237 0.003140.3

40.3 bi-weekly payments = approx 1yr 7months

1

350

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1. Base formula

iPMTFV ni)(1 1[ ]

2. To isolate n, divide both sides by PMT PMTPMT

…continued…

Developing the Formula

PMTFV

Formula iPMT

FV i*

1ln

1ln[ ]n

+

iPMTFV

ni)(1 1[ ]

i ni)(1 1[ ]

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…from 2. PMTFV

i ni)(1 1[ ]

(a) Multiply both sides by i 3. Continue to isolate n …

PMTFV

i ni)(1 1[ ] *i *i

[ ]PMT FV

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

n ln(1+ i) ln[ ]PMT * iFV1 +

(d) Solving for n… divide both sides by

ln(1+i) ln(1+i) ln(1+i)

n ln(1+ i) ln[ ]PMT * iFV1 +

PMTFV * i

i1ln

1 +lnn

[ ]

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You already have

$10000 saved for your down

payment. If you save $350 per pay,

how long would it take to have an additional $15000?

Assume you can earn 8%

compounded monthly on all of your savings.

12

26

Already entered

N = 37.25

350

10000

25000

8

37.5 bi-weekly payments = approx 1 yr 5months

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You already have


payment. If you save $350 per pay, for the next 2 years, find the size of your available down

payment. Assume you can earn 8%


Already entered

12

26

FV = 31430.12

10000

52

8

Formula solution

350

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3 Steps

Formula Solution

This is more complicated to solve when

using algebraic equations!

1.

2.

3.

Find the FV of the $10 000 in 2 years

Find the FV of the $350 per pay

Add totals together

The $10 000 continues to earn interest during the new savings period!

You already have





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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)52 –1].0031

= $19701.24 19,701.2431,430.12Total

PMTFV

i ni)(1 1[ ]

You already have





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A life insurance company advertises that $50,000 will purchase a 20-year annuity

paying $341.13 at the end of each month.

What nominal rate of return does the annuity investment earn?

1

12

341.13240

050000

C/Y = 1I/Y = 5.54

The annuity earns 5.54% pa

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…to solve for i without

a financial calculator

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This completes Chapter 11

Business Mathematics Jerome Chapter 11

Education

Transcript of Business Mathematics Jerome Chapter 11