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Chapter 14 of

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Calculate

Learning Objectives

…the principal balance after any payment using both the Prospective Method and the Retrospective Method

After completing this chapter, you will be able to:

… the principal and interest components of any payment

And…

… the final loan payment when it differs from the others

LO 1.

LO 2.

LO 3.

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Learning Objectives

Calculate

LO 4.

LO 5. … mortgage loan balances and amortization periods to reflect

prepayments of principal

… mortgage payments for the initial loan and its renewals

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A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its

20-year amortization period. (1) Calculate the monthly payment.

(2) Using the monthly payment from part (1), calculate the PV of all payments. (3) Why does the answer in (2) differ from $20,000?

24090

20 000

PMT = -179.9512

n =12* 20 = 240PV = $20000 FV = 0

1.

2. & 3.

LO 1.

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(2) Using the monthly payment from part (1), calculate the PV of all payments. (3) Why does the answer in (2) differ

from $20,000?

2.

3.

179.95

179.95

n =12*20 = 240PV = ? FV = 0 PMT = 179.95

PV = 20,000.5345

The difference of $0.5345 is due to rounding the monthly payment to the nearest cent!

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Calculate the exact balance after 5 years assuming the final payment will be adjusted for

the effect of rounding the regular payment.

A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its

20-year amortization period.

Calculate the exact n for monthly payments of $179.95 to repay a $20,000 loan...

20 000N = 239.982

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Calculate the exact balance after 5 years assuming the final payment will be adjusted for

the effect of rounding the regular payment.

A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its

20-year amortization period.

After 5 years, 239.982 – 60 = 179.982 payments remain. Therefore, balance (after 5 years)

= PV of 179.982 payments of $179.95

60

N = 239.982N = 179.9821P/V = 17,741.05

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An Original Loan =

Consider that…The PV of ALL of the Payments(discounted at the contractual rate of interest on the loan)

Also, that…A Balance = The PV of the remaining Payments

(discounted at the contractual rate of interest on the loan)

Then…

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…this can be expressed as …the Statement of Economic Equivalence

(Original Loan)

Focal Date…

PV of first x Payments

PV of the Balance just after the xth Payment

For a focal date of the original date of the loan,

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of the xth payment, the Statement of Economic Equivalence becomes…

Retrospective Method for Loan BalancesRetrospective

Retrospective

Suppose we locate the Focal Date…

Balance

This is now rearranged to isolate the “Balance”

Balance

FV of the Original Loan

FV of the Payments

already made

FV of the Original Loan

FV of the Payments

already made

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Retrospective Method for Loan BalancesRetrospective

Retrospective

… is based on PAYMENTS ALREADY MADE!`

Prospective Method for Loan Balances

… is based on PAYMENTS YET to be MADE!`

Application

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Calculate the exact balance after 5 years.

A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95

during its 20-year amortization period.

Solve using…Retrospective Method

Prospective Method

Then compare…

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Retrospective Method for Loan Balances

Calculate the exact balance after 5 years.

A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95

during its 20-year amortization period.

Balance = FV of $20,000 – FV of first 60 payments

60179.95

12

9

20,000

12 * 5 Years

FV= 17,741.05

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Prospective Method for Loan Balances

Calculate the exact balance after 5 years.

A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95

during its 20-year amortization period.

12* 20 Years = 240Total payments =

180179.9512

9

PV= 17,741.88

0

- 60 made = 180 remaining

Balance = PV of remaining 180 payments

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Difference ($0.83) is because the Prospective Method assumes that the final payment is the same as all the others.

The Retrospective Method is based on payments already made.

FV= 17,741.05 Retrospective Method

for Loan Balances

PV= 17,741.88 Prospective Method for Loan Balances

Comparison of Methods

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Final Payment = (1+i) * (Balance after 2nd to last payment)Balance after 239 payments =

FV of $20,000 after 239 months – FV of 239 payments

239

179.95

12 9 FV= - 175.42

20,000

Final Payment = (1+0.09/12) * 175.42

= $176.74

Calculate the size of the final payment.

A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95

during its 20-year amortization period. LO 2.

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Meditech Laboratories borrowed $28,000 at 10%, compounded quarterly, to purchase new testing equipment.

Payments of $1,500 are made every 3 months.A. Calculate the balance after the 10th payment.

B. Calculate the final payment.

Balance after 10 payments = FV of $28,000 after 10 quarters – FV of 10 payments

101500

4

10

FV= - 19,037.29

28,000

A.

B. 2.1. 3. Needed

Balance after 10 payments

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Meditech Laboratories borrowed $28,000 at 10%, compounded quarterly, to purchase new testing equipment.

Payments of $1,500 are made every 3 months.A. Calculate the balance after the 10th payment.

…the number of payments1. Calculate

0

N = 25.457 FV = -673.79

25

…the balance after the 2nd to last payment2. Calculate

3.

B. Calculate the final payment.

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…the final payment3. Calculate

Meditech Laboratories borrowed $28,000 at 10%, compounded quarterly, to purchase new testing equipment.

Payments of $1,500 are made every 3 months.A. Calculate the balance after the 10th payment.

B. Calculate the final payment.

Final Payment = (1+0.10/4) * 673.79

= $690.63

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A $9,500 personal loan at 10.5% compounded monthly is to be

repaid over a 4-year term by equal monthly payments.

A. Calculate the interest and principal components of the 29th

payment.

B. How much interest will be paid in the second year of the loan?

LO 3.

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A $9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term

by equal monthly payments.A. Calculate the interest and principal components

of the 29th payment. B. How much interest will be paid in the second year of the loan?

First: … find the size of the monthly paymentPV = n = i =9500 12(4) = 48 .105/12

4812

10.5

PMT = - 243.23

9500 0

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A $9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term

by equal monthly payments.A. Calculate the interest and principal components

of the 29th payment.

A.

First: … find the balance after the 28 payments

28

PMT = - 243.23

243.23

FV = -4445.06

Interest Component of Payment 29 = 0.105/12* 4445.06 = $38.89

= i * Balance after 28th payment

Principal Component = PMT – Interest Component= $243.23 - $38.89= $204.34

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A $9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term

by equal monthly payments.B. How much interest will be paid in the

second year of the loan?

First:… find the balance after 1 Year, and the balance after 2 Years

12

FV = -7483.53

Total Principal paid in year 2 = $7,483.53 - $5,244.84

= $2,238.69

24

FV = -5244.84

Total Interest paid in year 2 = 12($243.23) - $2,238.69

= $680.07

Balance after 1 year

Balance after 2 years

B.

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This completes Chapter 14