Section 5.3 Present Value of an Annuity; Amortization Page | 1

Section 5.3

Present Value of an Annuity; Amortization

Example 1. Tim Wilson and Carol Britz are both graduates of the Brisbane Institute of

Technology(BIT). They both agree to contribute to the endowment fund of BIT. Tim says that

he will give $500 at the end of each year for 9 years. Carol prefers to give a lump sum today.

What lump sum can she give that will equal the present value of Tim’s annual gifts, if the

endowment fund earns 7.5% compounded annually?

Present Value of an Ordinary Annuity

The present value P of an annuity of n payments of E dollars each at the end of consecutive

interest period with interest compounded at a rate of interest i per period is

𝑷 = 𝑬 [𝟏 − (𝟏 + 𝒊)−𝒏

𝒊]

F = Future value

E = Equal periodic payment

i = m

r

r = interest rate

m = compounding periods per year

n = mt

t = time in years

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Example 2. A car costs $19,000. After a down payment of $2000, the balance will be paid off in

36 equal monthly payments with interest of 6% per year on the unpaid balance. Find the amount

of each payment.

Amortization Payment

Amortization is the process of paying off a debt with equal periodic payments made over a

specified period of time that includes a portion of the principal and interest.

The periodic payment E on a loan of P dollars to be amortized over n periods with interest

charged at the rate of i per period is

𝑬 =𝑷𝒊

𝟏 − (𝟏 + 𝒊)−𝒏

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Example 3. The Perez family buys a house for $275,000 with a down payment of $55,000. They

take out a 30-year mortgage for $220,000 at an annual interest rate of 6%.

(a) Find the amount of the monthly payment needed to amortize this loan.

(b) Find the total amount of interest paid when the loan is amortized over 30 years.

(c) Find the part of the first payment that is interest and the part that is applied to reducing the

debt.

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Example 4. Kassy Morgan borrows $1000 for 1 year at 12% annual interest compounded

monthly. Verify that her monthly loan payment is $88.8488, which is rounded to 88.85. After

making three payment, she decides to pay off the remaining balance all at once. How much

must she pay?

Example 5. In Example 4, Kassy Morgan borrowed $1000 for 1 year at 12% annual interest

compounded monthly. Her monthly payment was determined to be $88.85. Construct an

amortization schedule for the loan and then determine the exact amount Kassy owes after three

monthly payment.

Section 5.3 Present Value of an Annuity; Amortization Page | 5

Summary (Formulas)

1. Present Value of an Ordinary Annuity (paid at the end of each time period)

𝑷 = 𝑬 [𝟏 − (𝟏 + 𝒊)−𝒏

𝒊]

2. Amortization Payment

𝑬 =𝑷𝒊

𝟏 − (𝟏 + 𝒊)−𝒏

F = Future value

E = Equal periodic payment

i = m

r

r = interest rate

m = compounding periods per year

n = mt

t = time in years