# The Wilson family buys a $325,000 home by putting a 20 % down payment and financing the...

## Question:

The Wilson family buys a {eq}$325,000 {/eq} home by putting a {eq}20 {/eq}% down payment and financing the balance with a {eq}30 {/eq} year fixed mortgage at {eq}4.35 {/eq}%.

What is the amount of their monthly loan payment to amortize the loan?

## Present Value of an Annuity

An annuity refers to a continuous series of a fixed number of cash inflows or outflows made at specific time periods. An annuity can be called an ordinary annuity or an annuity due. The time period of an annuity may be per year, per month, per week, per 2 years, and so on. For instance, repayments of a loan, pension amount received, the annual contribution to provident fund, all are examples of annuity.

The present value of any annuity refers to the discounted value of all cash flows to be received in the future,.

## Answer and Explanation: 1

Let {eq}EMI {/eq} be the amount of monthly loan payment to be made to amortize the loan.

The time period of amortization is given as 30 years. As the payment is being made per month, the number of payments made can be calculated as :

{eq}n=30\times 12 =360 {/eq}

Interest rate is given as {eq}4.35\% \: \: p.a. {/eq}. As the interest is compounded monthly, the effective interest rate per month can be calculated as follows:

{eq}r=\dfrac{4.35}{100\times 12}=0.003625 {/eq}

The principal amount of the loan is given as {eq}\$325,000 {/eq}.

A down payment of 20% has been made. This implies that the pending amount of the loan is {eq}P= (1-0.20)\times \$325,000 = \$260,000 {/eq}

The monthly installment amount can be calculated as follows:

{eq}\begin{aligned} EMI&=\dfrac{P\times r\times (1+r)^n}{(1+r)^n-1} \\&=\dfrac{\$260,000\times 0.003625 \times {(1+0.003625)}^{360}}{{(1+0.003625)}^{360}-1} \\&=\dfrac{\$3467.4555}{2.678998} \\&=\$1,294.31 \end{aligned} {/eq}

So, the amount of monthly payment is **$1,294.31**.

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Chapter 8 / Lesson 3Learn how to find present value of annuity using the formula and see its derivation. Study its examples and see a difference between Ordinary Annuity and Annuity Due.