Amortized loans

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Amortized loans are loans where interest and principal are repaid with a constant series of payments, which are paid at regular time intervals. Mortgage loans are an example.

The first subsection shows how to calculate the payments in an amortized loan and the second presents the loan amortization schedule, which helps you solve for the amount of interest paid with each payment and the principal owing after each payment.

Amortized Loan Payments Example

Let’s reconsider the $1,000 loan to Ernest Defraud. Instead of a balloon loan, assume that the loan is structured as an amortized loan with three annual, end-of-year payments as shown in the following timeline.

Amortized loans example

Figure 1: Amortized Loan Timeline

What set of end-of-year payments does the lender require? Here is a hint: Wouldn’t the lender be happy to exchange the principal for the payments if the present value of the payments is the same as the principal? The hint suggests that the focal point is time 0.

Notice that the payments look like an ordinary annuity. We know how to find the present value of annuity payments. The equation of value for an amortized loan is the equality between the principal and the present value of ordinary annuity payments.

\[\begin{matrix}Principal=PMT\times PVIF{{A}_{i,n}} & {} & \left( 1 \right) \\\end{matrix}\]

Where

\[PVIF{{A}_{i,n}}=\frac{1-\frac{1}{{{\left( 1+i \right)}^{n}}}}{i}\]

We call Equation 1 the amortized loan equation. The equation involves four variables: the principal, the payments, the interest rate, and the number of payments. We can use it to solve for any one variable if the other three are given.

In the Ernest Defraud example, the unknown is the annuity payment (PMT). We can rearrange Eq. 1 to solve for payments as follows.

\[\begin{matrix}PMT=\frac{\text{Principal}}{PVIF{{A}_{i,n}}} & {} & \left( 2 \right) \\\end{matrix}\]

For the loan to Ernest, the interest rate is 5% (i=5) and there are three payments (n=3).

So,

\[PVIF{{A}_{i,n}}=\frac{1-\frac{1}{{{\left( 1+i \right)}^{n}}}}{i}=\frac{1-\frac{1}{{{\left( 1.05 \right)}^{3}}}}{0.05}=2.723248\]

The amortized loan payments are

\[PMT=\$1,000/2.723248=\$367.21\]

Example Amortized Loans, Interest, and the Reinvestment Rate

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Your friend, Ernest Defraud, wants to borrow $1,000 for 3 years. He has proposed an amortized loan with payments of $367.21. What is the future value of the payments if you can reinvest the first and second payments at 5%? How much interest do you earn from reinvestment and how much interest is included in the payments themselves?

At year 3, the future value of the payments can be calculated by finding the future value of a three-period ordinary annuity at a rate of 5%. The future value of the annuity is

\[\begin{align}& F{{V}_{annuity}}=PMT\times \frac{{{\left( 1+i \right)}^{n}}-1}{i} \\& =367.21\times \frac{{{\left( 1.05 \right)}^{3}}-1}{0.05}=\$1157.63\\\end{align}\]

The amount of interest earned due to reinvestment is just the difference between the future value of the three payments and the sum of the three payments. The sum is what you would have at year 3 if you didn’t earn any interest by reinvesting. For example, if you put the payments in your mattress!

\[Reinvestment~interest=\$1,157.63-(3\times\$367.21)=\$56\]

The amount of interest that is included in the three payments is the difference between the sum of the three payments and the principal of the loan.

\[Interest~in~payments=(3\times \$367.21)-\$1,000=\$101.63\]

This example highlights two important points about amortized loans:

  1. Interest on amortized loan payments arises in two forms: (1) each payment comprises interest and (2) the lender gets the payments before the end of the term and so can make interest by investing them.
  2. The future value of the amortized loan payments is only equal to the balloon payment if the lender can reinvest intermediate payments at the loan rate. Consequently, lenders only earn i% on amortized loans if they can reinvest at i%. This is called the reinvestment rate assumption.
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