The first type of debt annuity we will examine is the (loan) — an annuity where we borrow an initial amount of money (PV) and repay the loan with a series of equal-sized payments (PMT), at regular intervals, over the course of a fixed time period. At the end, we owe nothing (FV = 0).

Note: PV and PMT have opposite signs. To better understand this: PV is the initial amount we receive (loan amount) and PMT is the repayments of that loan that we must pay back after receiving the loan. The interest adds to the amount owed (we are charged interest each period on what we owe).

See the sections below for key formulas, tips and examples related to loans and down payments.

the time Required to Repay a Loan & Key Questions

For loans, you can be asked to calculate the time, rate, initial amount borrowed or size of the loan payments. In the example below, we will calculate the amount of time required for Zhang Min to repay her line of credit that she took out to pay for school.

EXAMPLE 5.3.1

Today, Zhang Min borrowed $10,000 from her line of credit. Her line of credit charges 3.75%, compounded monthly. She can afford to pay $300 per month on her line of credit with the first payment one month from today. How many years will it take her to repay her line of credit?

Before we determine what to enter into your BAII Plus, let’s ask a few important questions:

This gives us the following values in the BAII Plus:

Zhang Min will make 35 full-sized payments and a smaller final payment at the end of 36 months.
To calculate the number of years, we use the following formula:

[latex]\textrm{Number of Years} = \frac{\textrm{N}}{\textrm{P/Y}} = \frac{36}{12} = 3\textrm{ years}[/latex]

Conclusion: It will take 3 years for Zhang Min to repay her line of credit.

Down Payments & Key Questions for Unknown Selling prices

A down payment is a lump-sum payment made before you take out the loan. Down payments are often required when taking out a car loan or mortgage. The down payment will save money in interest charges over the duration of the loan. This is because it reduces the amount borrowed (PV):

[latex]\textrm{Amount Borrowed (PV)} = \textrm{Selling Price} - \textrm{Down Payment}[/latex]

If we are asked to calculate the selling price when we know the value of the down payment and amount borrowed (PV), we can rework the above equation to solve for the selling price:

[latex]\textrm{Selling Price} = \textrm{Amount Borrowed (PV)} + \textrm{Down Payment}[/latex]

EXAMPLE 5.3.2

Raj wants to buy an All Wheel Drive Tesla Model S. He can take out a 5-year loan with Tesla Lending. He must make a $10,000 down payment followed by monthly payments of $2,074/month with the first payment one month after the car is purchased. Tesla Lending charges Raj 3.75% effective on the loan. What is the selling price of the car?

Before we determine what to enter into your BAII Plus, let’s ask a few important questions:

From the above answers and the values given in the problem, enter the following in the BAII Plus:

Because PV equals to the amount borrowed, we know that Raj will borrow $113,484.44. We still need to calculate the selling price. In order to do this, we use the following formula:

[latex]\begin{align*}\textrm{Selling Price} &= \textrm{Amount Borrowed (PV)} + \textrm{Down Payment} \\ &= \$113,484.44 + \$10,000 \\&= $123,484.44 \end{align*}[/latex]

Conclusion: The selling price is $123,484.44 for the All Wheel Drive Tesla Model S.

Cost of Financing on Loans & Practice Exercise

We call the interest charged on loans the cost of financing. The same interest formula is used as before:

[latex]\begin{align*} \textrm{Interest} &= \textrm{Money Out} - \textrm{Money In} = \textrm{\$ OUT} - \textrm{\$ IN} \end{align*}[/latex]

We consider the amount borrowed, PV, to be “money in” since we are receiving this money at the start of the loan. We consider the regular payments, PMT to be “money out.” To calculate the total amount paid from the regular payments, calculate PMT×N since we will make N payments of size PMT. Finally, FV equals zero because nothing is owed at the end of a loan.

This gives us the following equation for cost of financing for loans:

[latex]\begin{align*} \textrm{Interest Earned} &= \textrm{\$ OUT} - \textrm{\$ IN}\\ &=\textrm{Regular Payments} - \textrm{Amount Borrowed}\\ &= \textrm{PMT}\times\textrm{N}-\textrm{PV} \end{align*}[/latex]

Notice that the down payment is not used in the above formula. Only the loan payment size and amount borrowed are used to calculate the cost of financing for a loan.

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