15.6: Installment Loans
- Page ID
- 185445
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- Find the payoff of an installment loan.
Installment Payment On a Loan
If a person or business needs to buy or pay for something now—such as a car, a home, college tuition, or equipment for a business—but doesn’t have the funds, they can borrow the money through a loan. The borrower receives the loan amount, called the principal (or present value), upfront and is obligated to repay it over a specified period (the term of the loan) through regular periodic payments that include interest. An installment loan is a loan with a fixed term, where the borrower pays a consistent amount each period until the loan is fully repaid. These periods are almost always monthly.
The payment per period (installment payment) can be calculated using the following formula:
The payment for each period (PMT) can be calculated using the following formula,
\[PMT =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{(-nt)}\right]} \label{1}\]
where
- \(P\) is the starting loan principal (the amount borrowed),
- \(r\) is the annual interest rate (APR),
- \(n\) is the number of payment periods per year, and
- \(t\) is the loan term in years.
Therefore, \(nt\) represents the total number of payments over the life of the loan.
Note: Payments to lenders are always rounded up to the next penny. Do not round numbers during the computation, as rounding introduces errors. You may round only once at the end to report the final answer.
The following examples show how to calculate the loan payment.
Find the monthly payment for a car costing $15,000 if the loan is amortized over five years at an interest rate of 9%.
Solution
The monthly is given by the PMT formula, and it is
\[ PMT=\frac{15,000\left(\frac{0.09}{12}\right)}{\left[1-\left(1+\frac{0.09}{12}\right)^{(-60)}\right]} \approx 311.38 \nonumber \]
Therefore, the monthly payment needed to repay the loan is $311.38 for five years.
Calculate the monthly payment for each of the following installment loans.
(a) A car loan taken out for $28,500 at an annual interest rate of 3.99% for five years.
(b) A home loan taken out for $136,700 at an annual interest rate of 5.75% for 15 years.
- Solution
-
(a) \(P =\$28,500, ~ r=0.0399\), and \(t=5\). Since the payments are monthly, \(n=12\). Substituting these values into the PMT formula, we find that the monthly payment needed is $524.75.
\[ PMT = \frac{28,500\left(\frac{0.0399}{12}\right)}{\left[1- \left(1+\frac{0.0399}{12}\right)^{(-12\times 5)}\right]} \approx 524.75 \nonumber\]
(b) \(P= \$136,000, ~ r = 0.0575,\) and \(t=15\). Monthly payments mean \(n=12\). Substituting these values into the PMT formula, we find the monthly payment needed is $1,135.18.
\[ PMT=\frac{136,000\left(\frac{0.0575}{12}\right)}{\left[1-\left(1+\frac{0.0575}{12}\right)^{(-12\times 15)}\right]} \approx 1,135.18 \nonumber \]
Calculate the monthly payment for each of the following loans.
(a) A home improvement loan taken out for $17,950 at an annual interest rate of 7.5% for 10 years.
(b) A solar panel installation loan taken out for $33,760 at an annual interest rate of 4.3% for 20 years.
- Answer
-
(a) $213.07 (b) $209.96
Kylie takes out a $16,780 loan from her credit union for a new car. The loan term is four years, with an annual interest rate of 6.77%. What are Kylie’s monthly payments?
- Answer
-
$400.03
Crissy and Jonesy take out a $13,200 loan for roof repairs. The loan is for 10 years at an annual interest rate of 7.15%. What are their monthly payments for this loan?
- Answer
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$154.29
An amount of $5,000 is borrowed for 15 months at an interest rate of 9%. Find the monthly payment to the nearest cent.
Solution
\(P=\$5,000, ~r=0.09, ~n=12, ~nt=15.\) Therefore,
\[PMT =\frac{5,000\left(\frac{0.09}{12}\right)}{\left(1-\left(1+\frac{0.09}{12}\right)^{(-15)}\right)} \approx \$353.68\nonumber\]
Vijay's tuition for the next year is $32,000. His parents have decided to pay the tuition by making nine monthly payments. If the interest rate is 6%, what will the monthly payment be?
- Answer
-
$3,645.04
A loan of $250,000 is to be paid in 15 years at 6.9% interest. What are the monthly payments for the loan?
- Answer
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$2,233.12
Glen borrowed $10,000 for his college education at an 8% interest rate, compounded quarterly. Three years later, after graduating and finding a job, he decided to start paying off his loan. If the loan is amortized over five years at a 9% interest rate, what will his monthly payment be for the next five years?
- Solution
-
To calculate Glen's monthly payment for the loan, we need to follow these steps:
1. Find the balance of the loan after 3 years: Since the loan was compounded quarterly at 8%, we first need to calculate the loan balance after 3 years using the compound interest formula:
\[ A = P\left(1+\frac{r}{n}\right)^{nt}=10,000\left(1+\frac{0.08}{4}\right)^{12} =10,000(1.02)^{12}\approx \$12,682.42 \nonumber \]
2. Calculate the monthly payment for the remaining loan balance: After finding the balance (\(=A\)), we will then use the PMT formula to find the monthly payment for the next 5 years at 9% interest. Note that \(P=\$12,682.42, ~r=0.09, t=5\), and \(n=12\).
\[PMT =\frac{12,682.42\left(\frac{0.09}{12}\right)}{\left(1-\left(1+\frac{0.09}{12}\right)^{(-60)}\right)} \approx 263.27 \nonumber \]
Thus, the monthly payment is $263.37.
Calculating Loan Payoff
Sometimes, a consumer may want to pay off a loan before the end of the loan term. For example, suppose you have a five-year car loan but wish to purchase a new car after owning your current car for four years. Since there is still one year remaining on the loan, you'll need to pay off the remaining balance before purchasing another car. However, this isn't as simple as just multiplying the monthly car payment by 12 to find the payoff amount. The reason is that each payment consists of both interest and principal. By solving the PMT formula (\ref{1}) for an installment loan for \(P\), the loan amount, we can calculate the payoff amount, which is the remaining principal.
The payoff amount (\(P\)) for an installment loan is given by
\[P=\frac{PMT\left[1-\left(1+\frac{r}{n}\right)^{(-U)}\right]}{\left(\frac{r}{n}\right)} \label{6.4.2} \]
where
- \(PMT\) is the installment payment,
- \(r\) is the annual interest rate (APR),
- \(n\) is the number of payments per year, and
- \(U\) is the number of remaining (future) payments.
Note: Payments to lenders are always rounded up to the next penny. Do not round numbers during the computation, as rounding introduces errors. You may round only once at the end to report the final answer.
A firm needs a piece of machinery with a useful life of five years. It has the option to lease it for $10,000 per year or buy it for $40,000 in cash. If the interest rate is 10%, which option is better?
- Solution
-
The cost to buy is $40,000 (this is the immediate cash outflow, so no calculation is needed for this option).
Let's calculate the present value of the lease payments and compare it with the cost of buying the machinery.
Since the lease payment is $10,000 per year for five years, \(PMT = 10,000\). To calculate the present value of the lease payments, we will use the formula for the present value of an annuity (the payoff formula for loans) with \(n=1\) and, \(t=5 \):
\[ P=\frac{PMT[1-(1+r)^{(-t)}]}{r} = \frac{10,000[1-(1+0.1)^{(-5)}]}{0.1} \approx 37,907.87 \nonumber\]The present value of the lease payments over five years is approximately $37,907.87, while the cost to buy the machinery outright is $40,000.
Therefore, leasing the machinery is the better option, as the present value of the lease payments ($37,907.87) is lower than the purchase price of the machinery ($40,000).
A manufacturing company buys a machine for $500 cash and $50 per month for the next three years. Find the present cash value of the machine today if money is worth 6.2% compounded monthly.
- Answer
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$2,138.67
A machine costs $8,000 and has a useful life of five years. It can be leased for $160 per month over five years, with a cash down payment of $750. If the current interest rate is 8.3%, is it cheaper to lease or to buy?
- Answer
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It is cheaper to buy.
Relationship Between the Payments and the Loan Term
An important relationship exists between the loan term and the total interest paid. To illustrate, consider a $30,000 loan at 15% APR to be repaid in monthly payments over 30 years. The monthly payment for this loan is $379.33. Total interest paid on an installment loan is found by multiplying the regular payment, \(PMT\), by the total number of payments, \(nt\), and then subtracting the loan principal \(P\):
\[ I = A - P =\$379.33 \times (12\times 30) - \$30,000 = \$136,558.80 - \$30,000 = \$106,558.80 \nonumber\]
In this case, the total interest paid is $106,558.80.
Increasing the monthly payment by 10% to $417.27 reduces the loan term by 48% to just over 15.36 years. Meanwhile, the total interest paid is reduced by 56% to $46,961.
The reduction in the loan term in response to an increased payment is not always so significant. For example, if the above loan had an 8% APR, the monthly payment would be $220.13 instead of $379.33. Increasing the monthly payment by 10% would decrease the loan term by only 27%, from 30 years to 21.93 years, and the total interest paid would be decreased by 32%, from $49,247 to $33,722.
It would be useful to understand how changing the loan term affects both the payment size and the total interest paid. It can be shown that as \(nt\) becomes large, the monthly payment approaches the interest per period, meaning the smallest possible payment equals the interest charged on the outstanding loan balance. If the borrower wishes to minimize their payment, the appropriate term is the one that allows repayment of only the interest. On the other hand, the shortest repayment period is one. Clearly, there is a trade-off between the size of the loan payment and the length of the loan term.
In most cases, the relationship between the size of the loan payment and the loan term is not one-to-one. In other words, a 1% increase in the loan payment amount rarely leads to a 1% drop in the loan term. The corresponding percent decline in the loan term is usually much larger. Hence, we find that when applying for loans, it pays to explore various terms and payment sizes to find the best match—the one with the optimal trade-off between term and liquidity.
Exercises
Calculate the monthly payment for each of the following loans.
1. The principal is $8,600, the annual interest rate is 6.75%, and the term is five years.
2. The principal is $19,400, the annual interest rate is 2.25%, and the term is six years.
3. The principal is $11,870, the annual interest rate is 3.59%, and the term is three years.
4. The principal is $41,900, the annual interest rate is 8.99%, and the term is 15 years.
5. The principal is $26,150, the annual interest rate is 11.1%, and the term is seven years.
6. The principal is $46,350, the annual interest rate is 2.9%, and the term is six years.
7. The principal is $175,800, the annual interest rate is 4.73%, and the term is 25 years.
8. The principal is $225,000, the annual interest rate is 5.06%, and the term is 30 years.
In the following exercises 9 & 10, compare the monthly payment and interest rate for each installment loan.
9. A loan for $20,000 is taken out for five years. Find the monthly payment if the interest rate is: (a) 2.9% (b) 3.9% (c) 4.9% (d) 5.9%.
Did the monthly payments increase by the same amount for each 1% jump in interest rate? Describe the pattern.
10. A loan for $50,000 is taken out for seven years. Find the payment if the interest rate is: (a) 3.5% (b) 4.5% (c) 5.5% (d) 6.5%.
Did the monthly payments increase by the same amount for each 1% jump in interest rate?
11. A $14,250, four-year loan is taken out at an interest rate of 9.15%. What are the monthly payments for the loan?
12. A loan of $42,000 is taken out for six years at an interest rate of 7.5%. What are the monthly payments for the loan?