Exercises: Loan amortisation

To celebrate the end of exams, a group of friends decides to finance the purchase of a sound system for their hangout, valued at $\$1500$. The bank offers them a loan at a nominal annual interest rate of $6 \%$, which they must repay in monthly installments over one year. The figure summarizes the situation. Identify the elements of the loan so you can apply the constant annuity formula:

• Initial capital ($C$)

• Number of payments per year ($k$)

• Total number of installments ($n$)

• Monthly interest rate ($i$) expressed as a decimal.

A high school student wants to buy an electric mountain bike costing $\$2,400$. The store offers financing at a nominal annual interest rate of $4.8 \%$ spread across monthly installments. Calculate what monthly interest rate ($i$) will be applied to each installment, expressing it both as a percentage and as a decimal.

Emma has taken out a loan of $\$3,000$ to pay for her college tuition and course materials. The bank applies the French amortization system (equal installments) with a monthly interest rate of $0.5 \%$. If the monthly installment is $\$133.06$, calculate what amount of the first installment corresponds exclusively to interest payment.

Lucas finances his new next-generation game console, costing $\$600$, in 12 equal monthly installments. At the end of the year, he realizes that he has paid a total of $\$636$. Calculate:

1. a) The amount of the monthly installment.
2. b) The total cost of the interest paid over the whole loan.

A language academy offers to finance the intensive summer course ($\$900$) via an interest-free 4-month loan, but charges an origination fee of $\$25$ on the first payment.

1. a) What will be the amount of the installments for the second, third, and fourth months?
2. b) How much will the student pay in total in the first month?

Given the following incomplete amortization schedule for a loan for a graphics tablet, calculate the missing values marked with letters, knowing that the monthly interest rate is $1 \%$:

To buy a moped costing $\$2,000$, Isabella pays a down payment of $\$500$ and finances the remainder through equal monthly payments over 2 years. If the nominal annual interest rate is $3 \%$, apply the amortization annuity formula, clearly stating the principal $C$ that she is actually going to finance.

David needs to buy computer equipment for his new job as a web designer and finances $\$1,800$ to be repaid over 18 months through constant monthly payments. The store offers him financing at a $6 \%$ nominal annual interest rate ($\text{TIN}$). To calculate how much he will pay per month, David writes down the following approach in his notebook:

When checking the figure, he multiplies $166.24 \times 18 = \$2,992.32$ and gets alarmed because he thinks he will pay almost $\$1,200$ in interest for a loan of just a year and a half.

1. a) Identify the conceptual error made by David in applying the constant payment formula and explain why his calculation is not consistent with the time units.
2. b) Correct the calculation: determine the appropriate monthly interest rate $i$ and the actual monthly payment $a$ that he must pay (rounded to the nearest cent).
3. c) Calculate the actual total cost in interest of the loan and compare the result with David's erroneous figure to explain why his intuition about the disproportion of the interest was correct.

Sofía is going to purchase an electric motorcycle for her urban commutes for an amount of $\$4,800$. She applies for a bank loan to be repaid over 2 years (24 months) using the French amortization system at a nominal annual interest rate ($\text{TIN}$) of $6 \%$.

1. a) Before applying the financial formula, reasonably estimate a lower bound for the monthly payment (calculating how much she would pay if the interest rate were $0 \%$) and an approximate upper bound (calculating the total simple interest for the 2 years on the initial principal, adding it to the principal, and dividing by 24 months).
2. b) Use the constant amortization payment formula to calculate the exact value of the monthly payment $a$ (with precision to the cent).
3. c) Check whether the exact result falls within the range estimated in part a) and assess whether it is a reasonable result. Then, calculate the total amount paid in interest over the 2 years.

To set up a small recording studio and start his YouTube channel, Mateo has taken out a loan of $4\,500$ dollars to be repaid over 2 years in equal monthly installments. The bank charges a nominal annual interest rate of $6\%$. Calculate the amount of the monthly installment that Mateo must pay.

A young entrepreneur applies for a loan of $12\,000$ dollars to buy a used van for her delivery business. The loan will be paid off over 4 years with quarterly payments and an annual interest rate of $8\%$. Determine the portion of the first payment that goes toward interest and how much is actually used to pay down the debt.

A group of friends decides to buy a professional mixing console that costs $1\,800$ dollars. They pay $300$ dollars in cash and finance the rest at an interest rate of $10\%$ per year (nominal, with a monthly compounding period, that is, a monthly rate $i=\frac{0.10}{12}$) to pay it back over 6 months with constant monthly payments.

Remember that the constant payment $a$ that repays a principal $C$ in $n$ periods with rate $i$ per period is:

$a = C\times\dfrac{i}{1-(1+i)^{-n}}$

Create the first two rows of the amortization table, stating in each one: outstanding principal, interest for the period, principal repaid and monthly payment. Round to the nearest cent.

To finance an end-of-year trip, a student association takes out a loan of $3\,000$ dollars at an annual compound interest rate of $5\%$. The agreement is to repay it in a single lump sum payment after 18 months. What will the total interest amount to? If they had chosen to amortize it monthly over the same period, would they pay more or less interest in total? Reason your answer without needing to calculate the complete schedule.

We compare two loans to buy a motorbike costing $3\,500$ dollars:

• Option A: Monthly amortisation over 2 years at $6\%$ nominal annual interest.

• Option B: Quarterly amortisation over 2 years at $6.2\%$ nominal annual interest.

Calculate the instalment for each option and determine in which of the two options more interest is ultimately paid at the end of the two years.

Given the following row from the amortization table of a loan at $12\,\%$ per year with monthly payments (the monthly interest is calculated by dividing the annual rate by $12$, that is, $1\,\%$ per month):

• Month: $10$

• Initial debt for the month: $14\,250,80$ dollars

• Monthly payment: $450$ dollars

Calculate:

1. a) The interest for month $10$ (round to the nearest cent).

1. b) The principal repaid in that month.

1. c) The outstanding debt at the start of month $11$.

According to data from the Mortgage Statistics of the National Statistics Institute (INE), the average amount of mortgage loans for home purchases in Spain is around $\$144,000$, with an average amortization term of 24 years at a fixed nominal annual interest rate ($\text{TIN}$) of $3.6 \%$ through constant monthly payments.

1. a) Model the described average mortgage: calculate the monthly interest rate $i$, the total number of payments $n$, and the constant monthly payment $a$ paid by the buyer.
2. b) Break down the first monthly payment ($k=1$): calculate how much is used to pay interest ($I_1$) and how much to amortize borrowed principal ($A_1$). Determine what percentage of the first payment corresponds to interest and what percentage directly amortizes debt.
3. c) The Bank of Spain recommends that the monthly mortgage payment should not exceed $35 \%$ of net household income. Determine what the minimum net monthly income of a household must be in order to take out this average mortgage while meeting this financial prudence criterion.

A 24-year-old female entrepreneur decides to open a graphic design studio. To acquire the necessary equipment (high-end computers, software licenses and furniture), she applies for a loan from a financial institution to be repaid over 5 years in equal monthly installments of $\$425.50$. The institution charges a nominal annual interest rate of $5.4 \%$.

Calculate the initial principal that the bank granted her to set up her business. Round the result to the nearest dollars cent.

A family takes out a 20-year mortgage of $\$160,000$ at a fixed annual interest rate of $3.25 \%$ with monthly payments. After exactly 10 years of faithfully paying the monthly installments, they receive an inheritance and consider paying off the outstanding debt.

Calculate the breakdown of the 121st installment, specifying which part corresponds to interest payment and which part to principal amortization.

We compare two loan options to finance a renovation of $\$30,000$ to be repaid over 4 years with monthly payments:

• Option A: Nominal annual interest of $5 \%$ with no initial fees.

• Option B: Nominal annual interest of $4.2 \%$ but with an origination fee of $1.5 \%$ on the principal requested, paid in cash at signing.

Calculate the total cost of each option (sum of all installments plus fees) and determine which is more advantageous for the client.

A mortgage loan of $\$200,000$ is taken out over a period of 25 years with monthly installments. For the first two years, the interest rate is fixed at $2.5 \%$ per year. From the start of the third year, the interest rate becomes variable, changing to $4.75 \%$ per year.

Calculate by how much the family's monthly installment will increase from the third year onwards due to this interest rate rise.