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 $1500 \text{ €}$. 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 Sixth Form student wants to buy an electric mountain bike costing $2.400 \text{ €}$. The shop offers financing at a nominal annual interest rate of $4,8 \%$ spread across monthly instalments. Calculate what monthly interest rate ($i$) will be applied to each instalment, expressing it both as a percentage and as a decimal.

Marta has taken out a loan of $3.000 \text{ €}$ to pay for her university tuition fees and course materials. The bank applies the French amortisation system (equal instalments) with a monthly interest rate of $0,5 \%$. If the monthly instalment is $133,06 \text{ €}$, calculate what amount of the first instalment corresponds exclusively to interest payment.

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

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

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

1. a) What will be the amount of the instalments 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 $2.000 \text{ €}$, Iratxe pays a deposit of $500 \text{ €}$ and finances the remainder through equal monthly instalments over 2 years. If the nominal annual interest rate is $3 \%$, apply the amortisation annuity formula, clearly stating the principal $C$ that she is actually going to finance.

Draw a timeline that represents the amortization process of a loan of $1.200 \text{ €}$ that is repaid in 4 quarterly installments.

Mark the instants 0, 1, 2, 3 and 4 (in quarters) on the line. At instant 0, indicate the loan payout with an upward arrow. At each time mark (quarter 1, 2, 3 and 4), draw a downward arrow and write the name given to the fixed amount of money that is paid to the bank.

You can do it on paper and send a photo with your answer.

David needs to buy computer equipment for his new job as a web designer and finances $1.800 \text{ €}$ 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 $166,24 \cdot 18 = 2.992,32 \text{ €}$ and gets alarmed because he thinks he will pay almost $1.200 \text{ €}$ 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 $4.800 \text{ €}$. 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, Marcos has taken out a loan of $4\,500$ € to be repaid over 2 years in equal monthly instalments. The bank charges a nominal annual interest rate of $6\%$. Calculate the amount of the monthly instalment that Marcos must pay.

A young entrepreneur applies for a loan of $12\,000$ € to buy a second-hand van for her delivery business. The loan will be amortised over 4 years with quarterly payments and an annual interest rate of $8\%$. Determine the portion of the first instalment corresponding to interest payment and how much is actually used to amortise the debt.

A group of friends decides to buy a professional mixing console that costs $1\,800$ €. They pay $300$ € in cash and finance the rest at an interest rate of $10\%$ per year (nominal, with 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:

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

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 $3\,000$ € 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 amortise 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.

Calculate the outstanding balance of a loan of $20\,000$ € taken out at $4\%$ per year after paying 12 monthly instalments. The loan was granted for a total of 5 years. (Hint: The outstanding balance is the present value of the remaining instalments to be paid).

A video rental shop wants to renew its screens and applies for a loan of $6\,000$ € at $9\%$ per year to be repaid over 3 years in equal monthly instalments. The contract includes an early redemption fee of $1\%$ of the outstanding balance. If at the end of the second year they decide to settle the remaining debt, what will be the total amount they must pay at that moment (outstanding balance plus fee)?

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

• 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: $14\,250,80$ €

• Monthly payment: $450$ €

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 $11$.

On the bank statement of a personal loan under the French amortization system with constant monthly payments, several lines have been erased due to a printing error. However, the data for two consecutive months can be clearly seen in the history:

• In month 5, the principal repayment part of the payment was $A_5 = 245,60 \text{ €}$.

• In month 6, the principal repayment part of the payment was $A_6 = 246,828 \text{ €}$.

• The fixed monthly payment amount paid by the customer is $320,00 \text{ €}$.

1. a) Knowing that the principal repayments follow the relationship $A_{k+1} = A_k \cdot (1+i)$, deduce the monthly interest rate $i$ and the nominal annual interest rate ($\text{TIN}$) of the loan.
2. b) From the fixed monthly payment and $A_5$, find the amount allocated to the payment of interest in month 5 ($I_5$) and calculate the outstanding balance remaining at the end of month 4 ($C_4$).
3. c) Determine the principal repayment amount for the first month ($A_1$) and deduce the total principal initially borrowed ($C_0$).

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 $144.000 \text{ €}$, 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 licences and furniture), she applies for a loan from a financial institution to be repaid over 5 years in equal monthly instalments of $425,50 \text{ €}$. 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 euro cent.

A university student wants to buy a second-hand car costing $9.500 \text{ €}$. He has some savings, but needs to finance the remaining $7.000 \text{ €}$. The bank offers him a personal loan at an annual interest rate of $6,75 \%$. The student analyzes his budget and concludes that he can only allocate a maximum of $180 \text{ €}$ per month to the instalment payment.

Using logarithms, determine the minimum number of months it will take to repay the loan so that the monthly instalment does not exceed his limit of $180 \text{ €}$.

A family takes out a 20-year mortgage of $160.000 \text{ €}$ at a fixed annual interest rate of $3,25 \%$ with monthly payments. After exactly 10 years of faithfully paying the monthly instalments, they receive an inheritance and consider paying off the outstanding debt.

Calculate the breakdown of the 121st instalment, specifying which part corresponds to interest payment and which part to capital amortisation.

A logistics technology company applies for a loan of $250.000 \text{ €}$ to renew its fleet of electric vehicles. The loan is agreed at an annual interest rate of $4,5 \%$ with quarterly payments over 6 years. The financial institution grants them a full grace period of 6 months (during the first two quarters no payment is made, but interest accumulates on the initial principal).

Calculate the amount of the quarterly instalments that they will have to pay from the third quarter until the end of the loan.

We compare two loan options to finance a renovation of $30.000 \text{ €}$ 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 arrangement fee of $1,5 \%$ on the capital requested, paid in cash at signing.

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

A mortgage loan of $200.000 \text{ €}$ is taken out over a period of 25 years with monthly instalments. 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 instalment will increase from the third year onwards due to this interest rate rise.

A self-employed worker applies for a microcredit of $12.000 \text{ €}$ at $8 \%$ per year to be repaid over 3 years in semi-annual instalments. After the first year (after paying the second instalment), he decides to make an early repayment of $3.000 \text{ €}$ to reduce the debt. The bank allows him to choose between reducing the loan duration or reducing the amount of the remaining instalments.

If he chooses to reduce the instalment amount while maintaining the original 3-year term, calculate what the new semi-annual instalment amount will be.

Consider a loan with principal $D$ to be repaid over $n$ periods with an interest rate per period of $i$. Prove step by step, using the geometric progression of outstanding balances, that if the interest rate were $0 \%$ ($i=0$), the formula for the equal instalment $a$ simplifies to the simple arithmetic division: $a = \frac{D}{n}$.

Note: For the proof, use the limit of the annuity formula as $i$ tends to zero or analyze the sum of the amortisation terms.

Elena took out a loan of $50.000 \text{ €}$ 4 years ago at $4,8 \%$ annual $\text{TIN}$ to be repaid over 10 years (120 constant monthly payments of $525,45 \text{ €}$). Just after paying payment 48, the outstanding balance to be repaid is $32.808,16 \text{ €}$. Elena receives a windfall and makes an early repayment of $10.000 \text{ €}$, reducing her outstanding debt to $22.808,16 \text{ €}$. The bank offers her two options for the remaining 6 years (72 periods):

• Option A: Keep the planned term (72 remaining months) and reduce the amount of the monthly payment.

• Option B: Keep the original payment of $525,45 \text{ €}$ and shorten the remaining amortization term.

1. a) Calculate the amount of the new monthly payment in Option A.
2. b) In Option B, set up the payment equation and solve using logarithms for the number of months it will take to pay off the remaining debt while keeping the payment of $525,45 \text{ €}$. Round up to the nearest whole number of months.
3. c) Calculate the total cost in interest that Elena will pay over the remainder of the loan in each of the two options. Justify which of the two methods is financially more advantageous in terms of interest savings, and explain under what personal circumstances it might be advisable to choose the other option.

In the French amortization system with constant payment $a$, at the beginning most of each payment is used to cover interest and very little to amortize principal, but this relationship gradually reverses as the debt decreases.

Carlos and Lucía apply for a mortgage of $120.000 \text{ €}$ at an annual $\text{TIN}$ of $4,8 \%$ with monthly payments over 20 years ($n = 240$ months).

1. a) Analytically prove that the first principal amortization payment can be expressed as $A_1 = a \cdot (1+i)^{-n}$, starting from the relationship $I_1 = C_0 \cdot i$ and the constant payment formula $a$.
2. b) For the principal amortization to exceed the interest for the first time in a given month $k$, it must hold that $A_k > \frac{a}{2}$. Using the expression $A_k = A_1 \cdot (1+i)^{k-1}$ and the result from part a), set up the corresponding inequality and solve analytically with logarithms for the condition on $k$. Determine in which year and month of the loan this crossover occurs exactly.
3. c) Analyze the expression obtained for $k$ in part b): does the crossover month depend on the initial borrowed principal $C_0$? If they had taken out the mortgage for 30 years ($n = 360$ months) at the same interest rate, calculate in which month the crossover would occur and explain what financial consequence extending the term has on the amount of time during which interest is predominantly paid.