Exercises: Calculation of annual and monthly instalments

Lucía is planning to attend a music festival when she finishes Baccalaureate in 3 years' time. To cover the expenses, she decides to open a youth savings account offering $3 \%$ annual compound interest. If she deposits $250 \euro$ at the beginning of each year, what total capital will she have accumulated for her trip at the end of the three years?

A 1st Baccalaureate student wants to buy a gaming computer costing $1.200 \euro$. A computer shop allows him to pay for it in 12 equal monthly instalments at a nominal annual interest rate of $6 \%$. Knowing that the monthly interest rate is calculated as $i = \frac{0,06}{12}$, use the loan repayment formula to calculate the monthly instalment he must pay at the end of each month.

To save up for his driving licence, Carlos decides to deposit $60 \euro$ per month into a bank account at the end of each period. The bank offers him $2,4 \%$ annual compound interest with monthly compounding. Calculate how much money he will have accumulated after 2 years (24 months) of making these deposits.

Note: Remember to convert the annual interest rate into a monthly interest rate by dividing by 12.

A family applies for a loan of $5.000 \euro$ to buy a second-hand motorbike. The bank grants them the loan at an annual interest rate of $5 \%$, which they must repay in 4 equal annual instalments payable at the end of each year. Determine the amount of this annual loan repayment.

A school e-sports club decides to save up to replace their chairs. They deposit $100 \euro$ at the beginning of each quarter into an account yielding $4 \%$ annual compound interest. If compounding is quarterly ($n=4$ periods per year), calculate the capital they will have saved after one full year (4 deposits).

Calculate the monthly instalment required to repay a debt of $2.400 \euro$ incurred from buying a professional camera, if the loan is repaid over 2 years at a nominal annual interest rate of $4,8 \%$ with monthly payments.

A young saver decides to deposit $500 \euro$ at the start of each year into an investment fund guaranteeing $3,5 \%$ annual compound interest. Use the formula for the future value of an annuity to find how much money they will have after 5 years.

Marta bought an electric scooter by financing the $450 \euro$ it costs with a loan. The bank charges her fixed monthly instalments of $40 \euro$ at the end of each month for one year. If the total interest paid is the difference between the sum of all instalments and the cash price, how much money has Marta paid solely in interest by the end of the year?

Sergio wants to finance the purchase of a $1.800$ € laptop through a $2$-year personal loan ($24$ months) with fixed monthly instalments payable at the end of each period. The bank applies an annual nominal interest rate (TIN) of $6\%$. To calculate his monthly instalment $a$, Sergio writes in his notebook:

and gets $a \approx 873,79$ € per month.

1. a) Without needing to calculate the actual instalment, multiply Sergio's instalment by the $24$ monthly payments in your head and explain with reasoning why the result is an obvious financial absurdity.
2. b) Identify and explain the two serious conceptual errors made by Sergio when formulating the equation.
3. c) Set up the correct equation that determines the ordinary monthly instalment that Sergio must pay, calculate its exact amount rounded to the nearest cent, and find the total amount he will end up paying solely in interest.

Elena is 18 years old and plans to save to pay for the deposit and move-in expenses of a shared flat when she finishes her vocational training programme in $3$ years ($36$ months). She is going to deposit $150$ € at the end of each month into an interest-bearing youth account with a nominal interest rate of $3\%$ per year ($i = \frac{0,03}{12} = 0,0025$ monthly).

1. a) Make a reasoned prior estimate before applying complex formulas: calculate how much she would accumulate if the interest rate were $0\%$, and justify whether, after adding the $3\%$ per year, the final accumulated capital will manage to exceed the $6.000$ € threshold or not.
2. b) Apply the future value formula for an ordinary constant savings annuity to calculate the exact amount she will have accumulated after the $36$ months have elapsed.
3. c) Check the exact result against your prior estimate and calculate what percentage of the final accumulated capital comes from the interest paid by the bank.

To cover the expenses of an end-of-year trip in two years' time, a group of students decides to open a savings account. They commit to depositing $80$ € at the end of each month. The bank offers them an interest rate of $3,6\%$ APR with monthly compounding.

1. a) Calculate the equivalent monthly nominal interest rate.
2. b) How much capital will the group have at the end of the $24$ months?
3. c) What is the total interest generated by the investment?

A family applies for a loan of $15.000$ € to renovate their kitchen and bathroom. The bank grants them the loan at an annual interest rate of $5\%$ to be repaid over $4$ years in fixed monthly instalments due at the end of each month.

1. a) Determine the monthly instalment amount they will have to pay.
2. b) Calculate the total amount they will end up paying back to the bank after $4$ years.
3. c) What percentage of the total amount paid corresponds to interest?

Lucas wants to buy an electric motorbike that costs $4.200$ €. In order not to deplete his savings, he decides to finance it at a nominal annual interest rate of $6\%$, committing to pay monthly instalments of $120$ €.

1. a) Use the loan repayment formula to calculate how many months it will take him to pay off the motorbike (you will need to use logarithms).
2. b) If he decides to pay off the motorbike in exactly $3$ years, what should the fixed monthly instalment be?

A family wishes to build up a savings fund for their child's university education. They plan to deposit $1.500$ € at the start of each year for the next $10$ years. The financial institution guarantees them a compound interest rate of $4,25\%$ per annum.

1. a) What capital will they have accumulated right after making the tenth deposit?
2. b) What capital will they have one year after the last deposit (at the end of the tenth year)?

Note: Take into account whether the annuity is due (in advance) or ordinary (in arrears).

A small business buys a delivery van valued at $22.000$ €. They make a deposit of $4.000$ € and finance the rest at a nominal annual interest rate of $4,8\%$. The contract stipulates payment by equal annual instalments over $5$ years.

1. a) Calculate the amount of the annual loan instalment.
2. b) Prepare the first two rows of the amortisation schedule (indicating for each year: outstanding balance, interest, principal repaid, and instalment).

A private investor wants to save $50.000$ € over a period of $8$ years. To do so, they will make quarterly deposits into a fund offering a nominal annual interest rate of $3\%$.

1. a) What is the applied quarterly interest rate?
2. b) How much money must they deposit each quarter to achieve their goal?
3. c) If after $4$ years the market interest rate rises to a nominal $4\%$ per annum, how will the quarterly deposit change for the remaining $4$ years to maintain the same final target?

Calculate the APR of a bank loan advertised with a nominal interest rate of $6\%$ per annum if the repayments are:

1. a) Monthly.
2. b) Half-yearly.
3. c) In which case is the actual cost to the customer higher? Justify your answer by comparing the results obtained.

A professional athlete decides to save part of their salary for when their sporting career ends. They deposit $500$ € at the end of each month into a savings account for $15$ years. The account guarantees a nominal annual interest rate of $4,5\%$ with monthly compounding.

1. a) What capital will they have accumulated at the time of their retirement from sport?
2. b) If once retired, they wish to receive a fixed monthly income for the following $20$ years using up all the accumulated capital (assuming the bank maintains the same interest rate of $4,5\%$), what will be the amount of this monthly payment?

A family in Spain needs to purchase a hybrid vehicle for $22.000$ €. After putting down $4.000$ € that they had in savings as a deposit, they request to finance the remaining $18.000$ € over a $5$-year term ($60$ months). The finance company applies a nominal interest rate of $7,2\%$ per year ($i = \frac{0,072}{12} = 0,006$ monthly) and offers them two repayment methods:

• Method A (Standard French loan): $60$ constant ordinary monthly instalments of amount $a$, fully clearing the debt at the end of month $60$.

• Method B (Reduced instalment with residual value): $59$ reduced monthly instalments of $320$ € and a final instalment in month $60$ (balloon payment) that settles the entire outstanding principal balance at that moment.

1. a) Model Method A mathematically: calculate the amount of the monthly instalment $a$ and the final total cost that the family will pay.
2. b) Model Method B mathematically: calculate how much the final instalment in month $60$ must be to fully amortise the remaining loan, knowing that the present value of all payments must equal the financed principal.
3. c) Compare both methods: calculate the difference in the total amount of interest paid and write a financial recommendation reasoning the trade-off between monthly relief and final extra cost.

The Bank of Spain recommends that a household's monthly mortgage payment does not exceed $35\%$ of its net income to avoid the risk of over-indebtedness. Marcos and Sara have a combined net income of $2.600$ € per month and are going to apply for a mortgage at a fixed nominal interest rate of $3,6\%$ per year ($i = \frac{0,036}{12} = 0,003$ monthly).

1. a) Determine the maximum monthly instalment they can afford according to the Bank of Spain guideline.
2. b) If they take out the mortgage for the standard term of $25$ years ($300$ months), set up the present value equation and deduce the maximum principal that the bank will grant them with this cap instalment.
3. c) If the home they want requires borrowing $220.000$ € and they do not want to exceed the maximum monthly payment calculated in section a), set up the corresponding equation, solve algebraically using logarithms for the minimum number of years required to make the operation viable, and reason whether the bank will approve the mortgage knowing that the usual maximum commercial term in Spain is $30$ years.

A 22-year-old entrepreneur decides to plan her financial independence. Her goal is to have saved $100.000$ € by the time she turns 35. To do this, she sets up the following savings plan in an account with annual compound interest of $4,5 \%$:

1. During the first 5 years, she will make monthly contributions of $a$ euros at the beginning of each month.

2. During the next 8 years, she will increase her savings capacity and contribute $2a$ euros at the beginning of each month.

Calculate the value of the initial monthly payment $a$ needed to achieve her goal.

A haulage company needs to renew its fleet by purchasing three lorries for a total value of $360.000$ €. To finance the purchase, the bank offers them two options:

• Option A: A 6-year loan with a fixed monthly instalment, at a nominal annual interest rate of $4,8 \%$ with monthly compounding.

• Option B: A loan with a "full grace period" during the first two years (nothing is paid, but interest accumulates on the initial capital). Over the remaining 4 years, the resulting debt is amortised through constant monthly instalments at a nominal annual interest rate of $5,2 \%$.

Determine which of the two options involves a higher total outlay for the company and calculate the difference between the two options.

Parents create a university fund for their daughter with annual compound interest of $6 \%$. They decide to make contributions of $2.400$ € at the beginning of each year. However, due to inflation, they decide that each annual payment will be $3 \%$ higher than that of the previous year (geometric progression of instalments).

Calculate the capital accumulated in the fund after 15 years.

Hint: You will need to sum the compounded values of each annual payment individually and observe that they form a new geometric progression.

A self-employed professional wishes to buy premises valued at $120.000$ €. He has two financial options:

1. Save for 10 years in an investment fund guaranteeing an AER / TAE of $4 \%$, making monthly payments at the beginning of each month, to buy it outright at the end of the period (assuming the price of the premises does not change).

2. Buy it today by taking out a 10-year mortgage loan at a nominal annual interest rate of $4 \%$, with monthly compounding and instalments paid at the end of each month.

Calculate the difference between the monthly payment required to save the capital (Option 1) and the monthly payment required to repay the loan (Option 2). Explain which of the two strategies is more efficient from the point of view of monthly cash flow.

A family wants to set up a solidarity fund to finance a long-term scholarship. Their plan is to deposit a fixed amount $a$ at the end of each month for 20 years into an account paying $5 \%$ annual compound interest (compounded monthly).

At the end of those 20 years, they want the accumulated capital to allow them to withdraw $1.500$ € per month for the following 15 years (the scholarship payout period), at which point the account balance should be zero.

Calculate the monthly savings deposit $a$ that they must contribute during the first phase.

Calculate the Annual Percentage Rate (APR) of a bank loan advertised at $6 \%$ nominal annual interest with monthly compounding, given that the bank also charges an arrangement fee of $1,5 \%$ of the loan amount and a compulsory life insurance policy of $250$ € paid upfront. Assume the loan amount is $20.000$ € to be repaid in a single payment after one year.

How would this APR change if the loan amount were $50.000$ € under the same conditions? Explain why the APR depends on the requested amount in this case.

$3$ years ago, Javier took out a $30.000$ € loan at $4,8\%$ annual nominal interest ($i = \frac{0,048}{12} = 0,004$ monthly) to be repaid over $8$ years ($96$ months) through constant ordinary monthly instalments of $376,87$ €. After paying instalment number $36$, the outstanding principal balance is $20.060,58$ €. At that moment, he makes an early partial repayment of $6.000$ € in cash, reducing his debt to $14.060,58$ €. The bank allows him to choose between two ways to restructure the remaining loan:

• Method 1 (Reduce instalment): Keep the remaining planned term ($60$ months) and calculate a new, reduced monthly instalment $a'$.

• Method 2 (Reduce term): Keep the original monthly instalment of $376,87$ € and shorten the loan duration by recalculating the number of months $n'$ needed to amortise the remaining balance.

1. a) Apply Method 1: find the new monthly instalment $a'$ and the total interest that Javier will pay over the remaining $5$ years.
2. b) Apply Method 2: set up the present value equation, solve using logarithms for the new term $n'$ in months (rounding up to the nearest integer), and calculate the total interest that will be paid until cancellation.
3. c) Compare both methods: calculate the interest savings achieved by each compared to not having made the early repayment, and justify mathematically why Method 2 is more efficient from the perspective of overall financial cost.

A user finances an extraordinary purchase of $C_0 = 2.500$ € using a credit card in revolving mode with a nominal interest rate of $19,2\%$ per year ($i = \frac{0,192}{12} = 0,016$ monthly). The card's advertisement claims: "Pay comfortably from just $35$ € a month".

1. a) Try to formally determine the number of months $n$ required to pay off the debt with a monthly payment of $a = 35$ € by applying the formula $n = \frac{-\ln(1 - \frac{C_0 \cdot i}{a})}{\ln(1+i)}$. Explain the mathematical impossibility that arises when evaluating the argument of the logarithm and reason financially why, by comparing the payment with the interest accrued in one month ($C_0 \cdot i$), the debt would grow indefinitely instead of decreasing.
2. b) Deduce what the theoretical minimum monthly payment is that would avoid generating additional debt, and calculate the fixed monthly payment that should be set to clear the loan within a reasonable term of $3$ years ($36$ months).
3. c) If the cardholder chooses a fixed monthly payment of $60$ €, calculate the number of months $n$ it will take to pay off the debt (rounding up) and the total amount paid. Using these calculations, demonstrate why instalments set very close to the threshold $C_0 \cdot i$ lead to the financial phenomenon known as 'captive debt'.

Topic challenge. Your emancipation plan: loan and inflation.

Imagine that you start working and need to finance a second-hand vehicle for $6000$ €, analysing the real impact of the cost of living. Prepare your financial proposal in a spreadsheet following these stages:

1. CPI and purchasing power: Check the annual CPI rate for the last $3$ years on the INE website and construct its index numbers. If you start from a salary of $1300$ € that grows by $2,5$ % annually (geometric progression), deflate said salary to analyse whether you gain or lose real purchasing power.

2. Loan simulation: Search for the conditions of a consumer credit on the Bank of Spain's banking customer portal (record the nominal interest rate [TIN] and APR [TAE]). Design the $3$-year amortisation schedule using the French method with monthly payments, distinguishing between the principal and interest portions of each monthly instalment.

3. Final report: Write a reasoned proposal (1 page) evaluating the total financial cost against expected inflation, deciding whether the operation is viable.

Assessment criteria:

• Rigorous collection of data from the INE and correct calculation of rates, indices, and deflated real salary.

• Mathematical accuracy in the spreadsheet when modelling the monthly instalment, APR, and amortisation schedule.

• Clarity and critical judgment in the final report when justifying economic viability.