Calculating annuities and monthly payments

To save up for his driving licence, Cian decides to deposit €60 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.

Aoife bought an electric scooter by financing the €450 it costs with a loan. The bank charges her fixed monthly instalments of €40 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 Aoife paid solely in interest by the end of the year?

Aoife 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 Post-Leaving Certificate (PLC) course 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.

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 family applies for a loan of $\text{€}\ 5,000$ 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 1st Baccalaureate student wants to buy a gaming computer costing $\text{€}\ 1,200$. 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.

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?

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).

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.

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 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 $n = \frac{-\ln(1 - \frac{C_0 \times i}{a})}{\ln(1+i)}$. 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 ($C_0 \times i$), 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 $C_0 \times i$ lead to the financial phenomenon known as 'captive debt'.

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

$3$ years ago, Darragh 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 Darragh 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 school e-sports club decides to save up to replace their chairs. They deposit $\text{€}\ 100$ 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).

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 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.