Annuity and Monthly Payment Calculations

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.

To save up for his driver's license, Carlos decides to deposit $\$1\,200$ 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 young saver decides to deposit $\$500$ 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.

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 $80$ pesos 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?

Lucía is planning to attend a music festival when she finishes high school in 3 years. To cover the expenses, she decides to open a youth savings account offering $3 \%$ annual compound interest. If she deposits $\$5\,000$ at the beginning of each year, what total capital will she have accumulated for her trip at the end of the three years?

A school e-sports club decides to save up to replace their chairs. They deposit $\$2\,000$ 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 $30\,000$ pesos 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 family applies for a loan of $\$100\,000$ to buy a used motorcycle. The bank grants them the loan at an annual interest rate of $5 \%$, which they must repay in 4 equal annual installments 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 $\$24\,000$. 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 $300\,000$ pesos 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?

Marta bought an electric scooter by financing the $\$9\,000$ it costs with a loan. The bank charges her fixed monthly installments of $\$800$ at the end of each month for one year. If the total interest paid is the difference between the sum of all installments and the cash price, how much money has Marta paid solely in interest by the end of the year?

Calculate the monthly installment required to repay a debt of $\$48\,000$ 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 small business buys a delivery van valued at $22,000$ pesos. They make a deposit of $4,000$ pesos 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).

Lucas wants to buy an electric motorbike that costs $4,200$ pesos. 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 $120$ pesos.

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 private investor wants to save $1\,000\,000$ pesos 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?

A professional athlete decides to save part of their salary for when their sporting career ends. They deposit $500$ pesos 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 self-employed professional wishes to buy premises valued at $120,000$ pesos. 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 $30\,000$ pesos 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.

A haulage company needs to renew its fleet by purchasing three lorries for a total value of $7\,200\,000$ pesos. 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.

A 22-year-old entrepreneur decides to plan her financial independence. Her goal is to have saved $100,000$ pesos 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 $a$ pesos at the beginning of each month.

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

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

Parents create a university fund for their daughter with annual compound interest of $6 \%$. They decide to make contributions of $48\,000$ pesos 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.

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 $5\,000$ pesos paid upfront. Assume the loan amount is $400\,000$ pesos to be repaid in a single payment after one year.

How would this APR change if the loan amount were $1\,000\,000$ pesos under the same conditions? Explain why the APR depends on the requested amount in this case.