Bank interest

Your grandparents have given you $\text{€}\ 2000$ for your 16th birthday and you decide to invest it in a bank product at compound interest of $3 \%$ per year for 3 years. Since it is compound interest, the interest is compounded annually (added to the capital to generate new interest).

Calculate the final capital ($C_f$) you will have after 3 years using the formula:

Cian has received a prize of $6\,000$ € and decides to invest it for 4 years. His bank offers him two different options to earn a return on his money:

• Option A: A simple interest deposit with an annual rate of $4.5$ %.

• Option B: A compound interest savings account with an annual rate of $4.2$ % and annual compounding of interest.

Calculate the final amount he would obtain with each option and explain which one is more financially advantageous for him at the end of the 4 years.

The "Futuro Seguro" bank advertises a savings account with a nominal annual interest rate of $5$ %, paying interest quarterly. Calculate the Annual Equivalent Rate (AER) of this financial product. If the bank decided to change the interest payment frequency to monthly while maintaining the same AER, what should the new nominal annual interest rate be?

A bank offers two types of accounts for young people:

• Account A: Simple interest of $3.1 \%$ per year.

• Account B: Compound interest of $3 \%$ per year.

If you are going to deposit $\text{€}\ 1000$ for just one year, which of the two accounts will give you a greater profit at the end of that year? Justify your answer by showing the necessary calculations.

To buy a state-of-the-art tablet costing $\text{€}\ 600$, you take out a small loan from a financial institution that charges a simple interest rate of $6 \%$ per year. If you agree to repay the money together with the interest in a single payment after 8 months, calculate:

1. a) The rate ($r$) expressed as a decimal.
2. b) The time ($t$) expressed in years.
3. c) The total interest you will have to pay for the loan.

A financial institution offers an investment plan with a nominal annual interest rate of $3.6$ %. To attract different saver profiles, it allows customers to choose the frequency with which interest is paid into the account. If a person invests an initial capital of $12\,000$ € for 5 years, determine the difference between the final capital obtained if the compounding is:

1. a) Semi-annual.
2. b) Monthly.

Express the results rounded to the nearest cent of a euro.

A young person who has just got their first job applies for a loan of $15,000$ € to buy a car. The bank offers two options:

• Option A: Nominal interest rate of $6.5\%$ per year with monthly compounding and an arrangement fee of $2\%$ that is added to the initial capital.

• Option B: Nominal interest rate of $7.2\%$ per year with quarterly compounding, with no arrangement fees, but with a fixed quarterly maintenance fee of $15$ €.

Calculate the real APR of both options for a period of one year and give a reasoned answer as to which is more advantageous if the loan is repaid in full at the end of the first year.

A Year 12 student decides to open a youth pension plan. To do so, they commit to depositing $100$ € at the beginning of each month for the next 10 years. If the financial institution guarantees an annual interest rate of $3$ % with monthly compounding of interest, what total capital will they have accumulated at the end of the 10 years?

A family decides to open a pension plan for their newborn daughter. They deposit $100$ € at the beginning of each month into an account with a nominal interest rate of $4.8\%$ per year compounded monthly. They want to know when the accumulated capital will exceed $50,000$ €. Use logarithms to calculate the exact number of months required and express the final result in years and months.

A $30$-year-old worker in Ireland decides to plan for retirement at age $65$ (in $35$ years). To do so, they will deposit a fixed amount $A$ at the beginning of each year into an investment plan that guarantees a compound interest of $4$ % per year with annual compounding.

1. a) Knowing that the first contribution generates interest for $35$ years, the second for $34$ years, and so on up to the last contribution (which generates interest for $1$ year), derive the algebraic expression for the accumulated final capital as the sum of the terms of a geometric progression, proving that:

1. b) If the set goal is to accumulate a final capital of $150,000$ € upon retirement, calculate the exact value of the annual contribution $A$ that must be made.
2. c) If instead of starting at age $30$ they had postponed the start of the plan until age $45$ (having only $20$ years of annual contributions at the same $4$ % interest rate to achieve the identical goal of $150,000$ €), calculate the new required annual contribution. In light of both results, reason mathematically about the impact of the time factor and the non-linearity of compound interest on capital accumulation.

To finance the purchase of an electric car costing $24\,000$ €, a family pays a deposit of $6\,000$ € and finances the rest through a bank loan at a nominal interest rate of $6$ % per year. The loan must be repaid over 5 years in equal monthly installments paid at the end of each month.

1. a) Calculate the amount of the monthly installment the family must pay.
2. b) What is the total amount of interest the family will end up paying to the bank after 5 years?

A mortgage of $180,000$ € is taken out over $25$ years at an interest rate of $2.25\%$ per year. The contract includes an "interest-only grace period" clause of $2$ years, during which the customer only pays the interest generated each month, without repaying any capital. Calculate:

1. The monthly installment during the first two years.

2. The final monthly installment for the remaining $23$ years.

To finance the purchase of a hybrid vehicle costing $28,000$ €, an environmentally conscious person takes out a $6$-year loan at an interest rate of $5.5\%$ per year. After paying exactly the first $36$ monthly installments, they receive an inheritance and decide to make a partial repayment of $8,000$ €. If they decide to keep the remaining duration of the loan, how much money will they save in total interest compared to the original plan?