Exercises: Bank interest

After working over the summer as a camp counsellor, you have managed to save $1500 \text{ €}$. You decide to open a savings account that offers a simple interest rate of $2,5 \%$ per year. If you keep the money in the bank for exactly one year without making any transactions:

1. a) What interest ($I$) will your capital generate at the end of the year?
2. b) What will be the final capital ($C_f$) available in your account?

To buy a state-of-the-art tablet costing $600 \text{ €}$, 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.

Your grandparents have given you $2000 \text{ €}$ 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:

Look at the following bank advertisement aimed at young savers:

"Grow your savings! Deposit $3000 \text{ €}$ with us and after two years we will give you back $3180 \text{ €}$."

Assuming that the bank applies annual simple interest, identify the following financial elements:

1. Initial capital ($C_0$).

2. Time of the transaction ($t$).

3. Total interest generated ($I$).

4. Annual interest rate ($R$) as a percentage.

Calculate what initial capital ($C_0$) we should deposit in a bank account so that, at a simple interest rate of $4 \%$ per year, we obtain exactly $120 \text{ €}$ in interest after one year.

An online microcredit platform offers fast loans at a monthly simple interest rate of $1 \%$. If a user requests $400 \text{ €}$ to repair their motorbike and plans to repay everything at the end of a quarter (3 months):

1. a) What is the interest they will pay each month?
2. b) What is the total amount (capital + interest) they will repay at the end of the three months?

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 $1000 \text{ €}$ 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.

Graph the growth of the accumulated interest of a $1000 \text{ €}$ investment at a simple interest rate of $5 \%$ per year over the first 4 years.

First complete the table of values (accumulated interest at the end of each year) before drawing the graph:

• Year 1: $I = \dots$

• Year 2: $I = \dots$

• Year 3: $I = \dots$

• Year 4: $I = \dots$

Then, draw a set of axes: time (in years, from 0 to 4) on the horizontal axis and interest (in €) on the vertical axis. Choose a suitable scale, mark the points from the table (plus the point for year 0) and join them.

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

Lucía wants to make a return on $2.500$ € by opening a $3$-year fixed-term deposit at a $4$ % nominal annual interest rate with quarterly interest settlement. When solving the exercise to find out how much money she will have at the end, she writes in her notebook:

1. a) Identify and explain with reasoning the conceptual error Lucía made in her formulation, indicating why the result obtained is inconsistent with the agreed terms.
2. b) Correct the calculation by applying the appropriate fractional compounding formula and determine the exact final capital she will receive.

A bank promotes a $5$-year savings account with a $3$ % nominal annual interest rate and monthly interest settlement, in which you decide to deposit $10.000$ € without making any withdrawals.

1. a) Without applying the fractional compound interest formula yet, make a quick estimate of the approximate total profit by calculating the interest using the simple interest formula. Justify whether the actual final capital obtained with monthly compound interest will be greater than, less than, or equal to your previous estimate.
2. b) Accurately calculate the final capital and the actual interest generated using the fractional compound interest formula.
3. c) Compare both results and reason whether monthly compounding represents a substantial or moderate financial difference compared to the simple calculation for this time period and interest rate.

Marcos 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.

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.

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?

Sara wants to go on an international volunteering trip in a few years. The estimated total cost of the project is $4\,500$ €. Currently she has $3\,200$ € deposited in an account offering $3,75$ % annual compound interest. Using logarithms, calculate how long (expressed in years and months) she will need to keep the money invested to reach exactly the amount needed for her trip.

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?

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 young entrepreneur applies for a microcredit loan of $8\,000$ € to buy IT equipment. The bank offers two repayment methods:

• Option 1: A single payment at the end of 3 years with compound interest of $7$ % per year.

• Option 2: Repayment through equal annual installments over 3 years at a nominal interest rate of $7,5$ %.

Calculate how much extra money is paid in interest under the option that turns out to be more expensive.

We invest $15\,000$ € in a financial product offering compound interest compounded quarterly. After two years, the capital has grown to $16\,254,47$ €. Determine the annual rate (nominal interest percentage) at which the operation was carried out. Round the final percentage to an integer.

Paula needs to gather exactly $12.000$ € in $4$ years to buy a second-hand van. To achieve this, she goes to two banks to make a single initial deposit today:

• Bank A: Offers her a deposit with a nominal interest rate of $3,6$ % per year with monthly compounding of interest.

• Bank B: Requires her to deposit exactly $10.000$ € today, guaranteeing she will reach $12.000$ € after $4$ years through annual interest compounding.

1. a) State the equation corresponding to Bank A and calculate the exact initial capital Paula must deposit today to achieve her goal.
2. b) For Bank B, state the equation relating the initial capital to the final capital and solve for the nominal annual interest rate ($r$) that is actually being applied. Round the resulting percentage to two decimal places.

According to data from the Bank of Spain on remunerated deposits for households, a family analyses two alternatives to invest $6.000$ € of their savings over a $2$-year period:

• Option A (Fixed-Term Deposit): Offers a compound interest of $2,8$ % per year with quarterly compounding and no fees of any kind.

• Option B (Loyalty Account): Offers a nominal interest rate of $3,4$ % per year with annual compounding, but charges a fixed maintenance and administration fee of $25$ € that is deducted at the end of each of the two years after settling the interest.

1. a) Mathematically model the evolution of the money in both options year by year and calculate the final available balance after $2$ years in each one.
2. b) Assume that the initial capital is an unknown variable $C$. Set up the inequality that allows you to deduce from what minimum initial amount Option B would be more advantageous than Option A for that same $2$-year period (considering annual compounding in both for the comparison). Interpret the result in terms of fixed costs versus percentage yield.

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 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.

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?

An investor analyses a savings product that offers compound interest of $3,2\%$ per year compounded daily (assume years of $360$ days). If the average inflation forecast for the next $5$ years is $2,5\%$ per year, calculate the net real yield (purchasing power gained) of an initial investment of $10.000$ € after that time. Hint: Compare the final amount obtained with the value the money would need to have to maintain its purchasing power against inflation.

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.

A young woman opens a savings deposit of $10.000$ € with the goal of gathering the down payment for her first flat. The bank agrees to return the principal plus accumulated interest in a single payment once the amount has tripled. If the agreed interest rate is $12\%$ per year with semi-annual compounding, how many years and months must pass for this payment to be made? Use logarithms in the process.

An investment fund guarantees an amount of $30.000$ € in $10$ years through fixed annual contributions at the beginning of each year. The interest rate is $3,5\%$ per year. However, due to a financial crisis, at the end of the fifth year the interest rate drops to $2\%$ per year for the remainder of the period. If the investor maintains their annual contributions equal to those originally calculated, what capital will they actually have at the end of the $10$ years?

Calculate the initial capital that a person must deposit in a financial institution offering a nominal annual interest rate of $4,2\%$ compounded monthly, so that their heirs can receive a monthly income of $1.500$ € in perpetuity (consider the limit as the number of years tends to infinity, or simply the capital that generates that interest each month without decreasing). What would be the AER of this operation?

A family applies for a personal loan of $30.000$ € at a $6$ % nominal annual interest rate to renovate their home, to be repaid over $5$ years ($60$ months). The bank offers them two different payment methods to repay the loan:

• Method 1 (Standard monthly amortization): From the first month, they pay a fixed monthly installment of $579,98$ €.

• Method 2 (Grace period during the first year): During the first $12$ months, they only pay the monthly interest generated by the outstanding debt (grace period). Starting from month $13$, and for the remaining $4$ years ($48$ months), they switch to paying a constant monthly installment of $704,55$ €.

1. a) For Method 1, calculate the breakdown for month 1: determine what portion of the installment goes towards interest, how much money pays down the principal debt, and what the outstanding debt is at the beginning of month 2.
2. b) For Method 2, calculate the fixed monthly installment they will pay during the first $12$ months of the grace period and find the total sum of money paid over the $5$ years in each of the two methods.
3. c) Justify in detail why Method 2 turns out to be more expensive overall, relying on the concept of outstanding debt and how amortization influences the calculation of interest in subsequent periods.

A $30$-year-old worker in Spain 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.