Bank interest
At 16 you can already open a savings account and, in a few years, think about a loan for a car or a mortgage for a flat. In every case the bank gives you an interest rate, but the small print (how often it is compounded, what charges apply) changes the final result a lot. This section puts those numbers in order.
Simple and compound interest: fractional compounding
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…
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:
$$C_f = C_0 \cdot (1 + r)^t$$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…
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:
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Option A: A simple interest deposit with an annual rate of $4,5$ %.
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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.
Loans: the interest you pay, not the interest you earn
Borrowing money uses the same formula the other way round: the bank hands you a capital and you pay back more, at the same interest rate on money you have not earned anything from.
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.…
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:
- a) The rate ($r$) expressed as a decimal.
- b) The time ($t$) expressed in years.
- 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…
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:
- a) Semi-annual.
- b) Monthly.
Express the results rounded to the nearest cent of a euro.
Saving with regular contributions
Instead of paying in all the money at once, many savings plans consist of paying in a fixed amount every month or every year. Each contribution earns interest from the moment it is paid in, so the first contributions build up more interest than the last ones.
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…
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.
Mortgages: a first idea of amortisation
In a mortgage with a constant monthly payment (the French system), each payment is split between interest on the capital still owed and amortisation (repayment) of that capital. At the start, the interest part is the largest of the whole life of the loan (with high rates or long terms it can be almost the whole payment); it then decreases over time, and at the end almost the whole payment is amortisation, because the outstanding debt keeps falling.
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…
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 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…
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.