Exercises: The annual equivalent rate

A youth savings account offers a nominal annual interest rate (TIN) of $3 \%$ with monthly compounding. Calculate the Annual Equivalent Rate (AER) of this account.

To finance an end-of-year trip, a group of students deposits their savings in a financial product offering $4 \%$ nominal annual interest compounded quarterly. Determine the AER of the investment.

A bank offers two types of one-year fixed deposits:

• Deposit A: $5 \%$ nominal interest rate with semi-annual interest payments.

• Deposit B: $5.05 \%$ AER with annual interest payment.

Calculate the AER of Deposit A to find out which of the two is more profitable.

Complete the following table indicating the value of $n$ (number of compounding periods per year) that you must use in the AER formula for each case:

A bank advertises a savings account with an AER of $2.5 \%$ and annual interest settlement. If the settlement period is a full year, what is the nominal annual interest rate (TIN) of this account? Explain why both values coincide in this case.

Calculate the AER corresponding to a nominal interest rate of $12 \%$ per year compounded quarterly. If we then decide to switch to monthly compounding while keeping the same TIN, will the AER increase or decrease? Verify this by carrying out the second calculation.

Two digital banks, NeoBank and BancaApp, want to attract your savings for a fund you are building for your end-of-year trip.

• NeoBank offers you a nominal annual interest rate of $3.5\%$ with monthly compounding of interest.

• BancaApp offers you a nominal annual interest rate of $3.55\%$ with semi-annual compounding.

Calculate the Annual Equivalent Rate (AER) of both options and determine which is more profitable for your savings.

You want to buy a graphics tablet costing $9\,000$ pesos and decide to finance it with a credit card from an electronics shop. The contract states that the monthly interest rate applied is $1.8\%$.

1. a) Calculate the nominal annual interest rate you are being charged.
2. b) Find the APR associated with this financial transaction.

A bank advertises a personal loan for young people with an APR of $7.2\%$. If interest is paid monthly and there are no additional fees, what is the nominal annual interest rate applied to you?

A financial institution offers a one-year fixed-term deposit with a nominal interest rate of $4\%$. However, the bank charges a fixed setup fee of $25$ pesos which is deducted at the start of the investment. If you decide to invest $2,000$ pesos:

1. a) What real capital are you investing after paying the fee?
2. b) What capital will you receive at the end of the year?
3. c) Taking into account that at the end of the year you receive the capital plus interest, calculate the real AER of the operation by considering the fee as a lower initial capital.

You have seen an advert for "fast microcredit loans" online. They lend you $6\,000$ pesos for an urgent expense and you must repay $6\,300$ pesos after exactly one month.

1. a) Calculate the monthly interest rate as a percentage.
2. b) Calculate the APR of this loan if the same conditions were maintained for a whole year. Do you think this interest rate is reasonable compared to a standard bank loan?

The AhorroFácil bank pays interest on its accounts every quarter. If a customer wants to obtain an annual equivalent rate (AER) of $5\%$, what nominal annual interest rate should the bank offer?

Compare the following three investment offers for a capital of $100\,000$ pesos for one year:

• Offer A: $4.2\%$ nominal annual interest, compounded quarterly.

• Offer B: $4.15\%$ nominal annual interest, compounded monthly.

• Offer C: $4.25\%$ simple annual interest (interest is not reinvested).

Calculate the AER for each one (in case C, the AER is equal to the simple interest as it is for one year) and rank them from best to worst.

A phone company offers to finance a high-end smartphone. The phone costs $1,100$ pesos cash. The financing consists of $12$ monthly instalments of $98$ pesos.

1. a) Calculate the total amount paid at the end of the year.
2. b) If the excess paid over the cash price is considered interest accrued at the end of the year, what annual simple interest rate would you be paying?
3. c) If the financial institution tells us that the APR of the transaction is $13.5\%$, check whether the final amount from investing $1,100$ pesos at that monthly compound interest approximately matches the total of the instalments paid.

A well-known technology retail chain offers a promotion to buy a state-of-the-art computer costing $\$1,500$. The offer consists of a loan over $12$ months with a nominal interest rate of $0 \%$ ($TIN$), but with an arrangement fee of $\$75$ paid with the first installment. If we compare this offer with a traditional bank loan that has no fees but applies monthly compound interest, what should the $TIN$ of this second loan be so that its Annual Equivalent Rate ($TAE$) is identical to that of the shop's offer?

A young investor is analysing two financial products for their savings. The first is a savings account offering a nominal annual interest rate of $4.5 \%$ with daily compounding. The second is an investment fund offering a guaranteed nominal annual interest rate of $4.6 \%$ but with semi-annual compounding. Prove algebraically which of the two products yields a higher annual real return by calculating their respective $TAE$ and determine the percentage difference between them on an annual basis.

An online "payday loan" offers $\$300$ to be repaid in a single payment after $15$ days. The lender charges $\$12$ as "management fees", but advertises that the interest rate is $0 \%$. Calculate the real $TAE$ of this financial transaction assuming a year has $365$ days. Reflect on why this figure is required by law in advertisements for these products.

A bank advertises a deposit account with an $TAE$ of $5.116125 \%$. If the bank pays interest on a monthly basis, calculate the Nominal Annual Interest Rate ($TIN$) being applied. Once obtained, calculate how much money a customer who deposits $\$200\,000$ for a quarter would receive if they decide not to reinvest the interest generated each month (simple interest over that period).

When applying for a mortgage loan of $\$180,000$ to be amortized over 20 years using the French amortization system, the bank offers you a nominal annual rate ($TIN$) of $3.2 \%$ with monthly compounding. However, to grant it, it requires you to purchase an annual life insurance policy of $\$250$ (paid in monthly instalments together with the loan payment throughout the 20 years) and to pay an arrangement fee of $0.5 \%$ on the requested capital. Calculate the $EAR$ ($TAE$) of the transaction, incorporating all these additional expenses into the effective cost of capital.

We compare two credit cards. The Platinum Card charges a nominal monthly interest of $1.8\,\%$ with no maintenance fee. The Gold Card charges a nominal monthly $1.5\,\%$ but has a fixed maintenance fee of $\$45$ per year. A user keeps a constant average debt of $\$1,200$ throughout the year.

1. a) Calculate the $TAE$ (APR) of the Platinum Card and the $TAE$ (APR) of the Gold Card's interest (not counting the maintenance fee).

1. b) Calculate the total annual cost of each card (interest and fee) for an average debt of $\$1,200$.

1. c) Write the annual cost of each card as a function of the average debt $D$ and determine from what amount of average annual debt the Gold Card would start to be more cost-effective.

A bank offers an investment product where interest is compounded quarterly. If a customer wishes for their initial capital to double in exactly $10$ years, what must be the minimum $TAE$ offered by the product? Express the result to four decimal places and also calculate the corresponding annual $TIN$ for this transaction.

A crowdlending platform allows people to lend money to SMEs. A company requests $\$100\,000$ and commits to repaying $\$112\,000$ after $9$ months in a single payment. Calculate the $TAE$ that investors will obtain if the platform charges a management fee of $1 \%$ on the final capital collected at maturity. Compare this result with the $TAE$ that would be obtained if there were no management fee.