The annual equivalent rate (AER)

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.

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.

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.

Compare the following three investment offers for a capital of $5,000$ € 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 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?

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?

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 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 $25$ € which is deducted at the start of the investment. If you decide to invest $2,000$ €:

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 $300$ € for an urgent expense and you must repay $315$ € 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?

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

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 $1,100$ € 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?

At an urban mobility dealership, they offer to finance an electric moped costing $\text{€}\ 3,000$ to repay the full amount in a single payment after $1$ year, presenting two alternative promotions:

• 'Interest-Free' Promotion: Advertised with a $\text{TIN}$ of $0\,\%$. However, the contract includes an arrangement and origination fee of $3\,\%$ on the vehicle price ($\text{€}\ 90$), which is added to the debt to be repaid in one year, and upfront management expenses of $\text{€}\ 60$ to be paid in cash at delivery ($t = 0$).

• 'Standard Financing' Promotion: $\text{TIN}$ of $4.50\,\%$ per annum with quarterly compounding interest settlement ($n = 4$), without any origination fees or upfront expenses, paying principal plus interest at the end of the year.

1. a) Model the cash flow of the 'Interest-Free' Promotion: determine the net amount of money that the buyer actually receives financed at $t = 0$ and the total amount disbursed after $1$ year. With these two amounts, calculate the effective $\text{TAE}$ of the loan.
2. b) Model and calculate the $\text{TAE}$ corresponding to the 'Standard Financing' Promotion.
3. c) Determine, with reasoning, which of the two offers is financially more advantageous for the buyer and reflect on why banking transparency regulations require highlighting the $\text{TAE}$ in any advertising offer, even when the advertised $\text{TIN}$ is $0\,\%$.