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.

Aoife wants to calculate the Annual Equivalent Rate ($\text{AER}$) of a savings account that advertises a nominal rate ($\text{NIR}$) of $3\,\%$ with monthly interest settlement and no fees. In her copy, she writes the following solution:

Aoife notices that the result obtained is exorbitant for a real savings account, but she cannot find the mistake.

1. a) Identify and explain in detail the error that Aoife made when entering the nominal rate into the formula.
2. b) Correct the approach and find the true $\text{AER}$ of the account, expressing the result as a percentage and rounding to two decimal places.
3. c) Explain why, conceptually, the correct $\text{AER}$ must be only slightly higher than the advertised $3\,\%$ and not such a distant value.

A bank markets a $1$-year fixed-term deposit with a $\text{TIN}$ of $4\,\%$, no fees, and quarterly interest settlement ($n = 4$).

1. a) Before carrying out any calculation with the formula, make a qualitative and reasoned estimate: do you expect the $\text{TAE}$ to be less than $4\,\%$, exactly equal to $4\,\%$, slightly higher (for example, between $4.05\,\%$ and $4.10\,\%$), or much higher (above $5\,\%$)? Justify your prediction based on the reinvestment mechanism of the interest generated throughout the year.
2. b) Apply the $\text{TAE}$ formula to calculate its exact numerical value (round to two decimal places as a percentage).
3. c) Compare the value obtained with your previous estimate and assess whether it makes economic sense that the difference between $\text{TIN}$ and $\text{TAE}$ is of that magnitude for moderate interest rates.

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 $450$ € 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 $25$ € 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?

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 $5,000$ € 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 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 digital bank operating in Ireland wants to launch an interest-bearing savings account with no maintenance fees. The marketing team aims to set an exact $\text{AER}$ of $3.50\,\%$, paying interest month by month ($n = 12$).

1. a) Write the equation that allows you to obtain the annual nominal rate ($r$) from the target $\text{AER}$ and the number of settlement periods per year.
2. b) Solve algebraically for the variable $r$ and calculate the nominal rate that the institution must publish in its pre-contractual documentation, rounding to two decimal places of percentage.
3. c) If the bank decided to change the payment frequency to daily ($n = 365$), would the nominal rate required to maintain that same $\text{AER}$ of $3.50\,\%$ have to be higher or lower than the one obtained in the previous part? Justify your answer by analysing the relationship between the compounding frequency and the accumulation of interest.

You have $\text{€}\ 10,000$ and are offered two options for a $1$-year fixed-term deposit with no fees:

• Option A: $\text{TIN}$ of $3.20\,\%$ with semi-annual compounding ($n = 2$).

• Option B: $\text{TIN}$ of $3.18\,\%$ with daily compounding ($n = 365$).

To decide which product to choose, two classmates propose different strategies:

• Method 1 (Via TAE): Directly calculate the $\text{TAE}$ of each alternative using its formula and choose the one with the higher percentage.

• Method 2 (Via Final Capital): Calculate the final capital accumulated at the end of the year by applying compound interest period by period to the $\text{€}\ 10,000$ and choose the one that provides more euros in profit.

1. a) Apply both methods numerically and confirm whether both lead to the same decision.
2. b) Justify which method is more efficient and generalisable if you had to advise a group of clients with very different amounts (from $\text{€}\ 500$ to $\text{€}\ 500,000$).
3. c) Imagine that Option A adds an account maintenance fee of $\text{€}\ 12$ at the end of the year. Explain why Method 1 would require adjusting the calculation and what the qualitative impact of that expense on the real effective return would be.

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 $APR$ 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 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 $\text{€}\ 10,000$ 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 $\text{€}\ 180,000$ 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 $\text{€}\ 250$ (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 $\text{€}\ 45$ per year. A user keeps a constant average debt of $\text{€}\ 1,200$ 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 $\text{€}\ 1,200$.

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 $\text{€}\ 5,000$ and commits to repaying $\text{€}\ 5,600$ 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.

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 $0\,\%$.

An investment brochure advertises a $1$-year structured product with a $\text{TAE}$ of $5.00\,\%$. A financial sales representative claims to a client: “If we agree to settle interest instantaneously and continuously instead of monthly, while keeping the same annual $\text{TIN}$, the effective return ($\text{TAE}$) will increase substantially, potentially reaching $6\,\%$ or even $7\,\%$.”

1. a) Assuming that the product offers a $\text{TAE}$ of $5.00\,\%$, calculate the $\text{TIN}$ ($r$) that would correspond if the compounding were monthly ($n = 12$) and the one that would correspond under continuous compounding using the limit expression $\text{TAE} = e^{r/100} - 1$. Round both values to three decimal places of percentage.
2. b) Now suppose that the bank sets a $\text{TIN}$ of $4.88\,\%$ in advance. Analytically prove that there is a finite upper bound for the $\text{TAE}$ that cannot be exceeded, no matter how large the number of settlement periods $n$ is. Determine the exact value of this maximum upper bound.
3. c) Based on the results of the previous parts, refute the sales representative's claim using mathematical arguments and explain what practical conclusion a saver should draw about the real influence of increasing the settlement frequency on the profitability of their money.