APR vs. APY: The True Annual Rate

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

A youth savings account offers a nominal annual interest rate (APR) of $3 \%$ with monthly compounding. Calculate the Annual Percentage Yield (APY) 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 APY (annual percentage yield) of the investment.

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

Marta wants to calculate the Annual Equivalent Rate ($\text{TAE}$) of a savings account that advertises a $\text{TIN}$ of $3\,\%$ with monthly interest settlement and no fees. In her notebook, she writes the following solution:

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

A bank offers a $1$-year certificate of deposit (CD) with a $\text{TIN}$ of $4\,\%$, no fees, and interest compounded quarterly ($n = 4$).

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.

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 APY 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 APR, will the APY 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 have $\$10,000$ and are offered two options for a $1$-year certificate of deposit (CD) 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 APY): Directly calculate the $\text{TAE}$ of each alternative using its formula and choose the one with the higher percentage.

• Method 2 (Via Final Balance): Calculate the final balance accumulated at the end of the year by applying compound interest period by period to the $\$10,000$ and choose the one that provides more dollars 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 generalizable if you had to advise a group of clients with very different amounts (from $\$500$ to $\$500,000$).
3. c) Imagine that Option A adds an account maintenance fee of $\$12$ 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.

You want to buy a graphics tablet costing $450$ dollars and decide to finance it with a store credit card from an electronics store. 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 compounded monthly and there are no additional fees, what is the nominal annual interest rate applied to you?

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

Compare the following three investment offers for a capital of $5,000$ dollars 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 APY for each one (in case C, the APY is equal to the simple interest as it is for one year) and rank them from best to worst.

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

1. a) Write the equation that allows you to obtain the annual $\text{TIN}$ ($r$) from the target $\text{TAE}$ and the number of settlement periods per year.
2. b) Solve algebraically for the variable $r$ and calculate the $\text{TIN}$ 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 $\text{TIN}$ required to maintain that same $\text{TAE}$ of $3.50\,\%$ have to be higher or lower than the one obtained in the previous part? Justify your answer by analyzing the relationship between the compounding frequency and the accumulation of interest.

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

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 APY of the operation by considering the fee as a lower initial capital.

You have seen an ad for "fast microcredit loans" online. They lend you $300$ dollars for an urgent expense and you must repay $315$ dollars 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 $TAE$ of this financial transaction assuming a year has $365$ days. Reflect on why this figure is required by law in advertisements for these products.

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 annual 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 annual 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 crowdlending platform allows people to lend money to small businesses. A company requests $\$5,000$ and commits to repaying $\$5,600$ 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.

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

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 $1,100$ dollars 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 Percentage Yield ($TAE$) is identical to that of the store's offer?

At an urban mobility dealership, they offer to finance an electric moped costing $\$3,000$ 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 ($\$90$), which is added to the debt to be repaid in one year, and upfront management expenses of $\$60$ 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 consumer lending transparency regulations require highlighting the $\text{TAE}$ in any advertising offer, even when the advertised $\text{TIN}$ is $0\,\%$.