The APR/APY and the Annual Percentage Yield (APY): CAT, GAT, and TAE
One bank offers a deposit at "6% per year" and another at "5.9% nominal annual rate compounded monthly." A store advertises a computer on installments with "0% interest" but charges a $\$75$ fee. How can you compare such different offers with a single number? With an annual rate that reflects what you really earn or pay in a year: the GAT if you invest, the CAT if you borrow.
CAT, GAT, and the nominal rate
In Mexico, when you take out a loan, the CAT (Costo Anual Total, Total Annual Cost) summarizes everything it costs in one annual percentage: interest and fees. When you make an investment, the GAT (Ganancia Anual Total, Total Annual Gain) summarizes what you earn; the real GAT also subtracts inflation. Both are the same idea seen from opposite sides: the effective annual rate.
In other countries and in the exercises on this page, that effective rate is written with the abbreviation TAE (tasa anual equivalente, annual equivalent rate), and the nominal rate with the abbreviation TIN. They are equivalent: here TAE means "the CAT or the GAT, depending on whether you borrow or invest," and TIN is the nominal interest rate that the offer advertises.
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: | Compounding…
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:
| Compounding period | Value of $n$ |
|---|---|
| Semi-annual | |
| Quarterly | |
| Monthly | |
| Daily (calendar year) |
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.
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…
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.
Fractional compounding: the effective rate formula
Values of $n$: semiannual $2$, every four months $3$, quarterly $4$, every two months $6$, monthly $12$, twice a month $24$, daily $365$.
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…
A bank offers two types of one-year fixed deposits:
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Deposit A: $5 \%$ nominal interest rate with semi-annual interest payments.
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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.
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…
Two digital banks, NeoBank and BancaApp, want to attract your savings for a fund you are building for your end-of-year trip.
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NeoBank offers you a nominal annual interest rate of $3.5\%$ with monthly compounding of interest.
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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…
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.
The other direction: from the effective rate to the nominal rate
Sometimes you are given the effective rate (the GAT or the TAE) and you need the nominal rate. You just solve in the formula: the $n$th root undoes the power.
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…
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 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…
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 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 ($TIN$)…
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).
One-period rate and terms other than one year
Credit cards and installment loans from retail stores usually advertise the monthly rate. To annualize it, you compound that period as many times as fit in a year.
If the term is not one year, you calculate the rate for the term and raise it to the number of times that term fits in a year:
$$\text{TAE} = \left(\frac{\text{final amount}}{\text{principal received}}\right)^{\text{(length of the year)/(term)}}-1$$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…
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 $1.8\%$.
- a) Calculate the nominal annual interest rate you are being charged.
- b) Find the APR associated with this financial transaction.
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…
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.
- a) Calculate the monthly interest rate as a percentage.
- 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?
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…
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.
Fees and charges: the real cost
The CAT incorporates the mandatory costs of the transaction. What matters are the amounts and dates of each flow of money: what you actually receive today and what you pay later.
When you pay in monthly installments
If you pay in monthly installments, each payment is "brought back" to today using the monthly rate $j$: a payment $P$ made $k$ months from now is worth $\dfrac{P}{(1+j)^k}$ today. The correct rate is the one that makes what you receive today equal to the sum of what you pay, brought back to today. For $12$ equal payments $P$ at the end of each month:
$$\text{Amount received}=P\cdot\frac{1-(1+j)^{-12}}{j}$$You can't solve for $j$ by hand: you try values until the equality holds (with a calculator, spreadsheet, or GeoGebra). Then $\text{TAE}=(1+j)^{12}-1$.
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…
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:
- a) What real capital are you investing after paying the fee?
- b) What capital will you receive at the end of the year?
- 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.
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. a)…
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.
- a) Calculate the total amount paid at the end of the year.
- 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?
- 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.
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…
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 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…
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?
Comparing offers
The CAT and the GAT (or the TAE) are the numbers you look at to compare two loans, two cards, or two investments, in addition to the term, the payments, and the total cost, always under comparable conditions.
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:…
Compare the following three investment offers for a capital of $100\,000$ pesos for one year:
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Offer A: $4.2\%$ nominal annual interest, compounded quarterly.
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Offer B: $4.15\%$ nominal annual interest, compounded monthly.
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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.
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 $1.5\,\%$…
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.
- a) Calculate the $TAE$ (APR) of the Platinum Card and the $TAE$ (APR) of the Gold Card's interest (not counting the maintenance fee).
- b) Calculate the total annual cost of each card (interest and fee) for an average debt of $\$1,200$.
- 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.
Back to the beginning: the deposit at $5.9\,\%$ nominal compounded monthly yields a GAT (TAE) of $(1+0.059/12)^{12}-1\approx 6.06\,\%$, so it beats the one at $6\,\%$ per year. And the "interest-free" computer with a $\$75$ fee costs, in a single payment after one year, a CAT of $75/1425\approx 5.26\,\%$.