The annual equivalent rate (AER)
One bank offers a deposit at "6 % per year" and another at "5.9 % nominal per year with monthly compounding". A loan advertises "0 % interest" but charges a 75 € fee. How can you compare such different offers using a single number? With the AER: the rate that truly reflects what you earn or pay in a year.
The nominal rate and the AER are not the same number
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 AER formula
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.
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 AER with fees and charges
The AER incorporates the calculable costs of the transaction according to its terms and the applicable regulations; this does not mean that every conceivable charge is included. What matters are the amounts and the dates of the cash flows. See the definition from the Bank of Spain.
An online "payday loan" offers $300 \text{ €}$ to be repaid in a single payment after $15$ days. The lender charges $12 \text{ €}$ as "management fees", but advertises…
An online "payday loan" offers $300 \text{ €}$ to be repaid in a single payment after $15$ days. The lender charges $12 \text{ €}$ 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 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…
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 $2.000$ €:
- 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.
Comparing offers with the AER: loans and mortgages
The AER is the number to look at when comparing two loans, two mortgages or two deposits, together with the payment schedule, the instalments and the total cost, under comparable conditions.
A well-known technology retail chain offers a promotion to buy a state-of-the-art computer costing $1.500 \text{ €}$. The offer consists of a loan over $12$ months with…
A well-known technology retail chain offers a promotion to buy a state-of-the-art computer costing $1.500 \text{ €}$. The offer consists of a loan over $12$ months with a nominal interest rate of $0 \%$ ($TIN$), but with an arrangement fee of $75 \text{ €}$ 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?
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?
Back to the start: The deposit at $5{,}9\,\%$ compounded monthly yields an $\text{TAE} = (1 + 0{,}059/12)^{12} - 1 \approx 6{,}06\,\%$, beating the one at $6\,\%$ per year; and the loan at "$0\,\%$" with a $75\text{ \euro}$ fee actually costs more than a $5\,\%$ AER.