Equivalent annual rate

In finance, the Annual Equivalent or Equivalency Rate (TAE) is an indicative reference of the effective annual cost or yield of a financial product regardless of its term. Its calculation includes the nominal interest rate, expenses, commissions, payments and income and allows for a homogeneous comparison of the performance of different financial products.

Financial entities use the Equivalent Annual Rate (TAE) and the Nominal Interest Rate (TIN) to present the profitability of financial operations.

The Equivalent Annual Rate allows you to homogeneously compare the interest rates of multiple financial operations with different capitalization periods, using the same annual time base. It allows homogenizing different nominal rates, expenses, commissions, settlement periods, etc.

Definition

It is the annual interest that is generated after deducting the expenses and commissions for one or several capitalizations at nominal interest.

A fixed annual nominal rate would correspond to different APR values if the number of capitalizations varies within a year or if expenses or commissions change.

However, the APR does not include expenses that the client can avoid (for example, funds transfer expenses), those that are paid to third parties or companies (brokerages, notary fees and taxes) or expenses for insurance or guarantees (except for premiums intended to guarantee the entity the reimbursement of the credit in the event of death, disability or unemployment, provided that the entity imposes its subscription for the granting of the credit). In Spain, it is mandatory that the APR appear in the documentation and advertising of both savings products and loans.

Calculation of APR

The calculation of the APR is simply the calculation of the annual interest rate according to the compound interest, where the interests obtained are remunerated at the same interest rate (they are not ignored or transferred in time). In addition, the calculation of the APR must include all payments (including commissions or other mandatory costs such as insurance). The payments to be included vary according to the banking product in question and are established, in Spain, by Circular 5/12 of the Bank of Spain.

It is calculated as the result of a standardized mathematical formula that takes into account the interest rate, bank commissions, frequency of payments (monthly, quarterly, etc.) and other expenses or income. To calculate the APR in so much per one from the TIN also expressed in so much per one, the following formula is used:

TAE=(1+rf)f− − 1{displaystyle TAE=left(1+{frac {r}{f}}}right)^{f}-1 }

Where:

• r, is the type of nominal interest (monthly, semi-annual...) expressed as much by one.
• f, rate of payments/interest charges: 1 (annual type), 2 (six-annual), 3 (quatrimonthly), 4 (trimonthly), 6 (bimonthly), 12 (monthly).

Examples of its use

• If you invest 100 € in a monthly fund, 7 % TAE, for one year, at the end of the year will have 107 €.

The Nominal Interest (TIN), according to the previous formula, is:

r=f((TAE+1)1/f− − 1)=12((0,07+1)1/12− − 1)=0,06785{displaystyle r=f(TAE+1)^{1/f}-1)=12((0,07+1)^{1/12}-1)=0,06785,!}

I mean, we have an annual Nominal Interest (TIN) of 6.785 %,

and therefore a nominal interest in each collection period (every month), of:

100(r/12)=100(0,06785/12)=0,5654% % {displaystyle 100(r/12)=100(0,06785/12)=0,5654%,!}

Split r for twelve, because you want to find out the nominal interest for a single month, and it multiplies by 100 to pass it to so much percent.

Thus, month by month, a 0.56 % would be obtained on the accumulated (if no money is removed from the deposit):

• 0,56 % on 100€ the first month: 100,56 €
• 0,56 % on 100,56€ the second month: 101,12 €, etc.

so that the last month would accumulate 107 €, thus obtaining 7 % TAE.

• The TAE is important in composite interest calculation. Example:

With a nominal interest of 6 % per year and 12 payments per year (monthly payments), it is a TAE 6.17 %:

(1+0,0612)12− − 1=0,0617{displaystyle left(1+{frac {0,06}{12}}}{12}{12}-1=0,0617,}

obtained at the end of the year, for 600 euros:

600⋅ ⋅ 1,0617=637{displaystyle 600cdot 1,0617=637,!} €