# APR – APY conversion

## Set paramaters

% % < %
% % < %
year
% % < %
1 day ≃ 1 / 365 year
% % < %
1 week ≃ 1 / 52 year
% % < %
≃ 1 month ≃ 1 / 12 year
% % < %
≃ 2 months ≃ 1 / 6 year
% % < %
≃ 3 months ≃ 1 / 4 year
% % < %
≃ 4 months ≃ 1 / 3 year
% % < %
≃ 6 months ≃ 1 / 2 year
% % < %
≃ 9 months ≃ 3 / 4 year
% % < %
≃ 2 years
% % < %
≃ 5 years
% % < %
≃ 10 years

## Explanations

@@\mathrm{APR}@@ (Annual Percentage Rate) is the rate of simple interest for one year. If interests is given in each period, with $N$ periods for one year, then each period gives @@\frac{\mathrm{APR}}{N}@@ interest, always computed on the initial amount.

@@\mathrm{APY}@@ (Annual Percentage Yield) is the rate of compound interest for one year. If interests is given in each period, with $N$ periods for one year, then for each period the @@\frac{\mathrm{APR}}{N}@@ interest is added to the total amount, and so interests also generate interest in next periods.

By example, for a number of periods @@N = 5@@, a simple interest for the year @@\mathrm{APR} = 10\%@@ and an initial amount @@A_0 = 1000€@@.

The rate for each period is @@\frac{\mathrm{APR}}{N} = \frac{10}{5}\% = 2\% = 0.02@@.

With

### simple interest

the @@5@@ periods during the year give successively:
@@\begin{array}{@{}l|l|l|l@{}} \text{Interest} & \text{Total profit} & \text{Total amount}\\ \hline & & A_0 = 1000€\\ \hline I_1 = 1000€ \times\ 0.02 = 20€ & P_1 = 0€ +\ 20€ = 20€ & A_1 = 1000€\\ I_2 = 1000€ \times\ 0.02 = 20€ & P_2 = 20€ +\ 20€ = 40€ & A_2 = 1000€\\ I_3 = 1000€ \times\ 0.02 = 20€ & P_3 = 40€ +\ 20€ = 60€ & A_3 = 1000€\\ I_4 = 1000€ \times\ 0.02 = 20€ & P_4 = 60€ +\ 20€ = 80€ & A_4 = 1000€\\ I_5 = 1000€ \times\ 0.02 = 20€ & P_5 = 80€ +\ 20€ = 100€ & A_5 = 1000€ & 1\text{ year}\\ \hline & P_{10} = 200€ & & 2\text{ years}\\ & P_{15} = 300€ & & 3\text{ years}\\ & P_{20} = 400€ & & 4\text{ years}\\ \end{array}@@

In general,
total profit after @@k@@ periods @@P_k = A_0 + \underbrace{\frac{\mathrm{APR}}{N} + \dots + \frac{\mathrm{APR}}{N}}_{k\text{ times}} = A_0 \times \frac{\mathrm{APR}}{N} \times k@@.

Total profit after one year (i.e. @@N@@ periods) @@P_N = A_0 \times \mathrm{APR}@@.

Total profit after @@i@@ years @@P_{i N} = A_0 \times \mathrm{APR} \times i@@.

With

### compound interest

the @@5@@ periods during the year give successively:
@@\begin{array}{@{}l|l|l@{}} \text{Interest} & \text{Total amount}\\ \hline & A_0 = 1000€\\ \hline I_1 = 1000€ \times\ 0.02 = 20€ & A_1 = 1000€ \times\ 1.02 = 1000€ +\ 20€ = 1020€\\ I_2 = 1020€ \times\ 0.02 = 20.4€ & A_2 = 1020€ \times\ 1.02 = 1020€ +\ 20.4€ = 1040.4€\\ I_3 = 1040.4€ \times\ 0.02 = 20.808€ & A_3 = 1040.4€ \times\ 1.02 = 1040.4€ +\ 20.808€ = 1061.208€\\ I_4 = 1061.208€ \times\ 0.02 = 21.22416€ & A_4 = 1061.208€ \times\ 1.02 = 1061.208€ +\ 21.22416€ = 1082.43216€\\ I_5 = 1082.43216€ \times\ 0.02 = 21.6486432€ & A_5 = 1082.43216€ \times\ 1.02 = 1082.43216€ +\ 21.6486432€ = 1104.0808032€ & 1\text{ year}\\ \hline & A_{10} \simeq 1218.99442 & 2\text{ years}\\ & A_{15} \simeq 1345.868338 & 3\text{ years}\\ & A_{20} \simeq 1485.947396 & 4\text{ years}\\ \end{array}@@

That gives the corresponding @@\mathrm{APY} = 0.1040808032 = 10.40808032\%@@.

In general,
total amount after @@k@@ periods @@A_k = A_0 \times \underbrace{\left(1 + \frac{\mathrm{APR}}{N}\right) \times \dots \times \left(1 + \frac{\mathrm{APR}}{N}\right)}_{k\text{ times}} = A_0 \times \left(1 + \frac{\mathrm{APR}}{N}\right)^k@@.

Total amount after one year (i.e. @@N@@ periods) @@A_N = A_0 \times (1 + \mathrm{APY}) = A_0 \times \left(1 + \frac{\mathrm{APR}}{N}\right)^N@@.

Total amount after @@i@@ years @@A_{i N} = A_0 \times \left(1 + \frac{\mathrm{APR}}{N}\right)^{i N}@@.

Note that for any @@N: \mathrm{APY} \lt \lim\limits_{k\rightarrow\infty} \left(1 + \frac{\mathrm{APR}}{k}\right)^k - 1 = e^{\mathrm{APR}} - 1@@

### Conversion between APR and APY

For annual percentage rate @@\mathrm{APR} \in \mathbb{R}@@, annual percentage rate @@\mathrm{APY} \in \mathbb{R}@@ and the number of periods @@N \in \mathbb{N}@@.
• @@\mathrm{APY} = \left(1 + \frac{\mathrm{APR}}{N}\right)^N - 1@@
• Reciprocally @@\mathrm{APR} =\left(\sqrt[N]{1 + \mathrm{APY}} - 1\right) N@@