# APR – APY conversion

% % < %
% % < %
= year
% % < %
= year
% % < %
= year

## 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\unicode{8364}@@.

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\unicode{8364}\\ \hline I_1 = 1000\unicode{8364} \times\ 0.02 = 20\unicode{8364} & P_1 = 0\unicode{8364} +\ 20\unicode{8364} = 20\unicode{8364} & A_1 = 1000\unicode{8364}\\ I_2 = 1000\unicode{8364} \times\ 0.02 = 20\unicode{8364} & P_2 = 20\unicode{8364} +\ 20\unicode{8364} = 40\unicode{8364} & A_2 = 1000\unicode{8364}\\ I_3 = 1000\unicode{8364} \times\ 0.02 = 20\unicode{8364} & P_3 = 40\unicode{8364} +\ 20\unicode{8364} = 60\unicode{8364} & A_3 = 1000\unicode{8364}\\ I_4 = 1000\unicode{8364} \times\ 0.02 = 20\unicode{8364} & P_4 = 60\unicode{8364} +\ 20\unicode{8364} = 80\unicode{8364} & A_4 = 1000\unicode{8364}\\ I_5 = 1000\unicode{8364} \times\ 0.02 = 20\unicode{8364} & P_5 = 80\unicode{8364} +\ 20\unicode{8364} = 100\unicode{8364} & A_5 = 1000\unicode{8364} & 1\text{ year}\\ \hline & P_{10} = 200\unicode{8364} & & 2\text{ years}\\ & P_{15} = 300\unicode{8364} & & 3\text{ years}\\ & P_{20} = 400\unicode{8364} & & 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\unicode{8364}\\ \hline I_1 = 1000\unicode{8364} \times\ 0.02 = 20\unicode{8364} & A_1 = 1000\unicode{8364} \times\ 1.02 = 1000\unicode{8364} +\ 20\unicode{8364} = 1020\unicode{8364}\\ I_2 = 1020\unicode{8364} \times\ 0.02 = 20.4\unicode{8364} & A_2 = 1020\unicode{8364} \times\ 1.02 = 1020\unicode{8364} +\ 20.4\unicode{8364} = 1040.4\unicode{8364}\\ I_3 = 1040.4\unicode{8364} \times\ 0.02 = 20.808\unicode{8364} & A_3 = 1040.4\unicode{8364} \times\ 1.02 = 1040.4\unicode{8364} +\ 20.808\unicode{8364} = 1061.208\unicode{8364}\\ I_4 = 1061.208\unicode{8364} \times\ 0.02 = 21.22416\unicode{8364} & A_4 = 1061.208\unicode{8364} \times\ 1.02 = 1061.208\unicode{8364} +\ 21.22416\unicode{8364} = 1082.43216\unicode{8364}\\ I_5 = 1082.43216\unicode{8364} \times\ 0.02 = 21.6486432\unicode{8364} & A_5 = 1082.43216\unicode{8364} \times\ 1.02 = 1082.43216\unicode{8364} +\ 21.6486432\unicode{8364} = 1104.0808032\unicode{8364} & 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 \left(1 + \frac{\mathrm{APR}}{N}\right)^N = A_0 \times (1 + \mathrm{APY})@@.

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@@ (this is the value in the APY limit row).

### 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@@