APR – APY conversion (DragonSoft DS / online tools)

Set parameters

APR	Number of payment periods	APY	APY limit
%	← →	%	< %

Interest	Starting amount	Compound interest	Compound interest
	After m periods
%	← →	%	< %
	= year
Until this %
%	← →	%	< %
	= year
		Until this %
%	← →	%	< %
	= 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.

For 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}{llll} Interest & Total profit & Total amount \\ A_{0} = 1000 € \\ 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 year \\ P_{10} = 200 € & 2 years \\ P_{15} = 300 € & 3 years \\ P_{20} = 400 € & 4 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}{lll} Interest & Total amount \\ A_{0} = 1000 € \\ 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 year \\ A_{10} ≃ 1218.99442 & 2 years \\ A_{15} ≃ 1345.868338 & 3 years \\ A_{20} ≃ 1485.947396 & 4 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@@

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