$$B(t) \quad \text{or} \quad B(\tau)$$ Loan balance vs time

# APY is a dishonest representation of APR

Photo credit to Marek Piwnicki

Here we will define APR and APY to explain why APY is a fundamentally dishonest marketing term.

## Defining APR on an investment

We have repeatedly seen the following equation for the growth of an investment $$V$$ growing with rate of return $$r$$.

\begin{align} \frac{\partial V}{\partial t} &= rV\\ \int_{V_0}^{V(t)} \frac{\partial V}{V} &= \int_{0}^{t} r\partial t\\ V(t) &= V_0 e^{rt}\\ \end{align}

This $$r$$ is the APR. Through all of this blog unless otherwise stated, use these equations with $$r$$ = APR.

## Defining APY on an investment

APY is a marketing term designed to inflate numbers on investment products. Its usage is sometimes justified by the general public's ignorance of calculus. To that end, it is defined as "The total return on an investment in 1 year."

\begin{align} \frac{V(t=1)}{V_0} - 1 = \text{APY} &= \left(1+r\Delta t \right)^\frac{t =1}{\Delta t} - 1\\ \text{APY} &= \left( 1 + \frac{r}{n} \right)^n - 1\\ \end{align}

Here $$n$$ is the number of maturation periods in one year. In the limit as $$n \rightarrow \infty$$, this reduces to an exponential:

\begin{align} \lim_{n \rightarrow \infty} \text{APY} &= \lim_{n \rightarrow \infty} \left(1+\frac{r}{n} \right)^n - 1 \\ \lim_{n \rightarrow \infty} \text{APY} &= \lim_{n \rightarrow \infty} e^ {n\log \left(1+\frac{r}{n} \right)} - 1 \\ \lim_{n \rightarrow \infty} \text{APY} &= \lim_{n \rightarrow \infty} e^ {\frac{\log \left(1+\frac{r}{n}\right)}{\frac{1}{n} }} - 1 \\ \lim_{n \rightarrow \infty} \text{APY} &= \lim_{n \rightarrow \infty} e^ {\frac{\frac{d}{dn} \log \left(1+\frac{r}{n}\right)} {\frac{d}{dn}\frac{1}{n} }} - 1 \\ \lim_{n \rightarrow \infty} \text{APY} &= \lim_{n \rightarrow \infty} e^ {\frac{-\frac{r}{n^2} \left(1+\frac{r}{n}\right)} {-\frac{1}{n^2} }} - 1 \\ \lim_{n \rightarrow \infty} \text{APY} &= \lim_{n \rightarrow \infty} e^ {r\left(1+\frac{r}{n}\right)} - 1 \\ \lim_{n \rightarrow \infty} \text{APY} &= e^r - 1\\ \end{align}

APY is therefore bounded between two values depending on the value of $$n$$. At $$n$$ = 1, APY = $$r$$ but as $$n \rightarrow \infty$$, APY $$\rightarrow e^r-1$$. It is in the interest of investment managers therefore to report of APY rather than APR, to brag about large APY numbers without affecting investment performance.

$$r \leq \text{APY} \leq e^r - 1$$

In practice, the differences are quite small so they only become important when deciding between otherwise comparable investment products. For example, 12% APR becomes 12.75% APY with $$n \rightarrow \infty$$. With $$n$$ = 12, the APY becomes 12.68%.

[Caption] APY vs APR by number of maturation cycles per year ($$n$$). Dashed lines represent APY vs APR as $$n \rightarrow \infty$$ (top) and at $$n$$ = 0 (bottom). Drag the purple dot to change the number of maturation cycles. Double click the purple dot to set $$n$$ = 12 (the most common value since most interest calculations involve monthly maturation giving 12 cycles per year). The difference between $$n$$ = 100 and $$n \rightarrow \infty$$ is negligible (the blue line completely obscures the dashed line).