### Table of Contents

# Constant Growth / Inflation / Compound Interest

*“The greatest shortcoming of the human race is our inability to understand the exponential function.”* ^{1)}

Compound interest and Inflation being two notable financial examples where consequences of a constant growth rate can be observed:

- Good investment: 15% growth over 5 years will double iniitial value
- Inflation: -4% over 50 years will eliminate 87% of value

### Calculator

Set negative number for “Interest” to look at effect of inflation

### Math

Calculator uses a formula assuming the interest is Compound annually (APR%):^{2)}

$$ A = P \left( 1+r \right)^{t} \label{compound_annual}\tag{1} $$
where:

$A$ = amount of money accumulated after t years, including interest

$P$ = principal amount

$r$ = annual rate of interest (as a decimal)

$t$ = number of years the amount is deposited or borrowed for

A broader overview of how you can describe constant growth can be found in the table below ^{3)}. The difference is essentially the interval used for compounding, from never (simple) to every instant (continuous growth $e^{r t}$).

Term | Formula | Description & Usage |
---|---|---|

Simple | $$P \cdot (1 + r \cdot t)$$ | Fixed, non-growing return (bond coupons) |

Compound (Annual) | $$P \cdot (1 + r)^t$$ | Changes each year (stock market, inflation) |

Compound(n times per year) | $$P \cdot \left(1 + \frac{r}{n}\right)^{nt}$$ | Changes each month/week/day (savings account) |

Continuous Growth | $$P \cdot e^{rt}$$ | Changes each instant (radioactive decay, temperature) |

where $n$ is the number of times the interest is compounded per year

Assuming a $15\%$ growth rate the below graphic demonstrates what the difference the different compounding intervals make over a 5 year period.

Other math based article on the topic.

## Compounded Annual Growth Rate - CAGR

How to judge the performance of an investment that runs over a number of years with a different annualized rate of return each year? What is the Compounded Annual Growth Rate for a given starting & final value over a fixed time period? Rearranging equation $\ref{compound_annual}$ above:

$$r = \left( \frac{A}{P} \right)^\frac{1}{t}-1 \label{CAGR}\tag{2} $$

Given a number of years $t$ in which the starting principle $P$ has grown to the final accumulated value $A$, then equation $\ref{CAGR}$ provides you with the Compounded Annual Growth Rate (CAGR) for this period.

^{1)}

^{3)}