====== Constant Growth / Inflation / Compound Interest ======
//"The greatest shortcoming of the human race is our inability to understand the exponential function."// ((Albert Bartlett - On population growth and sustainability)) \\
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%):(([[http://en.wikipedia.org/wiki/Compound_interest|Compound Interest - Wikipedia 8.8.2012]]))
$$ A = P \left( 1+r \right)^{t} \label{compound_annual}\tag{1} $$
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 (([[https://betterexplained.com/articles/a-visual-guide-to-simple-compound-and-continuous-interest-rates/|A Visual Guide to Simple, Compound and Continuous Interest Rates]] //24 Dec 2016//)). 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.\\
{{:calculate:pasted:20161226-133517.png?400}}
Other math based [[http://www.math.hawaii.edu/~ramsey/CompoundInterest.html |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.