Exponentiation and Compounding Interest Rate
Exponential Function
- Exopential Function: Functions where the variable x appears as an exponent:
- a is the base of Exponential function.
Remember that
- if , we can also write, , for example, , and
Exponential Function Graphs?
- Note that the domain of exponential function includes positive and negative x, and the exponential function will always be positive.
- If base is below 1, then the curve is monotonically downward sloping
- If base is above 1, then the curve is monotonically upwards sloping
- If base is above 1, higher base leads to steeper curvature.
title('Exponential Function Graph with different bases')
legend(['base=',num2str(a1)], ['base=',num2str(a2)],['base=',num2str(a3)]);
Infinitely Compounding Interest rate
with 100 percent interest rate (APR), if we compound N times within a year, interest we pay at the end of the year is
Suppose (You can also think of this as a loan with interest rate of % for every days), then we pay % interest rate by the end of the year. What if we do more and more compounding, if we say interest rate compounds , , times over the year, what happens? With APR at 100%, the total interest rate you pay at the end of the year does not go to infinity, rather, it converges to this special number e, Euler's number, . This means if every second the interest rate is compounding, with an APR of 100%, you end up paying 272% of what you borrowed by the end of the year, which is 172% interest rate. We can visualize this limit below
f_compoundR = (1 + r/N)^N;
fplot(f_compoundR, [1,100])
ylabel({'Principle and Interests at End of Year Given 100% APR' 'for 1 dollar Borrowed, given infinite compounding'})
xlabel('Number of Evenly-divided Times to Compound Interest Rate in a Year')
double(subs(f_compoundR,[1,2,3,4,5,6,7,8,9,10]))
2.0000 2.2500 2.3704 2.4414 2.4883 2.5216 2.5465 2.5658 2.5812 2.5937
Infinitely compounding Interest rate, different r (APR r)
Given:
What is
- ?
We can replace N by This gives the base e exponential function a financial interpretation.
title('Exponential Function Graph with base e')
xlabel('r = interest rate');
ylabel({'Principle and Interests at End of Year' 'for 1 dollar Borrowed, given infinite compounding'})