Exponentiation and Compounding Interest Rate

back to Fan's Intro Math for Econ, Matlab Examples, or MEconTools Repositories
See also: Exponential Function and Log Function.

Exponential Function

Remember that

Exponential Function Graphs?

syms x
a1 = 0.5;
f_a1 = a1^(x);
a2 = 1.5;
f_a2 = a2^(x);
a3 = 2.5;
f_a3 = a3^(x);
figure();
hold on;
fplot(f_a1, [-2, 2]);
fplot(f_a2, [-2, 2]);
fplot(f_a3, [-2, 2]);
line([0,0],ylim);
line(xlim, [0,0]);
title('Exponential Function Graph with different bases')
legend(['base=',num2str(a1)], ['base=',num2str(a2)],['base=',num2str(a3)]);
grid on;

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.
r = 1.05;
N = 5;
(1 + r/N)^N - 1
ans = 1.5937
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
r = 1;
syms N
f_compoundR = (1 + r/N)^N;
figure();
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')
grid on;
grid minor
double(subs(f_compoundR,[1,2,3,4,5,6,7,8,9,10]))
ans = 1×10
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.
syms x
f_e = exp(x);
figure();
hold on;
fplot(f_e, [-3, 3]);
line([0,0],ylim);
line(xlim, [0,0]);
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'})
grid on;