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.

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• if , we can also write, , for example, , and
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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.

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

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.

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)

• ?

We can replace N by

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;