3.5 Applications of Exponential Functions

There are many applications of exponential functions in daily life. One of the biggest applications is in finance, specifically with regard to interest. When you invest money into a certificate of deposit (CD), for example, you earn interest on your savings. If you take out a loan or mortgage, you have to pay interest. The most basic type of interest is called simple interest. Simple interest is interest paid only on the principal amount. Suppose you deposit the principal, $P$ dollars, into a bank account with an annual interest rate of $r$ (expressed as a decimal). After $t$ years, you withdraw $A$ dollars from the account. The formula to calculate the total amount $A$ after $t$ years is

$A=P+Prt=P(1+rt)$

For example, if you save $100 today at the annual interest rate of 0.12, in five years you will have

$A=100(1+0.12\times 5)=\mathrm{\$}160$

While this is an easy calculation, simple interest is not often used in the finance world. More common is compound interest, which is the interest that is incurred not only on the principal but also on the interest. Suppose that you deposit $P$ dollars today into an account at an annual interest rate of $r$ (expressed as a decimal). You save the money for $t$ years without making any other withdrawals or deposits. How much do you have in the account in $t$ years?

Let’s start with one year. In one year, you will have

$A_1=P(1+r)$

How about in two years? Since you will earn interest on the amount $A_1$, the amount you will have in two years ($A_2$) is

$A_2=A_1(1+r)=P(1+r)(1+r)=P(1+r{)}^2$

Do you see the pattern? In three years, you will have

$A_3=A_2(1+r)=P(1+r{)}^2(1+r)=P(1+r{)}^3$

Following this pattern, we can calculate the amount in the savings account in $t$ years:

$A_t=P(1+r{)}^t$

If you save $100 today in an account that gives an annual interest rate of 12%, in five years you will have

$A_5=100(1+0.12{)}^5=\mathrm{\$}176.23$

This example has shown the case when the interest rate $r$ is compounded annually. However, different compounding frequencies may be used. The interest may be compounded semiannually, quarterly, monthly, or daily. There is also something called continuous compounding.

Suppose that you again deposit $P$ dollars into a bank account with an annual interest rate $r$ for $t$ years. Further, suppose that the interest is compounded $n$ times per year ($n$ = 2 if semiannually, 4 if quarterly, 12 if monthly, and 365 if daily). How much ($A_t$ dollars) will you have in the account in $t$ years? The formula to find this value is

$A_t=P(1+\frac{r}{n}{)}^{nt}$

As you can see, we divide the annual interest rate by the number of times interest is compounded per year to find the period interest rate. The number of times interest is compounded per year times the years invested gives us the total interest compounding time. For example, if you save $100 today at the rate of 0.12 compounded monthly for five years, you will have

$A_5=100{(1+\frac{0.12}{12})}^{5\cdot 12}=\mathrm{\$}181.67$

If the annual interest is compounded continuously, then after $t$ years, you will have

$A_t=P{e}^{rt}$

For example, if you saved $100 today at the rate of 0.12 compounded continuously for five years,

$A_5=100e^{0.12\cdot 5}=\mathrm{\$}182.21$

Notice that the more frequently interest is compounded, the faster the principal grows. In other words, if you save the same amount of money today in two accounts with the same annual interest rate for the same period of time, you will have a greater dollar amount in the account with more frequent compounding.