Simple interest
The
simple interest on an investment (or loan) of $PV$ at an annual interest rate of $r$ for a period of $t$ years is
The
future value $FV$ of such an investment is obtained from the present value by adding the interest:
$FV = PV + INT = PV(1+rt)$
We can also solve for the present value to get
Examples: Simple interest
For an 8.5% simple interest 4-year \$20,000 loan, the total interest is
$INT = PVrt = (20\,000)(0.085)(4) = \$6\,800,$
The future value is
$FV = PV + INT = 20\,000 + 6\,800 = \$26\,800.$
Practice:
Simple interest: Alternative formula
It is sometimes more convenient to use a formula based on periods of time other than years:
The simple interest on an investment (or loan) of $PV$ at an interest rate of $i$ per period for $n$ periods is
The future value $FV$ of such an investment is obtained from the present value by adding the interest:
$FV = PV + INT = PV(1+in)$
As above, we can also solve for the present value to get
Examples: Simple interest: Alternative formula
A city issues 10-year bonds that pay 2.4% every six months. Upon maturity, \$20,000 worth of bonds yields an interest of
$INT = PVin = (20\,000)(0.024)(20) = \$9\,600,$
The future value is
$FV = PV + INT = 20\,000 + 9\,600 = \$29\,600.$
Practice:
Compound interest
The future value $FV$ of an investment of $PV$ at an interest rate of $i$ per period for $n$ periods is
We can also solve for the present value to get
$PV = \frac{FV}{(1+i)^n}$.
Alternative formula based on years The future value $FV$ of an investment of $PV$ at an annual interest rate of $r$ for a period of $t$ years compounded $m$ times per year is
$FV = PV\left(1+\frac{r}{m}\right)^{mt}$.
Examples: Compound interest
For 4-year investment of \$20,000 earning 8.5% per year, with interest reinvested each month, the future value is
| $FV$ | $= (1+i)^n$ |
| $= 20\,000\left(1 + \frac{0.085}{12}\right)^{48}$ | $i = 0.085/12, \ n = 4 \times 12 = 48$ |
| $= \$28,065.30$ |
Practice:
Compound interest calculator
Effective interest rate
The
effective interest rate, $r_{\text{eff}}$, of an investment is the percentage return it gives at the end of a year. The
nominal interest rate, $r_{\text{nom}}$, is the stated interest rate of the investment regarless of the compounding period. To calculate the effective rate, we use the formula
$r_{\text{eff}} = \left(1 + \frac{r_{\text{nom}}}{m}\right)^m - 1$
Examples: Effective interest rate
For an investment earning 5% per year compounded quarterly, the nominal interest rate is
and the effective interest rate is
| $r_{\text{eff}} $ | $= \left(1 + \frac{r_{\text{nom}}}{m}\right)^m - 1$ |
| $= \left(1 + \frac{0.05}{4}\right)^4 - 1 \approx 0.05095$
|
Practice: