Chapter 11.3, Problem 5CFU

To determine

To find: the difference between interest compound continuously and interest compounded monthly.

Answer to Problem 5CFU

The interest compounded monthly is a discrete function and continuously compounded interest is a continuous function.

Explanation of Solution

Given:

Continuous interest compound and monthly interest compound.

Concept used:

Formula for discrete compounding:

The discrete compounding if interest rate is simple:

  $FV=P\left(1+\frac{r}{m}\right)mt$ .

The discrete compounding if interest rate is compound:

  $FV=P\left(1+\frac{r}{m}\right)mt$ .

  $FV=P\times e^{rt}$ .

Formula for continuous compounding:

  $FV=P\times e^{rt}$.

Calculation:

Continuous compounding and discrete compounding:

In discrete compounded interest id calculates and added to the principal at specific intervals.

Example: annually, monthly or weekly.

Continuous compounding used a natural log-based formula to calculate and add back accrued interest at the smallest possible intervals.

Interest can be compound discretely at many different times intervals discrete compounding explicitly defines the number of and the distance between compounding periods.

Example, an interest that compounds on the first day of every month is discrete.

Calculating discrete compounding if interest rate is simple:

  $FV=P\left(1+\frac{r}{m}\right)mt$ .

Where:

  $FV$ = future value.

  $\left(\frac{r}{m}\right)$ =interest rate.

  $mt$ =time period.

Calculating discrete compounding if interest rate is compound:

  $FV=P\left(1+\frac{r}{m}\right)mt$ .

Where:

  $t=$ the term of the contract in years

  $m$ = the number of compounding periods per year.

Continuous compounding:

Calculating compounding introduces the concept of the natural logarithm. This is the constant rate of growth for all naturally growing processes. It’s a figure that developed out of physics.

The natural log is typically represented by the letter $e$ .

  $FV=P\times e^{rt}$ .

Hence, interest compounded monthly is a discrete function and continuously compounded interest is a continuous function.