Chapter 1.1, Problem 15E

Reminder Round all answers to two decimal places unless otherwise indicated.

Note Some of the formulas below use the special number $e$, which was presented in the Prologue.

What if Interest is Compounded More Often than Monthly? Some lending institutions compound interest daily or even continuously. (The term continuous compounding is used when interest is being added as often as possible-that is, at each instant in time.) The point of this exercise is to show that, for most consumer loans, the answer you get with monthly compounding is very close to the right answer, even if the lending institution compounds more often. In part $1$ of Example $1.2$, we showed that if you borrow $\$7800$ from an institution that compounds monthly at a monthly interest rate of $0.67\%$ (for an APR of $8.04\%$ ), then in order to pay off the note in $48\text{months}$, you have to make a monthly payment of $\$190.57$.

a. Would you expect your monthly payment to be higher or lower if interest were compounded daily rather than monthly? Explain why.

b. Which would you expect to result in a larger monthly payment, daily compounding or continuous compounding? Explain your reasoning.

c. When interest is compounded continuously, you can calculate your monthly payment $M=M\left(P,r,t\right)$ in dollars, for a loan of $P\text{dollars}$ to be paid off over $t$ months using

$M=\frac{P\left(e^r-1\right)}{1-e^{-rt}}$,

where $r=\text{APR}/12$ if the APR is written in the decimal form. Use this formula to calculate the monthly payment on a loan of $\$7800$ to be paid off over $48\text{months}$ with an APR of $8.04\%$. How does this answer compare the result in Example $1.2$?