Chapter 5.7, Problem 17AYU

In Problems $15-22$, find the principal needed now to get each amount; that is, find the present value.

To get $\$1500$ after $2\frac{1}{2}$ years at $1.5\%$ compounded daily.

To determine

To calculate: The principal amount, if $\$1500$ is received after $2\frac{1}{2}$ years at $1.5\%$ compounded daily.

Answer to Problem 17AYU

Solution:

The principal amount needed, if interest compounded daily is $1444.79$

Explanation of Solution

Given information:

Future value $A=1500$

Compounded interest rate per annum $r=0.015$

Number of times the interest compounded per annum $n=365$

Amount compounded for number of years $t=2\frac{1}{2}=\frac{5}{2}$

Formula used:

The principal amount $P$ of $A$ dollars to be received after $t$ years, assuming a per annum interest rate $r$ compounded $n$ times per year is $P=A{\left(1+\frac{r}{n}\right)}^{-nt}$

Calculation:

Use the present value formula with $A=1500$, $r=0.015$, $n=365$ and $t=2\frac{1}{2}=\frac{5}{2}$ to find $P$.

$P=A{\left(1+\frac{r}{n}\right)}^{-nt}$

$P=1500{\left(1+\frac{0.015}{365}\right)}^{\left(-365\cdot \frac{5}{2}\right)}$

$P=1500\cdot {(1+0.0000410958904)}^{-912.5}$

$P=1444.79$

Hence, the principal amount needed to get $\$1500$ after $2\frac{1}{2}$ years at $1.5\%$ interest rate, if compounded daily is $P=1444.79$.