MATH SOLVE

5 months ago

Q:
# 4.Find the present value of the annuity.Amount Per Payment: $4,725Payment at End of Each: 6 monthsNumber of Years: 15Interest Rate: 10%Compounded: Semiannually$72,634.83$35,938.73$32,242.03$68,951.03

Accepted Solution

A:

To solve this we are going to use the present value of annuity formula: [tex]PV=P[ \frac{1-(1+ \frac{r}{n})^{-kt} }{ \frac{r}{n} } ][/tex]

where

[tex]PV[/tex] is the present value

[tex]P[/tex] is the periodic payment

[tex]r[/tex] is the interest rate in decimal form

[tex]n[/tex] is the number of times the interest is compounded per year

[tex]k[/tex] is the number of payments per year

[tex]t[/tex] is the number of years

We know from our problem that [tex]P=4725[/tex] and [tex]t=15[/tex]. To convert the interest rate to decimal form, we are going to divide it by 100%:

[tex]r= \frac{10}{100} [/tex]

[tex]r=0.1[/tex]

Since the interest is compounded semiannually, it is compounded 2 times per year; therefore, [tex]n=2[/tex]. Similarly, since the payment is made at the end of each 6 months, it is made 2 times per year; therefore, [tex]k=2[/tex].

Lest replace the values in our formula:

[tex]PV=P[ \frac{1-(1+ \frac{r}{n})^{-kt} }{ \frac{r}{n} } ][/tex]

[tex]PV=4725[ \frac{1-(1+ \frac{0.1}{2})^{-(2)(15)} }{ \frac{0.1}{2} } ][/tex]

[tex]PV=72634.83[/tex]

We can conclude that the correct answer is $72,634.83

where

[tex]PV[/tex] is the present value

[tex]P[/tex] is the periodic payment

[tex]r[/tex] is the interest rate in decimal form

[tex]n[/tex] is the number of times the interest is compounded per year

[tex]k[/tex] is the number of payments per year

[tex]t[/tex] is the number of years

We know from our problem that [tex]P=4725[/tex] and [tex]t=15[/tex]. To convert the interest rate to decimal form, we are going to divide it by 100%:

[tex]r= \frac{10}{100} [/tex]

[tex]r=0.1[/tex]

Since the interest is compounded semiannually, it is compounded 2 times per year; therefore, [tex]n=2[/tex]. Similarly, since the payment is made at the end of each 6 months, it is made 2 times per year; therefore, [tex]k=2[/tex].

Lest replace the values in our formula:

[tex]PV=P[ \frac{1-(1+ \frac{r}{n})^{-kt} }{ \frac{r}{n} } ][/tex]

[tex]PV=4725[ \frac{1-(1+ \frac{0.1}{2})^{-(2)(15)} }{ \frac{0.1}{2} } ][/tex]

[tex]PV=72634.83[/tex]

We can conclude that the correct answer is $72,634.83