Question

Vanessa can afford a $1405-per-month house loan payment. If she is being offered a 20-year house loan with an APR of 4.8%, compounded monthly, which of these expressions represents the value of the most money she can borrow?

Answer 1

To solve this we are going to use the present value formula: $PV=P[ \frac{1-(1+ \frac{r}{n})^{-kt} }{ \frac{r}{n} } ]$
where 
$PV$ is the present value
$P$ is the periodic payment
$r$ is the interest rate in decimal form
$n$ is the number of times the interest is compounded per year 
$k$ is the number of payments per year
$t$ is the number of years

We know for our problem that she can afford a $1405-per-month house loan payment, so $P=1405$ and $k=12$. We also know that the loan is a 20-year house loan, so $t=20$. To convert the interest rate to decimal form, we are going to divide the rate by 100% $r= \frac{4.8}{100}=0.048$. Since the interest rate is compounded monthly, it is compounded 12 times per year; therefore, $n=12$. Lets replace those values in our formula:

$PV=P[ \frac{1-(1+ \frac{r}{n})^{-kt} }{ \frac{r}{n} } ]$
$PV=1405[ \frac{1-(1+ \frac{0.048}{12})^{-(12)(20)} }{ \frac{0.048}{12} } ]$
$PV=216501.09$

We can conclude that the expression that represents the value of the most money she can borrow is: $PV=1405[ \frac{1-(1+ \frac{0.048}{12})^{-(12)(20)} }{ \frac{0.048}{12} } ]$. Evaluating the expression we can prove that she can borrow $216,501.09

Answer 2

$\frac{(1405)((1+0.004)^2^4^0-1)}{(0.004)(1+0.004)^2^4^0}$ Apex