Present Value of an Annuity

[cruz52]

Some friends tell you that they paid $25,000 down on a new house and are to pay $525 per month for 30 yrs. If interest is 7.8% compounded monthly, what was the selling price of the house? How much interest will they pay in 30 years?

I began by adding 25,000 + 189,000 (I multipled 525 by 30 years) ... The answer I received is 214,000

i = .0065
m = 12
t = 30
n = 360
214,000(1 - (1.0065)^-360/.0065 = 29,727,569.02 ... Except the back of the book that has the answer says it comes out to $97,929.78 .... SO what am I doing wrong??

 

[Denis]

.... SO what am I doing wrong??

Sorry Cruz...but I don't think you're ready for this; you're doing everything wrong :wink:

Anyway, what you need to do is find the Present Value of a monthly annuity of $525,
where n=360 and i=.0065; this will give you the amount that was borrowed.

 

[soroban]

Hello, cruz52!

This happens about once a week.
Someone is assigned an Amorization problem and does not know the Amortization Formula.
How come?

Some friends tell you that they paid $25,000 down on a new house and are to pay $525 per month for 30 yrs.
If interest is 7.8% compounded monthly, what was the selling price of the house?
How much interest will they pay in 30 years?

\(\displaystyle \text{Amortization Formula: }\;A \;=\;P\,\frac{i(1+i)^n}{(1+i)^n-1}\)

. . . \(\displaystyle \text{where: }\;\begin{Bmatrix}A &=& \text{periodic payment} \\ P &=& \text{pricipal borrowed} \\ i &=& \text{periodic interest rate} \\ n &=& \text{number of periods} \end{Bmatrix}\)


\(\displaystyle \text{Solving for }P\!:\;\;P \;=\;A\,\frac{(1+i)^n-1}{i(1+i)^n}\)

\(\displaystyle \text{We are given: }\:\begin{Bmatrix}A &=& 525 \\ i &=& 0.0065 \\ n &=& 360 \end{Bmatrix}\)


\(\displaystyle \text{Hence: }\;P \;=\;525\,\frac{(1.0065)^{360}-1}{0.0065(1.0065)^{360}} \;=\;72,\!929.78381 \;\approx\;\$72,\!930\)

\(\displaystyle \text{The selling price of the house is: }\:72,\!930 + 25,\!000 \:=\:\boxed{\$97,930}\)



\(\displaystyle \text{They will pay \$525 per month for 360 months: }\;525 \times 360 \;=\;\$189,\!000\)

\(\displaystyle \text{The total interest is: }\;\$189,000 - 97,\!930 \;=\;\boxed{\$91,\!070}\)