CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 13. Finance

The present value of a series of equal paymentsRwith interest rateiper period fornperiods is:
PV=R·^1 −(^1 +ii)−n
Example A
What is the monthly payment of a $1,000,000 house loan over 30 years with a nominal interest rate of 6% convertible
monthly?
Solution:PV=$1, 000 , 000 ,R=?,i= 0. 005 ,n= 360

PV=R·^1 −(^1 +i)

−n

i

1 , 000 , 000 =R·^1 −(^1 +^0.^005 )

− 360

0. 005

R= 11 −,^000 ( 1 +,^0000. 005 ·^0 .)^005 − 360 ≈ 5995. 51

It is remarkable that in order to pay off a $1,000,000 loan you will have to pay $5,995.51 a month, every month, for
thirty years. After 30 years, you will have made 360 payments of $5995.51, and therefore will have paid the bank
more than $2.1 million, more than twice the original loan amount. It is no wonder that people can get into trouble
taking on more debt than they can afford.
Example B
How long will it take to pay off a $20,000 car loan with a 6% annual interest rate convertible monthly if you pay it
off in monthly installments of $500? What about if you tried to pay it off in monthly installments of $100?
Solution:PV=$20, 000 ,R=$500,i=^012.^06 = 0. 005 ,n=?

PV=R·^1 −(^1 +i)

−n

i

20 , 000 = 500 ·^1 −(^1 +^0.^005 )

−n

0. 005

0. 2 = 1 −( 1 + 0. 005 )−n

( 1 + 0. 005 )−n= 0. 8

n=−ln 1ln 0. 005.^8 ≈ 44. 74 months

For the $100 case, if you try to set up an equation and solve, there will be an error. This is because the interest on
$20,000 is exactly $100 and so every month the payment will go to only paying off the interest. If someone tries to
pay off less than $100, then the debt will grow.
Example C
It saves money to pay off debt faster in order to save money on interest. As shown in Example A, interest can more
than double the cost of a 30 year mortgage. This example shows how much money can be saved by paying off more
than the minimum.
Suppose a $300,000 loan has 6% interest convertible monthly with monthly payments over 30 years. What are the
monthly payments? How much time and money would be saved if the monthly payments were larger by a fraction
of^1312? This is like making 13 payments a year instead of just 12. First you will calculate the monthly payments if
12 payments a year are made.