Question

Dusty has the choice of taking out a 25-year loan for $165,000 at 9.1% interest, compounded monthly, or the same loan at 20 years for a higher monthly payment. how much more is the monthly payment for the 20 -year loan than the monthly payment for the 25-year loan?

Answer 1

It is $99.04 more per month.

The payment is calculated by P = A/D, where A is the amount of the loan and D is the discount factor.

D = (((1+r)^n)-1)/(r(1+r)^n), where r is the annual interest rate as a decimal divided by 12, and n is the number of months he will be paying.

Since the rate is 9.1%, r = (9.1/100)/12 = 0.091/12 = 0.0076
For the 25 year loan, n = 25*12 = 300:

D = (((1+0.0076)^300)-1)/(0.0076(1+0.0076)^300) = 118.004
P = A/D = 165000/118.004 = 1398.26 per month

For the 20 year loan, n = 20*12 = 240:

D = (((1+0.0076)^240)-1)/(0.0076(1+0.0076)^240) = 110.198
P = A/D = 165000/110.198 = 1497.30 per month

The difference between payments is
1497.30 - 1398.26 = 99.04

Answer 2

The difference between the monthly payments is
`\$99.18`

Explanation:

We know that,

`\text{PV of annuity}=P\left[(1-(1+r)^(-n))/(r)\right]`

Where,

PV = Present value of annuity,

P = payment per period,

r = rate of interest per period,

n = number of period.

Monthly payment for 25 years.

`\Rightarrow 165000=P\left[(1-(1+(0.091)/(12))^(-25* 12))/((0.091)/(12))\right]`

`\Rightarrow P=(165000)/(\left[(1-(1+(0.091)/(12))^(-300))/((0.091)/(12))\right])=\$1395.99`

Monthly payment for 20 years.

`\Rightarrow 165000=P\left[(1-(1+(0.091)/(12))^(-20* 12))/((0.091)/(12))\right]`

`\Rightarrow P=(165000)/(\left[(1-(1+(0.091)/(12))^(-240))/((0.091)/(12))\right])=\$1495.17`

Therefore, the difference between the monthly payments is
`1495.17-1395.99=\$99.18`