A person takes out a monthly installment loan of $2000 with an annual percentage rate of 13.0% if the person can afford no more than $70 per month, then which of the following terms are feasible? a) 8 months b ) 12 months c) 18 months d) 24 months e) 36 months
Massaging the formula for the monthly payment
M = Pr/(1-(1+r)^-n) gives
n = log(M/(M-Pr)) / log(1+r)
With r = .13/12 = 0.010833333
M = 70
P = 2000
M/(M-Pr) = 70/(70 - 2000 * 0.01083333) = 1.448
log 1.448/log 1.010803333 = 34.3, so the 36 month term looks good.
You can also ballpark this by estimating:
13%/year is about 1%/month. 1% of 2000 is 20.
So, the payments start out with $20 interest, leaving $50 going to principal.
2000/50 = 40, which is bigger than 36, but later payments include less interest, so the term would be less than 40. 36 is the best choice.
To determine which of the given terms are feasible, we need to calculate the monthly payment for each term and check if it is affordable.
First, let's calculate the monthly interest rate by dividing the annual percentage rate (APR) by 12 (the number of months in a year):
Monthly interest rate = 13.0% / 12 = 0.0108
Now, we can calculate the monthly payment using the loan amount, interest rate, and number of months:
Monthly payment = Loan amount / Number of months + (Loan amount * Monthly interest rate)
Let's calculate the monthly payments for each term:
a) 8 months:
Monthly payment = $2000 / 8 + ($2000 * 0.0108) = $250 + $21.60 = $271.60
b) 12 months:
Monthly payment = $2000 / 12 + ($2000 * 0.0108) = $166.67 + $21.60 = $188.27
c) 18 months:
Monthly payment = $2000 / 18 + ($2000 * 0.0108) = $111.11 + $21.60 = $132.71
d) 24 months:
Monthly payment = $2000 / 24 + ($2000 * 0.0108) = $83.33 + $21.60 = $104.93
e) 36 months:
Monthly payment = $2000 / 36 + ($2000 * 0.0108) = $55.56 + $21.60 = $77.16
Now, let's check which terms are feasible based on the person's affordability of no more than $70 per month:
a) 8 months: Not feasible (monthly payment is $271.60)
b) 12 months: Not feasible (monthly payment is $188.27)
c) 18 months: Not feasible (monthly payment is $132.71)
d) 24 months: Not feasible (monthly payment is $104.93)
e) 36 months: Feasible (monthly payment is $77.16)
Therefore, the only feasible term is e) 36 months.