Question

You are considering the purchase of a new car, the reborn VW Beetle, and you have been offered two different deals from two different dealers. Dealer A offers to sell you the car for $20,000, but allows you to put down $2,000 and pay back $18,000 over 36 months (fixed payment each month) at a rate of 8% compounded monthly. Dealer B offers to sell you the car for $19,500 but requires a down payment of $4,000 with repayment of the remaining $15,500 over 36 months at 10% compounded monthly. Find the monthly discount rate that makes both deals equally attractive

Answer 1

annual interest rate = 9.336%, monthly 0.778%

Step-by-step explanation:

Dealer A: down payment of $2,000 + 36 monthly payments of $564.05

Dealer B: down payment of $4,000 + 36 monthly payments of $500.14

we must find the interest rate at which both dealers' offers have the same present value:

$2,000 + PV monthly payments A = $4,000 + PV monthly payments B

PV monthly payments A = payment A x {1/r - 1 /[r x (1 + r)³⁶}

PV monthly payments B = payment B x {1/r - 1 /[r x (1 + r)³⁶}

we must use trial and error:

for r = 0.8% monthly, annually = 9.6%

PV monthly payments A = $564.05 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $17,582.76

PV monthly payments B = $500.14 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $15,590.54

the difference between them = $17,582.76 - $15,590.54 = $1,992.22 ≤ $2,000, so r must be a little lower

for r = 0.78% monthly, annually = 9.36%

PV monthly payments A = $564.05 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $17,644.46

PV monthly payments B = $500.14 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $15,645.24

the difference between them = $17,644.16 - $15,645.24 = $1,998.92 ≤ $2,000, so r must be a little lower

for r = 0.77% monthly, annually = 9.24%

PV monthly payments A = $564.05 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $17,675.42

PV monthly payments B = $500.14 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $15,672.70

the difference between them = $17,675.42 - $15,672.70 = $2,002.72 ≥ $2,000, so r must be a little higher

we continue until we find that r = 0.778% monthly, annually 9.336%

PV monthly payments A = $564.05 x {1/0.00778 - 1 /[r x (1 + r)³⁶} = $17,650.64

PV monthly payments B = $500.14 x {1/0.00778 - 1 /[r x (1 + r)³⁶} = $15,650.70

the difference between them = $17,650.64 - $15,650.70 = $2,000.06 ≈ $2,000, so that is our r