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tangare [24]
2 years ago
12

You are considering the purchase of a new car, the reborn VW Beetle, and you have been offered two different deals from two diff

erent 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
Business
1 answer:
erastovalidia [21]2 years ago
8 0

Answer:

annual interest rate = 9.336%, monthly 0.778%

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

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As the question says that the company will maintain its existing after tax return on capital invested next year, hence that means that the net income for the next year remains the same, which is $140.

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