A car dealer asks you to make a payment of $7,000 for a car today and pay $5,000 at the end of year one, pay $4,000 at the end of year two and $3,000 at the end of year three. Assuming that the interest rate you can earn on a safe investment is 4%, what is the present value of your total payments for the car?[Fill in the blank]单项选择题
A
a. $18,172.91
B
b. $19,000.00
C
c. $18,094.67
D
d. $18,538.46

Question

A car dealer asks you to make a payment of $7,000 for a car today and pay $5,000 at the end of year one, pay $4,000 at the end of year two and $3,000 at the end of year three. Assuming that the interest rate you can earn on a safe investment is 4%, what is the present value of your total payments for the car?[Fill in the blank]单项选择题

A

a. $18,172.91

B

b. $19,000.00

C

c. $18,094.67

D

d. $18,538.46

BlackTom AI · Accepted Answer

Present value equals the sum of each payment discounted to today, so PV = 7000 + 5000/(1.04) + 4000/(1.04)^2 + 3000/(1.04)^3.

1) Relevant concepts
- Time value of money: a dollar received (or paid) in the future is worth less today; discount future amounts by (1 + r)^t where r is the interest rate and t is years.
- Present value of multiple cash flows is the sum of individually discounted cash flows.

2) Step-by-step calculation (showing intermediate steps)
- Cash flows: today (t=0) = 7000; end of year 1 (t=1) = 5000; t=2 = 4000; t=3 = 3000. Interest rate r = 4% = 0.04.
- Discount factors:
  - 1/(1.04) = 0.9615384615 → 5000/(1.04) = 5000 × 0.9615384615 = 4807.6923077
  - 1/(1.04)^2 = (1/(1.04))^2 = 0.9615384615^2 = 0.9245562130 → 4000/(1.04)^2 = 4000 × 0.9245562130 = 3698.224852
  - 1/(1.04)^3 = (1/(1.04))^3 = 0.8889964212 → 3000/(1.04)^3 = 3000 × 0.8889964212 = 2666.989263
- Sum the present values:
  - PV = 7000 + 4807.6923077 + 3698.224852 + 2666.989263
  - PV = 18172.906423 ≈ $18,172.91 (rounded to cents)

3) Why this answer is correct
Discounting converts each future payment to its equivalent value today; adding the undiscounted $7,000 paid today plus the discounted future payments yields $18,172.91, which matches the principle of present value with r = 4%.

4) Why the other choices are incorrect
- b. $19,000.00 equals the simple (undiscounted) sum 7000+5000+4000+3000; it ignores time value of money.
- c. $18,094.67 is off due to incorrect discounting or rounding errors in one or more cash flows.
- d. $18,538.46 is also incorrect—likely the result of applying an incorrect rate or misplacing the timing of the $7,000 payment.

Conclusion: The correct present value is $18,172.91, which matches answer (a).

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