Question20 You have just leased a car that has monthly payments of $375 for the next 3 years with the first payment due today. If the APR is 7.08 percent compounded monthly, what is the value of the payments today?Select one alternative: $12,202.12 $12,130.55 $12,627.07 $11,792.84 ResetMaximum marks: 3 Flag question undefined单项选择题
A
$12,202.12
B
$12,130.55
C
$12,627.07
D
$11,792.84

Question

Question20 You have just leased a car that has monthly payments of $375 for the next 3 years with the first payment due today. If the APR is 7.08 percent compounded monthly, what is the value of the payments today?Select one alternative: $12,202.12 $12,130.55 $12,627.07 $11,792.84 ResetMaximum marks: 3 Flag question undefined单项选择题

A

$12,202.12

B

$12,130.55

C

$12,627.07

D

$11,792.84

BlackTom AI · Accepted Answer

Because the first payment is due today and the remaining payments are monthly, this is an annuity-due problem. Use the annuity-due present-value formula.

1) Key concepts
- Periodic rate r = APR/12 because interest is compounded monthly.
- Number of payments n = years × 12.
- Present value of an annuity due: PV = PMT × [(1 − (1 + r)^−n)/r] × (1 + r). Equivalently PV = first payment (at t=0) + PV of the remaining (n − 1) payments treated as an ordinary annuity.

2) Step-by-step calculation
- PMT = $375
- APR = 7.08% so monthly rate r = 0.0708/12 = 0.0059 (0.59% per month)
- n = 3 years × 12 = 36 payments

Using the annuity-due formula:
(1) Compute (1 + r)^36. With r = 0.0059, (1.0059)^36 ≈ 1.2358703.
(2) Compute its reciprocal: (1 + r)^−36 ≈ 0.8088367.
(3) Compute the ordinary-annuity factor: [1 − (1 + r)^−36]/r ≈ (1 − 0.8088367)/0.0059 ≈ 0.1911633/0.0059 ≈ 32.4006.
(4) Convert to annuity-due by multiplying by (1 + r): 32.4006 × 1.0059 ≈ 32.53899.
(5) Multiply by PMT: PV ≈ 375 × 32.53899 ≈ $12,202.12.

(Alternative decomposition: PV = 375 + 375 × [1 − (1.0059)^−35]/0.0059 gives the same result to rounding.)

3) Intermediate steps shown above (exponents, reciprocals, divisions, and multiplications) are the arithmetic used to reach the PV.

4) Why this answer is correct
- The annuity-due formula accounts for the immediate first payment (t=0) by multiplying the ordinary-annuity factor by (1 + r), which increases PV slightly versus an ordinary annuity. Plugging in the exact monthly rate and 36 months yields $12,202.12.

5) Why the other choices are incorrect
- $12,130.55 and $11,792.84 are too low; they would result from treating the stream as an ordinary annuity or using a slightly different rate/term.
- $12,627.07 is too high; it overstates the PV (equivalent to using a much smaller discount rate or more payments).