Libby Company purchased equipment by paying $5,000 cash on the purchase date and agreeing to pay $5,000 every six months during the next four years. The first payment is due six months after the purchase date. Libby's incremental borrowing rate is 8%. (FV of $1, PV of $1, FVA of $1, and PVA of $1)

The equipment reported on the balance sheet as of the purchase date is closest to:

Note: Use the appropriate factor(s) from the tables provided.

Question

Libby Company purchased equipment by paying $5,000 cash on the purchase date and agreeing to pay $5,000 every six months during the next four years. The first payment is due six months after the purchase date. Libby's incremental borrowing rate is 8%. (FV of $1, PV of $1, FVA of $1, and PVA of $1)

The equipment reported on the balance sheet as of the purchase date is closest to:

Note: Use the appropriate factor(s) from the tables provided.

BlackTom AI · Accepted Answer

Convert the 8% annual rate to a 4% semiannual rate and compute the present value (PV) of the eight semiannual $5,000 payments, then add the $5,000 cash down payment.

1. Relevant knowledge points and concepts
- When payments occur semiannually, use the semiannual interest rate: 8% annual ÷ 2 = 4% per period.
- The PV of an ordinary annuity (payments at the end of each period) is: PVA = PMT × [1 − (1 + r)^−n] / r.
- The equipment’s recorded cost = immediate cash paid + PV of future payments (liability measured at present value).

2. Step-by-step reasoning
- There are four years of semiannual payments → n = 4 × 2 = 8 payments.
- Semiannual rate r = 0.08 / 2 = 0.04.
- Each future payment PMT = $5,000 (first at 6 months, so ordinary annuity).

3. Calculations and intermediate steps
- Compute (1 + r)^n = 1.04^8 = 1.3685690504052736.
- Compute (1 + r)^−n = 1 / 1.3685690504052736 ≈ 0.730691.
- PVA factor = [1 − (1 + r)^−n] / r = (1 − 0.730691) / 0.04 = 0.269309 / 0.04 ≈ 6.732725.
- PV of the eight payments = $5,000 × 6.732725 ≈ $33,663.63.
- Add the immediate cash payment: Total recorded cost = $5,000 + $33,663.63 = $38,663.63 ≈ $38,664.

4. Why the correct answer is correct
The equipment should be recorded at the amount paid plus the present value of the contractual future payments discounted at the company’s incremental borrowing rate (semiannual). The calculation above precisely discounts the eight $5,000 payments at 4% per period and adds the immediate $5,000, giving $38,663.63, rounded to $38,664.

5. Why the other options are incorrect
- $45,000: This is the undiscounted sum of payments (5,000 immediate + 8×5,000 = 45,000), ignoring the time value of money.
- $40,000: This would reflect an incorrect count or treatment of payments (e.g., treating only seven future payments plus the down payment) and ignores proper discounting.
- $33,664: This is essentially the PV of the eight future payments alone (≈ $33,663.63) but omits the $5,000 cash down payment, so it understates the equipment cost.

6. Conclusion
The equipment reported on the balance sheet as of the purchase date is $38,664, which matches the provided correct answer.

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