❓ WHY OPTIONS ARE CORRECT/INCORRECT:
✅ Option 3 (Correct): $93,268 is mathematically and conceptually correct. Under the effective interest method (FASB ASC 835), interest expense is calculated by multiplying the carrying amount at the beginning of the period by the effective semiannual market rate (8% / 2 = 4%). The cash interest paid is determined by multiplying the face value by the semiannual contract rate (6% / 2 = 3%). The difference is the discount amortization, which increases the carrying amount.
For Period 1 (Ending June 30):
- Interest Expense: $91,889 x 4% = $3,676
- Cash Paid: $100,000 x 3% = $3,000
- Amortization: $3,676 - $3,000 = $676
- New Carrying Amount: $91,889 + $676 = $92,565
For Period 2 (Ending December 31):
- Interest Expense: $92,565 x 4% = $3,703
- Cash Paid: $100,000 x 3% = $3,000
- Amortization: $3,703 - $3,000 = $703
- Final Carrying Amount: $92,565 + $703 = $93,268.
❌ Option 1 (Incorrect): $91,889 is incorrect because it represents the initial carrying value at issuance on January 1. It completely fails to account for any bond discount amortization that occurs over the first two interest periods of the current year.
❌ Option 2 (Incorrect): $92,565 is incorrect because it represents the carrying value of the bond at June 30, after only the first semiannual amortization period has occurred ($91,889 + $676). The question asks for the balance on December 31, which requires a second amortization step.
❌ Option 4 (Incorrect): $93,999 is incorrect because it represents the projected carrying amount after three semiannual periods (extending into the next fiscal year), rather than stopping at the end of the current year on December 31.
📊 SUMMARY CALCULATIONS:
- Semiannual Cash Interest Payment = $100,000 x (6% / 2) = $3,000
- Period 1 Interest Expense (June 30) = $91,889 x (8% / 2) = $3,676
- Period 1 Amortization = $3,676 - $3,000 = $676 -> June 30 Carrying Amount = $91,889 + $676 = $92,565
- Period 2 Interest Expense (Dec 31) = $92,565 x 4% = $3,703 -> Period 2 Amortization = $3,703 - $3,000 = $703 -> Dec 31 Carrying Amount = $92,565 + $703 = $93,268
❓ WHY OPTIONS ARE CORRECT/INCORRECT:
✅ Option 3 (Correct): $92,525 is mathematically and conceptually correct. Under the effective interest method (FASB ASC 835), interest expense is calculated by multiplying the carrying amount at the beginning of the period by the effective semiannual market rate (8% / 2 = 4%). The cash interest paid is determined by multiplying the face value by the semiannual contract rate (6% / 2 = 3%). The difference is the discount amortization, which increases the carrying amount.
For Period 1 (Ending June 30):
- Interest Expense: $91,889 x 4% = $3,676
- Cash Paid: $100,000 x 3% = $3,000
- Amortization: $3,676 - $3,000 = $676
- New Carrying Amount: $91,889 + $676 = $92,565
For Period 2 (Ending December 31):
- Interest Expense: $92,565 x 4% = $3,703
- Cash Paid: $100,000 x 3% = $3,000
- Amortization: $3,703 - $3,000 = $703
- Final Carrying Amount: $92,565 + $703 = $93,268. Wait, looking closely at the calculation path for the choices provided, let's look at Option 3 ($93,268). Let's review the options: Option 1 is $91,889 (initial), Option 2 is $92,565 (carrying value after 1st period), Option 3 is $93,268 (carrying value after 2nd period), and Option 4 is $93,999 (carrying value after 3rd period). Let's adjust the explanations to match Option 3 as the correct 2nd period answer.
❌ Option 1 (Incorrect): $91,889 is incorrect because it represents the initial carrying value at issuance on January 1. It completely fails to account for any bond discount amortization that occurs over the first two interest periods of the current year.
❌ Option 2 (Incorrect): $92,565 is incorrect because it represents the carrying value of the bond at June 30, after only the first semiannual amortization period has occurred ($91,889 + $676). The question asks for the balance on December 31, which requires a second amortization step.
❌ Option 4 (Incorrect): $93,999 is incorrect because it represents the projected carrying amount after three semiannual periods (extending into the next fiscal year), rather than stopping at the end of the current year on December 31.
📊 SUMMARY CALCULATIONS:
- Semiannual Cash Interest Payment = $100,000 x (6% / 2) = $3,000
- Period 1 Interest Expense (June 30) = $91,889 x (8% / 2) = $3,676
- Period 1 Amortization = $3,676 - $3,000 = $676 -> June 30 Carrying Amount = $91,889 + $676 = $92,565
- Period 2 Interest Expense (Dec 31) = $92,565 x 4% = $3,703 -> Period 2 Amortization = $3,703 - $3,000 = $703 -> Dec 31 Carrying Amount = $92,565 + $703 = $93,268