The firm has an average collection period of 34 days. Current practice is to factor all receivables immediately at a discount rate of 1.8%. Assume that default is extremely unlikely. What is the effective annual interest rate on this arrangement? A) 23.87 percent B) 24.35 percent C) 21.11 percent D) 21.53 percent E) 19.32 percent
The correct answer and explanation is:
Correct Answer: A) 23.87 percent
To calculate the Effective Annual Rate (EAR) on this factoring arrangement, we treat the 1.8% discount as an interest cost for borrowing the amount of the receivable for the period of the average collection time (34 days).
Step-by-step calculation:
- Periodic rate:
Since 1.8% is charged for 34 days, this is the periodic interest rate. Periodic rate=1.8%=0.018\text{Periodic rate} = 1.8\% = 0.018 - Number of periods per year: Number of periods=36534≈10.735\text{Number of periods} = \frac{365}{34} \approx 10.735
- Use the EAR formula: EAR=(1+periodic rate)number of periods−1\text{EAR} = (1 + \text{periodic rate})^{\text{number of periods}} – 1 EAR=(1+0.018)10.735−1\text{EAR} = (1 + 0.018)^{10.735} – 1
- Calculate: EAR=(1.018)10.735−1≈1.2387−1=0.2387\text{EAR} = (1.018)^{10.735} – 1 \approx 1.2387 – 1 = 0.2387
- Convert to percentage: EAR=23.87%\text{EAR} = 23.87\%
Explanation:
The effective annual interest rate accounts for the compounding effect of multiple short-term borrowing periods throughout the year. In this scenario, factoring receivables means the firm gets immediate cash at the cost of a 1.8% discount, which effectively resembles paying interest for borrowing money for 34 days. Since this borrowing repeats approximately 10.735 times in a year, the true cost over the year is more than simply multiplying 1.8% by the number of periods.
The formula used captures the compounding nature of the cost. As a result, the actual yearly rate is higher than a simple annualized figure. Therefore, the effective annual interest rate is 23.87 percent, making option A correct.