a firm offers terms \( 2 / 20 \), net 50 . what effective annual rate does the firm earn if the discount period is increased to 25 days?

The idea is to compare the benefit a customer now gets by paying early (taking a 2% discount) with the “penalty” for waiting longer (up to the net date). When a customer takes the discount, they pay 98% of the invoice; if they delay past the discount period they pay 100% at day 50. In effect, by not taking the discount a customer is "paying" an extra   (100% − 98%) / 98% ≈ 2%/98% ≈ 0.02041 or about 2.041% extra for waiting the extra time beyond the discount period. Originally the discount period ends at 20 days so that the extra delay is 50 − 20 = 30 days. Now if the discount period is extended to 25 days, then the extra delay is only 50 − 25 = 25 days. The extra cost incurred by not taking the discount remains 2.041%, but it is now imposed for a 25‐day period. To find the effective annual rate the firm earns, we “annualize” this 25‐day rate. Step 1. Calculate the effective 25–day “penalty rate” (r):   r = 0.02/0.98 ≈ 0.02041 Step 2. Find how many 25–day periods there are in a year. Assuming a 365–day year:   n = 365/25 ≈ 14.6 Step 3. The effective annual rate (EAR) is then given by compounding over these periods:   EAR = (1 + r)^n − 1 = (1 + 0.02041)^(14.6) − 1 Step 4. Compute the EAR:   Take logs for an approximate calculation:    ln(1.02041) ≈ 0.0202,   Multiply by 14.6: 0.0202 × 14.6 ≈ 0.2949,   Then exponentiate: exp(0.2949) ≈ 1.343,   So, EAR ≈ 1.343 − 1 = 0.343, or about 34.3%. Thus, if the discount period is extended to 25 days (with net still at 50 days), the effective annual rate earned by the firm for not taking the discount is approximately 34%.