Final answer:
The justified trailing P/E ratio calculated from the Gordon growth model and assumed discount rate is 17.33. The justified leading P/E ratio is 16.67. The discount rate was assumed as 10% since it was not provided in the question.
Step-by-step explanation:
The calculation of the justified price-to-earnings (P/E) ratio involves using the Gordon growth model to incorporate the growth of dividends over the long term. The model assumes that the dividend grows at a constant rate, which is sustainable in the long-term. Given that we have a price-to-sales ratio of 2.0 times, a net profit margin of 5%, and a long-term growth rate of 4%, we can calculate the justified trailing P/E and leading P/E ratios.
First, we need to calculate the firm's earnings by applying the net profit margin to the sales:
Earnings = Price-to-Sales Ratio × Net Profit Margin
Earnings = 2.0 × 0.05 = 0.1 (or 10%)
Next, we use the Gordon growth model to calculate the trailing P/E ratio based on the earnings and the long-term growth:
Trailing P/E = (1 + Growth Rate) / (Discount Rate - Growth Rate)
Discount Rate is typically the expected return; however, it's important to know that it has not been provided in the question. In the absence of a discount rate, the calculation of the exact trailing P/E ratio cannot be completed. If we consider expected return to be higher than the growth rate, let's assume a discount rate of 10% for illustration.
Trailing P/E = (1 + 0.04) / (0.10 - 0.04) = 1.04 / 0.06 = 17.33
For the leading P/E ratio, we need to use next year's earnings and also apply the growth rate to the denominator:
Leading P/E = 1 / (Discount Rate - Growth Rate)
Using the assumed discount rate:
Leading P/E = 1 / (0.10 - 0.04) = 1 / 0.06 = 16.67
Thus, the final answer in a two-line explanation in fewer than 200 words: Using the Gordon growth model, the justified trailing P/E ratio, assuming a discount rate of 10%, is 17.33, while the leading P/E ratio is 16.67. It is worth noting that the actual values would depend on the actual discount rate applied which we have assumed to be 10% for this calculation.