Using the Merton Model, the equity of a firm can be valued as a European call option on the firm's assets.

Given:

• Firm value ($V$) = $200 million
• Debt face value ($K$) = $180 million
• Time to maturity ($t$) = 3 years
• Risk-free rate ($r$) = 5% = 0.05
• Volatility of firm value ($\sigma$) = 10% = 0.10

Step 1: Calculate $d_1$ and $d_2$ $d_1 = \frac{\ln(V / K) + (r + \sigma^2 / 2)t}{\sigma \sqrt{t}}$ $d_1 = \frac{\ln(200 / 180) + (0.05 + 0.10^2 / 2) \times 3}{0.10 \sqrt{3}}$ $\ln(200 / 180) \approx 0.10536$ $(0.05 + 0.005) \times 3 = 0.165$ $d_1 = \frac{0.10536 + 0.165}{0.10 \times 1.732} = \frac{0.27036}{0.1732} \approx 1.56$

$d_2 = d_1 - \sigma \sqrt{t} = 1.56 - 0.1732 \approx 1.3868 \approx 1.39$

Step 2: Find $N(d_1)$ and $N(d_2)$ from the standard normal table $N(1.56) \approx 0.9406$ $N(1.39) \approx 0.9177$

Step 3: Calculate the value of equity ($E$) $E = V \times N(d_1) - K \times e^{-rt} \times N(d_2)$ $E = 200 \times 0.9406 - 180 \times e^{-0.05 \times 3} \times 0.9177$ $e^{-0.15} \approx 0.8607$ $E = 188.12 - 180 \times 0.8607 \times 0.9177$ $E = 188.12 - 180 \times 0.7898 = 188.12 - 142.16 = 45.96$ million

Rounding to the nearest million, the value of equity is approximately $46 million.