Financial Risk Manager Part 2

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Explanation:

Using the given formula for the estimated default probability variable: $\frac{\ln\left(\frac{K}{V}\right) - \delta \tau + 0.5 \sigma^2 \tau}{\sigma \sqrt{\tau}}$

Given: $K = 300$ million $V = 380$ million $\tau = 5$ years $\delta = 20\% = 0.20$ $\sigma = 30\% = 0.30$

Substitute the values: Numerator: $\ln(300/380) - (0.20 \times 5) + 0.5 \times (0.30)^2 \times 5$ $= -0.2364 - 1.0 + 0.5 \times 0.09 \times 5$ $= -1.2364 + 0.225 = -1.0114$

Denominator: $0.30 \times \sqrt{5} = 0.30 \times 2.236 = 0.6708$

Calculate the ratio: $-1.0114 / 0.6708 = -1.5077 \approx -1.51$

The default probability is $N(-1.51)$ . Using standard normal distribution tables: $N(-1.51) = 1 - N(1.51) = 1 - 0.9345 = 0.0655 = 6.55\%$

This is closest to 6.6%.

Explanation:

Using the given formula for the estimated default probability variable: $\frac{\ln\left(\frac{K}{V}\right) - \delta \tau + 0.5 \sigma^2 \tau}{\sigma \sqrt{\tau}}$

Given: $K = 300$ million $V = 380$ million $\tau = 5$ years $\delta = 20\% = 0.20$ $\sigma = 30\% = 0.30$

Substitute the values: Numerator: $\ln(300/380) - (0.20 \times 5) + 0.5 \times (0.30)^2 \times 5$ $= -0.2364 - 1.0 + 0.5 \times 0.09 \times 5$ $= -1.2364 + 0.225 = -1.0114$

Denominator: $0.30 \times \sqrt{5} = 0.30 \times 2.236 = 0.6708$

Calculate the ratio: $-1.0114 / 0.6708 = -1.5077 \approx -1.51$

The default probability is $N(-1.51)$ . Using standard normal distribution tables: $N(-1.51) = 1 - N(1.51) = 1 - 0.9345 = 0.0655 = 6.55\%$

This is closest to 6.6%.

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Q.42 Tenacity Limited, a leveraged firm in the United States, has, in its capital structure, a five-year zero-coupon bond with a face value of USD 300 million. The remaining capital is composed of equity. Currently, the market value of the firm’s assets is USD 380 million, and the expected rate of change of the firm’s value is 20%. Analysts have estimated annual volatility of the firm’s assets at 30%. The firm’s risk management unit (RMU) estimates the default probability using the Merton model, i.e.,

$\frac{\ln\left(\frac{K}{V}\right) - \delta \tau + 0.5 \sigma^2 \tau}{\sigma \sqrt{\tau}}$

Assume that firm value takes on the lognormal distribution with constant volatility. The estimated default probability is closest to:

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Last updated: July 5, 2026 at 07:21

22.4%

6.6%

12.5%

24.8%