Question

Credit risk measures using the structural model:  assume a company has the following characteristics.

Time t value of the firm’s assets:   At = $2,800

Expected return on assets:  u = 0.050 per year

Risk-free rate:  r = 0.025 per year

Face value of the firm’s debt:  K = $2,000

Time to maturity of the debt (tenor):  T – t = 1 year

Asset return volatility:  σ = 0.350 per year

 

(a)  Calculate the probability that the debt will default over the time to maturity.

            

(b)  Calculate the expected loss.

 

(c)  Calculate the present value of the expected loss.

Answer 1

Part (a)

The physical probability of default at time \(\mathrm{T}\), measured at time t, is

$$ P_{t}[\tau=T]=P_{t}\left[A_{T} \leq K\right]=N\left[-d_{2}^{P}\right] $$

where \(d_{2}^{P}=(\sigma \sqrt{T-t})^{-1}\left(\log \left(A_{t} / K\right)+\left(\mu-\sigma^{2} / 2\right)(T-t)\right)\)

Hence, \(\mathrm{d}_{2}^{\mathrm{P}}=\left(0.35 \times 1^{1 / 2}\right)^{-1}\left(\log (2,800 / 2,000)+\left(0.05-0.35^{2} / 2\right) \times 1\right)=0.92921\)

Hence, the probability that the debt will default \(=\mathrm{N}\left[-\mathrm{d}_{2} \mathrm{P}\right]=\mathrm{N}(-0.92921)=0.1764=17.64 \%\)

Part (b)

Loss given default \(=\mathrm{At}-\mathrm{K}=2,800-2,000=800\)

Expected loss = probability of default \(x\) loss given default \(=17.64 \% \times 800=141.11\)

Part (c)

PV of expected loss \(=\) Expected \(\operatorname{loss} x e^{-r(T-t)}=141.11 \times e^{-0.025 \times 1}=137.63\)