Credit Risk: Coursework
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INSTRUCTIONS: Submissions should be in the style of a small report answering each sub questions and including annotated tables and figures where required. There is no page limit but the submission should not be too long (use your judgement). The coursework requires coding. Use whatever language you like but include the code in an appendix at the end of the submission. Courseworks will be scanned for plagiarism (both the text and the code).
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Question 1
Compute the value of the firm (A) for a 20-year period using the Merton model. Use the following inputs:
• Risk-free rate = 3.10%
• Face value of debt at maturity equals D = 80
• Debt maturity T = 20
• Asset volatility 15% for the first 10 years and 25% for the last 10 years
• V(0)=100.
1. Simulate the value of the firm using the above assumptions and a daily time step for 5 random paths.
2. Default occurs when the value of the firm goes below the debt value. Simulate 10,000 paths and plot the distribution of residual firm values (A(T)−D) at maturity assuming the firm can only default at T.
3. What is the default probability assuming that the firm can only default at maturity T (model 1)?
4. Derive the analytical default probability under the risk neutral measure. Compare the simulation result to the analytical default probability for model 1.
5. What is the date t = 0 default probability assuming that the firm can default at any (simulated) time up to maturity T (model 2)? This is the simulated version of the Black-Cox Model. Note that you should take into account the discount factor to obtain the default threshold before maturity)
6. Are the default probabilities assuming model 1 and model 2 significantly different? Explain the results.
Question 2
Consider a standard Merton model in which the risk-neutral dynamics of a firm’s assets is given by
dAt =rtdt+σAdWtQ, (1) At
where rt is the potentially time-varying risk-free rate, σA is the volatility, and WQ is a
Brownian motion under the risk-neutral measure Q. The firm has issued zero-coupon debt
with a face value of D which matures at time T. Denote B the time t value of the zero- t
coupon defaultable debt claim and y = − log its promised yield-to-maturity. Begin
by assuming a constant risk free rate rt = constant = r and denote the price of default free
debt as Bt = e−rτ . Recall the credit spread is then give by sτ = yτ − r.
1. Using the following inputs: r = 2.80%, D = 95, and σA = 23%. complete the following table with the credit spread as a function of time to maturity. Spreads should be shown in basis points and rounded up to the nearest integer.
At =80 At =90 At =125 At =180
0.5 yr 1.0 yr .
9.0 yr 9.5 yr 10.0 yr
2. Next, complete the following table with the credit spread as a function of time to maturity for various levels of the risk free rate, for the model inputs: At = 120, D = 95 and σA = 18%. Spreads should be shown in basis points and rounded up to the nearest integer.
0.5 yr 1.0 yr .
9.0 yr 9.5 yr 10.0 yr
3. Confirm the findings in the table above analytically by computing the partial derivative of the credit spread with respect to the risk free rate. What is the effect of the level of the risk free rate on credit spreads? Explain.
Question 3
Now consider the following extension to the Merton model in which the risk-neutral dynamics of a firm’s assets is given by
dAt =rdt+σ dWA,Q, (2) AtAt
dr =κ(θ−r)dt+σ dWr,Q (3) ttrt
⟨dWA,Q,dWr,Q⟩ = ρ (4) tt
dWA,Q =dWA,P +ΘAdt (5) tt
dWr,Q =dWr,P +Θrdt (6) tt
In a few paragraphs discuss the main empirical failings of the benchmark Merton model. The discuss how the extension to the Merton model given by equations 2 – 3 class can resolve these failings. Full marks for this question will be awarded for a discussion that replicates Table 1 and Table 2 above for the Merton model with stochastic interest rates, and discusses the economic mechanism of the model with equations and derivations. Note: use parameters for the Vasicek short rate similar to those used in the lecture slides.
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