i. Explain the results. Full marks for this question require you to derive the credit spread in the limit τ → 0
(c) Complete the table below with the credit spread as a function of time to maturity and the risk-free rate r equal to 3% and 9%. The other inputs are: firm value = 120, debt value = 95 and asset volatility = 18%. Spreads should be shown in basis points and rounded up to the nearest integer. What is the effect on the spreads and yields? Explain.
r=3% r=9%
0.5 yr 1.0 yr .
9.0 yr 9.5 yr 10.0 yr
TABLE 2
(iii) Merton Model Extensions:
(a) In a few paragraphs discuss the main empirical failings of the benchmark Merton model and then discuss the extent to which extensions discussed class can resolve these failings. Full marks for this question will be awarded for a discussion that replicates Table 1 above for either the Merton model with stochastic volatility, stochastic interest rates or the Merton model with Jumps. Closed form solutions or simulation based approaches can be employed.
(b) What about equity? People often talk about credit risk in the context of corporate debt. But since assets = debt + equity there is a link between credit risk and equity. In the context of your answer to part (iii) − a discuss the link with equations and provide any derivations that are important. Full marks will be awarded for simulated equity return findings.
Question 2: Implied risk-neutral default probabilities from CDS data
This exercise uses observed CDS premiums for different maturities on a given day to back out risk-neutral default probabilities. We will use a formula for the fair swap premium, that uses the following assumptions:
• Protection payments are paid yearly
• Ifacompanydefaultsbetweenyeartandyeart+1wetreatthedefaultasoccurring at the midpoint of the interval. This means that the protection payment takes place at the midpoint, and the protection buyer also pays for insurance for the half year by paying half the annual premium at the midpoint
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• The recovery rate is fixed and equal to 40%
• The riskless interest rate is constant and equal to 2%
Let S(t) denote the probability of survival past time t. Let P (0, t) = exp(−rt) denote the discount factor at time 0 corresponding to maturity t. Let C(T) denote the annual premium payment on a CDS with maturity T. The fair premium on the swap is the value C(T) which equates the value of the protection leg with the value of the premium leg.
The value of the premium leg as a function of time to maturity is pb T C(T) T 1
π (T)=C(T) P(0,t)S(t)+ 2 P 0,t−2 (S(t−1)−S(t)) (1) tt
The value of the protection leg is
ps T1
π (T)=(1−δ) P 0,t−2 (S(t−1)−S(t)) (2) t
where the probability of survival is
S(t) = exp −
t
λ(s)ds (3)
0
The spreadsheet CDS data.xls contains CDS swap spread quotes for 7 companies for maturities between 1 and 10-years. Missing quotes for any maturity should be linearly interpolated. The following questions require you to compute the value of paying a CDS premium of 1 until a given maturity. Code up a routine that takes in a term structure of CDS quotes and numerically minimises the squared deviation between the fitted value of the protection leg and the value of the premium leg. The intensities should converge to values ensuring a virtually zero value of the objective function. The routine should output the fitted intensities and fitted default probabilities.
Produce a small written answer for each of the following questions.
(i) Pick a firm and implement your CDS fitting code. What happens if you change the recovery rate? Try setting it to zero. Can you set the recovery rate arbitrarily between 0 and 1 and always get a solution? Why or why not?
(ii) Run the calibrations for all other data sets that are provided in the sheet ’Data’. Are there recovery restrictions needed to be made for the most extreme cases?
(iii) For all firms, compute conditional default probabilities between year t and t+1 for t = 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; i.e. compute the probability of defaulting in
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the period between t and t + 1 given survival up to time t: Is there a connection between slopes of CDS curves and these values? Explain.
(iv) Compare the conditional default probabilities you estimate with the values of the intensities for the same period.
(v) What happens if you change the riskless rate? Are the implied default probabili- ties affected strongly or hardly at all?
Question 3: Dependence modelling and Gaussian / t-copulas
(i) Consider a portfolio with 30 assets. Denote P (D ≥ 8, τ ≤ 1.5y) the probability of 9 or more defaults within the first 18 months; P(D = 0,τ > 2y) the probability of zero defaults within the first 2 years, etc.
Generate Xˆ1, Xˆ2, . . . Xˆ30 which are jointly normal distributed variables with means zero, standard deviations one and correlation 0%,15% and 35%, respec- tively. Generate a covariance matrix Σ and compute
X ∼ N(0,Σ)
and return Ui = Φ(Xi) where Φ is the univariate cumulative normal distribution function. Calculate default times (τ) using the following assumptions for the λ = 15% and N = 250, 000 simulations. Populate the Gaussian copula default probabilities in the table below (in percent, 2 decimals).
(ii) Now generate Xˆ1, Xˆ2, . . . Xˆ30 which are jointly Student t distributed random variables with correlation 0%, 15% and 35%, respectively, and degrees of freedom ν = 4. Let Ui = tν(Xi) where tν is the is the univariate cumulative Student t distribution function. Again, calculate default times (τ) using the following assumptions for the λ = 15% and N = 250,000 simulations. Populate the t- copula default probabilities in the right hand side of the table below (in percent, 2 decimals).
(iii) Discuss the differences between the results in the table.
(iv) Redo the above exercise, but this time X1, X2, · · · X10 have intensity λ = 5%, and
X11,X12,···X20 have λ = 15%, and X21,X22,…X30 have λ = 25%.
(v) Interpret the differences.
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P(D ≥ 8,τ ≤ 1.5y) · P(D ≥ 10,τ ≤ 1.5y) · P(D ≥ 12,τ ≤ 1.5y) · P(D = 0,τ ≤ 2.0y) · P(D = 0,τ ≤ 3.0y) · P(D = 0,τ ≤ 3.5y) ·
· ·
· ·
· ·
· ·
· ·
· ·
· · · · · · · · · · · · · · · · · ·
Gaussian copula Student t-copula 0% 15% 35% 0% 15% 35%
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