Exercise 6 - Credit Risk 1

First we load the packages

# Package loading import pandas as pd import numpy as np

We then create the data set to be used and define the recovery rate, R.

# Define R R = 0.40 # Create data arrays = [['10-Jul_07']*5+['12-Jun_08']*5, ['1y','3y','5y','7y','10y']*2] data = pd.DataFrame(data = None, index = pd.MultiIndex.from_arrays(arrays, names=('Year', 'Maturity')), columns = ['s(bps)', 'lambda', 'PD(%)']) data['s(bps)'] = [16, 29, 45, 50, 58, 397, 315, 277, 258, 240] # Populate lambda according to eq (19.3) data['lambda'] = data['s(bps)'].values/10000/(1-R) # Show data data

1) Default intensity function

Fit a piece-wise linear Default Intensity Function for the two dates (Example 19.3 in the book/slides is useful here)

# A function to get intensity using a linear piecewise function def DefaultIntensity(data): # Subfunction to do it for one pair def _get_intermediate(sub_data): index_years = [int(i[:-1]) for i in sub_data.keys()] values_years = sub_data.values return (index_years[1]*values_years[1]-index_years[0]*values_years[0])/(index_years[1]-index_years[0]) # Apply to all pairs return [_get_intermediate(data.iloc[i:(i+2)]) for i in range(len(data)-1)]

# Define subdata data07 = data.loc['10-Jul_07','lambda'] data08 = data.loc['12-Jun_08','lambda']

y_keys = [data07.keys()[i]+'-'+data07.keys()[i+1] for i in range(len(data07.keys())-1)]

linear_default07 = pd.Series(data = DefaultIntensity(data07), index = y_keys) linear_default07

linear_default08 = pd.Series(data = DefaultIntensity(data08), index = y_keys) linear_default08

2) Cumulative probability of default

Calculate the probability of default

# Calculate the probability of default using eq. (19.2) in (%) def CumulProb(lamb, t): return (1-np.exp(-lamb*t))*100

# Use the function for 2007 t = [int(i[:-1]) for i in data07.keys()] lamb = data07.values data.loc['10-Jul_07','PD(%)'] = [CumulProb(l, t) for l, t in list(zip(lamb,t))] # Use the function for 2008 t = [int(i[:-1]) for i in data08.keys()] lamb = data08.values data.loc['12-Jun_08','PD(%)'] = [CumulProb(l, t) for l, t in list(zip(lamb,t))]

Below the probability of default is shown.

data

Below we tried to carry out the calculation with the second method (using the fitted piecewise linear function) but failed to obtain the same results. Instead we got slightly higher results

100*np.append(data.loc[('10-Jul_07', '1y'), 'lambda'], data.loc[('10-Jul_07', '1y'), 'lambda'] + np.cumsum(time_frames*linear_default07.values))

100*np.append(data.loc[('12-Jun_08', '1y'), 'lambda'], data.loc[('12-Jun_08', '1y'), 'lambda'] + np.cumsum(time_frames*linear_default08.values))

3) Compare with CDS spread

Comparing with calculations using the CDS spread, which rating should Lehman have had on the two days.

Looking at table on slide 10, it is seen that the data from 2007 lie between BAA and BA. Data from 2008 we believe can be mapped to B even though it is approaching Caa-C. We believe it would have had B due to the fact that ratings are based on stability (top of page 432 in Hull), and it probably would have been in a dropping phase as it had a better rating in 2007.

4) Default intensities using rating information

Fit piece-wise linear default intensities using the rating information in Table 19.1.

Tab191 = pd.DataFrame(data = np.array([ [0.000, 0.011, 0.011, 0.031, 0.085, 0.195, 0.386, 0.705, 0.824], [0.021, 0.060, 0.110, 0.192, 0.298, 0.525, 0.778, 1.336, 2.151], [0.055, 0.165, 0.345, 0.536, 0.766, 1.297, 2.224, 3.876, 5.793], [0.177, 0.461, 0.804, 1.216, 1.628, 2.472, 3.925, 7.006, 10.236], [0.945, 2.583, 4.492, 6.518, 8.392, 11.667, 16.283, 23.576, 29.733], [3.573, 8.436, 13.337, 17.828, 21.908, 28.857, 36.177, 43.658, 48.644], [10.624, 18.670, 25.443, 30.974, 35.543, 42.132, 50.258, 53.377, 53.930] ]), index = ['Aaa', 'Aa', 'A', 'Baa', 'Ba', 'B', 'Caa-C'], columns = [1,2,3,4,5,7,10,15,20]) Tab191

cols_ar = np.array(Tab191.columns) pd.DataFrame(data = ((Tab191.values)[:,1:]-(Tab191.values)[:,:-1]) / (cols_ar[1:]-cols_ar[:-1]), index = Tab191.index, columns = [str(cols_ar[i])+'-'+str(cols_ar[i+1]) for i in range(len(cols_ar)-1)])

This table emphazises the fact that the risk of defaulting is rising for good rated companies while it drops for bad rated companies. Moreover, it is in most cases seen that the intensity drops for the last interval (15-20 years) compared to 10-15 years. This must be because that if a company survives for 15 years into the future it must do something right.

5) Lehman Brothers' most recent rating

Find information on Lehman Brothers’ most recent ratings: https://www.reuters.com/article/lehman-moodys-idUSLF2328620080915

Lehman Brothers' ratings were slashed on september 15, about 5 days before they declared bankruptcy.

Both Moody's and Fitch changed their ratings drastically. Moodys cut Lehmann Brothers Holdings from A2 to B3 and the group’s subordinated debt to Caa2 from A3. Fitch changed their rating from A+ to D (Default).

While these ratings make sense, given the bankruptcy 5 days later, they seem to have been changed very late.

6) Information on actual recovery rate

Find information on the actual recovery rate – how does this change your conclusions

From https://libertystreeteconomics.newyorkfed.org/2019/01/creditor-recovery-in-lehmans-bankruptcy.html it is seen that the actual recovery rate is 21%. This means that the $\lambda$ values become smaller as we then divide by $(1-0.8)$. This means that the risk of going default would be lower. However, the difference is not large enough for the rating to be e.g. A.

Looking at Business Snapshot 19.1 in the book, the actual recovery rate for Lehman Brothers is at 8%. Then $\lambda$ becomes very small as we divide with $(1-0.92)$. However, as stated above, this does not bring the Lehmann Brothers' rating up to A but closer.

Exercise 6 - Credit Risk 1

1) Default intensity function

2) Cumulative probability of default

3) Compare with CDS spread

4) Default intensities using rating information

5) Lehman Brothers' most recent rating

6) Information on actual recovery rate

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