Credit Risk

# Package loading import pandas as pd import numpy as np

# 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

	s(bps)int6416 – 397	lambdafloat640.003 – 0.066	PD(%)object Missing100%
('10-Jul_07', '1y')	16	0.002666666666666667	nan
('10-Jul_07', '3y')	29	0.004833333333333334	nan
('10-Jul_07', '5y')	45	0.0075	nan
('10-Jul_07', '7y')	50	0.008333333333333333	nan
('10-Jul_07', '10y')	58	0.009666666666666667	nan
('12-Jun_08', '1y')	397	0.06616666666666667	nan
('12-Jun_08', '3y')	315	0.052500000000000005	nan
('12-Jun_08', '5y')	277	0.04616666666666667	nan
('12-Jun_08', '7y')	258	0.043000000000000003	nan
('12-Jun_08', '10y')	240	0.04	nan
10 rows × 3 columns

# 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

# 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))]

data

	s(bps)int6416 – 397	lambdafloat640.003 – 0.066	PD(%)object 0.266311426949905610% 1.439538126761763510% 8 others80%
('10-Jul_07', '1y')	16	0.002666666666666667	0.2663114269499056
('10-Jul_07', '3y')	29	0.004833333333333334	1.4395381267617635
('10-Jul_07', '5y')	45	0.0075	3.6805582279178184
('10-Jul_07', '7y')	50	0.008333333333333333	5.666455012650784
('10-Jul_07', '10y')	58	0.009666666666666667	9.214142477263543
('12-Jun_08', '1y')	397	0.06616666666666667	6.402514451800112
('12-Jun_08', '3y')	315	0.052500000000000005	14.572318639152048
('12-Jun_08', '5y')	277	0.04616666666666667	20.612823302897965
('12-Jun_08', '7y')	258	0.043000000000000003	25.99222272532924
('12-Jun_08', '10y')	240	0.04	32.967995396436066
10 rows × 3 columns

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))

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

	1float64	2float64	3float64	4float64	5float64	7float64	10float64	15float64	20float64
Aaa	0	0.011	0.011	0.031	0.085	0.195	0.386	0.705	0.824
Aa	0.021	0.06	0.11	0.192	0.298	0.525	0.778	1.336	2.151
A	0.055	0.165	0.345	0.536	0.766	1.297	2.224	3.876	5.793
Baa	0.177	0.461	0.804	1.216	1.628	2.472	3.925	7.006	10.236
Ba	0.945	2.583	4.492	6.518	8.392	11.667	16.283	23.576	29.733
B	3.573	8.436	13.337	17.828	21.908	28.857	36.177	43.658	48.644
Caa-C	10.624	18.67	25.443	30.974	35.543	42.132	50.258	53.377	53.93
7 rows × 9 columns

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)])

	1-2float64	2-3float64	3-4float64	4-5float64	5-7float64	7-10float64	10-15float64	15-20float64
Aaa	0.011	0	0.02	0.054000000000000006	0.055	0.06366666666666666	0.0638	0.023799999999999998
Aa	0.03899999999999999	0.05	0.082	0.10599999999999998	0.11350000000000002	0.08433333333333333	0.1116	0.16299999999999995
A	0.11000000000000001	0.17999999999999997	0.19100000000000006	0.22999999999999998	0.26549999999999996	0.3090000000000001	0.3303999999999999	0.3834000000000001
Baa	0.28400000000000003	0.343	0.4119999999999999	0.4119999999999999	0.42200000000000004	0.4843333333333333	0.6162000000000001	0.6460000000000001
Ba	1.6380000000000003	1.9089999999999998	2.026	1.8739999999999997	1.6375000000000002	1.538666666666667	1.4586	1.2314
B	4.8629999999999995	4.901	4.491	4.080000000000002	3.474499999999999	2.44	1.4962000000000004	0.9971999999999994
Caa-C	8.046000000000001	6.773	5.530999999999999	4.568999999999999	3.2944999999999993	2.708666666666668	0.6237999999999999	0.11059999999999945
7 rows × 8 columns