## import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

Credit risk

In this notebook we are going to speak about credit risk, what is credit risk, how to compute his main indicators.

Suppose that your society A loan some money to another society B, A face the risk that B go bankrupt and then become unable to reimburse all or just a part of the loan, B is in a default. This is the credit risk, loan money with some chances to not get reimburse totally or not at all.

To evaluate your Expected loss in a case of counterparty default you need to know:

Your exposition at default EAD this is the total amount of loan that is exposed to credit risk
You loss given default LGD that is the expected amount of your loan that you are probably not going to recover. It is expressed in percentage of the EAD
The probability of default of the counterparty, or the probability that he become unable to reimburse his debt or the probability that you do not get reimbursed this PD probability of default.

Then your expected loss due to a default of a counterparty is $EL = PD \times LGD \times EAD$

There are different way to model the default risk for PD and LGD estimation :

A poisson process, very simple based on rate spread (real or historical probability).
The merton model if you have the informations on the counterparty balance sheeet (risk- neutreal probability).

HAZARD RATES OR DEFAULT INTENSITY

The hazard rate or default intensity $ \lambda (t) $ is the default probability at time $t$ with the condition of no default between time zero and time $t$.

The hazard rate between time $t$ and $ \Delta t $ is $ \Delta \lambda (t) $ conditional on no default between time zero and $t$.

So when we get conditionnal default probability other a period it is the hazard rate.

GET THE PROBABILITY OF DEFAULT FROM THE HAZARD RATE.

Assume that $V(t)$ is the cumulative probability of a company surviving from time zero to time $t$ and $V(t+ \Delta t)$ is the cumulative probability of a company surviving from time zero to time $t + \Delta t$.

So that $1-V(t)$ and $1-V(t + \Delta t)$ are the cumulative default probabilities to time $t$ and time $ t + \Delta t $.

So the default probability from time $ t $ and time $ t + \Delta t $ condition on no default between time zero to $t$, is the hazard rate $ \Delta \lambda (t) $ :

$\frac{(1-V(t + \Delta t)) - (1-V(t))}{V(t)} = \frac{1-V(t + \Delta t) - 1+V(t)}{V(t)} = \frac{V(t) - V(t + \Delta t)}{V(t)} = \lambda (t) \Delta t$ $\frac{V(t) - V(t + \Delta t)}{\Delta t} = \lambda (t) V(t)$ $\frac{V(t + \Delta t) - V(t)}{\Delta t} = - \lambda (t) V(t)$ $\frac{dV(t)}{V(t)} = - \lambda (t) dt$

Resolving that differential equation :

$\int_0^t \frac{1}{V(s)} dV(s) = \int_0^t - \lambda (s) ds = - \int_0^t \lambda (s) ds$ $ln(V(t)) - ln(V(0)) = - \int_0^t \lambda (s) ds$ $V(t) = e^{-\int_0^t \lambda (s) ds}$

Then $Q(t)$ the probability of default by time $t$ is :

$Q(t) = 1 - V(t) = 1 - e^{-\int_0^t \lambda (s) ds}$

$Q(t) = 1 - e^{-\overline{\lambda}(t)dt}$

Where $ \overline{\lambda}(t) $ is the average hazard rate between time zero and time $t$.

Probability of default (PD) knowing $ \lambda (t) $

Knowing the hazard rate over a periode, it is possible to deduce the probability of default over any periode.

Suppose that the hazard rate per year is 2%.

PD over one month of a counterparty

$Q(T) = 1- e^{-0.02*(1/12)}$

print("The probabilty of default over one month time horizon : ", 1-np.exp(-0.02*1/12))

The probabilty of default over one month time horizon :  0.0016652785490612887

PD over 10 months of a counterparty

$Q(T) = 1- e^{-0.02*(10/12)}$

print("The probabilty of default on a 10 months time horizon : ", 1-np.exp(-0.02*10/12))

The probabilty of default on a 10 months time horizon :  0.01652854617838251

PD over 20 years of a counterparty

$Q(T) = 1- e^{-0.02*(20*1)}$

print("The probabilty of default on a 10 months time horizon : ", 1-np.exp(-0.02*20))

The probabilty of default on a 10 months time horizon :  0.3296799539643607

References

Risk Management and Financial Institutions, 4th Edition, John C. Hull, ISBN: 978-1-118-95594-9, Mar 2015, WILEY