Merton Probability of Default Calculator

This calculator outputs risk-neutral probabilities of default for any entity quoted on the stock market provided that it is also possible to access data on its short-term liabilities or any other default threshold.

Instructions:

To install run the following command which will pull and install the latest commit from this repository, along with its Python dependencies:

pip install git+https://github.com/mcherculano/merton_pds.git

Add the following line to your .py script:

import merton_pds.merton_pds as pds

Python Usage Example:

import merton_pds.merton_pds as pds
import pandas as pd
from matplotlib import pyplot as plt 

df = pd.read_excel(wc + '\data.xlsx',index_col=0)
rate = 0.05
output = pds.merton_pds(df.iloc[:,0].values*10**6, df.iloc[:,1].values*10**3, rate)
pds = pd.DataFrame(output[0], df.index)
plt.plot(pds)

Required Inputs:

• Equity: Market value of the firm's equity.
• Liabilities: Liability threshold of the firm.
• Rate: Risk-free interest rate
• **kwargs:
  • 'Maturity': default is 1, should be consistent with implied assumption on maturity of default threshold (liabilities)
  • 'Drift': default is 'rate'
  • 'NumPeriods': typically number of trading periods in a year. default 255
  • 'Tolerance': Used by the Solver. default is 1e-6)
  • 'MaxIterations' Used by the Solver. default is 500

Theory:

This program calculates Probabilities of Default for a a set of N firms across a number of time periods T.

The market value of the firm’s underlying assets $Va$ follows the stochastic process:

$$dVa = \mu *Va *dt + \sigma *Va *dz$$

If X is the book value, then

$$ Ve = Va *N(d1) - exp(-rT) *X *N(d2) $$

where Ve is the market value of the firm’s equity, and

$$ d_1 = [ln(Va/X) + (r+\sigma^2/2)T ]/ [\sigma*\sqrt{T}] $$

$$ d_2 = d1 - \sigma*\sqrt{T} $$

where $r$ is the risk-free interest rate. Thus,

$$ PD = N [(ln(Va/X)+( \mu -\sigma^2) *T)/(\sigma *\sqrt{2})] $$

Notes:

• This can be done for any quoted firm. As an example this code draws on a random set of US banks. Data sourced from Datastream.
• Default threshold defined as current liabilities (short-term debt plus current portion of long-term debt) (see References).

REFERENCES:

[1] Gray, D. F., Merton, R. C., and Bodie, Z. (2007). Framework for Measuring and Managing Macrofinancial Risk and Financial. NBER Working Paper Series, pages 1{32}.

[2] Gupton, G. M., Finger, C. C., and Bhatia, M. (2007). CreditMetrics - Technical Document.

[3] Crosbie,P.J. and Bohn,J.R. (2003) "Modeling Default Risk", available online: http://www.defaultrisk.com/pp_model_35.htm

[4] Herculano, C. M. (2020) "Systemic Risk and the Macroeconomy", PhD thesis University of Glasgow.