mertonmodel

Estimates probability of default using Merton model

Description

example

[PD,DD,A,Sa] = mertonmodel(Equity,EquityVol,Liability,Rate) estimates the default probability of a firm by using the Merton model.

example

[PD,DD,A,Sa] = mertonmodel(___,Name,Value) adds optional name-value pair arguments.

Examples

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Equity    = MertonData.Equity;
EquityVol = MertonData.EquityVol;
Liability = MertonData.Liability;
Drift     = MertonData.Drift;
Rate      = MertonData.Rate;
MertonData
MertonData=5×6 table
ID          Equity      EquityVol    Liability    Rate    Drift
__________    __________    _________    _________    ____    ______

{'Firm 1'}    2.6406e+07     0.7103         4e+07     0.05    0.0306
{'Firm 2'}    2.6817e+07     0.3929       3.5e+07     0.05      0.03
{'Firm 3'}     3.977e+07     0.3121       3.5e+07     0.05     0.031
{'Firm 4'}     2.947e+07     0.4595       3.2e+07     0.05    0.0302
{'Firm 5'}     2.528e+07     0.6181         4e+07     0.05    0.0305

Compute the default probability using the single-point approach to the Merton model.

[PD,DD,A,Sa] = mertonmodel(Equity,EquityVol,Liability,Rate,'Drift',Drift)
PD = 5×1

0.0638
0.0008
0.0000
0.0026
0.0344

DD = 5×1

1.5237
3.1679
4.4298
2.7916
1.8196

A = 5×1
107 ×

6.4210
6.0109
7.3063
5.9906
6.3231

Sa = 5×1

0.3010
0.1753
0.1699
0.2263
0.2511

Input Arguments

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Current market value of firm’s equity, specified as a positive value.

Data Types: double

Volatility of the firm's equity, specified as a positive annualized standard deviation.

Data Types: double

Liability threshold of firm, specified as a positive value. The liability threshold is often referred to as the default point.

Data Types: double

Annualized risk-free interest rate, specified as a numeric value.

Data Types: double

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [PD,DD,A,Sa] = mertonmodel(Equity,EquityVol,Liability,Rate,'Maturity',4,'Drift',0.22)

Time to maturity corresponding to the liability threshold, specified as the comma-separated pair consisting of 'Maturity' and a positive value.

Data Types: double

Annualized drift rate (expected rate of return of the firm's assets), specified as the comma-separated pair consisting of 'Drift' and a numeric value.

Data Types: double

Tolerance for convergence of the solver, specified as the comma-separated pair consisting of 'Tolerance' and a positive scalar value.

Data Types: double

Maximum number of iterations allowed, specified as the comma-separated pair consisting of 'MaxIterations' and a positive integer.

Data Types: double

Output Arguments

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Probability of default of the firm at maturity, returned as a numeric value.

Distance-to-default, defined as the number of standard deviations between the mean of the asset distribution at maturity and the liability threshold (default point), returned as a numeric value.

Current value of firm's assets, returned as a numeric value.

Annualized firm's asset volatility, returned as a numeric value.

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Merton Model Using Single-Point Calibration

In the Merton model, the value of a company's equity is treated as a call option on its assets and the liability is taken as a strike price.

mertonmodel accepts inputs for the firm's equity, equity volatility, liability threshold, and risk-free interest rate. The mertonmodel function solves a 2-by-2 nonlinear system of equations whose unknowns are the firm's assets and asset volatility. You compute the probability of default and distance-to-default by using the formulae in Algorithms.

Algorithms

Unlike the time series method (see mertonByTimeSeries), when using mertonmodel, the equity volatility (σE) is provided. Given equity (E), liability (L), risk-free interest rate (r), asset drift (μA), and maturity (T), you use a 2-by-2 nonlinear system of equations. mertonmodel solves for the asset value (A) and asset volatility (σA) as follows:

$E=AN\left({d}_{1}\right)-L{e}^{-rT}N\left({d}_{2}\right)$

${\sigma }_{E}=\frac{A}{E}N\left({d}_{1}\right){\sigma }_{A}$

where N is the cumulative normal distribution, d1 and d2 are defined as:

${d}_{1}=\frac{\mathrm{ln}\left(\frac{A}{L}\right)+\left(r+0.5{\sigma }_{A}^{2}\right)T}{{\sigma }_{A}\sqrt{T}}$

${d}_{2}={d}_{1}-{\sigma }_{A}\sqrt{T}$

The formulae for the distance-to-default (DD) and default probability (PD) are:

$DD=\frac{\mathrm{ln}\left(\frac{A}{L}\right)+\left({\mu }_{A}-0.5{\sigma }_{A}^{2}\right)T}{{\sigma }_{A}\sqrt{T}}$

$PD=1-N\left(DD\right)$

 Zielinski, T. Merton's and KMV Models In Credit Risk Management.

 Löffler, G. and Posch, P.N. Credit Risk Modeling Using Excel and VBA. Wiley Finance, 2011.

 Kim, I.J., Byun, S.J, Hwang, S.Y. An Iterative Method for Implementing Merton.

 Merton, R. C. “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance. Vol. 29. pp. 449–470.