mertonmodel

Estimates probability of default using Merton model

Syntax

[PD,DD,A,Sa] = mertonmodel(Equity,EquityVol,Liability,Rate)

[PD,DD,A,Sa] = mertonmodel(___,Name,Value)

Description

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

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

Examples

Compute the Probability of Default Using the Single-Point Approach to the Merton Model

Load the data from MertonData.mat.

load MertonData.mat
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

Equity — Current market value of firm's equity

positive numeric value

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

Data Types: double

EquityVol — Volatility of firm's equity

positive numeric value

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

Data Types: double

Liability — Liability threshold of firm

positive numeric value

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

Data Types: double

Rate — Annualized risk-free interest rate

numeric value

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

Data Types: double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

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

Maturity — Time to maturity corresponding to liability threshold

1 year (default) | positive numeric value

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

Data Types: double

Drift — Annualized drift rate

risk-free interest rate defined in Rate (default) | numeric value

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 — Tolerance for convergence of the solver

1e-6 (default) | positive scalar

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

Data Types: double

MaxIterations — Maximum number of iterations allowed

500 (default) | positive integer

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

Data Types: double

Output Arguments

PD — Probability of default of firm at maturity

numeric value

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

DD — Distance-to-default

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.

A — Current value of firm's assets

numeric value

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

Sa — Annualized firm's asset volatility

numeric value

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

More About

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 (σ_E) is provided. Given equity (E), liability (L), risk-free interest rate (r), asset drift (μ_A), and maturity (T), you use a 2-by-2 nonlinear system of equations. mertonmodel solves for the asset value (A) and asset volatility (σ_A) as follows:

$$E=AN(d_1)-L{e}^{-rT}N(d_2)$$

$${\sigma }_E=\frac{A}{E}N(d_1){\sigma }_A$$

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

$$d_1=\frac{\ln \left(\frac{A}{L}\right)+(r+0.5{\sigma }_A^2)T}{{\sigma }_A\sqrt{T}}$$

$$d_2=d_1-{\sigma }_A\sqrt{T}$$

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

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

$$PD=1-N(DD)$$

References

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

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

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

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

Version History

Introduced in R2017a