Estimates probability of default using Merton model
[PD,DD,A,Sa] = mertonmodel(Equity,EquityVol,Liability,Rate)
[PD,DD,A,Sa] = mertonmodel(___,Name,Value)
Load the data from
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
Equity— Current market value of firm’s equity
Current market value of firm’s equity, specified as a positive value.
EquityVol— Volatility of firm's equity
Volatility of the firm's equity, specified as a positive annualized standard deviation.
Liability— Liability threshold of firm
Liability threshold of firm, specified as a positive value. The liability threshold is often referred to as the default point.
Rate— Annualized risk-free interest rate
Annualized risk-free interest rate, specified as a numeric value.
comma-separated pairs of
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
[PD,DD,A,Sa] = mertonmodel(Equity,EquityVol,Liability,Rate,'Maturity',4,'Drift',0.22)
'Maturity'— Time to maturity corresponding to liability threshold
1year (default) | positive numeric value
Time to maturity corresponding to the liability threshold, specified
as the comma-separated pair consisting of
a positive value.
'Drift'— Annualized drift rate
Rate(default) | numeric value
Annualized drift rate (expected rate of return of the firm's
assets), specified as the comma-separated pair consisting of
a numeric value.
'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
'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
PD— Probability of default of firm at maturity
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.
A— Current value of firm's assets
Current value of firm's assets, returned as a numeric value.
Sa— Annualized firm's asset volatility
Annualized firm's asset volatility, returned as a numeric value.
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
mertonmodel function solves a
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.
Unlike the time series method (see
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
system of equations.
mertonmodel solves for the
asset value (A) and asset volatility (σA)
where N is the cumulative normal distribution, d1 and d2 are defined as:
The formulae for the distance-to-default (DD) and default probability (PD) are:
 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.