Main Content

Estimates probability of default using Merton model

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, *d _{1}* and

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

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

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

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

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

[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.