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Estimate default probability using time-series version of Merton model

Given the time series for equity (*E*), liability
(*L*), risk-free interest rate (*r*),
asset drift (*μA*), and maturity (*T*), `mertonByTimeSeries`

sets
up the following system of nonlinear equations and solves for a time
series asset values (*A*), and a single asset volatility
(σ_{A}). At each time
period *t*, where *t* = `1`

...*n*:

$$\begin{array}{l}{A}_{1}=\left(\frac{{E}_{1}+{L}_{1}{e}^{-{r}_{1}{T}_{1}}N({d}_{2})}{N({d}_{1})}\right)\\ {A}_{t}=\left(\frac{{E}_{t}+{L}_{t}{e}^{-{r}_{t}{T}_{t}}N({d}_{2})}{N({d}_{1})}\right)\\ \mathrm{...}\\ {A}_{n}=\left(\frac{{E}_{n}+{L}_{n}{e}^{-{r}_{n}{T}_{n}}N({d}_{2})}{N({d}_{1})}\right)\end{array}$$

where *N* is the cumulative normal distribution.
To simplify the notation, the time subscript is omitted for *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*) at each time period
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] Loffler, 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.