# mertonByTimeSeries

Estimate default probability using time-series version of Merton model

## Syntax

``````[PD,DD,A,Sa] = mertonByTimeSeries(Equity,Liability,Rate)``````
``````[PD,DD,A,Sa] = mertonByTimeSeries(___,Name,Value)``````

## Description

example

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

example

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

## Examples

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Load the data from `MertonData.mat`.

```load MertonData.mat Dates = MertonDataTS.Dates; Equity = MertonDataTS.Equity; Liability = MertonDataTS.Liability; Rate = MertonDataTS.Rate;```

Compute the default probability by using the time-series approach of Merton's model.

```[PD,DD,A,Sa] = mertonByTimeSeries(Equity,Liability,Rate); plot(Dates,PD)```

Load the data from `MertonData.mat`.

```load MertonData.mat Dates = MertonDataTS.Dates; Equity = MertonDataTS.Equity; Liability = MertonDataTS.Liability; Rate = MertonDataTS.Rate;```

Compute the plot for the default probability values by using the time-series approach of Merton's model. You compute the `PD0` (blue line) by using the default values. You compute the `PD1` (red line) by specifying an optional `Drift` value.

```PD0 = mertonByTimeSeries(Equity,Liability,Rate); PD1 = mertonByTimeSeries(Equity,Liability,Rate,'Drift',0.10); plot(Dates, PD0, Dates, PD1)```

## Input Arguments

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Market value of the firm's equity, specified as a positive value.

Data Types: `double`

Liability threshold of the 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 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] = mertonByTimeSeries(Equity,Liability,Rate,'Maturity',4,'Drift',0.22,'Tolerance',1e-5,'NumPeriods',12)`

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`

Number of periods per year, specified as the comma-separated pair consisting of `'NumPeriods'` and a positive integer. Typical values are `250` (yearly), `12` (monthly), or `4` (quarterly).

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.

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 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 for Time Series

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.

Given a time series of observed equity values and liability thresholds for a company, `mertonByTimeSeries` calibrates corresponding asset values, the volatility of the assets in the sample's time span, and computes the probability of default for each observation. Unlike `mertonmodel`, no equity volatility input is required for the time-series version of the Merton model. You compute the probability of default and distance-to-default by using the formulae in Algorithms.

## Algorithms

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 (σA). At each time period t, where t = `1`...n:

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

where N is the cumulative normal distribution. To simplify the notation, the time subscript is omitted for d1 and d2. At each time period, 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) at each time period are:

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

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

## References

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

[2] Loeffler, 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