Equity-based credit models

• Equity based models are discussed for public listed companies
• Equity can be seen as options with strike at the debt face value and the firm's value as the underlying
• Probability of default can be obtained from the correct interpretation of the option mentioned above.

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Hello all - hope you are all fine and safe.

In this post, I would like to briefly discuss some very interesting models for the quantification of credit score using the equity value of public listed companies.

Equity-based models

Suppose an on-going company needs additional resources to fund determined endeavor. In principle, such additional resources can be funded basically either issuing new debts in the form of bonds, FRNs, and other liability instruments, or selling fractions of ownership. Here we considered the case of companies whose stocks are negotiated in an stock exchange. Moreover, equity-based models assumes efficient market hypothesis (EMH). Basically,  EMH assumes that the stock price follows a random walk whose probability of movement is entirely based on the current information that is shared universally rather than on past trends.

Considering the simplistic assumptions presented above, the firm value is given by:

$F(t) = E(t) + C(t)$

where $E(t)$, $F(t)$, and $C(t)$ are correspondingly the time-dependent equity, firm, and debt values.

Suppose a company issues bonds aiming to fund some determined endeavor. In the case everything goes well, the company will profit from the endeavor and will be able to repay the bondholders in addition to create value for the equity holders. However, if the endeavor fails, the company might be in a unfavorable situation to repay bondholders. In this case, assets owned by the company would be used to repay such debts. Equity holders would be paid with what is left of company assets, likely reducing equity value. Under this point of view, equity value can be seen as a call option whose underlying is the firm value and the strike embeds the company debt. Put in other words, this call option describes the obvious situation in which the positive return for equity holders relies on the company health with respect to its capital structure.

Merton's model (1974)

In this model, it is assumed equity value at the debt maturity $T$ can be written as a payoff of a call option whose strike is a face value $D$ of the debt (with maturity at $T$):

$E(F, T) = \left[ F(T) - D \right] ^+$

It is also interesting to notice that one can use the capital structure proposed above to obtain the time-dependent debt value by observing that at maturity:
$C(F, T) = min \left[ F(T), D \right] = D -\left[ D-F(T) \right] ^+$

Both time-dependent equity and debt values are the future expectation of the "payoffs" shown above. In order to calculate such expectation one assumes that company assets (and so the firm value) follows a Geometric Brownian Motion:

$dF = r Fdt + \sigma_F FdW$

where $r$ is a deterministic risk free rate, so $dW$ is under the risk neutral measure. A function $V(F, t)$ will follow a Black-Scholes  like equation. For the equity value, one obtains:

$E(F, t) = F(t)N(d_1) - e^{-r(T-t)}DN(d_2)$

and for the debt:

$C(F, t) = F(t)N(-d_1) + e^{-r(T-t)}DN(d_2)$

where $d_{1,2}=\frac{ln(D/F) \pm (r-0.5 \sigma_F^2)(T-t)}{\sigma_F \sqrt(T-t)}$ and $N(\bullet)$ is the cumulative normal distribution.

Let's take a moment to evaluate our steps so far. Merton's model provides a very interesting insight on how to relate the capital structure of a listed company with the quoted equity value. One assumes that the debt has no convertible features, no coupons payments are considered, and the maturity is collapsed at $T$. Moreover, this model assumes that the dynamics of the firm's value $F(t)$ is calibrated through the equity value, which allow us to obtain the probability of default.
In the next steps, we will walk through further considerations that allows us to a default probability which is one of the most interesting metrics for credit scoring.

Merton's KMV Model 

From the point of view of the equity holder, the worst case scenario occurs when $F \leq D$. This is represented by the optionality on which the equity value is written in the Merton's model. In this sense, it is intuitive to say that the probability of default ($PD$) will be given by: $PD = 1-N(d_1)$.
It is quite reasonable to assume $E(t)$ as a Geometric Brownian motion, so:
$dE = \mu_E Edt + \sigma_E EdW$
If we apply Itô's lemma in $E(F, t)$ we will write:
$dE = \frac{\partial E}{\partial t}dt + \frac{\partial E}{\partial F}dF + \frac{1}{2}\frac{\partial^2 E}{\partial F^2}dF dF$
so:
$dE = \frac{\partial E}{\partial t}dt + \frac{\partial E}{\partial F}(r F dt + \sigma_F F dW) + \frac{\sigma_F^2 F^2}{2} \frac{\partial^2 E}{\partial F^2}dt$
and finally:
$dE = \left[ \frac{\partial E}{\partial t} + r F \frac{\partial E}{\partial F} + \frac{\sigma_F^2 F^2}{2} \frac{\partial^2 E}{\partial F^2}\right] dt + \sigma_F F \frac{\partial E}{\partial F}dW$
when comparing expresion above to $dE = \mu_E Edt + \sigma_E EdW$ we obtain the following interesting relation:

$\sigma_E E = \sigma_F F \frac{\partial E}{\partial F}$

The partial derivative of $E$ with respect to $F$ is simply the Delta $\Delta$ of the option discussed in the previous section. Therefore:

$\sigma_E E = \sigma_F F N(d_1)$

Expression above is required to obtain $PD$, because one still needs to determine $\sigma_F$, since $F$ can be directly calculated (a question for the reader: How can we obtain $F$?). The value of $E$ can be directly observed in the market. $\sigma_E$ can be obtained through the quotes of vanilla options with maturity coinciding with the one of the face debt value, that is, at $T$.

Modeling improvement opportunities

Modeling described so far is highly dependent on how the value of $D$ is estimated. Depending on how liability of a company is structured, it is extremely difficult to determine $D$. In addition, one should notice that coupons are not considered, but in reality a company that fails to pay a single coupon is already in default.
Another interesting fact is that the debt value changes according to the credit score of the company and the risk free interest rate, which may induce some changes in the capital structure if the company has issued callable (or puttable) instruments. This is because the issuer (holder) might be benefit of a reduction (increase) of the company credit spread or interest rate. (Actually, in terms of interest rate one is usually interested in the corresponding term-structure)
Model described so far only assumes only common equities and zero-coupon debts. However, more asset classes should be considered in order to obtain a more realistic model, named preference equities, convertible and callable instruments. Dividend distributions should be included as well.
Another challenge in terms of the modeling is the inclusion of a probability associated with the reorganization of the capital structure of the companies prior to $T$. Some companies might to choose to issue debts with longer maturity aiming to roll out some of the short term liabilities. This can be the case for companies that have a good credit score and need to postpone the repayment for an invested endeavor. In this sense, the interpretation proposed by the Merton's model that equities can be seen as an European call options is actually not appropriate. A more realistic interpretation would be to say that equities are like perpetuals down-out options whose strike changes from time-to-time, for not saying continuously.
In principle, the most known model in the market able to describe the capital structure more realistically is the Moody´s proprietary VK model which is a generalization of the Merton´s KMV model. Here you can find a very interesting research paper released by Moody's analytics. This model relies on large data bases that allows relaxing the efficient market hypothesis. I also suggest this paper that brings an interesting discussion on equity based models.

In a future post , I'll try to bring more information on the Moody's model. Pretty sure it is very interesting.

Hope you enjoyed the post. Leave your comments and share.

Peace profound for all of you.

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Diogo Pedroso

LinkedIn: www.linkedin.com/in/diogomourapedroso