The first thing that should be said about the model

is that it's as simple as it is beautiful.

Even though it's rarely directly used by practitioners because

it's a little bit too simplistic for actually doing business based on that,

it's extremely popular as a great conceptual framework.

Here's how it works.

A firm is run by equity holders,

to finance their activity,

they issue a bond with the face value D,

and sell it to bond holders with a promise to pay back the debt D,

at a future time T. For example,

in one year or five years from now,

if the total assets of the firm V, at time T,

also known as the firm value is larger than D,

the equity holders pay the debt D,

at time T, and keep the remaining amount.

If on the other hand,

the firm value VT,

at time T, is below D,

then equity holders are unable to pay their debt,

and bond holders take over the firm and immediately sell the assets.

The payoff to the bond holders in all cases will therefore be minimum of D, and VT.

In other words, the bond holders either get their money back,

or they get the assets.

The payoff to the equity holders will then be the difference VT minus D,

but only as long as this quantity is positive.

When the final value of the asset VT,

is less than the debt D,

equity holders simply hand the keys to the bond holders and walk away.

This means that the equity holders have a limited liability.

One important point here is that in the Merton model,

what happens with the asset before time T, doesn't matter.

For example, the firm rather can go below the debt value at

some earlier time and then they go back to levels above the default barrier later,

what matters is only the final value of the firm VT.

This of course not very realistic,

and there are tractable extensions of

the Merton model that enable the full triggering at intermediate times,

and not only at the maturity T. We will not

go into this as our goal is not more to incorporate defaults,

but rather to explain the Merton model itself.

Now we have two payoffs,

to the bond holders and equity holders at a future time T,

while we are currently at time t,

small t. So, we have to compute time t discounted

expected values of these payoffs to get the prices of the corporate bond and equity.

These expectations should be with respect to

all possible realizations of the future from the value VT,

under the so-called risk-neutral measure,

where the firm expected return is equal to

the risk-free rate R. To define and compute this expectation,

the Merton model assumes that the firm value VT,

follows a geometric Brownian motion with volatility Sigma V. In this case,

the firm value at time T,

will have a normal distribution and expectations can be computed in closed form.

For our purposes, the most relevant formula of the Merton model is shown here.

It gives the probability of default,

that is the probability that the final value VT,

will be below D. It's given by the cumulative normal distribution of d2,

where d2, is this expression shown here.

Formulas of these type are sometimes called probabilistic structural models.

They word structural here,

refers to the fact that this model is not a statistical model,

but it's rather based on the very well-defined model of the world.

It's also interesting to take a look at the structure of the expression for d2.

Essentially it tells us that,

there is only a single financial ratio VT to D,

that matters for this problem.

The only other firm-specific characteristic that impacts their result,

is the asset volatility Sigma V. It's a well-known fact that

financial institutions are difficult to model within Merton framework,

and there are actually a few reasons for that.

One of them is that the complex debt structure of a financial institution.

Another reason, is that unlike corporations,

financial institutions typically have a higher leverage,

that is higher debt to equity ratio than corporations.

Whereas, simplifying assumptions of

the Merton model are not working well anymore in this situation.

Therefore, a simple structural or model-based approach to

prediction of bank failures would probably not be

adequate to the complexity of the prediction problem itself.

So, while a classical approach to bank failures might struggle,

what about machine learning?

In the next video,

we will see how it works for the present problem.