Estructura Apoyada-empotrada

There are sophisticated models of a company’s financial structure based on derivatives, namely options. Let us consider a company that is financed partly by debt. The debt has a form of a zero-coupon bond with one year maturity T. The nominal value of the bond is denoted by F. It means that shareholders are obliged to pay F to the debt holders at time T, otherwise the company goes un- der. Let us denote the value of a company at time T by V(T). Once the debt is paid off (on the condition that the value of the company exceeds F ), the share- holders are left with V(T) – F. Otherwise, the company goes bankrupt and will be taken over by the debt holders. Here is a set of possible scenarios of what the shareholders get:

It resembles payoffs of a call option. Hence, the present value of equity is the same as a call premium. Shareholders have the right, but not obligation, to buy back the company (from a bank or debt holders) at the nominal price of debt. Another option-related concept that might be useful is a put call parity:

where S(0) is the present value of the underlying instrument, C and P are call and put premiums respectively, K is the strike price, and r is a risk free rate of return. The value of the underlying instrument is then:

and a call option represents equity:

Both are entered into the put call parity and yield:

The present value of debt can be found as:

If there is no risk (the value of the company exceeds F at T), the call option is worthless and the value of debt is represented by F. If there is some risk in- volved, the value of the debt is decreased by the value of the put premium. In a nutshell, it is possible that a bank will never recover the money loaned to the company, and the bank’s exposure seems bigger than the value of assets. If the option is deep in-the-money and volatility low, the value of the put is little.

In this context, the validity of Miller-Modigliani proposition (MM argue that in a world without taxes, both the value of a firm and its WACC would be unaffected by its capital structure) is worth emphasizing. It can be easily proved that the right side of the equation below does not depend on K.

At first sight, it seems ridiculous since K is obviously there. However, there is no contradiction because both put and call premiums change together with the value of K and the changes balance each other out.

Seeing that the values of both options grow with risk, we are able to explain the often observed conflict between shareholders and debt holders. The former can be interested in carrying out risky projects, since it results in increased eq- uity value at the expense of the value of debt (P grows too). Here is a short illus- tration of this scenario.

Suppose a company is close to bankruptcy. At a nominal debt of 100, the value of the company is below this number. Still, there is a chance to avoid bankruptcy, since the debt is to be paid off in one year’s time. The owners have the following scenarios to choose from:

1. To operate safely. The result of this is that the value of the company at time T will reach 110. At maturity the shareholders are left with 10. If the present value of debt is 100, then at a 10% risk free rate of return, the value of debt will reach 100e-10%_{= 90.48, and the value of equity is 9.52.}

2. To take extreme risk. The risky activities are represented by two possi- bilities: the value of the company may be 50 or 150 at the end of the year, both with a 50% probability. Then, the company will either go bankrupt (making debt holders very unhappy since then they will receive only half of the money due), or survives and then shareholders get 50. Thanks to the call option, the expected value for the shareholders is significantly higher than in scenario 1.

The risky scenario may mean accepting a negative NPV project, which in turn means a deterioration in the value of the company (high risk is reflected in the high cost of capital, which lowers NPV). And yet, in spite of accepting bad projects, the value of the company may skyrocket. Let us analyze a numerical example.

Let us suppose a company is worth 100 today (out of which 80 can be allo- cated to a risky project). The expected cash flows at the end of the year are 100. When the cost of capital is higher than 25%, it clearly means a negative NPV. If we let the cost of capital be 30%, then:

(we use continuous compounding to be in line with put call parity) and the value of the company is 94.08. The value of equity can be represented as the present value of expected cash flows:

Debt holders are left with 71.96. Let us compare the result with the safe sce- nario: the value of the company has dropped, while at the same time the value of equity has gone up (at the expense of debt holders though).

It is worth mentioning that the above is the structural approach to credit risk estimation in the simplest possible case. It may also be, and often is, the starting point for creating software used by the financial industry to value equity.