1 - Structural models

Summary

Consider a company launched at time 0, when some assets are purchased for V(0). Funding comes from two sources. Shareholders contribute E(0), referred to as equity. The remaining amount D(0) = V(0) − E(0), called debt, is either borrowed from a bank or raised by selling bonds issued by the company.

We consider this company over a time interval from 0 to T, during which the assets are put to work in order to generate some funds, which are then split between the two groups of investors at time T. The debt is first repaid with interest to the debt holders, who have priority over the equity holders. Any remaining amount goes to the equity holders.

The simplest way to raise money to make these payments is to sell the assets of the company. We begin our analysis with this case, by making the necessary assumption that the assets are tradeable.

Traded assets

We assume that there is a liquid market for the assets, and V(t) for t ∈ [0, T] represents their market value. We also assume that the assets generate no additional cash flows. A practical example of such assets would be a portfolio of traded stocks that pay no dividends, the company being an investment fund.

Payoffs

Suppose that the company has to clear the debt at time T, and that there are no intermediate cash flows to the debt holders. The interest rate applying to the loan will be quoted by the bank or implied by the bond price. We denote this loan rate by k_D with continuous compounding, and by K_D with annual compounding. The amount due at time T is

(Throughout this volume we take one year as the unit of time.) One of the goals here is to find the loan rate that reflects the risk for the debt holders.

At time T we sell the assets and close down the business, at least hypothetically, to analyse the company's financial position at that time. It is possible that the amount obtained by selling the assets is insufficient to settle the debt, that is, V(T) < F. In this respect, we make an important assumption concerning the legal status of the company: it has limited liability.