Contenidos del programa de intervención para 2º de Educación Primaria

The Merton model assumes the asset value of a firm to follow some stochastic process (Vt)t≥0. There are only two cases of securities: equity and debt. It is

assumed that equity receives no dividends and that the firm cannot issue new debt. The model assumes that the company’s debt is given by a zero coupon bond with face value B that will become due at a future time T. The firm defaults if the value of its assets is less than the promised debt repayment at time T. In the Merton model default can occur only at the maturity T of the bond. Denote the value at time t of equity and debt by St and Bt. In a

frictionless market (there are no taxes or transaction costs), the value of the firm’s assets is given by the sum of debt and equity, i.e.,

Vt=St+Bt, 0≤t≤T.

At maturity there are two possible scenarios:

1. VT > B: the value of the firm’s assets exceeds the debt. In this case

the debtholders receive BT = B, the shareholders receive the residual

value ST =VT −B, and there is not default.

2. VT ≤ B: the value of the firm’s assets is less than its debt. Thus the

firm cannot meet its financial obligations and defaults. In this case, the debtholders take ownership of the firm, and the shareholders are left with nothing, so we have BT =VT, ST = 0.

Hence, combining the above two results, the payment to the shareholders at timeT is given by

ST = max(VT −B,0) = (VT −B)+,

and debtholders receive

This shows that the value of the firm’s equity is the payoff of an European call option on the assets of the firm with strike price equal to the promised debt payment. The Merton model treats the asset valueVtas any underlying.

It assumes that under the real world probability measure _P the asset value process follows a geometric Brownian motion of the form

dVt =µVVTdt+σVtdWt, 0≤t≤T, (2.1)

for constants µV ∈R, σV >0 and a standard Brownian motion (Wt)t≥0.

Further, it makes all simplifying assumptions of the Black-Scholes option pricing formula. The solution at timeT of the stochastic differential equation (2.1) with initial value V0 is given by

VT =V0·exp µV − 1 2σ 2 V T +σVWT ,

which in particular implies that

ln(VT)∼Φ ln(V0) + µV − 1 2σ 2 V T, σ2_VT .

Hence the market value of the firm’s equity at maturity T can be de- termined as the price of a European call option on the asset value Vt with

exercise priceB and maturity T. The risk neutral pricing theory then yields that the market value of equity at time t < T can be computed as the discounted expectation of the payoff functionST,

St =E e−r(T−t)·(VT −B)+|Ft , and it is given by St =Vt·Φ(dt,1)−B·e−r(T−t)·Φ(dt,2), where dt,1 = ln(Vt B) + (r+ 1 2σ 2 V)(Tt) σV · √ T −t and dt,2 =dt,1−σV · √ T −t.

Herer denotes the risk-free interest rate, assumed to be constant.

According to the equation for BT, we are able to value the firm’s debt at

time t≤T as Bt=Ee−r(T−t)(B −(B−VT)+)|Ft =B·e−r(T−t)− B·e−r(T−t)Φ(−dt,2)−Vt·Φ(−dt,1) .

The default probability of the firm by time T is the probability that the shareholders will not exercise their call option to buy the assets of the company for B at timeT, and it can be computed as

P(VT ≤B) =P(ln(VT)≤ln(B)) = Φ _ln(B/V 0)−(µV − 1₂σ2V)·T σV √ T . (2.2)

The last equation shows that the default probability is increasing in B, decreasing in V0 and µV and, for V0 > B, increasing in σV, which is all

perfectly in line with economic intuition. Under the risk-neutral measure _Q we have Q(VT ≤B) =Q _ln(B/V 0)−(r− 1₂σV2)·T σV √ T ≤ −d0,2 = 1−φ(d0,2).

Hence the risk-neutral default probability, given information up to time

t, is given by 1−φ(dt,2).

Remark 1.

The Merton model can also incorporate credit migrations and, thus, is not limited to the default-only mode as presented above. Therefore, we consider a firm which has been assigned to some rating category at time t0 = 0. The

time horizon is fixed T > 0. Assume that the transition probabilities p(r) for a firm are available for all rating grades 0 ≤ r ≤ R. The transition probability thus denotes the probability that the firm belongs to rating class

r at time horizon T. In particular, p(0) denotes the default probability of the firm.

Suppose that the asset-value processVtof the firm follows the model given

in 2.1. Let define thresholds

−∞=b0 < b1 < . . . < bR< bR+1 =∞,

such that _P(br < VT ≤ br+1) = p(r) forr ∈ {0, . . . , R}, this is, the prob-

ability that the firms belongs to rating r at the time horizon T equals the probability that the firm’s value at timeT is between br and br+1. Hence we

have translated the transition probabilities into a series of thresholds for an assumed asset-value process. We recall that b1 denotes the default thresh-

old, i.e. the value of the firm’s liabilities B. The higher thresholds are the asset-value levels marking the boundaries of higher rating categories.

Although the Merton model provides a useful context for modeling credit risk, and practical implementations of the model are used by many financial institutions, it also has some drawbacks.

It assumes the firm’s debt financing consists of a one-year coupon bond. For most firms, however, this is an oversimplification. Moreover, the simplify- ing assumptions of the Black-Scholes model are questionable in the context of corporate debt. In particular, the assumption of normally distributed losses can lead to an underestimation of the potential risk in a loan portfolio.

Finally, and this might be the most important shortcoming of the Merton model, the firm’s value is not observable which makes assigning values to it and its volatility problematic.