2.3 QUANTITATIVE ANALYSES
2.3.1 Structural models
In the structural approach, assets and liabilities of a company are modeled simultaneously. Thus, structural models are based on fundamental com- pany data, focusing on its balance sheet and asset value. Default occurs when the value of the firm’s assets falls below its liabilities. Consequently, the required inputs comprise the firm’s liabilities, usually taken from its balance sheet, market value of equity and (implied) equity volatility. Since equities are typically more liquid than corporate bonds, one may argue that equity prices tend to reflect the value of a company’s assets more accurately.
Using information from the equity markets allows fixed income instru- ments to be priced independently, without requiring credit spread informa- tion from related fixed income instruments. However, if equity prices become irrationally inflated or deflated, as we have experienced during the equity hype of the late 1990s, they may be misleading indicators of actual asset values. Generally it is assumed that one can reasonably infer asset val- ues from equity prices. An option pricing model is then used to derive the volatility of the firm’s assets. Although it is generally possible to model financial institutions in the structural framework they should be treated with caution, since it is difficult to assess their assets and liabilities.
Furthermore, since financial institutions are highly regulated, default may not occur even if the value of assets falls below the firm’s liabilities.
The Black–Scholes (1973) option pricing model and Merton’s work on the pricing of corporate debt (1974) lay the foundations for the structural model. Merton’s model establishes a relationship between the market value of a firm’s assets and the market value of its equity. Consider a firm whose operations are financed exclusively by a zero coupon bond maturing at timeTwith a face value of X, and equity. Denote the market value of the firm’s assets at time TbyV␣(T). Then the company pays off its liabilities in full, if the market value of its assets exceeds the face value of the zero coupon bond, that is V␣(T)ⱖX. In this case, the shareholder’s value is
Table 2.1 Stylized balance sheet
Assets Liabilities
Assets Debt
Equity Source:Union Investment
V␣(T)⫺X. Conversely, if the value of the firm’s assets is lower than the face value of debt, V␣(T)⬍X, the company cannot repay its liabilities in full and defaults. The creditors take over the firm and the equity becomes worthless.
Figure 2.5 shows the pay-off profiles for debt- and equity holders at maturity of the liabilities.
Thus, the zero coupon bond is equivalent to a long position in a risk-free zero coupon bond and a short put on the assets of the company. Similarly, equity may be considered as a call option on the assets of the firm. The strike price of both options equals the face value of debt. Using the Black–Scholes option pricing model we are able to derive the market value and the volatility of the firm’s assets. According to Black–Scholes the mar- ket value of equity, V, and the market value of assets are related by
V⫽V␣N(d1)⫹e⫺rTXN(d2) with
and
d2⫽d1⫺␣兹T,
d1 ⫽ ln (V␣/X) ⫹ (r ⫹ (␣2/2))T
␣兹T
Figure 2.5 Pay-off patterns for debt- and equity holders at maturity of the liabilities
Source: Union Investment
Face value of debt (X) Asset value of the company X
Pay-off level
Equity holder
Debt holder
where ␣ denotes asset volatility, r is the risk-free rate, and N is the normal distribution. For equity volatility the condition
holds. Simple balance sheet theory suggests that V␣⫽D⫹V, where Drep- resents the market value of debt. Consequently, if asset value is known, the market value of debt can be inferred easily. Since the face value of debt and its maturity are known, too, we can now determine the yield of the zero coupon bond and its spread over the risk-free rate. However, it should be noted that the original Merton model tends to underestimate short-term spreads because of the assumption that asset value follows a continuous lognormal process. Using this assumption the probability of falling below the default threshold in a short period of time is usually very low.
Therefore, commercial models make adjustments in order to model short- term spreads more accurately.
To estimate the probability that the market value of assets will be lower than the face value of debt, we need to make an assumption about the behavior of the firm’s asset value. Black and Scholes posit that the market value of the firm’s asset follows a stochastic process:
dV␣⫽V␣dt⫹␣V␣dZ,
where denotes the drift of the market value of assets, and dzrepresents a Wiener process. Using to denote the random component of the firm’s return at time twe obtain
The probability of default is then described by the expression
where
ln (V␣/Xt) ⫹ ( ⫺ (␣2/2))t
␣兹t
⫽ P冤⫺ ln (V␣/Xt) ⫹␣兹(t⫺ (␣2/2))tⱖ冥,
P冤lnV␣ ⫹ 冢 ⫺ 2␣2冣t ⫹ ␣兹tⱕlnXt冥
lnV␣t ⫽ lnV␣ ⫹ 冢 ⫺ 2␣2冣t ⫹ ␣兹t.
⫽ V␣
VN(d1)␣
represents the number of standard deviations that asset value is separated from the default barrier (see Figure 2.6). KMV calls this expression “dis- tance to default”. In the Black–Scholes–Merton framework the expression above is equivalent to d2. Consequently, the risk neutral probability of default is N(⫺d2). It is also possible to compute the expected recovery rate under the risk-neutral measure. Conditional on the default of the company, debtholders may expect to recover an asset value of
As we have pointed out for short-term spreads expected recovery value may be too high for short-term liabilities due to the assumptions of the model. Commercial implementations of the original structural model are more sophisticated, in order to produce more accurate spreads, default probabilities and recovery rates even for short time horizons. While in pure diffusion models with a barrier overnight debt is quasi-riskless, the intro- duction of a jump process captures the fact that default could be triggered by a sudden, unexpected event. Thus, the jump process is appropriate to model the possibility of the firm defaulting instantaneously due to the arrival of negative information with respect to, for example, litigation or fraud.
Additional features of commercial applications include, for example, time-varying default barriers, continuous monitoring of the default thresh- old, the use of alternative option pricing models and different assumptions
V␣N(⫺ d1) N(⫺ d2).
Figure 2.6 Projected and realized asset value in the Merton model
Source: Union Investment
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Time
Asset value
Actual asset value Projected asset value
Default barrier
for the behavior of asset values. They typically incorporate short-term and long-term liabilities, convertible debt, preferred equity and common equity, although this substantially enhances complexity of the model. Hence, struc- tural models are well suited for handling different securities of the same issuer, including bonds of various seniorities and convertible bonds. Even the behavior of the company’s management can be incorporated into a structural model. A typical example is a “target leverage” model, in which the initial capital structure can be adjusted. The level of debt fluctuates over time depending on changes in the firm’s value, so that the ratio of debt to assets is mean-reverting. Nevertheless, it is hard to model a firm that is close to its default threshold, since management often chooses to adjust the capital structure in this situation. Over one-year horizons, however, com- mercial implementations of structural models have a consistently good track record with respect to the prediction of defaults.