Credit market participants often follow stock prices to get some direction on credit-default swap pricing. Although there is extensive research on swap pricing, it is hard to find explicit examples of how to compute a simple relationship between the two. There are two popular approaches to modeling credit risky instruments, the structural approach and the reduced-form approach. The structural approach treats corporate liabilities as contingent claims on the issuer's assets. Assuming a simple capital structure composed of one zero-coupon bond and a layer of equity, an event of default occurs when the firm's asset value falls below the face value of the zero-coupon bond at maturity. Equity is equivalent to a call option on the assets of the firm, whose strike is equal to the face value of the debt.
Conversely, the reduced-form approach assumes an event of default to be an exogenous process. The occurrence of default is typically modeled as a Poisson process, in which a default is always an unexpected event. To price credit risky instruments, one must estimate the hazard rate, a conditional probability of default that is calibrated from the debt term structure.
Why The Structural Approach?
The main drawback of the structural approach is that theoretical spreads go to zero for a shorter debt maturity because default never occurs as a sudden event.
Empirical evidence clearly shows corporate debt spreads do not trend to zero even for very short maturities. In defense of the structural approach, one could argue that actual spreads are likely to account for other types of risk besides default risk, such as liquidity and market risks and other factors such as taxes and bankruptcy costs, which are not usually treated in these models.
Despite its shortcomings, the structural approach is widely studied and used in practice because it offers a sound economic basis for default risk, whereas the reduced-form approach offers no equivalent. In addition, structural modeling provides a natural framework to analyze all types of corporate liabilities, including corporate equity, within a self-consistent framework. For these reasons, we use the structural framework to formulate a relationship between equity and credit-default swap prices.
Valuing Equity As A Barrier Option
A rigorous computation of the equity value in a structural framework is a daunting computational task. Publicly traded companies usually issue a wide variety of debt instruments that have a large range of maturities, different priorities and an array of embedded derivatives. A rigorous valuation of the equity entails valuing the company's entire debt structure, which can be carried out, in principle, by solving one or more partial differential equations with complex boundary conditions. However, if we use the stylized zero-coupon debt structure, then the value of the equity is simply computed using the Black-Scholes formula on call options. Since our goal is to find a back-of-the-envelope approach for equity valuation, it makes sense to turn an actual debt structure into an effective zero-coupon bond.
We also allow for an event of default to occur at any time, instead of only at maturity as in the original Merton model. Relaxing this assumption leads to a more realistic model and does not add relevant complexity. To model an early default we use a constant barrier option model, where an event of early default occurs if the asset value falls below a preset barrier level. The equity can be modeled as a European down-and-out call on the assets of the issuing firm. Barriers can mimic debt covenants. For example, a bank may require the borrowing firm to maintain a minimum level of net worth at all times and may force an event of default if this covenant is breached.
We also assume that the barrier level H is less than or equal to the strike K. Since the strike K is equal to the debt face value, the barrier assumption allows for a fractional debt recovery rate at default time.
The analytical formula for the down-and-out call is composed of three terms: a European call option term, the cost due to early default and a rebate term where the amount R is paid out to the equity holders if the covenant is breached before maturity. The rebate component is positive if stockholders can recoup part of their investment at default time.
The formula for the down-and-out call as applied to equity valuation is:
S( V, K, H, r, T,s ) = CBS( V, K, r, T, s ) ( H/V)(2l-2)
CBS ( H2/V, K , r, T, s ) + R* Reb
where CBS is the standard Black-Scholes call option formula, V is the market value of the assets of the firm, s is the annual asset volatility, r is the annual riskless rate of return and T is the time to debt maturity and l = r/ s2 + 0.5.
The rebate factor is Reb = (H/V)(2l-1) N(d) + (V/H)
N(d-2ls(check)T)
where d = ( ln(H/V) + ( r +s2/2) T ) / (s(check)T) and N is the cumulative normal distribution integral.
CDS Pricing Using Barrier Options
A default swap contract is usually entered by both parties at zero value, just like a plain vanilla interest swap. Pricing a credit-default swap consists of finding a spread at which the present values of the fixed leg and floating leg cancel each other out. The floating leg is the contingent payment made by the credit protection seller to the buyer upon an event of default. To value the floating leg we can use the rebate factor formula multiplied by the loss L given default.
FLT = L * Reb, where L = CDS Notional * (1-Recovery Rate).
The fixed leg is a series of quarterly payments made by the credit protection buyer until the swap matures or a default occurs. To value this leg each payment is discounted at the risk-free rate and multiplied by its risk-neutral probability of no default until the payment is made.
FIX = periodic payment * (sum) d(i) P(i)
where the periodic payment = Notional * CDS spread * time factor, d(i) is the discount factor for payment, µi = (r-s2/2)t(i), HV = ln(H/V) and P(i) is the risk-neutral no-default probability defined as
P(i) = N( (-HV + µi)/ s(check)t(i) )-exp(2 HV µi /s2)
N( (HV + µi)/ s(check)t(i) )
The default swap value to a credit protection seller is
CDS value = FIX-FLT.
Case Study: AOL Time Warner's Stock
And CDS Price
The next step is to test the equity and CDS models using both the market and accounting information of a public company. On Feb. 7, AOL's total equity market value was USD47.6 billion. To compute the total AOL asset value, we estimated the total present value of liabilities and added it to the equity market value. To estimate the value of AOL liabilities we used the 2002 fourth quarter balance sheet statement, which states that the total value of AOL liabilities is USD57.6 billion. The weighted average cost of AOL debt was estimated at 5.4%. We also estimated the average duration of the total debt to be approximately seven years. The present value of liabilities was calculated to be USD39.9 billion, where we made use of the average duration and average debt cost to discount the total value of liabilities. The total value of the assets is therefore USD39.9+USD47.6=USD87.5 billion.
The barrier level was set to the present value of liabilities in accordance with our assumption that the barrier level is less than or equal to the strike price. We derived a riskless rate of return of 4% from the seven-year swap market.
We also chose the rebate to be zero, which assumes that upon default the stock price will be equal to zero.
Trial-And-Error
To determine asset volatility, we used a trial-and-error method to find the implied volatility that is consistent with the total equity market value. The implied annual volatility was found to be 32%, a high value but not unreasonable. Once the equity inputs were determined we moved to the CDS model. To price a swap, we needed to find a spread and a debt recovery rate that are consistent with all equity variables. The spread was obtained from market quotes on AOL: an annual spread of 250 basis points for a five-year contract. Since there is one equation with one unknown, we computed the implied recovery rate that matches the spread. The implied recovery rate is 64.4%. Alternatively, one could have used a different recovery rate, perhaps derived from fundamental analysis, and computed a different spread from the actual market spread. Once all variables for both equity and models were determined, we plotted the spread as a function of the stock price by varying the asset value around its equilibrium value and by recalculating the stock price and default swap spread for each asset value sample. This relationship was computed keeping all other variables constant. In the real world, all input variables are constantly fluctuating due to market and business forces. For instance, if a firm starts risky projects, its asset volatility will increase; or, if it negotiates different debt covenants, then its debt value and expected recovery will change accordingly, and so on. The models must be recalibrated as new information arrives.
Our example shows how to get a quick and dirty relationship between credit-default swap prices and stock prices. More empirical research is needed to assess the size of the error due to a stylized capital structure.
This week's Learning Curve was written by Luigi Vacca,a director in structuring and analytics at American Capital Access in New York.
