Debt Valuation

Debt Valuation In the enterprise model of valuation, the firm's equity value is calculated by subtracting the value of the firm's debt from the enterprise value. Debt valuation then becomes an important component of a valuation of the firm's equity. A company's debt is valued by calculating the payoffs that debt holders can expect to receive, taking into account the risk of default. The default risk is addressed by considering the probability of default and the amount that could be recovered in that event. For modeling purposes, one may assume that the cash flow from the recovered amount is realized at the end of the year of default. Debt valuation may take one of the following two approaches: 1. Discount the expected cash flow at the expected bond return; or 2. Discount the scheduled bond payments at the rating-adjusted yield-to-maturity. Debt Valuation - Method 1 Discount the expected cash flow at the expected bond return Under this method, the value of the bond is the sum of the expected annual cash flows discounted at the expected bond return: Value = the sum for each year t of E(cash flow)t / ( 1 + rdebt )t where E(cash flow)t = expected cash flow in year t. For a one year bond: Value = E(cash flow) / [1 + E(rd)] The expected bond return is the risk-adjusted discount rate, rdebt. The expected cash flow is the cash flow considering the probability of default: E(cash flow) = π ( 1 + C ) F + ( 1 - π ) λ F where

π λ C F

= = = =

probability of no default recovery rate in case of default, (percentage of face value) annual coupon rate of the bond face value of the bond

rdebt can be calculated using the CAPM: rdebt = rf + βdebtΠS&P500

where ΠS&P500 = risk premium for the market portfolio βdebt = covariance between rdebt and the market return; rf = yield to maturity on a risk-free bond having the same maturity. If βdebt is not known, it can be found using ordinary least squares regression. If π = 1 (no default risk), then rdebt = yield to maturity. The difference in rdebt and YTM reflects the default risk. Debt Valuation - Method 2 Discount the scheduled bond payments at the rating-adjusted yield-to-maturity For this method, estimate the rating-adjusted yield-to-maturity (RAYTM) by averaging the market yield-to-maturities (YTM) of bonds in the same group. The promised cash flows then are discounted at this rate that already has factored in the default risk. Markov Chain Representation A firm's debt rating can change over time, and the value of future cash flows should take into account the possibility of one or more rating changes. In this regard, bond valuation can be modeled as a Markov Chain problem in which a transition matrix is constructed for the probabilities of the firm's debt moving from one rating to another. For example, if there are five possible ratings: A, B, C, D, E, and F; and πxy represents the probability of moving from state x to state y, then the transition matrix would look like the following:

πAA πBA πCA πDA πEA

πAB πBB πCB πDB πEB

πAC πBC πCC πDC πEC

πAD πBD πCD πDD πED

πAE πBE πCE πDE πEE

For multiple periods, the transition matrices for each period must be multiplied in order to calculate the multi-period probabilities. This multiplication easily can be performed by spreadsheet software.