QuickMBA / Finance /
**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:

- Discount the expected cash flow at the expected bond return; or
- 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 + r_{debt} )^{t}

where E(cash flow)_{t} = expected cash flow in year **t**.

For a one year bond: Value = E(cash flow) / [1 + E(r_{d})]

The expected bond return is the risk-adjusted discount rate, r_{debt}.

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 = probability of no default | |

λ = recovery rate in case of default, (percentage of face value) | ||

C = annual coupon rate of the bond | ||

F = face value of the bond |

r_{debt} can be calculated using the CAPM:

r_{debt} = r_{f} + β_{debt}Π_{S&P500}

where

Π_{S&P500} = risk premium for the market portfolio | |

β_{debt} = covariance between r_{debt} and the market return; | |

r_{f} = yield to maturity on a risk-free bond having the same maturity. |

If β_{debt} is not known, it can be found using ordinary least squares regression.

If π = 1 (no default risk), then r_{debt} = yield to maturity.

The difference in r_{debt} 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 π_{xy} represents the probability of moving from state *x* to state *y*, then the transition matrix would look like the following:

π_{AA} | π_{AB} | π_{AC} | π_{AD} | π_{AE} |

π_{BA} | π_{BB} | π_{BC} | π_{BD} | π_{BE} |

π_{CA} | π_{CB} | π_{CC} | π_{CD} | π_{CE} |

π_{DA} | π_{DB} | π_{DC} | π_{DD} | π_{DE} |

π_{EA} | π_{EB} | π_{EC} | π_{ED} | π_{EE} |

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.

**Recommended Reading**

Simon Z. Benninga and Oded Sarig, *Corporate Finance* *: A Valuation Approach*

QuickMBA / Finance /
**Debt Valuation**

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