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 + 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_debtΠ_S&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  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:

πAA	πAB	πAC	πAD	πAE
πBA	πBB	πBC	πBD	πBE
πCA	πCB	πCC	πCD	πCE
πDA	πDB	πDC	πDD	πDE
πEA	πEB	πEC	πED	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.