1. Chapter 12: On the Pricing of Corporate Debt
XingluanHuang
Department of Finance, XMU
2018/11/23

2. 12.1 Introduction
The value of corporate debt depends on essentially on
The required rate of return of riskless debt;
The various provisions and restrictions contained in the indenture;
The probability of default.
The purpose of this chapter is to present a theory of the risk structure of interest rates.
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3. 12.2 On the pricing of corporate liabilities
Some assumptions:
There are no transactions costs, taxes, or problems with indivisibilities of assets.
There are sufficiently many investors with comparable wealth levels.
There exists an exchange market for borrowing and lending at a same rate of interest.
Short-sales of all assets, with full use of the proceeds is allowed.
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4. Assumptions (con.) Trading in assets takes place continuously in time.
The MM theorem obtains.
The term structure is “flat” and known with certainty.
The dynamics for the value of the firm, V, can be describer as:
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5. Where is the instantaneous expected rate of return on the firm per unit time;C is the total dollar payouts by the firm per unit time to either its shareholders or liabilities-holders if positive, and it is the net dollars received by the firm from new financing if negative; is the instantaneous variance of the return on the firm per unit time;and dz is a standard Gauss-Wiener process.
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6. Suppose there exists a security whose market value Y =F(V, t).
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7. Comparing terms in (12.2) and (12.1)
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8. Forming a three-security portfolio
Let:
W1 be the number of dollars of the portfolio invested in the firm;
W2 be the number of dollars invested in the particular security;
W3 (= -W1+W2) be the number of dollars invested in riskless debt.
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9. If dx is the instantaneous dollar return to the portfolio ,then
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10. Choosing W*
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11. Substituting for
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12. Remark on (12.7)
Equation (12.7) must be satisfied by any security whose value can be written as a function of the value of the firm and time.
Note that F does not depend on the expected rate of return on the firm nor on the risk preferences of investors nor on the characteristics of other assets available to investors beyond the three mentioned.
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13. 12.3 On the pricing of “risky” discount bond
Suppose the corporation has two classes of claims: (a) A single homogeneous class of debt and (b) The residual claim, equity.
And suppose :
The firm promises to pay a total of B dollars on date T.
In the event that this payment is not met, the bondholders take over the company.
The firm cannot issue any new senior claims on the firm nor can it pay cash dividends or do share repurchase prior to the maturity of the debt.
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14. If F is the value of the debt issue and f is the value of the equity, we have that:
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15. The value of equity and boundary conditions (12.9a) and (12.9b)
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16. Comparing (12.10) with B-S-M differential equation
Inspection of the B-S equation or Merton shows that (12.10) and (12.11) are identical with the equation for a European call option on a non-dividend-paying common stock where firm value in (12.10) -(12.11) corresponds to stock price and B corresponds to the exercise price.
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17. Black-Scholes formula
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18. From (12.12) and F=V-f, we have that:
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19. is the yield to maturity.
It is common in discussions of bond pricing to talk in terms of yields rather than price, we can rewrite (12.13) as:
is the yield to maturity.
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20. Derivation of (12.13) and (12.14)
we get (12.13) and will have that (12.14)
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21. Review on (12.14)
It seems reasonable to call a risk premium in which case (12.14) defines a risk structure of interest rates.
For a given maturity, the risk premium is a function of only two variables:(a) the variance of the firm’s operations; and (b) the ratio of the present value of the promised payment to the current value of the firm.
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22. 12.4 A comparative statics analysis of the risk structure
The value of debt can be written as
we can show that F is a first-degree homogeneous concave function of V and B. Further we have that :
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23. The implications of (12.15)
The value of the debt is an increasing function of the current market value of the firm and the promised payment at maturity, and a decreasing function of the time to maturity, the business risk of the firm, and the riskless rate of interest.
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24. From (12.13) we have that:
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25. Derivation
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26. Another ratio
From the no-arbitrage condition, we have that
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27. Derivation of (12.19)
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28. Study P(d,T)
We can rewrite (12.17) and (12.18) in elasticity form in terms of g as
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29. Statics analysis If we define , then from (12.14) we have that
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30. 0 “Quasi” Debt-to-Firm Value Ratio d
Figure 12.1:Term Premium vs “Quasi” Debt-to-Firm Value Ratio
“Quasi” Debt-to-Firm Value Ratio d
Term Premium
R-r
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31. Figure 12.2:Term Premium vs Variance of the Firm
R-r
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32. Figure 12.3:Term Premium vs Time to Maturity
R-r
Term until Maturity
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33. H and r
To complete the analysis of the risk structure as measured by the term premium, we show that the premium is a decreasing function of the riskless rate of interest, i.e.
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34. An important question
Can one assert that if R-r is larger for one bond than for another, then the former is riskier than the latter?
To answer this question, one must first establish an appropriate definition of “riskier”.
The natural choice as a measure of risk is the standard deviation of the return on the bond,
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35. From the definition of G and (12.19), we have that:
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36. 12.5 On the MM theorem with bankruptcy
The fundamental equation for pricing of corporate debts (12,7) is derived on the assumption that MM theorem held. Because of bankruptcy costs or corporate taxes, the MM theorem does not obtain and the value does depend on the debt-equity ratio, then the formal analysis of the chapter is still valid. However, the linear property of (12.7) would be lost, and a nonlinear simultaneous solution would be required.
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37. Examining a debt issue for single firm
As discussed in section 12.4, the correct measure of this relative riskiness is
From (12,16) and (12.19), we have that
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38. Implications
The debt of the firm can never be more risky than the firm as a whole, and as a corollary, the equity of a levered firm must always be at least as risky as the firm. In particular, from (12.13) and (12.31), when As
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39. Statics analysis of g
Noting that ,we have from(12.19) and (12.27) that
That is, the relative riksiness of the debt is an increasing function of d and
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40. Further study of g Further, we have that
Thus, for d=1, independent of the business risk of the firm or the length of time until maturity, the standard deviation of the return on the debt equals half the standard deviation of the return on the whole firm.
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41. A classic problem in corporate finance
Given a fixed investment decision, how does the required return on debt and equity change, as alternative debt-equity mixes are chosen?
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42. Definition
Define the market debt-to-equity to be X which equal to F/f=F/(V-F).
From (12.20), the required expected rate of return on the debt:
Thus, for a fixed investment policy,
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43. Derivation of (12.38)
Form the definition of X and (12.13), we have that
Since we have from (12.32),(12.36), and (12.37) that
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44. Further analysis of (12.38)
starts out as a convex function of X, passes through an inflection point where it becomes concave and approaches asymptotically as X tends to infinity.
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45. Determine the path of the required return on equity
Has a slope of at X=0 and is a concave function bounded from above by the line
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46. 0 Market debt to equity ratio X
Expected return
0 Market debt to equity ratio X
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47. 12.6 On the pricing of risky coupon bonds
In the usual analysis of default-free bonds in term structure studies, the derivation of a pricing relation for pure discount bonds for every maturity would be sufficient. However, no such simple formula exists for risky coupon bonds.
The apparatus developed in the previous sections is sufficient to solve the coupon bond problem.
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48. Assumption
Assume the same simple capital structure and indenture conditions as in section 12.3 except modify the indenture condition to require payments at a coupon rate per unit time, .
From the indenture restriction( c ),we have that, in equation (12.7),
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49. New PDE
Hence the coupon bond value will satisfy the partial differential equation
subject to the same boundary conditions (12.9). The corresponding equation for equity f will be:
subject to boundary conditions (12.9a),(12.9b),and (12.11)
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50. Solution of (12.41)
Equation(12.41) is identical with equation for the European option value on a stock which pays dividends at a constant rate per unit time of
Using the identity F=V-f, we can write the solution for the perpetual risky coupon bond as:
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51. A case of callable bonds
Assume the same capital structure but modify the indenture to state that “the firm can redeem the bonds at its option for a stated price of dollars” where K may depend on the length of time until maturity.
Equation (12.40) and boundary conditions (12.9a) and (12.9c) are still valid. However, instead of the boundary condition(12.9b), we have another new boundary condition:
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52. Solution of F
Since the function of is not known, we couldn't solve for F. fortunately, economic theory is rich enough to provide us with an answer.
See the detail on page 411.
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