# The pricing of risky corporate debt to be issued at par value.

I. INTRODUCTION

Building on the rational option pricing theories of Black and Scholes (1973) and Merton (1973) suggesting that the option pricing model can be used to price elements of corporate capital structure, Merton (1974) presents the systematic development of a theory for pricing discount bonds when there is a significant probability of default at maturity. In addition, Merton shows that one can derive a risk-structure of interest rates as a function of the debt-to-equity ratio, a measure of the riskiness of the assets of the firm, and the riskless debt rates.

Merton (1974,p. 454) shows that the value of risky discount bond debt can be written:

D(d) = B exp(-rt) {P(d)}

where d = B exp (-rt) / V (a quasi debt-to-asset ratio where the promised payment on debt is valued at the riskless rate of return, r, and P (d) is the per dollar price of risky debt. B exp (-rt) is the present value of the promised payment, B; b is the par value of the bond.

Merton noted that the function P (d) decreases from 1 to 0 as d increases from 0 to [infinity] (its derivative is always negative). Hence if the proceeds at maturity. B were accumulated at the riskless rate of return r, i.e. B = b exp (-rt), then the market value

D(d) = b exp(rt) exp(-rt) P(d) = bP(d) < b

This just states the financially plausible fact that the market value of a risky discount bond is less than par if it promised the investor only the riskless rate of return. In order to obtain market value equal to par value Merton showed that the yield to maturity on risky debt R is given by:.

R = r-(1/t)1n(P(d)

R - r = -(1/t)1n(P(d)

Then if B = b exp(Rt) = b exp(rt)/P(d), it is in fact the case that D(d)= B exp(-rt)P(d) = b exp(rt) exp(-rt)P(d)/P(d) = b as desired. But there is a circular element in this analysis. To compute R in Equation 3 requires the value of d; but d = B exp(-rt)/V can only be computed once B is specified. To put it differently, the quotient P(d)/P(d) is not one since the same value of d does not appear in both the numerator and denominator. A simple example will clarify this paradox and also illustrate the application of Equations 1 and 3. Example:

Let [sigma.sup.2] = .1, t = 5, r = 10%, b = 50 anv V = 100. The volatility of the firm's assets [sigma.sup.2] is a parameter in the computation of the function P(d) and must be specified if we expect to compute the market value of risky debt. If we assume the bond promises the investor only the riskless rate of 10%, then the proceeds at manturity B = 50 exp(.10*5) = 82.436. The quasi debt-to-asset ratio is d = B exp(-rt)/V = B exp(rt) exp(-rt)/V = b/V = 50/100 = .5.P(.5) = .917 so that equation 1 gives a market value:

D(d) = 82.436 exp(-.10*5) (.917) = 45.84

which is less than the par value of 50. From Equation 3

R = .10-(1/5)1n(.917) = .11736

or the risk premium is 1.736%. The proceeds B at the risky rate of 11.736% are equal to 50 exp(.11736*5) = 89.811. When we substitute this value from B in Equation 1 we still do not get the par value of 50 for the market value D(d) = B exp(-rt)P(d) because although the factor B exp(-rt) = 89.911 exp(-.1*5) = 54.433 is now larger, the factor P(d) is now smaller than .917 since d is no longer equal to .5 but rather:

d = B exp(-rt)/V = 54.533/100 = .54533

This larger value of d yields a smaller value of P(d) since recall P(d) is a decreasing function. The final value for D(d) that is obtained is close to 49; this is larger than the original value of 45.84 but still less than the par value of 50. Hence the question remains: can we find a value for the proceeds B at maturity of the risky bond debt such that for this value B the market value D(d) given by equation 1 will exactly equal b, the par value of the risky debt?

II. THE METHODOLOGY OF SUCCESSIVE APPROXIMATIONS OR

ITERATIONS

The key to our procedure is the observation that using the risky rate R = 11.736% lead to a larger value of the proceeds B, which in turn lead to a larger market value D(d) when substituted in Equation 1. Why not repeat this algorithm? Use the larger value of d(=.54533) to compute a new (larger) value of R in Equation 3, which in turn should yield a still larger value of the proceeds B at maturity. Substitute this B into Equation 1 to get a larger market value D(d), which, however, will still be less than par value. Note also that the larger B value led to a larger d (quasi debt-to-asset) ratio, which in turn can be used to generate a still larger value of R from Equation 3. Continue this iterative algorithm. In our example, just the 6th iteration produces the following excellent approximations: R(6) = 12.222% B(6) = 92.126; d(6) = .55877; P(d(6)) = .895 and D(6) = 49.9999, which is only .0001 less than par value of 50. Thus the risky discount bond must promise a yield of 12.222% or a risk premium of 2.222% in order to issue the bond at par. (These results are consistent with Merton (1974) Table 1 and Figure 1, accounting for the value of the quasi debt/asset value ratio, d. For d = .55877, ceteris paribus, Merton's model would show R - r = 2.222%. Figure l illustrates the successive iteration process for [sigma] = .5 and t = 36. The convergence of the successive approximations is somewhat slower than in our simple example but nevertheless still inexorable. D(20) is essentially equal to par value of 50. Figure 2 illustrates the monotonically decreasing sequence P(d(n)) of price per dollar of risky bond debt for an increasing sequence debt-to-asset ratios. Figure 3 illustrates the Risk Premium as a function of d-Ratio and Figure 4 as a function of the Bond at Par in % of Assets.

[TABULAR DATA OMITTED]

Formal Iterative Procedure

The formal iterative procedure will now be briefly described.

CASE 1.

Step 1: Choose d[0] = 0

Step 2: For n [greater than or equal to] 1, choose

R[n] = r-(1/t)1n(P(d[n-1]))

B[n] = b exp(R[n]t)

d[n] = B[n] exp(-rt)/V

D[n] = B[n] exp(-rt) P(d[n]) = bP(d[n])/P(d[n - 1])

It is possible to rigorously proof that each of the sequences R[n], B[n], d[n] and D[n] is monotonically increasing and convergent. The key point is to show that d[n] converges to the fixed point of a contraction mapping. Once it is established that d[n] [right arrow] d#, it easily follows that both Rn] and B[n] converge and finally that the successive iterations of the market price D[n] satisfy

D[n] [right arrow] bP(d#)/P(d# = b as n [right arrow] [infinity]

III. SOME APPLICATIONS

The primary purpose of this method is to show how to apply Merton's model for pricing risky corporate debt in order to issue a discount bond at par. There are other interesting applications of the results. We mention two applications here, using Table l which is calculated based on the iterative methodology described above.

APPLICATION I. The Implied Volatility of Untraded Securities.

There are times when it is useful to estimate the volatility of the assets of a firm whose securities are not publicly traded. If you believe that commercial banks (arguably) efficiently price their loans to small or untraded business firms, then you can use the results of a computer run to estimate sigma. Here is a hypothetical example. Suppose a small finn has a bank loan that is 20% of their assets. The bank offers a time loan for five years at 1.875% over prime and prime is 2% over the stripped 5-year Treasury zero coupon note. This is, in effect, a 3.875% risk premium. Using Table 1 the implied sigma is about.6.

Case 1 - Selling subordinated debt. Suppose a publicly traded firm has total debt at par that is 50% of asset value and wants to sell an additional 10% to retire equity. If its sigma is .2, then the additional debt, subordinated to all other debt, would require a risk premium of about 3.42%. One need use only Table 1 and solve for x

10(x + r) = 60(.868 + r) - 50(.356 + r)

Case 2 - High yield bond. Suppose a takeover is to be financed by a relatively large subordinated bond. If the current debt is 40% of assets, and an additional 20% debt is needed to fund the takeover, and if the firm has sigma of .3, then the new debt should promise about 7.7% premium. This is in the ball park for a junk bond yield. To obtain this result, again use Table 1 and solve:

20(x +_r) = 60(3.211 + r) - 40(.937 + r)

Case 3 - Commercial paper. For publicly traded firms with sigma in the .1 to .3 range, it is seen that they can offer commercial paper with a very small premium over the riskless rate for relatively high amounts of debt that has a high priority position in case of financial distress.

IV. SUMMARY AND CONCLUSIONS

This paper builds on the seminal work of Merton (1973, 1974) to extend the pricing of risky corporate debt to determine the promised payment and the promised yield to maturity in order to issue a discount bond at par.

We use an iterative method. The sequence of successive approximations for the quasi debt-to-asset value of the firm is convergent, and the resulting sequence of successive approximations for the market value of a risky discount bond approaches par value within a reasonable value, say .0001, for a finite number of approximations. The rate of return required to issue the risky discount bond at par is calculated.

A similar method of successive approximations can be extended to determine the required promised coupon rate for coupon-bearing securities.

APPENDIX

Assumptions Concerning the Valuation of Securities

1. Every individual acts as if he can buy or sell as much of any security as he wishes without affecting the market price. 2. There exists a riskless asset paying a known interest rate, r. 3. Individuals may take short positions in any security including the riskless asset, and receive the proceeds of the sale. 4. Trading takes place continuously. 5. There are no taxes, indivisibilities, bankruptcy costs, transactions costs, or agency costs. 6. The value of the assets of the firm follows a diffusion process with instantaneous variance proportional to the square of the value.

NOTES

(*) Direct all correspondence to: Franklin Lowenthal, California State University, Department of Accounting, Hayward, CA 94542.

REFERENCES

Black, Fischer and Myron Scholes. 1976. "The Pricing of Options and Corporate liabilities." Journal of Political Economy, 81(May-June): 637-659. Black, Fischer and John Cox. 1976. "Valuing Corporate Securities: Some Effects of Bond Indenture Provisions." Journal of Finance, 31(May): 351-368. Cox, John C. and Mark Rubinstein. 1985. Options Markets. Englewood Cliffs, NJ. Prentice-Hall. Kellison, Stephen G. 1975. Fundamentals of Numerical Analysis. Homewood, IL. Irwin. - 1974. "On The Pricing of Corporate Debt." Journal of finance, 29(may): 449-470. Merton, Robert C. 1973. "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science,