Question

Horus is now a mature firm and decides to raise funds using debt for the first time. It decides to issue 35,000 senior bonds and 20,000 junior bonds. Both types of bonds are zero-coupon, have a maturity of 1 year and a face value of $1,000. The table below describes the return of the market portfolio rM and Horus’ asset value as a function of the state of the economy. The risk-free rate is 3%. In case of default, that is, if the value of Horus’ assets in one year is too low to cover payments to bondholders, Horus’ assets are liquidated at their market value. 1. Suppose that both categories of bonds carry no systematic risk. How much can Horus raise with this bond issue? (i.e., what price would investors be willing to pay for these bonds?) (10 points) 2. Is it correct to assume that both categories of bonds have no systematic risk? Explain (no computation) (6 points) 3. Taking into account systematic risk, how much can Horus raise with this bond issue? (10 points) Hint: let V denote the value of one bond (for one of the two categories). Compute the β of this bond as a function of V . Then use the CAPM equation. This is a tougher question...

Answer 1

We do not have the asset value of Horus provided in the question. Hence, we assume asset value of A0 as of today and A1 as of 1 year from now.

1. No systemic risk.

When a security has not systemic risk, it could be considered as a risk free security. Hence, the investors will be willing to treat both bond issues as risk free bonds. The present value of each bond is as follows-

Senior bond issue	35,000
Face value	1,000
Senior bond size	35,000,000
Discount rate (Rf)	3%
1 year discounted price for ZCB	33,980,583	=35,000,000/(1+3%)^1
Subordinate bond issue	20,000
Face value	1,000
Subordinate bond size	20,000,000
Discount rate (Rf)	3%
1 year discounted price for ZCB	19,417,476	=20,000,000/(1+3%)^1

Hence, investors would be willing to pay lower of $33.980Mn and firm asset value after 1 year A1 for the senior issue. This is because, in case of liquidation, the firm should have enough assets to cover the liability, even for a risk free bond
Additionally, investors would be willing to pay lower of $19.417Mn and firm asset value after 1 year minus senior debt value i.e. (A1 - 35Mn) for the senior issue. This is because, in case of liquidation, the firm should have enough assets to cover the liability in the order of seniority, even for a risk free bond.

2.
As seen from above case, even if both bond issues have no systemic risk, they are not at par from risk perspective. This is because the firm raising the bonds is not treating them at par and there is a seniority on the claims in case of default. Moreover, whenever there is chance of default, the investment is not truly risk free. If the securities are truly risk free, they must be backed by an asset or entity that has no default or credit risk. Since this is not the case here, the securities can not be treated as risk free. Hence, the bond issues can not be treated as having no systemic risk.

3.
The firm has no debt before the bond issue. Hence the beta of the firm before the debt issue is the delivered beta. Let this be denoted by Bu. This is also called as asset beta.
We have to find the levered beta or equity beta - Be. This will be the sensitivity of returns of the firm after raising the debt and adding leverage
Equity beta= Asset Beta * [1+ (1-tax rate)*(debt/equity)]
Be= Bu * (1+(1-T)*(V/E))
Where- T= tax rate,
V= value of debt

Hence, expected return for the firm by CAPM=
Rp= Rf+ Be*(Rm - Rf)
Where- Rf= risk free rate,
Rm= return on market portfolio,
Be= Bu * (1+(1-T)*(V/E))

Cost of equity-	Re= Bu * (Rm - Rf)
Cost of total capital (levered firm)-	Rp= Rf+ Be*(Rm - Rf)
Portion of debt=	D/(D+E)
Portion of Equity=	E/(D+E)
WACC calculation-	=Re*E/(D+E) + (Rd*1-T)*D/(D+E)
Where, Rd= return of debt
WACC for the firm can be found be using CAPM for levered beta Be
Hence,	Rp=Re*E/(D+E) + (Rd*1-T)*D/(D+E)
i.e.	[Rf+ Be*(Rm - Rf)]=[Re*E/(D+E) + (Rd*(1-T))*D/(D+E)]

We will find the expected cost of debt or return on debt asset from above WACC calculation as-
Rd= {[Rf+ Be*(Rm - Rf)]- [Re*E/(D+E)]}/{(D/(D+E))*(1-T)}

Use the above discount rate Rd to find the PV of the maturity of the bond (V).
PV= V/(1+Rd)^1
The PV calculated above is the price of the bond today.