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Lecture note 5

Valuation of Corporate Securities

A corporate security has three possible sourcesof value:

Terminal value of the security on itsmaturity date;

Interim cash flows, such as coupon ordividend payments;

Value of the security in the event ofrecapitalization of the firm.

We look at all three sources of value in anoptions context.

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Basic Securities:

Stocks and Zero-Coupon Bonds

Consider a firm that has only two securitiesoutstanding: zero-coupon bonds maturing attime Tand common stock. The total currentmarket value of the firm, V, is

V = S + B,

where S is the current market value of the stockon which no dividends are paid before thematurity date of the bonds, T, and B is thecurrent market value of the zero-coupon bondswith a face value Fpayable at date T.

The contractual payoff of these securities at date

Tcan be stated as

BT= min [VT, F],

= F max [F VT, 0]ST= max [VT F, 0]

= VT F + max [F VT, 0]

Option to default

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Example: Suppose that the current market valueof the firm, V, is 100. The rate of return on theassets of the firm, over one period, is either

u 1 = 20% with probabilityp or d 1 = 20%with probability 1 p. The risk-free rate is 10%.The face value, i.e., the promised payoff at theend of one period, of the (risky) discount bond

issued by the firm is 100. What are the marketvalues of the stock and bond of the firm?

Solution

120

100

80 = [(1 + r) d] / (u d)

= (1.1 0.8) / (1.2 0.8) = 0.75

Bond value = (0.75 x 100 + 0.25 x 80) / 1.1

= 86.36

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Stock value = 100 86.36 = 13.64

= (0.75 x 20 + 0.25 x 0) / 1.1

YTM = 100 / 86.36 1 = 15.79%

Default premium = 15.79% 10% = 5.79%

Riskless bond value = 100 / 1.1 = 90.91

=> Default premium = 90.91 86.36 = 4.55

Coupon Bonds

If a firm has coupon bonds outstanding withmore than one period left until maturity, we canview the common stock as a compound optionor an option on an option.

- Option 1: On the maturity date, the stockholdershave the option to buy the firm from thebondholders for the face value plus the lastcoupon of the bonds

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- Option 2: One period before the maturity date,

the stockholders have the option to buy option 1

by making the second to last coupon payment.

Option 3 Option 2 Option 1

N 2 N 1 N

P Coupon Coupon MatureCoupon + face value

Example: Suppose that the current market valueof a firm, V, is 150. The firm lasts for two periodswith three dates, 0, 1, and 2. The rate of returnon the assets of the firm, over one period, iseither u 1 = 20% with probabilityp, or d 1 =

20% with probability 1 p. The risk-free rate is5% in each period. If any cash payout, couponand/or dividend, is made at time 1, then the ex-payout firm value will change by either u 1 ord 1 over the second period. The firm pays nodividends on the stock at time 1 and paysliquidating dividends at time 2.

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What is the price of a coupon bond that has a

face value of 100 and a coupon rate of 6.02%,

and matures at time 2?

Solution

rf= 5% u = 1.2 d= 0.8

Risk neutral probability = (1.05 0.8) / (1.2 0.8) = 0.625

Bond

180 6.02 = 208.78 106.02

173.98

(100.97) 139.18 106.02

150(100) 120 6.02 = 136.78 106.02

113.98

(95.67) 91.18 91.18

Bond values in (.)

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Value of the coupon bond

YTM = 6.02%

Default premium = 1.02% per period

Value of an otherwise equivalent riskless bond

Default premium = 101.9 100 = 1.9

Callable Bonds

Many corporate bonds have a call provision that

allows the firm to repurchase (i.e., call) the bond

during a specified period (call period) for a pre-

specified price (call price) plus the accrued

coupon since the last coupon date.

The firm calls its bonds either to refinance the

debt at a lower interest cost or to obtain

operating flexibility by removing restrictive

covenants that may impede M&A or other major

investment projects.

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1) general interest rate decreases

2) firms prospects improve => defaultpremium decreases

Immediate call: The call period startsimmediately after the issue.

Rolling call: The call period starts after a callprotection period has elapsed.

Call premium: The call price over and above theface value of the bond.

Declining call schedule: The call premium isdeclining over the life of the bond to reach theface value by maturity.

Under the optimal call strategy, the ex-couponvalue of a callable bond will never exceed thecall price: B < K (A constraint that has to besatisfied throughout the life of the bond.)

If B > K, the firm can sell new bond at a price, B,and buy back its existing bond at K. The firm hasan immediate gain, B K.

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Example: Suppose that the firm pays no

dividends on the stock at time 1. Consider a

callable bond that has the coupon rate of 6.02%

and the face value of 100, and matures at the

end of the second period. The call price on an

ex-coupon basis is 100. What is the market price

of this callable bond?

180 6.02 = 208.78

173.98

139.18

150

120 6.02 = 136.78

113.98

91.18

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Called 106.02 106.99 208.78 106.02

Bu 139.18 106.02

100.97

B (106.02/1.05) 136.78 106.02

99.43

Bd 91.18 106.02

95.67Not called 106.02 101.69 (default)

Call option value = 100 99.43 = 0.57

For par pricing, the coupon rate = 6.56%

Additional coupon = 6.56% 6.02% = 0.54%

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180 6.56 = 208.13

173.44

138.75

150

120 6.56 = 136.13

113.44

90.75

Called 106.56 108.05 208.13 106.56

Bu 138.75 106.56

101.49B (106.56/1.05) 136.13 106.56

100

Bd 90.75 106.56

95.84

Not called 106.56 102.40 (default)

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Convertible Bonds

A convertible bond offers investors the right toconvert the bond into the stock of the issuingfirm during the conversion period at the specifiedconversion terms.

Conversion ratio: The number of shares ofcommon stock into which a bond can beconverted.

Conversion price: The face value of the bond

divided by the conversion ratio. Dilution factor, a: The fraction of the post-

conversion equity value that accrues to thebondholders.

Suppose the firm has 40 shares outstandinginitially and the conversion ratio of theconvertible bond is 60 shares.

The dilution factor is a = 60 / (40 + 60) = 60%

Under the optimal conversion strategy, the valueof the convertible bond will be max [aV, B + c],i.e., the conversion value or the cum-couponvalue of the convertible bond, whichever ishigher (Presumption: Conversion does not carryan accrued coupon).

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Example: Suppose that the firm pays no

dividends on the stock. Consider a convertible

bond that has the coupon rate of 6.02% and the

face value of 100, and matures at time 2. The

bond can be converted into 60% of firm value

either at time 1 or at time 2 (maturity). What is

the market price of this convertible bond?

180 6.02 = 208.78

173.98

139.18

150120 6.02 = 136.78

113.98

91.18

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(No conversion) 108.00 118.45 (converted) 125.27 208.78 106.02

Bu (No conversion) 83.51 139.18 106.02

112.43

B 82.07 136.78 106.02

106.82

Bd 54.71 91.18 106.02

95.67

(No conversion) 72 101.69 (default)

Value of conversion option = 106.82 100

= 6.82

If the firm pays dividend, D, at time 1.

D = 25. D + C= 25 + 6.02 = 31.02

180 31.02 = 178.78

148.98

119.18150

120 31.02 = 106.78

88.98

71.18

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Let n be the number of shares initially

m be the number of warrants

Wbe the warrant price

Xbe the exercise price

Since VT= nST, we have

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= dilution factor

= number of warrants per share outstanding upon

exercise of warrants

Example: Suppose that the firm has only one

share on which no dividends are paid at time 1.

The firm issues one European warrant with an

exercise price of 150. The proceeds are paid out

as dividends to shareholders. The European

warrant expires at time 2.

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Warrant X = 150

19.64

11.69

(0.5 x 23.38) 0