Corporate Valuation and Financing Convertibles and Warrants

Profs. André Farber & Hugues Pirotte

2 Prof H. Pirotte

H. Pirotte 3

Remember the binomial model for bond prices…

492.1 teu 670.1

ud

462.670.0492.1

67.05.11

du

drp

f

Original Data Market Value of Unlevered Firm: 100,000 Risk-free rate per period: 5% Volatility: 40%

Contract Company issues 1-year zero-coupon Face value = 70,000 Proceeds used to pay dividend or to buy back shares

f

du

r

fppff

1

)1(

V = 100,000 E = 34,854 D = 65,146

V = 67,032 E = 0 D = 67,032

V = 149,182 E = 79,182 D = 70,000

∆t = 1

Binomial option pricing: review Up and down factors:

Risk neutral probability :

1-period valuation formula

05.1

032,67538.0000,70462.0 D

0.462 79,182 0.538 0

1.05E

From there... The “binomial tree” technique can be used to articulate any final

payoff function based on the same underlying, i.e. the value of the firm

Since any financing instrument is a contract defining a payoff sharing function of the assets of the firm between the various claimholders, we might use this technology to value: » Subordinated debt

» Convertibles

» Warrants

» Etc...

Prof H. Pirotte 4

Two special mezzanine products Convertible bonds

» You can convert your bonds into equity, based on a predefined “strike”

» The option to convert is “embedded” into the product

Bonds + warrants » You can trade separately the warrants from the bond.

» The warrant is in this case a call option like any other...

Special difficulty » Exercising the convertibles or the warrants implies an issuance of new

shares, so some “dilution” that has to be valued into the convertibles.

Prof H. Pirotte 5

Payoff functions Convertibles

» 3 payoffs potentially at the end:

High case: A fraction of VT

Medium case: F

Default case: V

» Take the max of 0, F, qVT where q = m/(m+n)

Warrants » Same idea, but the value of the bond itself must be considered separately

and prior to the warrant.

» Take the max of 0, qVT - F

Prof H. Pirotte 6

LET’S START WITH WARRANTS…

• Be careful about the “warrant” terminology:

• A call option attached to a bond issue, issued directly by the firm: its exercise assumes the creation of extra shares.

• A call or put option on the stock of a firm issued by a third party and cash-settled, with no direct link to the underlying firm itself: its exercise is analog to a bet organized by an independent third-party. It is just an option issued by an intermediary as a bet on the original firm.

Prof H. Pirotte 7

H. Pirotte 8

Warrants Give to its owners the right to buy new shares issued by the

company during a period of time at a price set in advance.

Most of the time, warrants are issued with bonds A price is the set for a “package” bond + warrant(s)

Later on, both components are traded separately

Warrants are similar to call option except for two differences: 1. Warrants are sold by companies

2. If exercised, new shares are created

Note: “warrants” are also long term (maturity 2-5 years) call options sold by financial institutions

H. Pirotte 9

Warrant issue

Company issues m = 50 warrants

Maturity = 2 years

Exercise price K = €120/share

Issue price = €8/warrant

Proceed of issue (400 = 50 * 8) paid out to shareholders as a dividend.

Assets Liabilities

Fixed Assets 10,000 Book Equity (n = 100 shares P0 = €100)

10,000

Initial Balance Sheet

Assets Liabilities

Fixed Assets 10,000 Book Equity (n = 100 shares P0 = €96)

9’600

Warrant 400

Final Balance Sheet

H. Pirotte 10

What happens at maturity? Suppose market value of company at maturity is VT = 15,000

If warrant exercised: » Company issues 50 new shares

» Receives 50 x 120 = 6,000 in cash

» Market value of company becomes:

VT + m * K = 15,000 + 6,000 = 21,000

» Allocation of shares Type Number Percentage Value

Old 100 2/3 14,000 New 50 1/3 7,000

» Gain for warrantholders = Value of shares – Price to pay

= m * PT - m * K

= 50 * 140 – 50 * 120

= 1,000 (20/warrant)

H. Pirotte 11

To exercise or not to exercise? If they exercise, warrantholders own a fraction q of the shares

» q = Number of new shares / Total number of shares

= m / (m+n)

They should exercise if the value of their shares is greater than the price they have to pay to get them: Exercise if: q (VT + m K) > m K

q VT > (1-q) m K

VT > n K

In previous example, exercise if: VT > 100 * 120 = 12,000

H. Pirotte 12

Value of warrants at maturity

nK

12,000 15,000

1,000

q = 1/3

VT

m WT

H. Pirotte 13

Warrants compared to call options Consider now 100 calls on the shares with exercise price 120.

They will be exercised if stock price > 120

Value of (all) warrants at maturity = 1/3 value of calls » 50 WT = (1/3) * Max(0, VT – 12,000)

In general: » m WT = q Max(0,VT – n K)

1,000

3,000

12,000 15,000 VT

100 Calls

50 Warrants

Proof:

m WT = Max[0, q(VT+mK)-mK]

= Max[0, qVT – m(1-q)K]

= q Max(0,VT – nK)

H. Pirotte 14

Valuing one warrant at maturity m WT = q Max(0,VT – n K)

» As: VT = n PT » and: q = m/(m+n) » we get:

The value of one warrant at maturity is equal to the value one call option multiplied by an adjustment factor to reflect dilution.

In the previous example, for VT = 15,000: » PT = 150 » CT = 150 – 120 = 30 » WT = (1 – 1/3) 30 = 20

(0, ) (1 )T T T

nW Max P K q C

n m

H. Pirotte 15

Current value of warrant 2 steps:

1. Value a call option

2. Multiply by adjustment factor 1-q

Back to initial example. Assume volatility of company = 22.3%

Use binomial option pricing with time step = 1 year

622.080.025.1

80.008.11 80.0

1 25.1223.0

du

drp

udeeu t

0 1 2 Call

156 36

125

100 100 0

80

64 0

Evolution of stock price

Call = (0.622)² (36)/(1.08)² = 11.94

Warrant = (1-q) C = 7.96

H. Pirotte 16

Issuing bonds with warrants Consider now issuing a zero-coupon bond with warrants.

» Face value 6,000

» Number of bonds 50

» Maturity 2 years

» 1 warrant / bond

Maturity 2 years

Exercise price 120

» Issue price 107

» Proceed from issue 5,350 (=50 * 107)

Suppose that the issue is used to buy new assets.

H. Pirotte 17

To exercise or not to exercise? Suppose VT = 21,000

If warrants exercised, value of equity after repaying the debt is: » VT – F + m K = 21,000 – 6,000 + 6,000 = 21,000

As previously, warrantholders own a fraction q (=1/3) of equity.

Their gain is: » q (VT – F + m K) – m K = (1/3)(21,000) – 6,000 = 1,000

Conclusion: exercise if: q (VT – F + m K) > m K

VT > [(1-q)/q] m K + F

VT > n K + F

H. Pirotte 18

Example In our example, warrant will be exercised if:

» VT > 100 * 120 + 6,000 = 18,000

The value of all warrants is equal to 1/3 of the value of 100 calls with strike price equal to 180 » m WT = q Max[0, VT – (nK+D)]

6,000

6,000

18,000 VT

1/3

Do not exercise Exercise Bonds + warrants

Valuation using binomial model 0 1 2

V = 23'984

E = 15'990

D = 6'000

mW = 1'995

V = 19'188

E = 12'483

D = 5'556

mW = 1'149

V = 15'350 V = 15'350

E = 9'544 E = 9'350

D = 5'144 D = 6'000

mW = 662 mW = 0

V = 12'280

E = 6'724

D = 5'556

mW = 0

V = 9'824

E = 3'824

D = 6'000

mW = 0

19 H. Pirotte

Bonds+Warrants = 5,806

Price / bond = 116

Issuing price (107)

undervalued

Market value of equity drops

accordingly

CONVERTIBLES…

• There is a whole theory about why firms issue convertibles…

• In the last decade, there is also a new reason why convertibles were “in the radar”; they were used by Hedge Funds for Convertible Arbitrage.

Prof H. Pirotte 20

H. Pirotte 21

Convertible bond A bond with a right to convert into a number of shares.

Similar to bond with warrants except: » Right to convert cannot be separated from the bond

» If converted, the bond disappears.

Back to previous example: » Current stock price = 100 (number of shares n = 100)

» Issue 50 zero-coupon convertible with face value 120

» Each bond is convertible into 1 share

Conversion ratio = # shares/ bond = 1

Conversion value = Conversion ratio * Stock price = 100

Conversion price = Face value/Conversion ratio = 120

Conversion premium = (Conversion price – Stock price)/(Stock price) = 20%

H. Pirotte 22

Valuing the convertible bond Valuation similar to valuation of bond with warrants.

Value 5,806 » Straight bond 5,144

» Conversion right 662

Yield to maturity on convertible bond: » Solve

Is this cheap debt?

%66.1)1(

000,6806,5

2

y

y

Binomial Valuation of Convertible Bond

0 1 2

V = 23.984

E = 15.990

D = 7.995

V = 19.188

E = 12.483

D = 6.705

V = 15.350 V = 15.350

E = 9.544 E = 9.350

D = 5.806 D = 6.000

V = 12.280

E = 6.724

D = 5.556

V = 9.824

E = 3.824

D = 6.000

23 H. Pirotte

No free lunch! If Firm Subsequently Does Poorly

If Firms Subsequently Prospers

Convertible bonds (CBs) Compared to:

No conversion because of low stock price

Conversion because of high stock price

Straight bonds CBs provide cheap financing because coupon rate is lower

CBs provide expensive financing because bonds are converted which dilutes existing equity

Common stock CBs provide expensive financing because firm could have issued common stock at high price

CBs provide cheap financing because firm issues stock at high price when bonds are converted.

24 H. Pirotte

Source: Ross, Westerfield, Jaffee Chap 22 Table 22.2

H. Pirotte 25

Conversion Policy Convertible bonds are very often callable by the firm.

If bond called, holder of convertible can choose between: » Converting the bond to common stock at the conversion ratio.

» Surrendering the bond and receiving the call price in cash.

Convert if conversion value greater than call price (force conversion)

In theory: » companies should call the bond when conversion value = call price

Empirical evidence: » Bonds called when conversion value >> call price

Force conversion: example

0 1 2

V = 23,984

E = 15,990

D = 7,995

V = 19,188

E = 12,792

D = 6,396

V = 15,350 V = 15,350

E = 9,722 E = 9,350

D = 5,628 D = 6,000

V = 12,280

E = 6,724

D = 5,556

V = 9,824

E = 3,824

D = 6,000

26 H. Pirotte

Assume convertible callable in year 1

Call price = 125

Total call value = 6,250

Firm’s decision:

If not called: D = 6,705 > 6,250

Firm calls CBs

Bonholder’s decision:

Convert: (1/3)(19.188) = 6,396 Receive call price: 6,250

Bondholders convert

Current values incorporate force conversion in year 1

Why Are Warrants and Convertible Issued?

Companies issuing convertible bonds » Have lower bond rating than other firms

» Are smaller with high growth opportunities and more financial leverage

Possible explanations: » Matching cash flows

Low intial interest costs when cash flows of young risky and growing company are low

» Lower sensitivity to volatility of firm

If volatility increases: straight bond but warrants

– Protection against mistakes of risk evaluation

– Mitigation of agency costs

27 H. Pirotte

28 Prof H. Pirotte

Convertible bond and volatility

29 H. Pirotte

H. Pirotte 30

Matching financial and real options Ref: Mayers, D., Why firms issue convertible bonds: the matching of

financial and real options, Journal of Financial Economics 47 (1998) pp.83-102

Sequential financing problem: investment option at future date

Providing fund up front for both initial investment and investment options difficult because of overinvestment (free-cash flow) problem

Issuing security is costly: avoid multiple issues

Convertible bonds are a solution: » Leaves funds in the firm if investment option valuable » Funds returned to bondholders if investment option not valuable » Call provision allows to force the financing plan when investment option

valuable

Empirical evidence: call of convertible debt by 289 firms 1971-1990 » Increase in investment and new financing at the time of the calls of

convertibles.

Convertible Bond Arbitrage How does it work?

Prof H. Pirotte 31

Other types Automatic convertibles

Prof H. Pirotte 32