Advanced Finance 2006-2007 Warrants-Convertible bonds Professor André Farber Solvay Business School Université Libre de Bruxelles

Advanced Finance2006-2007 Warrants-Convertible bonds

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

Advanced Finance 2007 Warrant & convertible |2April 18, 2023

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


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.

Initial Balance Sheet

Fixed Assets 10,000 Book Equity 10,000n = 100 sharesPrice per share P0 = €100

Final Balance Sheet

Fixed Assets 10,000 Book Equity 9,600n = 100 shares P0 = €96

Warrant 400


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 ValueOld 100 2/3 14,000New 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)


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


Value of warrants at maturity

nK12,000 15,000

1,000

q = 1/3

VT

m WT


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 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)


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

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

TTT CqKPMaxmn

nW )1(),0(


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 year622.0

80.025.1

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


Issuing bonds with warrants

• Consider now issuing a zero-coupon 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.


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


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 ExerciseBonds + warrants


Valuation using binomial model

0 1 2V = 23.984E = 15.990D = 6.000mW = 1.995

V = 19.188E = 12.483D = 5.556mW = 1.149

V = 15.350 V = 15.350E = 9.544 E = 9.350D = 5.144 D = 6.000mW = 662 mW = 0

V = 12.280E = 6.724D = 5.556mW = 0

V = 9.824E = 3.824D = 6.000mW = 0

Bonds+Warrants = 5,806

Price / bond = 116

Issuing price (107) undervalued

Market value of equity drops accordingly


Convertible bond

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

• Similar to bond with warrants except:

1. Right to convert cannot be separated from the bond

2. If converted, the bond disappear.

• 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%


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 2V = 23.984E = 15.990D = 7.995

V = 19.188E = 12.483D = 6.705

V = 15.350 V = 15.350E = 9.544 E = 9.350D = 5.806 D = 6.000

V = 12.280E = 6.724D = 5.556

V = 9.824E = 3.824D = 6.000


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.

Source: Ross, Westerfield, Jaffee Chap 22 Table 22.2


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


0 1 2V = 23,984E = 15,990D = 7,995

V = 19,188E = 12,792D = 6,396

V = 15,350 V = 15,350E = 9,722 E = 9,350D = 5,628 D = 6,000

V = 12,280E = 6,724D = 5,556

V = 9,824E = 3,824D = 6,000

Force conversion: example

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) = 6,396Receive 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


Convertible bond and volatility

0

20.000

40.000

60.000

80.000

Variance

Non convertible bond

Convertible bond

Warrants


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 firmes 1971-1990

– Increase in investment and new financing at the time of the calls of convertibles.

Documents

Advanced Finance 2006-2007 Warrants-Convertible bonds Professor André Farber Solvay Business School Université Libre de Bruxelles