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

2. 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:
Warrants are sold by companies
If exercised, new shares are created
Note: “warrants” are also long term (maturity 2-5 years) call options sold by financial institutions
Advanced Finance 2006 Warrant & convertible

3. Warrant issue Initial Balance Sheet
Fixed Assets 10,000 Book Equity 10, n = 100 shares Price per share P0 = €100
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.
Final Balance Sheet
Fixed Assets 10,000 Book Equity 9, n = 100 shares P0 = € Warrant
Advanced Finance 2006 Warrant & convertible

4. 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 = 21,000
Allocation of shares
Type Number Percentage Value Old /3 14,000 New /3 7,000
Gain for warrantholders = Value of shares – Price to pay
= m * PT - m * K = 50 * 140 – 50 * 120
= 1,000 (20/warrant)
Advanced Finance 2006 Warrant & convertible

5. 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
Advanced Finance 2006 Warrant & convertible

6. Value of warrants at maturity
m WT
q = 1/3
1,000
nK 12,000 VT
15,000
Advanced Finance 2006 Warrant & convertible

7. 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)
100 Calls
3,000
1,000
50 Warrants
12,000
15,000 VT
Proof: m WT = Max[0, q(VT+mK)-mK] = Max[0, qVT – m(1-q)K] = q Max(0,VT – nK)
Advanced Finance 2006 Warrant & convertible

8. 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
Advanced Finance 2006 Warrant & convertible

9. Current value of warrant
2 steps:
Value a call option
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
Evolution of stock price 1 2
Call
156 36
125
100 80 64
Call = (0.622)² (36)/(1.08)² = 11.94
Warrant = (1-q) C = 7.96
Advanced Finance 2006 Warrant & convertible

10. 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
Exercise price 120
Issue price 107
Proceed from issue 5,350 (=50 * 107)
Suppose that the issue is used to buy new assets.
Advanced Finance 2006 Warrant & convertible

11. 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 = 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
Advanced Finance 2006 Warrant & convertible

12. Example In our example, warrant will be exercised if:
VT > 100 * ,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)]
Bonds + warrants
Do not exercise
Exercise
1/3
6,000 VT
18,000
6,000
Advanced Finance 2006 Warrant & convertible

13. Valuation using binomial model
Bonds+Warrants = 5,806
Price / bond = 116
Issuing price (107) undervalued
Market value of equity drops accordingly
Advanced Finance 2006 Warrant & convertible

14. 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 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%
Advanced Finance 2006 Warrant & convertible

15. Valuing the convertible bond
Valuation similar to valuation of bond with warrants.
Value 5,806
Straight bond 5,144
Conversion right
Yield to maturity on convertible bond:
Solve
Is this cheap debt?
Advanced Finance 2006 Warrant & convertible

16. Binomial Valuation of Convertible Bond
Advanced Finance 2006 Warrant & convertible

17. 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
Advanced Finance 2006 Warrant & convertible

18. 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
Advanced Finance 2006 Warrant & convertible

19. 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) = 6,396 Receive call price: ,250
Bondholders convert
Current values incorporate force conversion in year 1
Advanced Finance 2006 Warrant & convertible

20. 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
Advanced Finance 2006 Warrant & convertible

21. Convertible bond and volatility
Advanced Finance 2006 Warrant & convertible

22. 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
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
Increase in investment and new financing at the time of the calls of convertibles.
Advanced Finance 2006 Warrant & convertible