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

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

Warrant issue Initial Balance Sheet Fixed Assets 10,000 Book Equity 10,000 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 9,600 n = 100 shares P0 = €96 Warrant 400 Advanced Finance 2006 Warrant & convertible

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) Advanced Finance 2006 Warrant & convertible

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

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

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

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

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

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

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 Advanced Finance 2006 Warrant & convertible

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)] Bonds + warrants Do not exercise Exercise 1/3 6,000 VT 18,000 6,000 Advanced Finance 2006 Warrant & convertible

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

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

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? Advanced Finance 2006 Warrant & convertible

Binomial Valuation of Convertible Bond Advanced Finance 2006 Warrant & convertible

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

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

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: 6,250 Bondholders convert Current values incorporate force conversion in year 1 Advanced Finance 2006 Warrant & convertible

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

Convertible bond and volatility Advanced Finance 2006 Warrant & convertible

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. Advanced Finance 2006 Warrant & convertible