corporate valuation and financing...h. pirotte 3remember the binomial model for bond prices… u...
TRANSCRIPT
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