高级公司金融学讲义(黄小平)

t=0 r V0=42 1-r
SWUFE 2006 James Huang
t=1 V1=63
V1=28
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Shareholders have the control over the assets of the firm. -When value of assets V1 <50 => Shareholders get nothing; everything will be paid out to the bond holders. -When value of assets V1 >50 => Every additional gain (above 50) on the value of the firm is given to the shareholders.
c = V0 N ( d1 ) − e − rt KN ( d 2 )
42 50
1 ln + (0.1 + 0.4352 ) 2 where d1 = = 0.046573 0.435 d 2 = d1 − 0.435 = −0.38843
=>
SWUFE 2006 James Huang
c=6.00
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Stockholders' Position
•If debt must be paid back immediately: => => Company is bankrupt (42<50) Bond holders receive 42 Shareholders receive 0 Shares value = 0
SWUFE 2006 James Huang
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Shareholders' Position
Payoff (shareholders)
150
Total assets V1
SWUFE 2006 James Huang
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As if the total assets are to be divided into 10 shares; the shareholders receive 10 calls (c1) on those shares with an exercise price of K1=150/10= 15 (per call). one share=one call v0=V0/10=165/10=16.5 s=0.40, t=1, K1=15
50
Total assets V1
A written call
SWUFE 2006 James Huang
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An Alternative View
Put-call parity: c=p+V0-PV(K) In the context of stockholders of a company debt financed, this becomes: Value of call on firm's assets = Value of put on firm's assets + Asset value - PV(bond) => Bond Value = Asset value - call value = PV(bond) - put value
1 ln 16.5 + (0.1 + 0.402 ) 15 2 d1 = = 0.688275 0.40 d 2 = d1 − 0.40 = 0.288275
=>
=>
SWUFE 2006 James Huang
c1 = v0 N ( d1 ) − e − rt K1 N ( d 2 ) = 16.5 * N (0.688275) − e −0.1 * 15 * N (0.288275) = 4.12
Payoff (shareholder)
B Total assets V1 -B
SWUFE 2006 James Huang
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Issuing Senior and Junior Debts
E.g., The company issues: -Senior (=preferential) debt: nominal value: 50, time to maturity: 1 year -Junior (=subordinated) debt: nominal value: 100, time to maturity: 1 year -Share capital: 10 shares Assume that the current value of total assets is 165, the standard deviation of the asset return is s=0.40 per year, and the risk free rate is r=10% per year.
Assume r=10% => r=(1.1-2/3)/(1.5-2/3)=0.52 c=0.52*(63-50)/1.1=6.15
SWUFE 2006 James Huang
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Value of the Call: Black-Scholes model V0=42; K=50; r=10% (continuous); s=0.435; t=1 =>
Payoff (shareholder)
Debt Book Value
SWUFE 2006 James Huang
total assets
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Value of the Call: Binomial Model t=0 r V0=42 1-r V0d=28 t=1 V0u=63 u=1.5 d=2/3 cu=13 cd=0
V0 – c = PV(B) - p
SWUFE 2006 James Huang
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Conclusion: The Bond holders have bought a safe bond and given the stockholders a put option to sell them the firm's assets for the amount of debt
Advanced Corporate Finance
Lecture 5
Valuation of Company Assets & Liabilities
James Huang SWUFE 2006
SWUFE 2006 James Huang
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Topics
Spotting the options Valuation of junior debt Valuation of senior debt Value of limited liability
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Bondholders' Position
50, Min{V1, 50} = V1 , if V1>50 if V1£50
Min{V1, 50} =V1 -Max{V1-50, 0}
Total assets Payoff Payoff to bondholders
50
Debt value: B0=V0-c=42-6=36
Reference: Robert C. Merton (1974), On the Pricing of Corporate Debt: the risk structure of Interest Rates, Journal of Finance, 29, p449-470.
SWUFE 2006 James Huang
SWUFE 2006 James Huang
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Spotting the Options
Balance Sheet
B a la n c e S h e e t (b o o k v a lu e ) B a la n c e S h e e t (m a rk e t v a lu e )
Assets: 100 Bonds: 50 Assets: 42 Bonds: 36 Shares: 50 Shares: 6 --------------------100 100 42 42
SWUFE 2006 James Huang
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The value of limited liability
Payoff
B
0
B Debt (book value)
Total assets
SWUFE 2006 James Huawk.baidu.comg
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Unlimited liability
If the liability of the shareholders is unlimited, then there is no put options given to the shareholders. If the value of the assets V1< 50, the shareholders have to pay back the debt of 50 partly by using private capital.
Payoff
Riskless bond
50
Firm’s bond
50 -50
Put option
Total assets V1
SWUFE 2006 James Huang
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Limited Liability of the Shareholders
The put sold by the bond holders to the stockholders assumes limited liability If the value of the assets V1<50 The shareholders will exercise their put option with an exercise price of 50. They pay back their debt (50) by giving the assets to the bond holders.
•If debt only has to be paid back within one year: => =>
SWUFE 2006 James Huang
There is a small chance that assets will rise in value Shares still have a small value
=>
Conclusion: Shareholders with limited liability of a company debt financed have a put option on the assets of the company with an exercise price equal to the promised payment to the bond holders
Payoff to the shareholder: Max{V1-50, 0}
SWUFE 2006 James Huang
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Conclusion: The shareholders of a company debt financed have a call option on the assets of the company with an exercise price equal to the promised payment to the bondholders (See, for example, Merton (1974)). In the example mentioned before, The shareholders have a 1year call on the assets of the company with an exercise price K=50.
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An example: Assume the value of total assets Vt follows a binomial distribution. That is, it either rises by one half to 63 or falls by one third to 28 in one year.