1
Finance School of Management
Chapter 16,The Valuation of
Contingent Claims
Objective
?To explain the theory and application
of the Contingent-claim
valuation method
2
Finance School of Management
Chapter 16 Contents
? Contingent Claims Analysis of Corporate Debt and
Equity
? Convertible Bonds
? Valuing Pure State-Contingent Securities
3
Finance School of Management
Objectives
?To show how the option pricing method can
be applied to pricing of contingent-claims
4
Finance School of Management
Valuation of Uncertain Cash Flows,
CCA / DCF
? The DCF approach discounts the expected cash
flows using a risk-adjusted discount rate
? The Contingent-Claims Analysis (CCA) uses
knowledge of the prices of one or more related
assets and their volatilities
5
Finance School of Management
An Example,Debtco Corp,
? Debtco is in the real-estate business
? It issues two types of securities,
– common stock (1 million shares)
– corporate bonds with an aggregate face value of $80
million (80,000 bonds,each with a face value of
$1,000) and maturity of 1 year
– risk-free interest rate is 4%
? The total market value of Debtco is $100 million
6
Finance School of Management
Debtco,Notation
– V be the current market value of Debtco?s assets
($100 million)
– V1 be the market value of Debtco?s assets a year
from now
– E be the market value of Debtco?s stocks
– D be the market value of Debtco?s bonds
7
Finance School of Management
Two Ways to Think about the Debtco?s
Market Value
? To think of the assets of the firm,real estates in
Debtco?s case,as having a market value of $100
million
? To imagine another firm that has the same assets
as Debtco but is financed entirely with equity,and
the market value of this all-equity-financed,twin”
of Debtco is $100 million
8
Finance School of Management
Pa y off s f or B on d and Stock I ssu es
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160 180 200
V a l ue of Fi rm ( M i l l i on s )
V
a
l
ue
of
B
o
nd
a
nd
S
t
o
c
k
(
M
i
l
l
i
o
ns)
B o n d V a lue
S t o c k V a lue
9
Finance School of Management
Prob al i l i t y D ens i t y of a Fi rm 's Va l ue
0, 0 0
0, 0 1
0, 0 2
0, 0 3
0, 0 4
0, 0 5
0, 0 6
0, 0 7
0, 0 8
0, 0 9
0 20 40 60 80 100 120 140 160 180 200
V a l ue of a Fi rm
P
rob
a
b
i
l
i
t
y
D
e
n
s
i
t
y
10
Finance School of Management
Debtco Security Payoff Table
($?000,000)
Security Pay o ff S tate A Pay o ff S tate B
Fi rm 140 70
Bo nd 80 70
Stoc k 60 0
11
Finance School of Management
Debtco?s Replicating Portfolio
?Let
– x be the fraction of the firm in the replication
– Y be the borrowings at the risk-free rate in the
replication
– The following equations must be satisfied
3 0 8,6 9 2,57$;
7
6
04.1700;04.11 4 060
??
?????
Yx
YxYx
12
Finance School of Management
Debtco?s Replicating Portfolio
($?000)
Posi ti o n Imm edi ate Case A Case B
6/ 7 assets - 85,714 120,000 60,000
B o nd ( RF ) 57,692 - 60,000 - 60,000
Total 28,022 60,000 0
13
Finance School of Management
Debtco?s Replicating Portfolio
? We know the value of the firm is $1,000,000,and
the value of the total equity is $28,021,978,so the
market value of the debt with a face of 80,000,000
is $71,978,022
? The yield on this debt is (80…/71…) - 1 = 11.14%
14
Finance School of Management
Another View of Debtco?s
Replicating Portfolio (?$000)
S ecu ri t y To t al
market
V alu e
Equivalen t
Amoun t o f
Fi rm
Equivalen t
Amoun t o f
Rf Debt
B o n ds 7 1,978 1 4,286
(1/ 7)
57,692
S t o ck 2 8,022 8 5,714
(6/ 7)
- 57,692
B o n ds +
S t o ck
100,000 100,000 0
15
Finance School of Management
Given the Price of the Stock
? Suppose,
– 1 million shares of Debtco?s stock outstanding,and the
market price is $20 per share
– two possible future value of for Debtco,$70 million
and $140 million
– the face value of Debtco? bonds is $80 million
– risk-free interest rate is 4%
16
Finance School of Management
Valuing Bonds
– We can replicate the firm?s equity using x = 6/7
of the firm,and about Y = $58 million riskless
borrowing (earlier analysis)
– The implied value of the bonds is then
$90,641,026 - $20,000,000 = $70,641,026 & the
yield is (80.00-70.64)/70.64 = 13.25%
0 2 6,6 4 1,90$76 3 0 8,6 9 2,570 0 0,0 0 0,20; ???????? x YEVYxVE
17
Finance School of Management
Given the Price of the Bonds
? Suppose,
– the face value of Debtco? bonds is $80 million,the
yield-to-maturity on the bonds is 10% (i.e.,the price of
Debtco bonds is $909.09)
– two possible future value of for Debtco,$70 million
and $140 million
– risk-free interest rate is 4%
18
Finance School of Management
Replication Portfolio
Pos ition Imm edia te
Ca s h Flow
Sc enar io A
V 1 = 7 0
Sc enar io B
V 1 = 1 4 0
Pur c ha s e x
of f ir m
- x V 70 x 140 x
Pur c ha s e
Y RF B ond
- Y Y (1,04) Y (1,04)
Total
Por tf olio
- x V - Y 70 80
19
Finance School of Management
Determining the Weight of Firm
Invested in Bond,x,and the Value of the
R.F.-Bond,Y
3 0 8,6 9 2,57$;
7
1
04.11 4 080
04.17070
??
?
?
?
?
??
??
Yx
Yx
Yx
20
Finance School of Management
Valuing Stock
– We can replicate the bond by purchasing 1/7 of
the company,and $57,692,308 of default-free 1-
year bonds
– The market value of the bonds is $909.0909 *
80,000 = $72,727,273
– The value of the stock is therefore E = V -D =
$105,244,753 - $72,727,273= $32,517,480
753,244,105$71 308,692,57273,727,72; ???????? x YDVYxVD
21
Finance School of Management
Convertible Bonds
?A convertible bond obligates the issuing
firm either to redeem the bond at par value
upon maturity or to allow the bondholder to
convert the bond into a prespecified number
of shares of common stock
22
Finance School of Management
An Example,Convertidett Corp,
? Convertidett has assets identical to those of Debtco
? Its capital structure consists of
– 1 million shares of common stock
– One-year zero-coupon bonds with a face value of $80
million (80,000 bonds,each with a face value of $1,000),
that are convertible into 20 shares of Convertidett stock at
maturity
– risk-free interest rate is 4%
? The total market value of Debtco is $100 million
23
Finance School of Management
Critical value of Convertidett for
Conversion
? Upon convertion,the total shares of stock will be
2.6 million
11 %5.616.26.1 VV ?
m i l l i o nV
m i l l i o nV
1 3 0$
80$6.26.1
*
1
*
1
?
?
24
Finance School of Management
C on v ert i bl e B on d
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120 140 160 180 200
V a l ue of t he Fi rm
V
a
l
ue
of
S
t
ock
a
nd
B
on
d
I
s
s
ue
C o n v e r t ible B o n d V a lue
D il u lt e d S t o c k V a lue
25
Finance School of Management
Payoff for Convertidett?s Stocks and
Bonds
Secur ity Pa yoff Sta te A Pa yoff Sta te B
Fir m 1 4 0,0 0 0,0 0 0 7 0,0 0 0,0 0 0
Bond s 8 6,1 5 3,8 4 6 7 0,0 0 0,0 0 0
Stocks 5 3,8 4 6,1 5 4 0
26
Finance School of Management
Convertidett?s Replicating Portfolio
?Let
– x be the fraction of the firm in the replication
– Y be the borrowings at the risk-free rate in the
replication
– The following equations must be satisfied
148,775,51$;077,923,76.
04.1700;04.1140154,846.53
??
?????
Yx
YxYx
27
Finance School of Management
Values of Convertidett?s Stocks and Bonds
929,147,25$148,775,51$077,923,76$ ???E
071,852,74$929,147,25$000,000,100$ ????? EVD
%88.665.935$ 65.935$000,1$ ???YTM
65.935$?P r i c eB on d
28
Finance School of Management
Decomposition of Convertidett?s
Stocks and Bonds
Se curity To ta l
mark et
V alue
Eq u iv alen t
Amo u n t o f
F ir m
Eq u iv alen t
Amo u n t o f
R f Debt
B o n ds 7 4,8 5 2,0 7 1 2 3,0 7 6,9 2 3
(.2 3 )
5 1,7 7 5,1 4 8
St o ck 2 5,1 4 7,9 2 9 7 6,9 2 3,0 7 7
(.7 7 )
- 5 1,7 7 5,1 4 8
B o n ds +
St o ck
1 0 0,0 0 0,0 0 0 1 0 0,0 0 0,0 0 0 0
29
Finance School of Management
Pure State-Contingent Securities
? Securities that pay $1 in one of the states and
nothing in the others
? For Debtco and Convertidett,if we know the
prices of the two pure state-contingent securities,
then we are able to price any securities issued by
the firms—stocks,bonds,convertible bonds,or
other securities
30
Finance School of Management
Valuing Pure State-Contingent
Securities
Securi ty Payoff
Scenari o a
Payoff
Scenari o b
Fi rm $70,000,000 $140,000,000
Co nti ngent
Securi ty # 1
$0 $1
Co nti ngent
Securi ty # 2
$1 $0
31
Finance School of Management
State-Contingent Security #1
967032467.0
04.1
1
000,000,70
000,000,100
000,000,1
04.1
1;
000,000,70
1
104.1000,000,140
004.1000,000,70
1
?????
????
?
?
?
??
??
YxP
Yx
Yx
Yx
#1 S,C,S.
32
Finance School of Management
State-Contingent Security #2
04.1
1$
538961.0$505494.0$033467.0$
495505494.0
04.1
2
000,000,70
000,000,100
000,000,1
04.1
2;
000,000,70
1
004.1000,000,140
104.1000,000,70
21
2
?????
??????
????
?
?
?
??
??
PP
YxP
Yx
Yx
Yx
#2 S,C,S.
33
Finance School of Management
Valuing Debtco?s Securities
S ecurity Pos s i bl e Pay off i n 1 Yea r
F i rm $ 1 4 0 m i l l i on $ 7 0 m i l l i on
Debtco S tock $ 6 0 0
Debtco B ond $ 1,0 0 0 $875
? Price of a Debtco stock = 60P1 = 60*$.4670329 = $28.02
? Price of a Debtco bond = 1,000P1 + 875P2
= 1,000*$.4670329 + 875*$.494505 = $899.73
34
Finance School of Management
Valuing Convertidett?s Securities
S ecuri ty Pos s i bl e Pay off in 1 Year
F i rm $ 1 4 0 m i l l i on $ 7 0 m i l l i on
Deb tco S tock $ 5 3,84 6 1 5 0
Deb tco B ond $ 1,07 6,92 3 $875
? Price of a Convertidett stock = 53.86415P1
= 53.86415*$.4670329 = $25.15
? Price of a Convertidett bond = 1,076.923P1 + 875P2
= 1,076.923*$.4670329 + 875*$.494505 = $935.65
35
Finance School of Management
Payoff for Debtco?s Bond Guarantee
Sec uri ty Sce nari o A Sce nari o B
F i rm $14 0,000,000 $70,000,000
B o nds $1,000 $875
G uarant ee $0 $125
36
Finance School of Management
SCS Conformation of Guarantee?s
Price
? Guarantee?s price = 125P2 = 125* 0.494505 = $61.81
Finance School of Management
Chapter 16,The Valuation of
Contingent Claims
Objective
?To explain the theory and application
of the Contingent-claim
valuation method
2
Finance School of Management
Chapter 16 Contents
? Contingent Claims Analysis of Corporate Debt and
Equity
? Convertible Bonds
? Valuing Pure State-Contingent Securities
3
Finance School of Management
Objectives
?To show how the option pricing method can
be applied to pricing of contingent-claims
4
Finance School of Management
Valuation of Uncertain Cash Flows,
CCA / DCF
? The DCF approach discounts the expected cash
flows using a risk-adjusted discount rate
? The Contingent-Claims Analysis (CCA) uses
knowledge of the prices of one or more related
assets and their volatilities
5
Finance School of Management
An Example,Debtco Corp,
? Debtco is in the real-estate business
? It issues two types of securities,
– common stock (1 million shares)
– corporate bonds with an aggregate face value of $80
million (80,000 bonds,each with a face value of
$1,000) and maturity of 1 year
– risk-free interest rate is 4%
? The total market value of Debtco is $100 million
6
Finance School of Management
Debtco,Notation
– V be the current market value of Debtco?s assets
($100 million)
– V1 be the market value of Debtco?s assets a year
from now
– E be the market value of Debtco?s stocks
– D be the market value of Debtco?s bonds
7
Finance School of Management
Two Ways to Think about the Debtco?s
Market Value
? To think of the assets of the firm,real estates in
Debtco?s case,as having a market value of $100
million
? To imagine another firm that has the same assets
as Debtco but is financed entirely with equity,and
the market value of this all-equity-financed,twin”
of Debtco is $100 million
8
Finance School of Management
Pa y off s f or B on d and Stock I ssu es
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160 180 200
V a l ue of Fi rm ( M i l l i on s )
V
a
l
ue
of
B
o
nd
a
nd
S
t
o
c
k
(
M
i
l
l
i
o
ns)
B o n d V a lue
S t o c k V a lue
9
Finance School of Management
Prob al i l i t y D ens i t y of a Fi rm 's Va l ue
0, 0 0
0, 0 1
0, 0 2
0, 0 3
0, 0 4
0, 0 5
0, 0 6
0, 0 7
0, 0 8
0, 0 9
0 20 40 60 80 100 120 140 160 180 200
V a l ue of a Fi rm
P
rob
a
b
i
l
i
t
y
D
e
n
s
i
t
y
10
Finance School of Management
Debtco Security Payoff Table
($?000,000)
Security Pay o ff S tate A Pay o ff S tate B
Fi rm 140 70
Bo nd 80 70
Stoc k 60 0
11
Finance School of Management
Debtco?s Replicating Portfolio
?Let
– x be the fraction of the firm in the replication
– Y be the borrowings at the risk-free rate in the
replication
– The following equations must be satisfied
3 0 8,6 9 2,57$;
7
6
04.1700;04.11 4 060
??
?????
Yx
YxYx
12
Finance School of Management
Debtco?s Replicating Portfolio
($?000)
Posi ti o n Imm edi ate Case A Case B
6/ 7 assets - 85,714 120,000 60,000
B o nd ( RF ) 57,692 - 60,000 - 60,000
Total 28,022 60,000 0
13
Finance School of Management
Debtco?s Replicating Portfolio
? We know the value of the firm is $1,000,000,and
the value of the total equity is $28,021,978,so the
market value of the debt with a face of 80,000,000
is $71,978,022
? The yield on this debt is (80…/71…) - 1 = 11.14%
14
Finance School of Management
Another View of Debtco?s
Replicating Portfolio (?$000)
S ecu ri t y To t al
market
V alu e
Equivalen t
Amoun t o f
Fi rm
Equivalen t
Amoun t o f
Rf Debt
B o n ds 7 1,978 1 4,286
(1/ 7)
57,692
S t o ck 2 8,022 8 5,714
(6/ 7)
- 57,692
B o n ds +
S t o ck
100,000 100,000 0
15
Finance School of Management
Given the Price of the Stock
? Suppose,
– 1 million shares of Debtco?s stock outstanding,and the
market price is $20 per share
– two possible future value of for Debtco,$70 million
and $140 million
– the face value of Debtco? bonds is $80 million
– risk-free interest rate is 4%
16
Finance School of Management
Valuing Bonds
– We can replicate the firm?s equity using x = 6/7
of the firm,and about Y = $58 million riskless
borrowing (earlier analysis)
– The implied value of the bonds is then
$90,641,026 - $20,000,000 = $70,641,026 & the
yield is (80.00-70.64)/70.64 = 13.25%
0 2 6,6 4 1,90$76 3 0 8,6 9 2,570 0 0,0 0 0,20; ???????? x YEVYxVE
17
Finance School of Management
Given the Price of the Bonds
? Suppose,
– the face value of Debtco? bonds is $80 million,the
yield-to-maturity on the bonds is 10% (i.e.,the price of
Debtco bonds is $909.09)
– two possible future value of for Debtco,$70 million
and $140 million
– risk-free interest rate is 4%
18
Finance School of Management
Replication Portfolio
Pos ition Imm edia te
Ca s h Flow
Sc enar io A
V 1 = 7 0
Sc enar io B
V 1 = 1 4 0
Pur c ha s e x
of f ir m
- x V 70 x 140 x
Pur c ha s e
Y RF B ond
- Y Y (1,04) Y (1,04)
Total
Por tf olio
- x V - Y 70 80
19
Finance School of Management
Determining the Weight of Firm
Invested in Bond,x,and the Value of the
R.F.-Bond,Y
3 0 8,6 9 2,57$;
7
1
04.11 4 080
04.17070
??
?
?
?
?
??
??
Yx
Yx
Yx
20
Finance School of Management
Valuing Stock
– We can replicate the bond by purchasing 1/7 of
the company,and $57,692,308 of default-free 1-
year bonds
– The market value of the bonds is $909.0909 *
80,000 = $72,727,273
– The value of the stock is therefore E = V -D =
$105,244,753 - $72,727,273= $32,517,480
753,244,105$71 308,692,57273,727,72; ???????? x YDVYxVD
21
Finance School of Management
Convertible Bonds
?A convertible bond obligates the issuing
firm either to redeem the bond at par value
upon maturity or to allow the bondholder to
convert the bond into a prespecified number
of shares of common stock
22
Finance School of Management
An Example,Convertidett Corp,
? Convertidett has assets identical to those of Debtco
? Its capital structure consists of
– 1 million shares of common stock
– One-year zero-coupon bonds with a face value of $80
million (80,000 bonds,each with a face value of $1,000),
that are convertible into 20 shares of Convertidett stock at
maturity
– risk-free interest rate is 4%
? The total market value of Debtco is $100 million
23
Finance School of Management
Critical value of Convertidett for
Conversion
? Upon convertion,the total shares of stock will be
2.6 million
11 %5.616.26.1 VV ?
m i l l i o nV
m i l l i o nV
1 3 0$
80$6.26.1
*
1
*
1
?
?
24
Finance School of Management
C on v ert i bl e B on d
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120 140 160 180 200
V a l ue of t he Fi rm
V
a
l
ue
of
S
t
ock
a
nd
B
on
d
I
s
s
ue
C o n v e r t ible B o n d V a lue
D il u lt e d S t o c k V a lue
25
Finance School of Management
Payoff for Convertidett?s Stocks and
Bonds
Secur ity Pa yoff Sta te A Pa yoff Sta te B
Fir m 1 4 0,0 0 0,0 0 0 7 0,0 0 0,0 0 0
Bond s 8 6,1 5 3,8 4 6 7 0,0 0 0,0 0 0
Stocks 5 3,8 4 6,1 5 4 0
26
Finance School of Management
Convertidett?s Replicating Portfolio
?Let
– x be the fraction of the firm in the replication
– Y be the borrowings at the risk-free rate in the
replication
– The following equations must be satisfied
148,775,51$;077,923,76.
04.1700;04.1140154,846.53
??
?????
Yx
YxYx
27
Finance School of Management
Values of Convertidett?s Stocks and Bonds
929,147,25$148,775,51$077,923,76$ ???E
071,852,74$929,147,25$000,000,100$ ????? EVD
%88.665.935$ 65.935$000,1$ ???YTM
65.935$?P r i c eB on d
28
Finance School of Management
Decomposition of Convertidett?s
Stocks and Bonds
Se curity To ta l
mark et
V alue
Eq u iv alen t
Amo u n t o f
F ir m
Eq u iv alen t
Amo u n t o f
R f Debt
B o n ds 7 4,8 5 2,0 7 1 2 3,0 7 6,9 2 3
(.2 3 )
5 1,7 7 5,1 4 8
St o ck 2 5,1 4 7,9 2 9 7 6,9 2 3,0 7 7
(.7 7 )
- 5 1,7 7 5,1 4 8
B o n ds +
St o ck
1 0 0,0 0 0,0 0 0 1 0 0,0 0 0,0 0 0 0
29
Finance School of Management
Pure State-Contingent Securities
? Securities that pay $1 in one of the states and
nothing in the others
? For Debtco and Convertidett,if we know the
prices of the two pure state-contingent securities,
then we are able to price any securities issued by
the firms—stocks,bonds,convertible bonds,or
other securities
30
Finance School of Management
Valuing Pure State-Contingent
Securities
Securi ty Payoff
Scenari o a
Payoff
Scenari o b
Fi rm $70,000,000 $140,000,000
Co nti ngent
Securi ty # 1
$0 $1
Co nti ngent
Securi ty # 2
$1 $0
31
Finance School of Management
State-Contingent Security #1
967032467.0
04.1
1
000,000,70
000,000,100
000,000,1
04.1
1;
000,000,70
1
104.1000,000,140
004.1000,000,70
1
?????
????
?
?
?
??
??
YxP
Yx
Yx
Yx
#1 S,C,S.
32
Finance School of Management
State-Contingent Security #2
04.1
1$
538961.0$505494.0$033467.0$
495505494.0
04.1
2
000,000,70
000,000,100
000,000,1
04.1
2;
000,000,70
1
004.1000,000,140
104.1000,000,70
21
2
?????
??????
????
?
?
?
??
??
PP
YxP
Yx
Yx
Yx
#2 S,C,S.
33
Finance School of Management
Valuing Debtco?s Securities
S ecurity Pos s i bl e Pay off i n 1 Yea r
F i rm $ 1 4 0 m i l l i on $ 7 0 m i l l i on
Debtco S tock $ 6 0 0
Debtco B ond $ 1,0 0 0 $875
? Price of a Debtco stock = 60P1 = 60*$.4670329 = $28.02
? Price of a Debtco bond = 1,000P1 + 875P2
= 1,000*$.4670329 + 875*$.494505 = $899.73
34
Finance School of Management
Valuing Convertidett?s Securities
S ecuri ty Pos s i bl e Pay off in 1 Year
F i rm $ 1 4 0 m i l l i on $ 7 0 m i l l i on
Deb tco S tock $ 5 3,84 6 1 5 0
Deb tco B ond $ 1,07 6,92 3 $875
? Price of a Convertidett stock = 53.86415P1
= 53.86415*$.4670329 = $25.15
? Price of a Convertidett bond = 1,076.923P1 + 875P2
= 1,076.923*$.4670329 + 875*$.494505 = $935.65
35
Finance School of Management
Payoff for Debtco?s Bond Guarantee
Sec uri ty Sce nari o A Sce nari o B
F i rm $14 0,000,000 $70,000,000
B o nds $1,000 $875
G uarant ee $0 $125
36
Finance School of Management
SCS Conformation of Guarantee?s
Price
? Guarantee?s price = 125P2 = 125* 0.494505 = $61.81
