Longstaff Schwartz _95_.xls
Document Sample
Formulas in Longstaff & Schwartz 1995 for Risky Debt
Source:
Longstaff, Francis A., Schwartz, Eduardo S. A simple approach to Valuing Risky Fixed and Floating Rate Debt. Journal of
Finance. Vol 3. July 1995.
Vasicek, O. 1977 "An Equilibrium Characterization of the term structure." Journal of Financial Economics 5: 177-188.
Firm asset value process: with parameters
dV m Vdt s VdZ 1
V: firm asset value
m: drift rate of asset process
dr z br dt hdZ 2
Interest rate process: r: short-term riskless interest rate
s: instanteaneous stdev of asset process (constant)
z: long-term equilibrium of mean reverting process (constant)
b: "pull-back" factor - speed of adjustment (constant)
h: spot rate volatility (constant)
Value of risky discount bond: dZ1,2 standard Wiener processes with correlation rdt
K: Bankruptcy threshold - financial distress if V falls below K
P( X , r, T ) D(r, T ) wDr, T QX , r, T
X: ratio of V/K - firm asset value as % of bankruptcy threshold
a: z plus constant (c) to represent market price of risk
T: time to maturity
w: writedown = 1 - recovery rate
D(r,T) is the value of riskfree (no credit risk) discount bond according to Vasicek (1977).
D ( r , T ) e A T B (T ) r )
2
A (T ) h
h 2 a 2 e bT 1 h
2 bT
with
a
2
T
b 3 e 1
2b 2
b b 4b 3
1 e bT
B (T )
a
Q(X,r,T) term can be interpreted as probability - under risk neutral measure - that default occurs.
n
q1 N a1
Q X,r,T,n i
( ) q
qi N ai q j N bi j , i 2,3,....., n
i 1
i1
j 1
N(.) denotes the cumulative standard normal distribution
with
ln X M iT n, T M jT n,T M iT n,T
ai bi j
S iT n S iT n S jT n
a rsh h 2
s 2
M (t , T )
t
b b 2
2
rsh h 2
exp b T exp b t 1
b 2 2b 3
a h
2
1 exp b t
r
b b 2 b 3
h 2
exp
b T 1 exp b t
2b 3
rsh h2 2
b b 2 s t
S (T )
rsh 2h 2
b 2 b 3 1 exp bt
h 2
2 b 3 1 exp 2 bt
Kurt Hess, kurthess@waikato.ac.nz 9/18/2011 8cdbd895-c4ea-48fa-a8a1-ebc684967b7f.xls Formulas
Risky Debt Valuation Model Longstaff & Schwartz 95 Risky Discount Bond Prices as a functio
Notation L&S 95
Rate r0 at t=0 r 7.0% 14
1.0
Maturity time (years) T 2.0 20 0.9
"Pullback" b 1.00 100
0.8
Instantaneous StDev. of short rate h 3.162%
0.7
h2 0.0010 10 0.6
a in L&S = z + constant a 0.060 60 0.5
V/K =X X 1.10 110 0.4
Writedown = 1 - Recovery Rate w 0.50 50
0.3
Volatility of asset value process s 20.00%
2 0.2
s 0.0400 40
0.1
Instantaneous correl. Asset/interest rate r - 0.25 75
-
Iterations for Q n 10
Risky Discount Bond
0
Value risk free discount bond (Vasicek) D #NAME? #NAME?
Yield riskfree bond #NAME? #NAME?
Probability of default (risk neutral) Q #NAME?
Value risky discount bond (L&S 95) P #NAME? #NAME?
Term Structure of Interest Risky Discoun
Yield risky discount bond #NAME? #NAME?
8%
Floating-rate Coupon Payment
Time of floating rate payment (<=T) t 2.0 set to T (testing) 7%
Expected value r at t (risk neutral process) R #NAME? 6%
Correlation adjustment G #NAME?
Value of floating rate payment at time tau F #NAME? 5%
Values of floating-rate coupon payments (for T=t) 4%
8.0% Value of floating rate 3%
payment at time tau
2%
6.0%
r at t=0 1%
4.0%
0%
2.0% 0 2
0.0%
0 2 4 6 8 10 12
Time to maturity
Risky Discount Bond Prices as a function of bond tenor (time to maturity)
Value risky discount bond (L&S
95)
Value risk free discount bond
(Vasicek)
2 4 6 8 10 12
Time to maturity
Term Structure of Interest Risky Discount Bonds
Floating-rate
Coupon Payment
r at t=0
Yield riskfree bond
2 4 6 8 10 12
Time to maturity
Data Table
#NAME? #NAME? #NAME?
0.01 #NAME? #NAME? #NAME? #NAME? #NAME?
0.25 #NAME? #NAME? #NAME? #NAME? #NAME?
0.5 #NAME? #NAME? #NAME? #NAME? #NAME?
0.75 #NAME? #NAME? #NAME? #NAME? #NAME? r at t=0
1 #NAME? #NAME? #NAME? #NAME? #NAME? 0 7.00%
1.5 #NAME? #NAME? #NAME? #NAME? #NAME?
2 #NAME? #NAME? #NAME? #NAME? #NAME?
2.5 #NAME? #NAME? #NAME? #NAME? #NAME?
3 #NAME? #NAME? #NAME? #NAME? #NAME?
4 #NAME? #NAME? #NAME? #NAME? #NAME?
5 #NAME? #NAME? #NAME? #NAME? #NAME?
6 #NAME? #NAME? #NAME? #NAME? #NAME?
7 #NAME? #NAME? #NAME? #NAME? #NAME?
8 #NAME? #NAME? #NAME? #NAME? #NAME?
9 #NAME? #NAME? #NAME? #NAME? #NAME?
10 #NAME? #NAME? #NAME? #NAME? #NAME?
List of iterations for q
1
2
5
10
20
50
100
200
500
1000
Risky Coupon Debt Model Longstaff & Schwartz 95 Risky Coupon Bond Prices as a func
Notation L&S 95
Rate r0 at t=0 r 4.0% 8
1.00
Maturity time (years) T 2.0 20
0.90
"Pullback" b 1.00 100
0.80
Instantaneous StDev. of short rate h 3.162%
0.70
h2 0.0010 10 0.60
a in L&S = z + constant a 0.060 60 0.50
V/K =X X 1.50 150 0.40
Writedown = 1 - Recovery Rate w 0.50 50
0.30
Volatility of asset value process s 20.00%
2 0.20
s 0.0400 40
0.10
Instantaneous correl. Asset/interest rate r - 0.25 75
-
Iterations for Q n 10
0
Risky Coupon Bond
Fixed coupon 8.00% 32
Risky (clean price) #NAME? Yield Curve Risky vs. Riskless Co
Risky (incl. accrued interest) #NAME?
Yield to Maturity Risky Bond #NAME? 5%
Riskless Coupon Bond
Riskless coupon bond (sigma=0) #NAME? 4%
Riskless (incl. accrued interest) #NAME?
4%
Yield to Maturity Riskless Bond #NAME?
Credit Spread (in bps) #NAME? 3%
Credit Spread (in bps) 3%
1 2%
1 2%
1 1%
0
1%
0
- 0%
0 2 4 6 8 10 12 0
Time to maturity
Formulas in Longstaff & Schwartz 1995 for risky debt
Source:
Longstaff, Francis A., Schwartz, Eduardo S. A simple approach to Valuing Risky Fixed and Floating Rate Debt. Journal of Finan
Firm asset value process:
dV m Vdt s VdZ 1
Interest rate process:
dV m Vdt s VdZ 1
dr z br dt hdZ 2
Value of risky discount bond:
P( X , r, T ) D(r, T ) wDr, T Q X , r, T
Testing
#NAME?
5%
#NAME?
0.1 0.045 0.044775562
0.6 0.045 0.043670049
1.1 1.145 1.083725494
1.172171105
Risky Coupon Bond Prices as a function of bond maturity
Risky (clean price)
Risky (incl. accrued interest)
Riskless coupon bond (sigma=0)
Riskless (incl. accrued interest)
2 4 6 8 10
Time to maturity
Yield Curve Risky vs. Riskless Coupon Bonds
Yield to Maturity Risky Bond
Yield to Maturity Riskless Bond
r at t=0
2 4 6 8 10
Time to maturity
and Floating Rate Debt. Journal of Finan ce. Vol 3. July 1995.
with parameters
V: firm asset value
m: drift rate of asset process
r: short-term riskless interest rate
s: instanteaneous stdev of asset process (constant)
z: long-term equilibrium of mean reverting process (constant)
b: "pull-back" factor - speed of adjustment (constant)
h: spot rate volatility (constant)
dZ1,2 standard Wiener processes with correlation rdt
X: ratio of V/K - firm asset value as % of bankruptcy threshold
a: z plus constant (c) to represent market price of risk
TABLE 1 TABLE 2
Risky Dirty Riskless Clean
Data Table Risky Clean Riskless Dirty
#NAME? #NAME? #NAME? #NAME?
0.001 #NAME? #NAME? #NAME? #NAME? 0.001
1 #NAME? #NAME? #NAME? #NAME? 0.1
1.01 #NAME? #NAME? #NAME? #NAME? 0.2
2 #NAME? #NAME? #NAME? #NAME? 0.3
2.01 #NAME? #NAME? #NAME? #NAME? 0.4
3 #NAME? #NAME? #NAME? #NAME? 0.5
3.01 #NAME? #NAME? #NAME? #NAME? 0.75
4 #NAME? #NAME? #NAME? #NAME? 1
4.01 #NAME? #NAME? #NAME? #NAME? 1.5
5 #NAME? #NAME? #NAME? #NAME? 2
5.01 #NAME? #NAME? #NAME? #NAME? 3
6 #NAME? #NAME? #NAME? #NAME? 4
6.01 #NAME? #NAME? #NAME? #NAME? 5
7 #NAME? #NAME? #NAME? #NAME? 6
7.01 #NAME? #NAME? #NAME? #NAME? 7
8 #NAME? #NAME? #NAME? #NAME? 8
8.01 #NAME? #NAME? #NAME? #NAME? 9
9 #NAME? #NAME? #NAME? #NAME? 10
9.01 #NAME? #NAME? #NAME? #NAME?
10 #NAME? #NAME? #NAME? #NAME?
List of iterations for q
1
2
5
10
20
50
100
200
500
1000
r at t=0
0 4.00%
YTM Risky YTM Riskless Spread
#NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
#NAME? #NAME? #NAME?
Term structure in Vasicek Model
Vasicec Term Structure of Inter
Hull L&S 95
Running time t n.a. 0 9%
Rate r0 at t=0 r0 r 8.0% 16 8%
Maturity time (T) T 7 2.0 20
7%
"Pullback" a b 0.07 7
6%
Long-run equilibrium b n.a. 8.0% 80
5%
Instanteanous StDev. of short rate s h 2.0% 20
4%
Market Price of Interest Rate Risk l n.a. 0.0% 0
Constant in L&S a z c = s * l =sxl c 0.00E+00 3%
a*b z 0.0056 2%
a in L&S = c + l in Hull a 0.0056
1%
Results: 0%
B in Vasicek Model (Hull) 1.87 0
A in Vasicek Model (Hull) 0.989838 -0.01021 0.039184
Infinitely-long Rate (Y) 3.92%
Vasicek Discount Factor 0.852554 #NAME? #NAME?
Vasicek Zero Rate 7.976%
Vasicec Discount Function
1.0
0.9 Vasicek Discount
Factor
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-
0 5 10 15 20 25 30 35
Time to maturity
Formulas in Longstaff & Schwartz for Vasicek Model
Interest rate process: dr z br dt h dZ 2
D ( r , T ) e A T B (T ) r )
Value of zero=coupon bond:
with
1 e bT
B (T )
b
AT h
2 a h2
a 2 ebT h2
2bT
( ) 2
T 3 1 3 e 1
2b b b b b
4
AT h
2 a h2
a 2 ebT h2
2bT
( ) 2
T 3 1 3 e 1
2 b b b b b
4
Source:
Longstaff, Francis A., Schwartz, Eduardo S. A simple approach to Valuing Risky Fixed and Floating Rate Debt. Journal of Finan
Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice -Hall. p. 567.
Model: Vasicek, O. 1977 "An Equilibrium Characterization of the term structure." Journal of Financial Economics 5: 177
AT h
2
a h2
a 2 e b h
T 2
( )
2 b2 T
b b3
b
1
b
4
3 e
L&S 95 Vasicek translated to Hull notation:
A (T ) s
2
2a 2
b c T s 3 b c 2 e aT 1 s
a
2
a a a
2
4a 3
Formula in Hull p. 567. The following transformations prove that A(T) is the same in Hull and L&S 95
2 s
1 e
2
1 e aT a b 2 s
2 aT 2
A (T )
T
a a2 4a 3
()
2 2b s2 2 2b s2
a a aT s 1 2
2
e aT aT
e2
2 2 2 3 1 e
AT T
a a 43
a
s2
AT 2 b T 3 1 e
( )
b s2
aT s2 12 aTe2aT
e
a 43
2a a 2 a
s s b aT s aT s seaT se2aT
2 2
2 2 2 2
bT
A ) 2 3 e 3 e 3 3 3
T
( 1 1
2a a a
2a 4a 2 a 4a
s s b aT seaT s s seaT se2aT
2 2 2
2 2 2 2
bT
A ) 2 3 e 3 3 3 3 3
T
( 1
2a a a
2 a a 2
a 2 4 a 4a
s s b
aT s 2aT
2 2 2
bT
T 2 3 e 3 e
A )
( 2 a a 1
2 1
a 4a
Formula in L&S p. 795 with market price of risk factor.
AT) s 2
s 2
e
aT
1
( b c T 2
2 2
b c
2a a 2a a a
AT) s 2
s 2
e aT
1
( b c T 2
2 2
b c
2a a 2a a a
A T) T
eaT 1 s2
eaT 1 s2
2 b
( c
2a a 2a
a a
ls ls c
Y b s
2
Infinitely long rate:
2 c
2a a a a
Formula for infinitely long rate with market price of risk from:
Craig Holden, Spreadsheet Modeling CD-ROM series, http://www.prenhall.co
Holden, Craig W. Holden. Spreadsheet Modeling in Investments, Prentice Ha
Vasicec Term Structure of Interest
9% Data Table
8% 0.852554
Vasicek Zero Rate
0.001 0.99992
7%
0.5 0.960797
6% Long-term 1 0.923175
5% equilibrium rate
2 0.852554
4% Infinitely long rate 3 0.787842
3% 4 0.728678
2% r at t=0 5 0.674666
6 0.625399
1%
7 0.580475
0% 8 0.539508
0 5 10 15 20 25 30 35 9 0.502134
Time to maturity 10 0.468013
15 0.336216
20 0.249459
25 0.190067
30 0.147827
This sheet has been added to compare the valuation of discount bonds
according the Vasicek (77) model both in the notation used by Longstaff &
Schwartz (95) and Hull in his well known derivatives textbooks (see
comment in cell A! for detailed reference)
with constants
z: long-term equilibrium of mean reverting process
b: "pull-back" factor - speed of adjustment
h: spot rate volatility
dZ2 standard Wiener process (N.B. Z1 is process for asset value)
Infinitely-long Rate (Y)
1 h
b 2 2bT
T
3 e 1
b
4
1 h
b 2 2bT
T
3 e 1
b
4
Fixed and Floating Rate Debt. Journal of Finance. Vol 3. July 1995.
Journal of Financial Economics 5: 177-188.
eb h
T 2 b
2 1 3 e
2 T
1
b b
4
c
a
e
2
aT
1 s 2
4a
e
3
2 aT
1
s2 e aT aT
1 2 e2
43
a
2
2
1 e aT
e 2aT
43
a
s seaT se2aT
2 2 2
3 3 3
4a 2a 4a
s seaT se2aT
2 2 2
3 3 3
4a 2a 4a
s
2
aT
3 e2 2 1
4a
e aT
1 s2
2
b c e 2aT1
2a a a 43
a
e aT
1 s2
2
b c e 2aT1
2a a a 43
a
c
a a
eaT 1 s2
2a2 s
2
4a3 e
aT
2
1
ls c
c ls
a a
ate with market price of risk from:
ROM series, http://www.prenhall.com/holden/
Spreadsheet Modeling in Investments, Prentice Hall (2002), p.49
Long-term equilibrium rate
8.000% 0 8.00%
7.998% 30 8.00%
7.994%
7.976% r at t=0
7.949% 0 8.00%
7.913%
7.871%
7.823% Infinitely long rate
7.770% 0 3.92%
7.714% 30 3.92%
7.654%
7.593%
7.267%
6.942%
6.642%
6.372%
