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Longstaff Schwartz _95_.xls

VIEWS: 25 PAGES: 19

									          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 )  wDr, T QX , 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
                                      i1

                                                                        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 )  wDr, 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 ebT  h2
                                               2bT
                                                                                                                                    
   ( )       2   
                  T     3          1      3 e    1
          2b   b  b     b              b
                                           4 
  AT  h
        2    a  h2
                         a 2 ebT  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 12 aTe2aT
                                   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 
             eaT 1 s2
                     
                                    
                                     eaT 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 2aT1
 2a     a                                    a      43
                                                    a
                               e  aT
                                         1 s2    
                                                                    
  2 
    b c                                          e 2aT1
2a     a                                a      43
                                               a

c 
  a     a
           
     eaT 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%

								
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