John Hull's Chapter10-23研究生金融工程课件

23.21
Independence Assumption
• The independence assumption states that the variables affecting the price of a derivative are independent of the variables determining defaults • This assumption (although not perfect) makes pricing for default risk possible
23.20
Quantifying the Cost of Default
Two Categories of Derivatives:
• Those that are always assets to one party and liabilities to the other (e.g., options) • Those that can become assets or liabilities (e.g., swaps, forward contracts)
23.9
The expected loss on a BBB bond between years 5 and 10 is 9.34% of the no - default value
23.10
Historical Data
Historical data provided by rating agencies are also used to estimate the probability of default
• Similarly the holder of a A-rated bond expects to lose
e
0.0550 2
e
0.052
e
0.052
0.009950
or 0.9950% in the first two years • Between years one and two the expected loss is 0.7453%
Notation (page 626)
h(T1,T2): Expected proportional loss between times T1 and T2 as seen at time zero y (T ): yield on a zero-coupon corporate bond maturing at time T y *(T ) yield on a zero-coupon Treasury bond maturing at time T P (T ): price of a zero-coupon corporate bond paying $1 at time T P*(T ): price of a zero-coupon Treasury bond paying $1 at time T
E0 V0 N (d1 ) De rT N (d 2 ) where
2 ln V0 / D (r sV 2) T d1 ; d 2 d 1 sV T sV T
23.19
Volatilities
V0 E sE sV E0 V
This equation together with the option pricing relationship enables V and sV to be determined from E and sE
23.17
Merton’s Model
• Merton’s model regards the equity as an option on the assets of the firm • In a simple situation the equity value is max(VT -D, 0) where VT is the value of the firm and D is the debt repayment required
Average Cumulative Default Rates (%)
(S&P Credit Week, April 15, 1996, Table 23.2, page 627)
Yrs AAA AA A BBB BB B CCC 1
0.00 0.00 0.06 0.18 1.06
23.11
2
0.00 0.02 0.16 0.44 3.48
23.1
Credit Risk
Chapter 23
23.2
Credit Ratings
• In the S&P rating system AAA is the best rating. After that comes AA, A, BBB, BB, B, and CCC • The corresponding Moody’s ratings are Aaa, Aa, A, Baa, Ba, B, and Caa • Bonds with ratings of BBB (or Baa) and above are considered to be “investment grade” • Traders regularly estimate zero curve for bond with different credit ratings
23.7
23.8
Estimating Default Statistics from Bond Prices
P * (T ) P(T ) h(0, T ) P * (T ) Because P ( T ) e h(0, T ) 1 e h(T1 , T2 ) e
y(T )T
5.20 11.00 19.79 26.92
15.95 19.40 31.63 35.97
23.12
Bond Prices vs Historical Default Experience
• The estimates of the probability of default calculated from bond prices are much higher than those from historical data • Consider for example A-rated bonds • These typically yield at least 50 bps more than Treasuries
Example 23.1 (Page 627)
Suppose that the spreads over Treasuries for the yields on 5 - and 10 - year BBB - rated zero coupon bonds are 130 and 170 basis points. In this case: h( 0,5) 1 e0.0135 0.0629 h( 0,10) 1 e0.01710 01563 . so that h(5,10) 01563 . 0.0629 0.0934.
23.15
A Key Theoretical Reason
• The default probabilities estimated from bond prices are risk-neutral default probabilities • The default probabilities estimated from historical data are real-world default probabilities • In the real world we earn an extra 40 bps per year
23.3
Example
( See Table 23.1, page 625)
Maturity Risk-free Corporate (years) yield bond yield 1 5% 5.25% 2 3 4 5 5% 5% 5% 5% 5.50% 5.70% 5.85% 5.95%
23.4
or at a 0.2497% discount • This indicates that the holder of a corporate bond expects to lose 0.2497% from defaults in the first year
0.051
23.5
Example continued
Example continued
• One-year Treasury bond (principal=$1) sells for
e 0.951229 • One-year corporate bond (principal=$1) sells for 0.05251 e 0.948854
23.14
Possible Reasons for These Results
• The liquidity of corporate bonds is less than that of Treasury bonds • Bonds traders may be factoring into their pricing depression scenarios much worse than anything seen in the last 20 years
23wk.baidu.com6
Summary of Results
Maturity (years) Cumul. Def. Prob Def. Prob During Yr (%)
1 2 3 4 5
0.2497 0.9950 2.0781 3.3428 4.6390
0.2497 0.7453 1.0831 1.2647 1.2962
23.13
Bond Prices vs Historical Default Experience
This means that we expect to lose at least 1 e0.0055 0.0247 or 2.47% of the bond' s value over a 5 - year period. Assume a low recovery rate of 30%. The probability of default is then 2.47 0.7 353% . This is over five times greater than the 0.67% historical probability
23.18
Equity vs Assets (Equation 23.4, page 631)
An option pricing model enables the value of the firm’s equity today, E0, to be related to the value of its assets today, V0, and the volatility of its assets, sV
3
0.07 0.12 0.27 0.72 6.12
4
0.15 0.25 0.44 1.27 8.68
5
0.24 0.43 0.67 1.78
7
0.66 0.89 1.12 2.99
10
1.40 1.29 2.17 4.34 17.73 29.02 45.10
10.97 14.46 21.88 25.14 40.15 42.64
and P (T ) e
*
y* ( T ) T
[ y ( T ) y * ( T )]T
Because h(T1 , T2 ) h(0, T2 ) h(0, T1 )
[ y ( T1 ) y * ( T1 )]T1
e
[ y ( T2 ) y * ( T2 )]T2
23.16
Real World vs Risk Neutral World
• When we infer default probabilities or expected losses from security prices they are “risk-neutral”. When we infer them from historical data they are “real-world” • When should each be used?