CIR 6 - Default Probability

Question 1

Survival and density functions for a piecewise constant hazard rate function h(t)

Answer 1

Question 2

what does the difference b/w risk-neutral and real-work default probabilities represent?

Answer 2

Represents the risk premium required by investors for default risk

Question 3

Challenge of Bond pricing formula in calculting bond price while discounting with risk-free rate

Answer 3

• does not take into risk-aversion
• in order to have bond pricing incorporate risk premium investors demand for bearing default risk, one must calculate the risk-neutral probability of default PD by using the bond pricing equation to solve for the default probability that yields the quoted bond price from the market.

Question 4

Formula of risk-neutral probability of default in terms of real-wold probability and firm’s assets sharp ratio

Answer 4

Question 5

Formula of risk-neutral probability of default in terms of real-wold probability and risk premium on the market return π

Answer 5

• because the formula using company sharp ratio not usuble since expected return on company asset is unkown
• Correlation b/w asset and market return can be obtained from a linear regression
• Vol of market return can be estimated from market data
• However, market risk premium is difficult to estimte statistically since it is time-varying
  
  • See formula interms of market Sharp ratio (SR)

Question 6

Formula of risk-neutral probability of default in terms of real-wold probability and market Sharp Ratio

Answer 6

• Theta and SR are obtained by calibrating the model to credit spread data

Question 7

Credit spread of a risky zero coupon bond in terms to calibrate market sharp ratio and time parameter in risk-neutral prob of default

Answer 7

Question 8

Default intensity in terms of default probablity assuming constant default intensity (exponential time to default) under quick and dirty method

Answer 8

Question 9

Cumulative default probablity (PD_t)for an intermediate time t ( t_ii+1)

Answer 9

Question 10

Slope of default term structure

Answer 10

• Investment Grade Bonds: Tend to have upward sloping default term structures
  
  • very low risk at issue
  • cannot become any less risky over time
  • only has a change to get more risky
• Speculative Grade Bonds: Tend to have downward sloping default term structures
  
  • Very risky at issue
  • if ther is no default, bond credit quality improved over time

Question 11

Overview of Credit migration matrices

Answer 11

• Model how corporations can experience changes in credit quality over time
• give likelihood of transitioning from one rating class to another
• fallow Markov provess:
  
  • Previous rating history won’t affect current prob of transitioning
• transition probs are independent of time
• time period of matrix is typically one year

Question 12

Important facts of a credit transition matrix

Answer 12

• entires are all non-negative
• All rows equal 1
• Last column of matrix contains default probabilities
• Default state is absorbing (one cannot escape from default state)

Question 13

What shortcoming of Transition matrices is overcome by Generator matrices?

Answer 13

• it is very difficult to have transition matrices sampled from historical data to
  satisfy all these requirements
• An alternative approach to match a transition matrix to sampled data but still fulfill the required properties is to use Generator matrices

Question 14

Describe first method to remove negative off-diagonal entries from a generator matrix

Answer 14

• Simply zero out any negative off-diagonal entries
• and then allocate it fully into the diagonal element of the same row

Question 15

Describe second method to remove negative off-diagonal entries from a generator matrix

Answer 15

• Zero out any negative off-diagonal entries
• Allocates the negative off-diagonal entries in proportion to the absolute values of the other entries in the same row with the correct sign

Question 16

One way to measure the difference b/w exp (Q) and M matrices

Answer 16

• Calculate L¹ norm of the difference b/w the two matrices II M - exp (Q) II

Question 17

Term of Structure of Default Probabilities Based on Market Spreads

Answer 17

Method constructs an implied default term structure using observable market information such as spreads and defaultable bond prices.

Question 18

Shortcomings of piecewise constant function of Hazard Rate Model

Answer 18

• Inconsistencies due to
  
  • data error
  • model error

Question 19

how does the parameter a₁ impact the slope of the default intensity function

Answer 19

• This parameter allows the default intensity function to be sloped.
  
  • a₁ > 0 : a downward sloping intensity function
  • a₁ < 0 : an upward sloping intensity function
• a₁ represents the current deviation of the default intensity from the long-term mean a₀

Question 20

Septs of Parameters Calibration of Hazard Rate Function

Answer 20

• Obtain the model price of bonds and credit spreads by plugging in the model formula for S(t) from Nelson-Siegel model into the dirty bond/CDS pricing
• Perform a nonlinear optimization to find the values of a₁, a₂, a₃ that minimizes the difference between the model and market bond prices and/or credit default spreads.
• Once the calibration is complete, use the calibrated survival function S(t) to calculate the probability of default at any time t.