Credit Risk Flashcards
What is default probabilitiy and how to obtain it?
Notation: Let τ denote the time of default
The default probability of a bond with maturity T is then given as: P(τ≤T)
1 Use historical data provided by rating agencies (real-world)
2 Use credit spreads (risk-neutral)
3 Use a structural credit risk model (e.g. Merton’s model) (real-world or
risk-neutral)
What is the conditional probability of default between time t and t+delta t and what is the default intensity?
The conditional probability of default between time t and t + ∆t is approximately λ(t)∆t
λ(t) := 1 lim∆t→0 P(τ ≤ t +∆t|τ > t) is the default intensity ∆t
The default intensity is also called the hazard rate
Intensities and probabilities are linked by the following formulas for survival probability:
P(τ>T)=E(e−RTλ(s)ds) 0
and default probability: P(τ≤T)=1−E(e−RTλ(s)ds)
0
What is Loss Given Default?
LGD = Difference between face value and recovery rate
In case of default, lenders (investors) do not lose everything but receive the recovery rate. According to Moody’s, the historical average recovery rate for senior unsecured corporate bonds for 1983-2019 wa 35.0% measured by post-default trading prices (what do you get if you sell the bond just after default) 47.0% measured by ultimate recovery (what do you get back if you hold on to the bond, potentially for several years)
When pricing derivatives such as Credit Default Swaps often market practitioners use 40%
Credit Ratings
In the S&P rating system, AAA is the best rating. After that comes: AA, A, BBB, BB, B, CCC, CC, and C
The corresponding Moody’s ratings are: Aaa, Aa, A, Baa, Ba, B, Caa, Ca, and C
Bonds with ratings of BBB (or Baa) and above are considered to be investment grade
BBB is the most common rating (”average rating”)
Average asset volatility of BBB firms is 25% (found by ”unlevering” equity volatility)
Difference between risk neutral default probability vs. real world
Default probabilities backed out of historical default data are real-world default probabilities. Default probabilities backed out of historical default data are real-world default probabilities
The risk-neutral default probabilities are higher than the estimates from Moody’s historical data
They should be: bonds and shares are two different claims on the same asset – the underlying firm – and therefore share the same risks
Put differently, when a firm is in trouble that firm’s share price and bond price go down at the same time
Both bond and equity holders require compensation for holding this risk. The compension is different in terms of expected excess return, but the same in terms of Sharpe ratios. Some argue that bonds have additional risk that equities don’t have – it’s difficult to diversify away non-systematic risk and bonds are illiquid – and these additional risks matter substantially for bond pricing. Others argue that these additional risks don’t seem to matter much.
Sovereign Credit Risk
Sovereigns can also default on their debt
Default risk is higher for countries that cannot print money (do not have their own currency) (European debt crisis)
Default risk is higher for sovereign debt denominated in foreign currency (e.g. USD-denominated debt / Argentina 2002)
Defaults can also happen for countries that can print money (e.g. Russia 1998)
Difference sovereign and corporate default:
Important differences between sovereign and corporate default:
No international bankruptcy court
Assets are not seized and sold at auction (lawsuits are possible)
Sovereign defaults are usually restructuring events (e.g. extending the maturity of the debt)
Creditors are not necessarily treated equally (e.g. ECB did not face losses from Greek restructuring)
Main sanction against sovereign borrower: Stop future lending to the borrower in case of default
How can the payoff and price of a risky bond can be expressed as riskfree and put?
Payoff risky bond = Payoff riskfree bond - Payoff put option
Price risky bond = Price riskfree bond - Price put option
Put option has firm value as underlying, strike equal to face value of debt, and maturity equal to bond maturity
What is leverage?
Book value of debt / (book value of debt + market value of equity)
Merton Model of Credit Risk
We want to understand how the credit spread of a 10-year ZCB bond vary with leverage. For a given leverage percent:
1. Normalize the firm value to 1/x and the face value of debt to 1
2. Calculate the price of a risk-free bond as P=e^(-0.03×10)=0.7408
3. Calculate the price of a put option as “put”=BS(K=1,S_0=1/x,σ=0.25,T=10 )
4. The price of the risky bond is P^risky=P-“put”
5. The yield spread on the bond is -log(P^risky )/10-r
To calculate default probabilities we need the drift/expected excess return on the assets, μ. The Sharpe ratio for all claims on the assets are the same. The typical Sharpe ratio for equity is around 0.22, SR=(μ-r)/σ=0.22 so μ=8.5%. In Black-Scholes the risk-neutral probability of the stock price ending below the strike price is N(-d_2). To get the actual probability just use μ instead of r in the calculation of d_2.
What is missing in the merton model of credit risk and how to solve?
- Several bonds outstanding: Use a default boundary during the life of the bond
- Bond pays out dividends and interest: Model these payments as a dividend yield
- Bankruptcy costs: Allow the value of assets to jump down at default
- Non-constant interest rates: Allow stochastic interest rates, but doesn’t matter too much
