Chapter 27
Credit Derivatives
Credit Default Swap
Company A buys default protection from B to protect against default on a reference bond issued by the reference entity, C.
A makes periodic payments to B
In the event of a default by C
A has the right to sell the reference bond to B for its face value, or
B pays A the difference between the market value and the face value
CDS Structure
Default
Protection
Buyer, A
Default
Protection
Seller, B
90 bps per year
Payment if default by reference entity,C
CDS Final Payments
Notation:
L: Face value of bond, notional value of CDS
A(t): Accrued interest on bond per $ of principal at time t
R: Recovery rate, market price as a percent of face value plus accrued interest
s: CDS payment rate per year. Annual payment = sL
: Time since last CDS payment
A pays sL and B pays L - RL[1 + A(t)]
Sample Quotes (Jan 2001)
Company Rating 3yr 5yr 7yr 10yr
Toyota Aa1/AAA 16/24 20/30 26/37 32/53
Merrill Lynch Aa3/AA- 21/41 40/55 41/83 56/96
Ford A+/A 59/80 85/100 95/136 118/159
Enron Baa1/BBB+ 105/125 115/135 117/158 182/233
Nissan Ba1/BB+ 115/145 125/155 200/230 244/274
CDS Valuation
T :
Life of credit default swap
pi:
Risk-neutral default probability density at time ti
u(t):
Present value of $1 per year on payment dates between time zero and time t
e(t) :
Present value of an accrual payment at time t
v(t):
Present value of $1 received at time t
w :
Total payments per year made by CDS buyer
s :
Value of w for which CDS value is zero
( :
Risk-neutral probability of no credit event
during the life of the swap
A(t):
Accrued interest on the reference obligation at
time t as a percent of face value
PV of CDS Payments per $1 of Notional
If default event occurs at t < T, PV of payments is
If no default event, PV of payments is
Expected PV is
PV of CDS Costs per $1 of Notional Principal
If default event occurs at t < T cost is
Expected cost is
Value of CDS to Buyer
Value is expected PV of payments less expected PV of costs
CDS Rate
CDS rate sets value to zero
CDS Rate continued
When default can happen at any time this becomes
Approximate CDS Spread
Let
y be the yield on bond issued by reference entity with maturity T
x be the yield on risk-free bond with maturity T
a be average value of A(t)
a* be average value for A(t) if reference bond is a par-yield bond with maturity T
Alternative Uses of the Formula
To calculate CDS spreads from the probabilities of default and expected recovery rate
To bootstrap the probabilities of default from CDS spreads and expected recovery rates
Sensitivity to Recovery Rate
Vanilla CDS is not very sensitive to the recovery rate providing the same recovery rate is used to estimate default probabilities and calculate payoffs
Binary swaps, which provide a fixed payoff in the event of a default, are much more sensitive to recovery rates
First-to-default swaps
Similar to a regular CDS
Several reference entities and reference bonds
First entity to default triggers a payoff
Settlement is same as ordinary CDS
Valuation
Must use Monte Carlo simulation
Each reference entity is simulated to determine when if ever it defaults
Valuation is sensitive to default correlation
A conservative (and easy) assumption for the seller is that all correlations are zero
The easiest way to build in non-zero correlations is with the Gaussian copula model
Seller Default Risk
The impact of seller default risk on a CDS swap can be calculated by jointly simulating the reference entity and the seller
Suppose Y=PV of payoff and C is PV of payments
What rules should the simulation have for calculating Y and C?
Total Return Swap
Company A agrees to pay B the total return earned on a reference bond issued by the reference entity, C, over some period of time.
Total return includes all coupon payments and any change in the price of the reference bond. (Usually the latter is made at the end)
B pays A LIBOR plus a spread on a notional equal to the initial value of the reference bond
The Structure
Total Return
Payer
Total Return
Receiver
Total Return on Bond
LIBOR plus 25bps
Uses of a TRS
Total Return Swaps are usually used a financing vehicles
Receiver wants to invest in bond
Payer (a financial institution) buys the bond and agrees to the swap
Payer has less credit exposure than if it had lent Receiver money to buy bond
Valuation of TRS
If there were no risk of default by receiver, the value of a TRS would be difference between value of reference bond and value of LIBOR bond
The spread above LIBOR would be zero
In practice the payer loses money if the receiver defaults at a time when the bond value has declined
Credit Spread Options
These provide a payoff dependent on movements in a particular credit spread.
There is usually no payoff in the event of a default on the reference asset
Payoff may be defined in terms of difference between actual spread and a strike spread or in terms of the difference between the price of an FRN and a strike price
Valuation
European options can be valued using Black’s model
This assumes that, conditional on no default, spread or FRN price is lognormal
Need a volatility for forward credit spread or forward FRN price
Must multiply Black’s formula by risk-neutral probability of no default
Collateralized Debt Obligation
A pool of debt issues are put into a special purpose trust
Trust issues claims against the debt in a number of tranches
First tranche covers x% of notional and absorbs first x% of default losses
Second tranche covers y% of notional and absorbs next y% of default losses
…
Collateralized Debt Obligation
Bond 1
Bond 2
Bond 3
Bond n
Average Yield
%
Trust
Tranche 1
1st 5% of loss
Yield = 35%
Tranche 2
2nd 10% of loss
Yield = 15%
Tranche 3
3rd 10% of loss
Yield = %
Tranche 4
Residual loss
Yield = 6%
CDOs continued
Note that average yield on tranches equals average yield on bonds less fee taken by trust manager
Often trust manager holds first tranche
CDO Applications
Can provide a range of credit quality debt objects
Can create high quality debt from low quality debt
Can create high yield debt from average risk debt
Can create artificial short by selling tranches before buying bonds
Valuing CDO Tranches
Depends on default correlation of bonds in portfolio
Must use Monte Carlo simulation
It is easiest to handle the default correlation with the Gaussian copula model
Quantifying the Cost of Default
on a stand-alone derivatives contract
Two Categories of Derivatives:
Those that are always assets to one party and liabilities to the other (., options)
Those that can become assets or liabilities (., swaps, forward contracts)
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
Notation
Contracts that are Assets
A Simple Interpretation
Use the “risky” discount rate rather than the risk-free discount rate when discounting cash flows in a risk-neutral world
Note that this does not mean we simply increase the interest rate in option pricing formulas
Credit Exposure for Contracts
That Can be Assets or Liabilities
Exposure
Contract value
The Impact of Defaults
Example
5 year fixed-for-fixed annual-pay currency swap where interest at 10% in £ is exchanged for interest at 5% in $
Principals are exchanged at the end of the life of the swap
Initial exchange rate =
Volatility of exchange rate = 15%
£ principal = 50 $ principal = 100
£ yield curve flat at 10% pa (ann comp) $ yield curve flat at 5% pa (ann comp)
1-, 2-, 3-, 4-, & 5-year zero-coupon bonds issued by the counterparty would have yields that are spreads of 25, 50, 70, 85, & 95 basis points above the risk-free rate
Defaults can occur only at the end of years 1, 2, 3, 4, & 5
Evaluating the Cost of Defaults
Maturity
when we receive $s
when we pay $s
ti
ui
vi
vi
uivi
1
2
3
4
5
Total
uivi
Example continued
The total cost of defaults on a matched pair of swaps with similar counterparties is +=% of principal.
This means that a bid-offer spread of 20 to 21 basis points is required to compensate for credit risk
Why do we have more credit risk when we are receiving dollars in this example?
From a credit perspective, is it better to receive fixed or floating in an interest rate swap when yield curve is upward sloping?
Why Cost of Defaults for
Currency Swaps > Interest Rate Swaps
Convertibles
A convertible bond is a corporate bond that can be exchanged for equity at certain times in the future at a predetermined exchange ratio (shares per bond)
Convertibles continued
One of the problems in valuing convertibles is that, in order to value the corporate bond correctly, it is necessary to take account of the chance of default in some way
Otherwise we are implicitly assuming it is a no-default Treasury bond
Valuing Convertible Bonds
The value at a node is
MAX[ MIN(Q 1, Q 2), Q 3 ]
where
Q 1 is the value given by the rollback
Q 2 is the call price, &
Q 3 is the value if conversion takes place.
Valuing Convertible Bonds
(continued)
We divide the value of the bond at each node into two components
a component that arises from situations where the bond ultimately ends up as equity
a component that arises from situations where the bond ultimately ends up as debt
Example
9-month zero-coupon bond with face value of $100
Convertible into 2 shares
Callable for $115 at any time
Initial share price = $50, volatility = 30%, no dividends
Risk-free rates all 10%
Yields on issuer’s non-convertible bonds = 15%
The Tree
(Numbers at each node in descending order are the stock price, equity component, debt component & total value)
