RMBI 4210 Quantitative Methods for Risk Management
Tutorial 6 – Credit default swaps (CDS) and Bond price based pricing
Credit default swaps (CDS)
The protection seller receives fixed periodic payments from the protection buyer in return for making a single contingent payment covering losses on a reference asset following a default.
In the lecture notes, it shows an example that CDS is beneficial for both protection seller and buyer (High credit rated institution seeking for protection from an institution with a slightly lower credit rating, buyer transfer the risk of having a risky asset while seller gets higher return than holding the risky asset itself).
HW Q11-12 CDS – CDO
Spring, 2021
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A synthetic CDO does not actually own the pool assets on which it has the risk; it only absorbs the economic risks but not the legal ownership. This property gives additional flexibility in bank balance sheet management. It helps banks to reduce regulatory capital charges and reduce economic risk while retaining ownership of the attendant assets.
Numerical example of replication of fixed-coupon bonds
Portfolio 1
One defaultable coupon bond C*=100, paying annual coupon rate c*=4%, maturity in 2 years +
One CDS on this bond with CDS spread s=1.98%. Portfolio 2
One default-free coupon bond C=100, with the same payment dates as the defaultable coupon bond and coupon rate c=c*-s=4%-1.98%=2.02%.
Comparison of cash flows of the two portfolios
1) In survival the cash flows of both portfolio are identical Portfolio 1
T=0 -100
T=1 4 -1.98
T=2 100+4-1.98
Portfolio 2 -100
2.02 100+2.02
2) If default at 𝜏, portfolio 1’s value =par = 100 (full compensation by the CDS). As the default-free bond is a par bond. If we sell the default-free bond at 𝜏, it also worths 100.
The value of CDS is 0 at time 0. Assume the annual interest rate is r=2%. Let’s define the zero- coupon default-free bond B(0, 2) and annuity A(0,2) as the following:
B(0,2) = 100*e-2*2% ; A(0,2) = 1*e-2*2%+ 1*e-2%.
C(0) = B(0,2) + c*A(0,2) =(100+2.02)*e-2*2%+ 2.02*e-2% =(100+4-1.98)*e-2*2%+ (4-1.98)*e-2% = B(0,2) + c*A(0,2) – s A(0,2) = C*(0)
∴𝑠=𝐵 0,2 +𝑐∗𝐴 0,2 −𝐶∗(0)=100𝑒34∗4% +4𝑒34∗4% +4𝑒34% −100=4−2.02 𝐴(0,2) 𝑒34∗4% + 𝑒34%
= 1.98
Bond price based pricing — Try to understand the derivation (some derivation relies on knowledge from MATH4512), how to represent or derive the credit spread/ implied probability of survival/ implied hazard rate based on market prices of zero-coupon bonds.
Spring, 2021
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