Case Study – First American Bank: Credit Default Swaps CapEx Unlimited (CEU), one of Charles Bank International’s (CBI) important clients, asked for $50 million to finance its network expansion. However, the new loan would put CBI over its credit exposure limit. CBIT contacted First American Bank (FAB) to establish a credit default swap, which would mitigate its credit risk from the new loan. What is the probability that CEU will default within two years?
In order to accurately price the credit default swap, we need to start with an assessment of credit risk – the probability of default. According to Exhibit 10b, the probability that CEU (rating B2) will default by the end of year 2 is 13.7%. But, this data only reflects historical information, which is not appropriate for derivatives pricing. Therefore, we use Merton Model to calculate CEU’s default probability. The Merton Model proposed by Robert Merton characterizes a company’s equity as writing a call option or buying a put option on the assets of the company with maturity T and a strike price equal to the face value of the debt. The implied volatility from options can be regarded as the expected probability of default. Currently, CEU’s market value of the firm equals to $10,900 million (S0) and the outstanding debt has a maturity of 5 years (T). CEU’s market value of debt is $4,100 million, so its face value of debt should be more than $4,100 million. For treasury STRIP with 5-year maturity (r=4.5% according to Exhibit 8), if its market value is $4,100 million, its face value will be $5109 million. Therefore, it is reasonable to estimate that CEU’s face value of debt is $5,200 million, which equals to option strike price X. If the volatility of equity (sd) is given, then we can easily get the price of option and the probability of default by using the formula below. (See table below an example) P0 = X*e-rt * [1-N(d2)] – S0*[1-N(d1)] Where
P(D) = N(-d2) Black Scholes Calculation Example
Exercise Price=Debt Face Value
X
5200
Time to Expiration of Option
T
5
Volatility of Equity
sd
30%
5 Year STRIP Yield
rf
4.50%
Market Value of Firm
So
10900
P0 116.26
S0
X
r
T
Sd
d1
d2
N(d1)
N(d2)
10900
5200
0.045
4
0.3
1.8186
1.2038
0.9655
0.8857
P(D)
0.11434
Swap Fee Rate Calculation
Now, assume we have an accumulated default probability of 10.52% within 5 years, which indicates a semiannual default probability of 1.052%. The notional amount equals to new loan $50 million with a recovery rate of 82% (Exhibit 14). The swap fee is paid semiannually coinciding with coupon of the bonds. At the time of the deal, the five-year risk-free rate was 4.5%. Set swap fee rate as s. As we know, in an efficient market, the cost of loss without swap should equal to fee payments. The calculation below is based on a notational principal of $1.
Year
Default
Survival
Probability
Probability
LGD
Expected
Expected
Cost at
Fee
Default
Payment
Discount Factors
PV
PV Expected Cost
Expected Fee Payment
0.5
1.05%
98.95%
18.00%
0.1894%
0.9895s
0.9753
0.00185
0.9650s
1
1.05%
97.90%
18.00%
0.1894%
0.9790s
0.9512
0.00180
0.9312s
1.5
1.05%
96.84%
18.00%
0.1894%
0.9684s
0.9277
0.00176
0.8984s
2
1.05%
95.79%
18.00%
0.1894%
0.9579s
0.9048
0.00171
0.8667s
0.00712
3.6310s
Total
Set total PV of expected loss cost equals to PV of expected swap fee. Then, s= =0.00712/3.631 = 0.00196, indicating the annual fee payment on a default swap with a notational principal of $1. Therefore, Annual Swap Fee = 2*s*$50,000,000 = $196,089 Swap Fee Rate Calculation
The table below shows the cash flows from the swap. Year
Default
Suvival
Probability
Probability
LGD
Expected
Expected
Discount
PV(Expect
PV(Expect
Cost at
Fee
Factors
ed Cost)
ed Fee
Default
Payment
Payment)
0.5
1.05%
98.95%
18.00%
94680
96210
0.9753
92342
93835
1
1.05%
97.90%
18.00%
94680
95187
0.9512
90062
90545
1.5
1.05%
96.84%
18.00%
94680
94165
0.9277
87839
87361
2
1.05%
95.79%
18.00%
94680
93142
0.9048
85670
84278
355914
356019
Total
