first american bank_case report

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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 AmericanBank (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 (S 0) 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 toestimate 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 S0 X r T Sd d1 d2 N(d1) N(d2)

116.26 10900 5200 0.045 4 0.3 1.8186 1.2038 0.9655 0.8857P(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 amountequals 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

Probability

Survival

Probability LGD

Expected

Cost atDefault

Expected

FeePayment

Discount

Factors

PV

ExpectedCost

PV

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

Total 0.00712 3.6310s

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 witha 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

Probability

Suvival

Probability

LGD Expected

Cost at

Default

Expected

Fee

Payment

Discount

Factors

PV(Expect

ed Cost)

PV(Expect

ed Fee

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

Total 355914 356019

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