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Topic

Various risks associated with investing in the bond market

Pricing Bond, pricing floating rate and inverse floating rate securities

Computing the yield on portfolio of bonds Various yield measures Sources of income from bond or bond

portfolio Credit risk spread, estimating default

probability, Loss given default

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Introduction

Base interest rate Yields On the run treasuries Risk premium Types of issuers Credit worthiness of the issuer Inclusion of options Taxability of the interest Expected liquidity of an issue Term to maturity

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Various risks Associated with investing in Bonds

Interest Rate RiskReinvestment riskcall riskdefault riskinflation riskexchange rate riskliquidity riskvolatility riskRisk risk

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Price Yield relationship

The price of a bond will change for the following reasons:

Change in credit quality of firm resulting a change in required yield

Change in required yield due to change in the market yield for comparable bonds

Change in price without a change in required yield as bond approaches its maturity (premium or discount bond)

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Issuer Coupon %

Maturity YTM %

Modified Duration

(Year)

DV01 $

Par Amount(million)

Full Price(000)

S&PCredit Rate

Treasury 6.5 2/15/10 4.55 6.56 50,109.87 67 76,387

Treasury 5.625 5/15/08 4.64 5.48 54,121.57 92 98,762

Treasury 5 8/15/11 4.57 7.77 203,838.95 84 87,192

Treasury Sector 6.56 172,095.69 262,341

Time Warner Enterprise

8.18 8/15/07 5.47 4.72 44,231.12 82 93,710 BBB+

Texas Utilities 6.375 1/1/08 6.19 5.06 15,523.06 30 30,678 BBB

Rockwell International

6.15 1/15/08 6.04 5.13 26,735.52 52 52,837 A

Transamerica Corporation

9.375 3/1/08 6.34 4.93 8,600.38 15 17,445 AA-

Coastal Corporation 6.5 6/1/08 6.76 5.25 15,830.32 30 30,153 BBB

United Airlines 6.831 9/1/08 5.99 5.51 22,011.89 38 39,949 A-

Burlington Northern Santa Fe

7.34 9/24/08 5.67 5.36 1,817.04 3 3,390 A+

News America Holding

7.375 10/17/08 6.56 5.34 17,243.92 30 32,292 BBB-

Litton Industries 8 10/15/09 6.70 5.81 38,830.55 60 66,834 BBB-

America Standard Inc.

7.625 2/15/10 7.59 6.10 25,212.52 41 41,332 BB+

Caterpillar Inc. 9.375 8/15/11 6.01 6.79 18,759.41 A+

Corporate Sector 5.39 235,137.67 436,248

Portfolio 5.83 402,030.38 698,589

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Pricing Floating rate and Inverse floating rate securuties Collateral price=floater’s price+ inverse

floater’s price

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Measuring default risk from Market price

Credit risk can be inferred from the market price of debt, equity, and credit derivatives whose values are affected by default.

P = 100/(1+y*) where y* is the yield to maturity in defaultable debt.

Using risk neutral pricing

P = 100/(1+y*)= [100/(1+y)](1-π) + [100/(1+y)](π)Where y is the yield to maturity for default free debt and

π is the probability of default. π = 1/(1- f) [1- (1+y)/(1+y*)]Y* ≈ y + π (1-f)(Y* - y) = π’ (1-f) + RP

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Example

Consider 10 year US treasury strip and 10 year zero coupon bond issued by Citigroup rated A by Moody’s and S&P. The respective yields on the two bonds are 5.5 and 6.25 percent assuming semiannual compounding. Assuming recovery rate on the risky bond is 45 percent. What does the credit spread imply for the probability of default?

π/(1-f) = [1- (1+y/2)^20/(1+y*/2)^20] π/(1-.45)= [1- (1+.055/2)^20/(1+.0625/2)^20] π = 7.27 percent π = 3.4 % historical default probability for an A rated

credit over 10 years (Moody’s 1920-2002) Factors contaminating spread, tax, liquidity, etc

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Portfolio exposure, default risk, and credit losses

Issuer Exposure Probability of default

A $25 .05

B $30 .10

C

Total

$45

$100 Million

.20

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Exposure, default and credit loss

Consider three bonds, A, B, and C with various default probabilities. Assume recovery in the event of default is zero. Default events are independent.

No default =(1-.05)(1-.10)1-.20)=.6840 Bond A defaults, while B and C do not. The probability of that is= .05(.90)(.80)=.036

and so on The probability of all three

default=.05x.10x.20=.001

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  default no default Exposure  

A 0.05 0.95 25000000  

B 0.1 0.9 30000000  

C 0.2 0.8 45000000  

Expected Loss= 13250000   red - default  

probability   Cumulative Loss Variance of loss

0.684 ABC 0.684 0 1.20085E+14

0.036 ABC 0.72 25,000,000 4.97025E+12

0.076 ABC 0.796 30,000,000 2.13228E+13

0.171 ABC 0.967 45,000,000 1.72379E+14

0.019 ABC 0.986 75,000,000 7.24482E+13

0.009 ABC 0.995 70,000,000 2.89851E+13

0.004 ABC 0.999 55,000,000 6.97225E+12

0.001 ABC 1 100,000,000 7.52556E+12

1       4.34688E+14

         

  VAR= 31,750,000 SDT= 20,849,160.65

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Transition Matrix for a BBB credit

spread,bps price % AAA 0.03 86 104.27 AA 0.24 126 102.53 A 3.87 180 100.25 BBB 82.52 203 98.92 BB 4.68 250 97.38 B 0.61 326 94.39 CCC 0.06 500 87.98 D 0.28 2036 50.24 WR 7.71 100 coupon yield maturity

6.50% 6.76% 5 years

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Expected credit loss= $13.25 M P(CL)>.95, = $45 Million Deviation from the mean= 45-13.25=31.8 This is credit VAR= $31.8 M Distribution of credit loss is highly skewed to

the left.

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Diversification of credit risk

Consider a loan to a single entity of $200 M. Assume default is 3 percent and zero recovery.

Expected loss= $.03x200= $6 M Variance=$1163.92 Standard deviation= $34.11 M Now consider 10 loan each $20 m, same

recovery and default. Expected loss= 10x.03x200/10=$6 M Variance = p(1-p)Nx($200/N)^2=116.4 Standard deviation=$10.78 M

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Credit Spread

The spread of default-free bond with that of the defaultable debt instruments provides valuable information in the following ways. -Conveys Probability of default -As a leading economic indicators -As an efficient allocator

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Example

What would be a fair price of a $10 million loan to a counterparty with a probability of default of 2 percent and recovery rate of 40 percent, assuming the cost of the fund for the lender is equal to LIBOR?

(y* - y) = π (1-f)

= .02 ( 1-.40)

= 120 bps

Fair price = L + 1.2%

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Credit spread

Credit spread is useful for estimation of default probability when there is a good bond market primary and secondary. This is rarely the case for number of reasons for many sovereign as well as other international debts. Absence of a well developed debt market The counterparty may not have publicly

traded debt The bond may not trade actively

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Merton Model

Merton model views stock price as call option on the value of the firm at strike price equal to the face value of debt.

Debt on the other hand can be viewed as risk free debt minus put option on the firm value.

Example:

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Credit Exposure

This is the amount of at risk during the life of the financial contract. For loans the credit exposure is close to notional. However, since the introduction of swaps, the measurement of credit exposure has become more sophisticated.

Credit exposure is the value of the asset at bankruptcy, that is positive like value of an option.

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Measurement of Current or Potential Exposure

Loans and bonds: are balance sheet assets whose current or potential exposure is close to notional amount.

Guarantees: are off-balance sheet contracts, such as LC. Standby facilities, acceptance.

Commitments: are off-balance sheet contracts to a future transactions, such as note issuance facility, where a minimum price is promised for notes issued by a borrower.

Swaps or forwards: are off-balance sheet contracts, as they are irrevocable commitment to buy or sell.

Long Options: are off-balance sheet contracts

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Exposure Modifiers

To limit exposure the financial industry has developed a number of methods to limit exposure. MTM has alleviated counterparty credit risk, however, MTM

introduces other risks: Operational risk Liquidity risk

Margins: as potential exposure is covered by setting a margin.

Collateral – the amount of collateral in excess of what is owed is called haircut, that reflect default and market risk

Exposure limits, with very little success Socio General Recouponing, requiring MTM at some fixed point, and

resetting coupon or exchange rate to prevailing market. Netting arrangements Credit triggers Time puts , termination option, Enron, LTCM

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Protection Buyers/Sellers

Survey conducted by BBA in 2003-4, reveals that : Banks account for 51% of protection buyers,

and 38% of protection sellers Insurance companies account for 1% of

protection buyers, and 21% of protection sellers

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AIG: Risk Management

Insurers in their main line of business are fairly diversified as the law of large numbers works for their advantage.

AIG sold default insurance on all kinds of bonds issued by major corporations.

AIG offset the potential losses stemming from bonds default, by taking short position on the underlying issuers stock.

AIG sold default insurance on CDO packaged by various underwriters, Countrywide, Fanni, and Freddi

Prudence required offsetting potential losses by selling the stocks of those companies short

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AIG: Why Did They Loose So Much?

They sold default insurance, credit default swap, whose payoff is a binary; 0 or (Par)(1-f), where f is recovery rate on the bond in the event of default or Material events as covered in the ISDA master agreement.

They could not or did not hedge their exposure to individual mortgagors with credit score at 650 or under in the sub-prime mortgage mess.

As foreclosures mounted the protection buyers default insurance on the par value of the debts, triggered by default became liabilities due.

AIG ran out of cash and unable to raise capital to pay of default insurance.

Government agreed to bail out, by taking 80 percent equity interest on AIG stock for providing $85 billion loan at L+8.5%

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Example

Treasury

Time Warner

Texas Utility

United Airline

Coupon

.05625

.0818

.06375

.0631

Yield Maturity

.0464 5

.0547 5

.06125 5

.0599 5

Par

125

85

36

69

Price Rate

130.384 --

98.84 BBB

36.37 BBB

69.93 BBB

Default rate in all 3 bonds is 5 percent with recovery of .48

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Example: continued

Create 2 tranches of floating rate note and Inverse floater, with 80/20 allocation from the portfolio of 3 corporate bonds rated BBB.

WAC=.0715 WAM= 5

WAC of 2-tranches= .07

Coupon of Floating rate note= L +2%

Assume 6-month LIBOR is 3.5 percent.

Coupon of Inverse floater= .27- 4(L) Both issues are priced at par initially

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Example continued

Floater is rated AAA, and has default rate of 1 percent

Inverse floater is rated B, and absorbs the first loss in the underlying collateral.

Market value weighted average of default of the 2-tranches has to be equal that of the collateral.

Tranche floater is less risky, therefore, the inverse floater must be more risky, with default rate of 21 percent.