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 4/22/2017

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 4/22/2017

Various risks Associated with investing in Bonds Interest Rate Risk Reinvestment risk call risk default risk inflation risk exchange rate risk liquidity risk volatility risk Risk risk 4/22/2017

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) 4/22/2017

4/22/2017 Issuer Coupon % Maturity YTM Modified Duration (Year) DV01 $ Par Amount (million) Full Price (000) S&P Credit Rate Treasury 6.5 2/15/10 4.55 6.56 50,109.87 67 76,387 5.625 5/15/08 4.64 5.48 54,121.57 92 98,762 5 8/15/11 4.57 7.77 203,838.95 84 87,192 Treasury Sector 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 6/1/08 6.76 5.25 15,830.32 30,153 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 5.34 17,243.92 32,292 BBB- Litton Industries 8 10/15/09 6.70 5.81 38,830.55 60 66,834 America Standard Inc. 7.625 7.59 6.10 25,212.52 41 41,332 BB+ Caterpillar Inc. 6.01 6.79 18,759.41 Corporate Sector 5.39 235,137.67 436,248 Portfolio 5.83 402,030.38 698,589 4/22/2017

Pricing Floating rate and Inverse floating rate securuties Collateral price=floater’s price+ inverse floater’s price 4/22/2017

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 4/22/2017

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 4/22/2017

Portfolio exposure, default risk, and credit losses Issuer Exposure Probability of default A $25 .05 B $30 .10 C Total $45 $100 Million .20 4/22/2017

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 4/22/2017

  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 1.20085E+14 0.036 0.72 25,000,000 4.97025E+12 0.076 0.796 30,000,000 2.13228E+13 0.171 0.967 45,000,000 1.72379E+14 0.019 0.986 75,000,000 7.24482E+13 0.009 0.995 70,000,000 2.89851E+13 0.004 0.999 55,000,000 6.97225E+12 0.001 1 100,000,000 7.52556E+12 4.34688E+14 VAR= 31,750,000 SDT= 20,849,160.65 4/22/2017

Transition Matrix for a BBB credit 4/22/2017

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. 4/22/2017

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 4/22/2017

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 4/22/2017

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% 4/22/2017

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 4/22/2017

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: 4/22/2017

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. 4/22/2017

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 4/22/2017

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 4/22/2017

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 4/22/2017

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 4/22/2017

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% 4/22/2017

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 4/22/2017

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 4/22/2017

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. 4/22/2017