1. 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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2. 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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3. 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
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4. 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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5. 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
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6. Pricing Floating rate and Inverse floating rate securuties
Collateral price=floater’s price+ inverse floater’s price
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7. 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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8. 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/( /2)^20]
π = 7.27 percent
π = 3.4 % historical default probability for an A rated credit over 10 years (Moody’s )
Factors contaminating spread, tax, liquidity, etc
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9. 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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10. 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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11. default
no default
Exposure A
0.05
0.95 B
0.1
0.9 C
0.2
0.8
Expected Loss=
red - default
probability
Cumulative
Loss
Variance of loss
0.684
ABC
E+14
0.036
0.72
25,000,000
E+12
0.076
0.796
30,000,000
E+13
0.171
0.967
45,000,000
E+14
0.019
0.986
75,000,000
E+13
0.009
0.995
70,000,000
E+13
0.004
0.999
55,000,000
E+12
0.001 1
100,000,000
E+12
E+14
VAR=
31,750,000
SDT=
20,849,160.65
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12. Transition Matrix for a BBB credit
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13. Expected credit loss= $13.25 M P(CL)>.95, = $45 Million
Deviation from the mean= =31.8
This is credit VAR= $31.8 M
Distribution of credit loss is highly skewed to the left.
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14. 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= $6 M
Variance=$
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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15. 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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16. 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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17. 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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18. 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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19. 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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20. 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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21. 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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22. Protection Buyers/Sellers
Survey conducted by BBA in , 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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23. 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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24. 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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25. Example Treasury Time Warner Texas Utility United Airline Coupon
.05625
.0818
.06375
.0631
Yield Maturity
Par
125 85 36 69
Price Rate
BBB
BBB
BBB
Default rate in all 3 bonds is 5 percent with recovery of .48
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26. 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= 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= (L)
Both issues are priced at par initially
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27. 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.
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