Risk Measurement and Management Week 11 –November 2, 2006

Measuring Holding-Period Risk Price sensitivity of bonds is measured in terms of a bond price elasticity This elasticity is called duration denoted d1, which is widely used by bond traders and analysts and is often available on quote sheets

Example of Duration Assume a 10-year 8% coupon bond is priced at 12% yield to maturity and has value of 77.4 and duration of 6.8 If yields changed immediately from 12% to 10%, that is a 2/112 or 1.8% change in gross yield The bond price should change about 1.8% * 6.8 = 12.1%

Duration as Time Measure In 1930’s, Macauley noted that maturity was not relevant measure of timing of payments of bonds and defined his own measure, duration The definition of duration is (19-4): 19-4

Duration has two interpretations Elasticity of bond prices with respect to changes in one plus the yield to maturity Weighted average payment date of cash flows (coupon and interest) from bonds Duration measure Can be modified to be a yield elasticity by dividing by (1+yield to maturity) can be redefined using term structure of yields (Fisher-Weil duration noted d2)

Duration Calculations Duration can be calculated for bonds: For level-payment loans (e.g. mortgages):

Duration is an Approximation Derivative is used in calculating duration Price (Par=1.0) Actual price change Change predicted by duration Yield to Maturity

Properties of Duration Can be interpreted as price elasticity or weighted average payment period Note when c=0 that d1= M When M is infinite d1= (1+i)/i Duration measure effect of parallel shift in interest rates Other economic risks are not assessed

Duration as Risk Measure Good Balances reinvestment yield risk against capital gains risk Widely used and clear mathematical expression assessing holding-period yield risk Bad Approximation and theoretical issues Convexity adjustment only approximate improvement

Asset Liability Management: Definitions Approach to balance sheet management including financing and balance sheet composition and use of off-balance sheet instruments Assessment or measurement of balance sheet risk, especially to interest rate changes Simulation of earnings performance of a portfolio or balance sheet under a variety of economic scenarios

History of ALM After World War II to mid 1960’s ASSET MANAGEMENT Interest rates stable, large post-war holdings of government bonds, deposit markets protected Mid 1960’s to late 1970’s LIABILITY MANAGEMENT Interest rates rising, global financial markets developing (e.g. Eurodollars), regulation binding (maximum deposit interest rates)

History of ALM (continued) Late 1970’s to present ASSET/LIABILITY MANAGEMENT Use balance sheet composition or off-balance sheet instruments to management interest rate and other economic risks Changing markets - increased competition from non-banks, foreign institutions Goverment concerns - S&L failures, Continental Bank and Texas banks, etc.

Measurement of Risk of Balance Sheet Maturity gaps are common way to assess the sensitivity of a balance sheet to changes in interest rates Assets and liabilities classified by maturity or repricing interval Cumulative gap calculated Not easy to interpret in terms of risk

Duration of Balance Sheet Duration of a number of assets is Duration of net worth in a portfolio is

Simulation Computer simulation can handle more complex economic changes Many simulations can assess sensitivity of earnings to changes Regulators require and consultants can apply

Managing Interest Rate Risk Change balance sheet composition Adjust assets and liabilities until dE is at acceptable level Use futures or options to adjust next exposure What is source of value added?

Can Risk Management Add Value? Return to risk-free portfolio is the risk-free rate Investors can manage their own interest rate risk Does risk distract management or prevent exploitation of competitive advantage? Pleasing regulators and better understanding may be biggest advantage of ALM

Risk Management Balance sheet management ALM Duration and immunization Off balance sheet Futures Options Swaps

Types of Derivative Contracts Three basic types of contracts Futures or forwards Options Swaps Many basic underlying assets Commodities Currencies Fixed incomes or residual claims

Managing Risk with Futures Offset price or interest rate risk with contract which moves in opposite direction “Cross diagonally in the box” Identify contract with price or interest rate which moves as close as possible with the price or interest rate exposure Imperfect correlation is basis risk Not using futures or forwards can be speculation

Hedging Bank Planning to Borrow Insurance Hedge Borrowing Insurance Company with Premiums

Interest-Rate Options Interest rates and asset values move in opposite directions Long cash means short assets Short cash means long (someone else’s) asset Basis risk comes from spreads between exposure and hedge instrument Problem with production risk

Caps, floors, and collars If a borrower has a loan commitment with a cap (maximum rate), this is the same as a put option on a note If at the same time, a borrower commits to pay a floor or minimum rate, this is the same as writing a call A collar is a cap and a floor

Collars: Cap 6%, floor 4% Profit 9400 9500 9600 Loss

Options and Product Pricing Option pricing is well established technology Black-Scholes approaches Present value approaches Simulation In interest rates, lattice models used which are consistent with interest rate movements Can model any cash flow with combinations of options “Rocket Science”

Replication Futures with Options Profit Profit Buy Call Long P0 P0 Loss Loss Write Put 25

Other option developments Credit risk options Casualty risk options Requirements for developing an option Interest Calculable payoffs Enforceable

Swaps Exchange of future cash flows based on movement of some asset or price Interest rates Exchange rates Commodity prices or other contingencies Swaps are all over-the-counter contracts Two contracting entities are called counter-parties Financial institution can take both sides

Interest Rate Swap: Plain vanilla, LIBOR@5.5% 1/2 5% fixed Company A (receive floating) Company B (receive fixed) $2.5mm $2.75mm 1/2 6-month LIBOR Notional Amount $100 mm

Issues in Hedging Micro-hedging versus macro-hedging Accounting Regulation Assumptions underlying hedging Market liquidity Covariance structure (second moments) Notorious examples PNC, IG Metall, Bankers Trust, Orange Cy, Long-Term Capital Mgmt (LTCM), BancOne

Overview of Credit Risk Usual interpretation of credit risk is default on a loan or bond New views of credit risk are focused on the change in the credit-worthiness of debt instruments as well as default Risk changes will be reflected in the value of a portfolio over time as write-downs or downgrades short of default reduce value of claims (mark-to-market view of risk)

Default Private debt (corporate and household) may not pay cash flows as promised Late payments Nonpayment of interest or principal Other default or credit events Violation of covenants and other creditor interventions in operations Change in risk of default (e.g. highly leveraged transactions)

Credit Events Probability of default (PD) can change affecting the value of default-risky securities Upgrades and downgrades reflecting changes in PD are credit events Recent progress has been made in quantifying these probabilities

Bond and Debt Ratings Rating agencies Standard and Poor’s (AAA to D) Moody’s (Aaa to C) Fitch and Duff and Phelps Migration of ratings, e.g. from BBB to BB (investment grade to below investment grade) represents credit risk For example, change from BBB to BB has historical probability of 5.3% (S&P, 1996)

Ratings and Defaults

Risk of Fixed Incomes Probability Maximum value=F Future Value of Debt

Credit Losses Three elements in credit losses Estimated default probability (PD) Loss given default (LGD) Exposure at default (EAD) Credit losses = PD*LGD*EAD Investors in debt securities will be concerned about all these elements in managing their risks

Credit Risk Analysis Credit risk has become a major focus of rating agencies, regulators, and investors Very important to capital market development (e.g. asset securitizations, loan syndications) Enron, Global Crossing, and GE exemplify different stages of concern with these issues Consulting industry in credit analysis RiskMetrics (formerly J.P. Morgan) KMV (academic based research) Others (KPMG, PricewaterhouseCoopers, etc.)

Credit Risk Assessment Default occurs when value of assets less than value of liabilities (insolvency) Example of analysis used by KMV uses simplified estimates of variables Must calculate market value of assets (market value of debt and equity) and variability of market value Identify book value of liabilities

Motorola: Debt and Equity Total Market Value

Distance to Default: Example Motorola 2001-II (billions) Value of long-term debt = $ 7.3 Book value of current liabilities = 12.9 Total value of liabilities = $20.2 Market value of assets = $56.6 Standard deviation of change in market value = 16.4% Market value  standard deviation of percent change = $9.3 billion

Reduced Probability of Default? Estimated default point in example is midway between book value of current liabilities and long-term debt Theory is that long-term debt does not require immediate payment, short-term liabilities may allow some flexibility KMV uses historical data to fine-tune this estimate

Estimated Distance to Default Market value to default point = $40.0 $20.2 $56.6 $12.9 CL CL+LTD TMV Default point (estimated as midpoint) = $16.6

Distance to Default: 12-31-01 Total Value of Assets (from “Capital Structure” and Financial Statements): E + LTD + CL = TA $33.9 + $ 8.1 + $9.7 = $ 51.7 Book value of LTD and CL $8.4 and $9.7 Midpoint estimate of default point = $13.9 Std Dev = 16.4% * $51.7 = $8.48

Probability of Default KMV has used historical data to relate distance from default to probability of default That measure is proprietary (not available) As example, Motorola is rated A3 by Standard and Poors, historically associated with a default rate of about .82% over next five years (.61% in Moody’s experience)

Private Firm Default Risk KMV estimates non-traded firm risk by using market-traded comparables Data base on 35,000 traded firms globally Valuations of private firms and risk estimated by using EBITDA/Assets ratios KMV estimates default probabilities for private firms based on data on 300,000 firms in 30 countries Estimates depend on EBITDA0

Credit Risk in Portfolios Individual assets have probability of default and risk and discussed last week Loans in portfolios will have an interdependent risk structure due to correlations in defaults Credit risk within portfolio context is a major advance in credit risk management Search for a summary measure of portfolio risk led to the concept of value at risk

Value at Risk (VAR) Value at risk (VAR) looks at risk of portfolio accounting for covariance of assets Risk is defined in terms of likelihood of losses

VAR and Capital B Capital

Portfolio Credit Risk Credit risk different than usual portfolio risk analysis Returns are not symmetric Concentrations of exposure complicate losses Major issue is correlation of defaults and losses given default We will discuss approach followed by CreditMetrics Other approaches exist (including KMV)

Credit Risk as Rating Changes Increased credit risk Default CCC B BB Same credit risk (BBB) BBB A AA AAA Less credit risk

Rating Migrations (BBB rating) Source: Standard & Poors

Two Bond Rating Migrations

Probability of Default: Two Firms Value of Firm B Probability = 1/2% Probability = 1/10% Probability = 1/100% Default Point B Value of Firm A Default Point A

Loss Given Default

Simplified “Road Map” Compute exposure profile Of each asset Compute the volatility Of value caused by Up (down)grades and defaults Compute correlations Portfolio value-at-risk due to credit Source: Introduction to CreditMetrics (1997)

Required Resources Default probabilities (or ratings) Migration probabilities Historical data requirements Approaches to estimating correlations Complete data on types of credits and estimations of losses given defaults Exposures to classes of risks Models and simulations of value changes given credit events

Credit Portfolio Risk One Asset Many Assets Frequency Frequency Return Return Return

Incremental Risk Introduction to CreditMetrics provides good examples (in Section 5) Importance portfolio risk is the marginal risk Marginal risk considers portfolio risk implications 10%  High risk and large size $ 10mm $ Credit Exposure

Example Portfolio Source: Creditmetrics Technical Document (April 2, 1997)

Credit Risk Management Derivatives: Single-name v. multi-name Types of credit derivatives Total return swap Credit risk swap Credit risk option Credit inter-mediation swap Credit spread derivative Default substitution swap Over $400 billion notional amount 2000-IV

Hedging Credit Risk Hedging Instrument Payoff Change in Portfolio Value Risky Outcomes

Example of Total Return Swap 3-year 8% coupon bond If default probability increases from 10 to 20%, bond return is 8% - 6.4537% = 1.5463% (coupon minus loss due to downgrade)

Total Return Swap (8-6.5437)% Company A (pay total return) Company B (pay fixed) $154,626 $750,000 7.5% Notional Amount $10 mm

Total Return Swap Difference between payments and receipts by total return receiver is compensation for risk Total return payer receives cash in case of downgrade as in example, subsidizing loss realized on balance sheet Can have other swap types, as in default swap

Limitations of Derivatives Market limited to single name and portfolio instruments Typically individual corporate borrowers Some portfolio of commercial loans Market not developed for consumer credit Growth of consumer market but most risk-management consists of sales of loans Global market ready for consumer-risk derivatives

For Next Classes Prepare First American Bank: Credit Default Risk case for November 9 Read Chapters 23 and 24 for discussion in class on November 9 and November 16 Read KMV paper and Creditmetrics paper before November 16 class Teams should schedule appointments with me 13