Risk Premiums in Interest Rates

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Risk Premiums in Interest Rates

• Week 8 October 12, 2005

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Types of Risk

• Holding-period yield risk
• Capital gains risk
• Reinvestment risk
• Default risk
• Non-payment of principal
• Delayed or reduced payments
• Sometimes viewed as embedded option (put)
• Call risks
• Call risks can be viewed as embedded options
  (call)
• Liquidity or marketability risk
• Often measured as bid-ask spread for traded
  securities

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Text Exhibit 8.6 Risky Rates
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Holding Period Yield Risk

• Bond price is function of expected cash flows and
  RADR, but usually bond traders define contractual
  cash flows and yields to maturity
• Relationship between YTM and p is a curve defined
  by equation from last week

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Bond Prices and Yields
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Holding Period Risk (contd)

• Zero coupon default risk free bonds held to
  maturity will earn yield to maturity
• If maturity equals holding period, no risk
• Future wealth from coupon bonds depends on income
  from reinvested cash at rates not known now
• Bonds which must be sold at end of holding period
  (maturity does not equal holding period) have
  risk of capital gains or losses

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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

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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

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Duration as Time Measure

• 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 (p. 192)

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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 (1yield to maturity)
• Can be redefined using term structure of yields
  (Fisher-Weil duration noted d2)

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Alternative Duration Calculation

• Duration is widely used by bond traders and fixed
  income portfolio managers
• Duration values are available from information
  services like Bloomberg
• Calculated is three ways
• Macauleys formula (but combersome)
• Calculate two prices and rates, divide changes
• Closed-form solution, e.g. Dietrich formulation

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Duration is an Approximation
Derivative is used in calculating duration
Price (Par1.0)
Actual price change
Change predicted by duration
0
Yield to Maturity
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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

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Default or Credit Risk
Example of 8 3-year bond with risk-adjusted
discount rate (RADR) of 12
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Effect of Credit Risk Change

• RADR constant at 12
• Note price and yield change
• Could be a change in one industry or firm
• Default risk could be diversified in this case

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Pricing Default-Risky Bond

• Discount expected cash flows (related to default
  probabilities) to obtain value
• Default probabilities may be related to bond
  ratings
• Change in default probability will change
  expected cash flows and yield to maturity
• Risk-adjusted discount rate (RADR) may or may not
  change with change in default risk

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Risk Premiums in RADRs

• Diversifiable or avoidable risks
• Hedge portfolios can reduce or eliminate
• Unsystematic risks are not priced (do not
  increase discount rate)
• Priced versus non-priced risk
• Systematic risks are unavoidable
• Risk aversion (declining marginal utility of
  wealth) is underlying assumption

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Short and Long Risk Premia
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Default Risk Premiums

• Vary over the business cycle
• Can be changes in default risk probabilities or
  in the market price of risk
• Current default risk premiums are very high
  relative to historical experience
• Development of bond markets internationally is
  believed to promise substantial growth and risk
  analysis of private borrowers will be very
  important

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Liquidity Risk

• Unusual securities in often cannot be sold
  readily
• Reflected in dealers spreads
• If no market makers, can only be estimated
• Thin markets require price concessions for quick
  transactions despite intrinsic values
• Examples
• Structured notes and Orange County
• Drexel Burnam and junk bonds in 1980s
• The Russian debt market collapse in 1998

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Developments in 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

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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)

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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

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Bond and Debt Ratings

• Rating agencies
• Standard and Poors (AAA to D)
• Moodys (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 (SP, 1996)

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Risk of Fixed Incomes
Probability
Maximum valueF
Future Value of Debt
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Credit Losses

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

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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.)

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Example of Steps to estimate PD

• 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

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Motorola Debt and Equity
Total Market Value
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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

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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

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Estimated Distance to Default
Market value to default point 40.0
56.6
20.2
12.9
TMV
CLLTD
CL
Default point (estimated as midpoint) 16.6
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Distance to Default 12-31-02

• Total Value of Assets (from Capital Structure
  and Financial Statements) E LTD CL
  TA33.9 8.1 9.7 51.7
• Book value of LTD and CL 8.4 and 9.7Midpoint
  estimate of default point 13.9Std Dev 16.4
  51.7 8.48

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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

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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

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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

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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)

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Credit Risk as Rating Changes

• Increased credit risk Default
  CCC B BB
• Same credit risk (BBB) BBB
• A AA
• AAA
• Less credit risk

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Rating Migrations (BBB rating)
Source Standard Poors
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Two Bond Rating Migrations
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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
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Loss Given Default
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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)
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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

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Credit Portfolio Risk
One Asset
Many Assets
Frequency
Frequency
0
0
Return
Return
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Incremental Risk

• Introduction to CreditMetrics provides good
  examples (in Section 5)
• Importance portfolio risk is the marginal risk
• Marginal risk considers portfoliorisk
  implications

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High risk and large size
?
10mm
Credit Exposure
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Next Week October 19, 2005

• Take-home midterm examination due 90-minute
  examination is open book and open note
• Examination must be completed without discussion
  with anyone and a declaration to that effect and
  time spent turned in with examination
• Prepare Chapter 9
• Arrange a meeting regarding project progress if
  you have not talked to me