Risk Premiums in Interest Rates

J. K. Dietrich - FBE 524 - Fall, 2005

Risk Premiums in Interest Rates

Week 8 – October 12, 2005


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


Text Exhibit 8.6 Risky Rates


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

mm yyy

cp

)1(

1)

)1(

11(0


Bond Prices and YieldsBond Prices and Yields

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.03 0.05 0.07 0.09 0.11 0.13 0.15

Yield to Maturity

Bo

nd

Pric

e (

Par =

1.0

)

c=8%, m=10 years c=8%, m=20 years c=6%, m=10 years c=6%, m=20 years


Holding Period Risk (cont’d)

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


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

dprice

i yield1

%

% ( )


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

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):

n

tt

t

n

tt

t

yIytI

dD

1

11

)1(

)1()(


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)


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– Macauley’s formula (but combersome)– Calculate two prices and rates, divide changes– Closed-form solution, e.g. Dietrich formulation:

11 )1(

111Mi

cMpi

Mi

id


Duration is an Approximation

Yield to Maturity

Pri

ce (

Par

=1.

0)

0

p

i

Derivative is used in calculating duration

Change predicted by duration

i

Actual price change


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


Duration Calculation J. K. Dietrich - 2005Coupon BondsFormula:

Calculation: i = 10.0%

M= 10 d1 = 6.759

p = 1.0000c = 10.0%

Level Payment LoansFormula

Calculation i = 10.0%

M= 30 d1 = 9.1762

1M)i1(

1cM

pi

1M

i

i1d

1)i1(

M

i

i1d

M


Default or Credit Risk

1 2 3($849.12) 80 80 1080

Probability of payment 0.9 0.9 0.9Default cash flow 0 0 500Expected cash flow 72 72 1022Price $849.12RADR 12.00%YTM=IRR 14.56%

Example of 8% 3-year bond with risk-adjusted discount rate (RADR) of 12%:


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

Probability Price YTM0.9 849.12 14.56%0.8 794.32 17.36%0.7 739.52 20.45%


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


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


Short and Long Risk Premia

0

1

2

3

4

5

72 74 76 78 80 82 84 86 88 90 92 94 96 98

BAARISKPR CPRISKPR

Baa and Commercial Paper Risk Premia


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


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


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


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)


Risk of Fixed Incomes

Future Value of Debt

Pro

babi

lity

Maximum value=F


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.)


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


Motorola: Debt and Equity

0

20000

40000

60000

80000

100000

120000

140000

91 92 93 94 95 96 97 98 99 00

MVETMV

LTDANDCLCL

Motorola Total Market Values 1991-1 to 2001-2

Total Market Value


Distance to Default: Example

Motorola 2001-II (billions)Value of long-term debt = $ 7.3Book value of current liabilities = 12.9Total value of liabilities = $20.2 Market value of assets = $56.6Standard 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

$56.6$20.2$12.9TMVCL+LTDCL

Default point (estimated as midpoint) = $16.6

Market value to default point = $40.0

3.43.9$

6.16$6.56$defaulttocetanDis


Distance to Default: 12-31-02

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 $9.7Midpoint estimate of default point = $13.9Std Dev = 16.4% * $51.7 = $8.48

5.448.8$

9.13$7.51$defaulttocetanDis


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


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

Future Value of Portfolio

Pro

babi

lity

Maximum value=F

Value at Risk


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

CCCBBB

Same credit risk (BBB) BBBAAA AAA

Less credit risk


Rating Migrations (BBB rating)

Year-End Rating Probability (%)AAA 0.02AA 0.33A 5.95

BBB 86.93BB 5.30B 1.17

CCC 0.12Default 0.18

Source: Standard & Poors


Two Bond Rating Migrations

AAA AA A BBB BB B C Default0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06

AAA 0.02 0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.00AA 0.33 0.00 0.04 0.29 0.00 0.00 0.00 0.00 0.00A 5.95 0.02 0.39 5.44 0.08 0.01 0.00 0.00 0.00

BBB 86.93 0.07 1.81 79.69 4.55 0.57 0.19 0.01 0.04BB 5.30 0.00 0.02 4.47 0.64 0.11 0.04 0.00 0.01B 1.17 0.00 0.00 0.92 0.18 0.04 0.02 0.00 0.00

CCC 0.12 0.00 0.00 0.09 0.02 0.00 0.00 0.00 0.00Default 0.18 0.00 0.00 0.13 0.04 0.01 0.00 0.00 0.00

Obligor #1 (BBB)

Obligor # 2 (Single-A)


Probability of Default: Two Firms

Value of Firm A

Val

ue

of F

irm

B

Default Point A

Def

ault

Poi

nt

B

Probability = 1/100%

Probability = 1/2%

Probability = 1/10%


Loss Given Default

Seniority Class Mean (%) Standard Deviation (%)Senior Secured 53.8 26.86Senior Unsecured 51.13 25.45Senior Subordinated 38.52 23.81Subordinated 32.74 20.81Junior Subordinated 17.09 10.9Source: Carty & Lieberman [96a] -- Moody's Investors Service


Simplified “Road Map”

Computeexposure profile

Of each asset

Compute the volatilityOf value caused by

Up (down)grades and defaults

Computecorrelations

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 RiskF

requ

ency

Return

Fre

quen

cy

Return

One Asset Many Assets

0 0


Incremental Risk

Introduction to CreditMetrics provides good examples (in Section 5)

Importance portfolio risk is the marginal risk

Marginal risk considers portfoliorisk implications

10%

$ Credit Exposure

High risk and large size

$ 10mm


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

Risk Premiums in Interest Rates

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