18th annual derivatives securities and risk management conference arlington, virginia linking credit...

18th Annual Derivatives Securities and Risk Management ConferenceArlington, Virginia

Linking Credit Risk Premia to the Equity Premium(1) (Tobias Berg, Christoph Kaserer – Technische Universität München)

Apr. 11th, 2008

(1) Working Paper available on http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1019279

2Linking Credit Risk Premia to the Equity Premium-11Apr08-Berg_V01_SV.ppt

Agenda

The equity premium

Model setup

Empirical findings


Agenda

The equity premium

Model setup

Empirical findings


Several approaches to estimate future equity premia have been proposed – none has gained ultimate acceptance

ApproachApproach

Historical Equity Premium Estimates (HEP)

Implicit Equity Premium Estimates (IEP)

Utility-function-based estimation (UEP)

Expert estimates

EstimatesEP / SR

EstimatesEP / SR

EP: 7-9 % SR: 40-50 %

EP: 1-9 % SR: 5-50%

EP: < 1 %SR: < 5 %

EP: 4-7 %SR: 25-40 %

RemarksRemarks

Assumption: Historical returns are a proxy for future returnsHEP is widely seen as upward biased (for the U.S.) in todays research

Based on DCF-valuation formulaeHigh dependency on terminal value / long-run growth assumptions

Only of minor importance for practical applications

Reliance on expert estimates not satisfactory from an academic perspective

Main LiteratureMain Literature

Ibbotson (yearly)Fama/French (2001)

Claus/Thomas (2001)Gebhardt/Lee/Swaminathan (2001)Ohlson/Juettner-Nauroth (2005), Easton (2004)

Mehra/Prescott (1985)

Welch (2000, 2001,2008)

EP: Equity Premium, SR: Sharpe ratioRemark(s): Not all studies mentioned above do report equity premia and market sharpe ratios. If, not, equity premia were converted using a market volatility of 15-20 %.


Agenda

The equity premium

Model setup

Empirical findings


Risk aversion does not only influence equity prices but credit prices as well

Theoretical indicationsTheoretical indications

Defaults are correlated and have a systematic driving factor(1)

• Usual assumption in credit portfolio management / CreditVaR-calculation

• Supported by historical default rates

EL: Expected Loss, bp: basis point(1) Ratio of risk neutral to actual expected loss(2) Cf. for example Hull/Predescu/White (2005), Green (1991), Fama (1993), Moody's (2007)(3) Based on US CDS from 2003-2007 and based on Moody’s ratings, see section “Empirical findings” for details

Empirical indications(3)Empirical indications(3)

2.07 35.34 2.59 34.91 3.32 33.83 4.07 28.20 6.03 26.30

Q-to-P(1) Δ (bp)

33.17 22.00 14.60 9.17 5.23

Average 5-y-EL p.a.(bp)

68.51 56.91 48.43 37.37 31.53

Average 5-y-CDS-spread (bp)

Baa3 Baa2 Baa1 A AA

Rating grade

risk neutral world real world credit risk premium

We will use credit risk premia together with structural models of default to estimate the market Sharpe ratio and the equity premium


Merton framework: We derive a simple formula for extracting market Sharpe ratios out of credit spreads

Risk neutral wolrdRisk neutral wolrd Real worldReal world

Asset value processAsset value process

Default mechanismDefault mechanism

Default probabilityDefault probability

Asset Sharpe ratioAsset Sharpe ratio

PPPt

P

tttdBVdtVdV QQQ

tQ

tttdBVdtrVdV

LV PT

Default occurs, if assets at maturity are below the default threshold L є lR

LV QT

T

TrVLPDQ

)2/1()/ln( 2

0

T

TVLPDP

)2/1()/ln( 2

0

T

PDPDSR

PQ

Assets

)()( 11

Market Sharpe ratio Market Sharpe ratio MarketAssets

PQ

MarketT

PDPDSR

,

11 1)()(

Default occurs, if assets at maturity are below the default threshold L є lR

Pt

t

BtP eVV )5.0(

0

2Q

t

t

BtrQ eVV

)5.0(

0

2

PDQ: cumulative risk neutral default probability, PDP: cumulative real world default probability, SR: Sharpe ratio, T: Maturity, ρ: Correlation(Assets, Market), Φ: cumulative standard normal distribution


Three key properties needed for empirical applications

MarketAssets,

P1Q1

Market ρ

1

T

)(PD)(PDSR

Input parameters must be available1

Estimator must be robust with respect to model changes2

Estimator must be robust with respect to noise in the input parameters3

PDQ: cumulative risk neutral default probability, PDP: cumulative real world default probability, SR: Sharpe ratio, T: Maturity, ρ: Correlation(Assets, Market), Φ: cumulative standard normal distribution


Input parameters can be derived from CDS-spreads, ratings and equity correlations

1

Input parameterInput parameter

PDQ

ExplanationExplanation

Risk neutral default probability

SourceSource

CDS-spreads

RemarkRemark

• λQ = spread ∙ (1-RR), PDQ = exp(-λQ∙ T)• Widely available• Very liquid (average bid/ask-spread of 4

bp for CDX.NA.IG-index)• CDS better suited than bond-spreads

due to risk-free rate problem

PDP Actual default probability

Ratings • Point-in-time ratings: EDFs, Altman• Ratings of rating agencies + historical

default probabilities per rating grade (cycle-problem)

• Bank internal ratings + masterscale (default criteria!)

T Maturity Maturity of CDS

ρAsset, Market Correlation between assets and market portfolio

Equity correlations • It can be shown, that equity correlations are a good proxy for asset correlations

There is no need to calibrate the t0-asset value, the asset volatility, the default barrier or the risk free rate

λQ = risk neutral default intensity, RR: Recovery rate, EDF: Expected default frequencies(1) Cf. for example Hull/Predescu/White (2005), Green (1991), Fama (1993), Moody's (2007)


Two further model classes examined: Merton style first passage time models and a model with incomplete information

Model ingredientsModel ingredients MertonMerton

Asset value in t=0Asset value in t=0

Asset value processAsset value process

Default boundaryDefault boundary

Default mechanismDefault mechanism

V0 є lR

Geometric Brownian Motion

Exogenous

Default only at maturity

2

SourceSource Merton (1974)

(1) In our application, we include all combinations of (V0, L), therefore all endogenous default models where the optimal liquidation time can be expressed as the first time that the asset value falls below a constant default barrier (which is the usual case) are implicitly included

Key characteristicsKey characteristics

First passage/Strategic defaultFirst passage/

Strategic default

V0 є lR


Exogenous/Endogenous(1)

First passage

Black/Cox (1976), Leland (1994), Leland/

Toft (1996) (among others)

• Allows for a default before maturity

• Strategic/Endogenous default models

Incomplete InformationIncomplete Information

V0 є (L , ∞)


Exogenous/Endogenous(1)

First passage

Duffie/Lando (2001)

• Consistent with reduced form credit pricing

• Realistic short term default probabilities

• First structural default model

• Simple framework


In contrast to the default probabilities itself, the Merton estimator is robust with respect to model changes

2

MertonMerton

First passageFirst passage

Duffie/LandoDuffie/Lando

ModelModel AF(2) AF(2)

1.00

1.06

0.99

1. Please note that not all parameters are needed for all models2. AF: Adjustment factor := market Sharpe ratio divided by Merton estimator for market Sharpe ratiocPDQ: cumulative risk neutral default probability, PDP: cumulative real world default probability, SR: Sharpe ratio, T: Maturity, ρ: Correlation(Assets, Market), Φ: cumulative standard normal distribution

Parameter Combinations(Representative Example)(1)

Parameter Combinations(Representative Example)(1)

V0=200

T=5

σ=15%

r=4%

SRMarket = 40%

L=100

α=30%

s=1 V-s=200

δ=2%ρ=0.5

MarketAssets,

1

T

)(PD)(PD P1Q1

40.00 %

37.89 %

40.48 %

PDPPDP

0.41 %

1.04 %

1.52 %

PDQPDQ

1.40 %

2.94 %

4.33 %

Model (and parameter) changes affect both PDP and PDQ in the same direction – the Sharpe ratio is the only parameter that solely has an influence on PDP


Merton-formula is robust with respect to model changes –adjustment factor close to one for all investm. grade ratings

(1) σ < 10% leads to larger adjustment factors. σ < 10 % is though only reasonable for financial services companies. Effect is rather technical and due to default timing. If an additional restriction concerning the default timing is introduced, than the formula is also robust for asset volatilities smaller than 10 % (see next slide).Parameter combinations: Asset volatility: σ = 10 – 30 %, Asset Sharpe ratio: SR = 10 – 40 %, Risk neutral drift (after payouts): m = 0 – 5 %, Asset value uncertainty: α = 0 – 30 %, Uncertainty time factor: s = 0 – 3 years, Default barrier: L = 100, Asset Value in t=0: All values that resulted in a rating between AA and B for any of the above combinations

2

Rule of thumb: Resulting error is on average smaller than 10% for all investment grade obligors

Minimum and maximum adjustment factor(1) (T=5, First passage and Duffie/Lando (2001), σ ≥ 10%(1))

Minimum and maximum adjustment factor(1) (T=5, First passage and Duffie/Lando (2001), σ ≥ 10%(1))

Large adjustment factors due to positive risk neutral asset vale drift relative to

default barrier

Small adjustment factors due to high asset value

uncertaintyAdjustment factor = 1 means that result is equal to the

Merton framework


The Merton-formula is robust with respect to noise in the input parameters

3

Example (for illustration)Example (for illustration)

Calculation of model-implied CDS-spread

Methodology• Merton framework• Based on relationship between PDQ and PDP

Parameters• Maturity: 5 years• Rating: BBB (Cumulative PDP = 2.17%)• Recovery rate: 50%

Resulting model-implied CDS-spread• Company Sharpe ratio=10%: 37 bp• Company Sharpe ratio=40%: 140 bp

High sensitivity of model-implied CDS-spread w.r.t. asset Sharpe ratio:

Low sensitivity of SR-estimator w.r.t. PDP and PDQ

Sensitivity analysisSensitivity analysis

0

50

100

150

200

250

300

350

400

0% 5% 10% 15% 20% 25% 30% 35% 40%

Asset sharpe ratio

CD

S sp

read

(bp

)

Aa

A

Baa

Ba


Agenda

The equity premium

Model setup

Empirical findings


Data sources: Appr. 20.000 US-Investment-Grade CDS from 2003-2007 were analyzed

Data sourcesData sources

Market: USInstruments: CDS (Senior)Obligors: CDX.NA.IG (125 most liquid IG CDS)Time frame: 01/2003 – 06/2007Frequency: weekly

GeneralGeneral

1. See our paper for methodological details

CDS-spreads / Risk neutral default probabilities

CDS-spreads / Risk neutral default probabilities

CDS-spreads• Source: CMA (through Datastream)• Maturity: 5 years• Only actual trades and firm quotes

Risk neutral default probabilities• Recovery rate used for determination of PDQ:50 %1

Actualdefault probabilitiesActualdefault probabilities

Two different sources were used:• EDFs (KMV) (monthly)• Moody's Senior unsecured ratings + historical defaults per rating grade

CorrelationsCorrelationsCorrelation

• 3-year weekly equity correlations with S&P500

ParameterParameter


Descriptive statistics: Mean CDS-spread of ~ 50 bp, mean cumulative PD of ~ 2 %, mean correlation ~ 0.5

Descriptive statisticsDescriptive statistics

0,600,42250,520,5119945Correlation

0,06%-0,11%11660-0,01%0,00%14743Δ (EDF, Moodys-PD)

2,51%0,93%751,63%2,01%14743Moodys PD5

0,18%0,05%1200,10%0,15%14743Moodys PD1

2,15%1,00%951,38%1,93%19945EDF5

0,15%0,05%1740,08%0,15%19945EDF1

18%11%3815%15%19945Asset Vol.

53544419945Δ (offer, bid) in bp

612677404919945CDS bid in bp

643073445319945CDS offer in bp

632875425119945CDS mid in bp

75th Pctl25th PctlCoeff of

VariationMedianMeanNVariable


Median market Sharpe ratio: 37% (EDF) and 35% (Moodys)

26,51%10,15%67,6317,79%18,70%14743Sharpe ratio company (Moodys)

53,03%21,50%75,7035,25%39,01%14743Sharpe ratio market (Moodys)

20,27%8,30%46,3213,95%14,76%19945

Sharpe ratio company (EDF), after tax adjustment

43,12%15,45%59,6427,39%32,13%19945

Sharpe ratio market (EDF), after tax adjustment

26,71%11,76%60,3219,04%19,56%19945Sharpe ratio company(EDF)

56,80%21,91%76,3337,23%42,46%19945Sharpe ratio market(EDF)

75th Pctl25th PctlCoeff of

VariationMedianMeanNVariable

Remark(s): (EDF) and (Moodys) denotes that the real world default probability was taken from EDFs or from Moody's Senior Unsecured ratings respectively(1) Furthermore, this result offers a line of thought for a solution to the credit spread puzzle, Working Paper "A solution to the Credit Spread Puzzle" available on request

Based on credit valuations, a Sharpe ratio of 40-50% corresponding to an historical equity premium of 7-9% seem to be too high(1)


Implicit market Sharpe ratio fluctuates between 30 % and 50 %Volatility of market Sharpe ratio approximately 50%

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

01/03 07/03 01/04 07/04 01/05 07/05 01/06 07/06 01/07

Date

Impl

icit

sha

rpe

rati

o

Company sharpe ratio

Market sharpe ratio

Downgradesof Ford and

GM

Subprimecrisis

De-coupling of spreads (increasing)and EDFs/Equity markets (decreasing

volas, slightly increasing prices)


Our calculations should determine an upper limit for the market Sharpe ratio / equity premium

Input parametersInput parameters Not considered in our calculationNot considered in our calculation

CDS-spreadsCDS-spreads

Recovery rateRecovery rate

CorrelationsCorrelations

• Implicit Options (delivery) have not been considered

• Part of spread may not be attributable to credit risk

• Recovery rate used (50%) is slightly higher than most estimates from historical averages

• Risk neutral recovery rate should be even lower than actual recovery rate

• Correlations could also be derived from historical PD-volatilities or from the Basel-II-framework

• Asset correlations may be lower than equity correlations

EffectEffect

• Our result is upward biased




• Our result is upward biased(1)

• Our result may be slightly downward biased

1. Results available on request


Summary

We derive a simple and convenient estimator for the market Sharpe ratio and the equity premium within the Merton framework which is based on credit valuations

• All input parameters (actual + risk neutral default probability, maturity, equity correlations) are available

• Noise in the input parameters does not have a large influence on the resulting Sharpe ratio estimation

• The approach offers a new line of thought which is not directly linked to current methods

The estimator is robust with respect to model changes

• Model classes analyzed: Merton framework, First passage time / Strategic default framework, Duffie/Lando (2001) framework with unobservable asset values

• Reason: The estimator uses the difference between risk neutral and actual default probabilities. In contrast to the default probabilities itself, this difference is quite robust with respect to model changes

Empirical results from U.S.-CDS-spreads (2003-2007, ~20.000 observations) indicate, that historical equity premia are upward biased

• Estimator yields an upper limit for the market Sharpe ratio between 30-40% (equivalent to an equity premium of appr. 5-7%) which is lower than historical market Sharpe ratios (~ 40-50%)

• Time series estimation for market Sharpe ratio was carried out, volatility of time series ~ 50%

• Results offer a possible solution to the Credit spread puzzle(1)

1. Working Paper "A solution to the Credit Spread Puzzle" available on request

18th annual derivatives securities and risk management conference arlington, virginia linking credit...

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