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18th Annual Derivatives Securities and Risk Management Conference Arlington, Virginia

18th Annual Derivatives Securities and Risk Management Conference Arlington, Virginia Linking Credit Risk Premia to the Equity Premium (1) (Tobias Berg, Christoph Kaserer – Technische Universität München). Apr. 11 th , 2008.

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18th Annual Derivatives Securities and Risk Management Conference Arlington, Virginia

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  1. 18th Annual Derivatives Securities and Risk Management Conference Arlington, 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

  2. Agenda The equity premium Model setup Empirical findings

  3. Agenda The equity premium Model setup Empirical findings

  4. Several approaches to estimate future equity premia have been proposed – none has gained ultimate acceptance Estimates EP / SR Approach Main Literature Remarks Historical Equity Premium Estimates (HEP) Implicit Equity Premium Estimates (IEP) Utility-function-based estimation (UEP) Expert estimates 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: 7-9 % SR: 40-50 % EP: 1-9 % SR: 5-50% EP: < 1 % SR: < 5 % EP: 4-7 % SR: 25-40 % Assumption: Historical returns are a proxy for future returns HEP is widely seen as upward biased (for the U.S.) in todays research Based on DCF-valuation formulae High 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 EP: Equity Premium, SR: Sharpe ratio Remark(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 %.

  5. Agenda The equity premium Model setup Empirical findings

  6. Theoretical indications • Defaults are correlated and have a systematic driving factor(1) • Usual assumption in credit portfolio management / CreditVaR-calculation • Supported by historical default rates Rating grade Average 5-y-CDS-spread (bp) Average 5-y-EL p.a.(bp) Δ (bp) Q-to-P(1) AA 31.53 5.23 26.30 6.03 A 37.37 9.17 28.20 4.07 Baa1 48.43 14.60 33.83 3.32 Baa2 56.91 22.00 34.91 2.59 Baa3 68.51 33.17 35.34 2.07 Risk aversion does not only influence equity prices but credit prices as well Empirical indications(3) 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 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

  7. Merton framework: We derive a simple formula for extracting market Sharpe ratios out of credit spreads Risk neutral wolrd Real world Asset value process Default mechanism Default occurs, if assets at maturity are below the default threshold L є lR Default occurs, if assets at maturity are below the default threshold L є lR Default probability Asset Sharpe ratio Market Sharpe ratio 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

  8. Three key properties needed for empirical applications Input parameters must be available 1 Estimator must be robust with respect to model changes 2 Estimator must be robust with respect to noise in the input parameters 3 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

  9. T Maturity Maturity of CDS Input parameters can be derived from CDS-spreads, ratings and equity correlations 1 Input parameter Explanation Source Remark PDQ Risk neutral default probability CDS-spreads • 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!) 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)

  10. First passage/Strategic default Incomplete Information Black/Cox (1976), Leland (1994), Leland/Toft (1996) (among others) Duffie/Lando (2001) V0 є lR V0 є (L , ∞) Geometric Brownian Motion Geometric Brownian Motion Exogenous/Endogenous(1) Exogenous/Endogenous(1) First passage First passage • Allows for a default before maturity • Strategic/Endogenous default models • Consistent with reduced form credit pricing • Realistic short term default probabilities Two further model classes examined: Merton style first passage time models and a model with incomplete information 2 Merton Model ingredients Source Merton (1974) Asset value in t=0 V0 є lR Asset value process Geometric Brownian Motion Default boundary Exogenous Default mechanism Default only at maturity Key characteristics • First structural default model • Simple framework (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

  11. Parameter Combinations (Representative Example)(1) PDP PDQ AF(2) 0.41 % 1.40 % 40.00 % 1.00 V0=200 σ=15%   T=5 ü r=4% 1.04 % 2.94 % 37.89 % 1.06 L=100 SRMarket = 40% ρ=0.5 ü δ=2%   α=30% 1.52 % 4.33 % 40.48 % 0.99 s=1 V-s=200 In contrast to the default probabilities itself, the Merton estimator is robust with respect to model changes 2 Model Merton First passage Duffie/Lando 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 1. Please note that not all parameters are needed for all models 2. AF: Adjustment factor := market Sharpe ratio divided by Merton estimator for market Sharpe ratioc 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

  12. Large adjustment factors due to positive risk neutral asset vale drift relative to default barrier Small adjustment factors due to high asset value uncertainty Merton-formula is robust with respect to model changes –adjustment factor close to one for all investm. grade ratings 2 Minimum and maximum adjustment factor(1) (T=5, First passage and Duffie/Lando (2001), σ ≥ 10%(1)) Adjustment factor = 1 means that result is equal to the Merton framework Rule of thumb: Resulting error is on average smaller than 10% for all investment grade obligors (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

  13. Sensitivity analysis High sensitivity of model-implied CDS-spread w.r.t. asset Sharpe ratio: Low sensitivity of SR-estimator w.r.t. PDP and PDQ The Merton-formula is robust with respect to noise in the input parameters 3 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

  14. Agenda The equity premium Model setup Empirical findings

  15. General Market: US Instruments: CDS (Senior) Obligors: CDX.NA.IG (125 most liquid IG CDS) Time frame: 01/2003 – 06/2007 Frequency: weekly 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 probabilities • Two different sources were used: • EDFs (KMV) (monthly) • Moody's Senior unsecured ratings + historical defaults per rating grade Correlations • Correlation • 3-year weekly equity correlations with S&P500 Data sources: Appr. 20.000 US-Investment-Grade CDS from 2003-2007 were analyzed Parameter Data sources 1. See our paper for methodological details

  16. Variable N Mean Median Coeff of Variation 25th Pctl 75th Pctl CDS mid in bp 19945 51 42 75 28 63 CDS offer in bp 19945 53 44 73 30 64 CDS bid in bp 19945 49 40 77 26 61 Δ (offer, bid) in bp 19945 4 4 54 3 5 Asset Vol. 19945 15% 15% 38 11% 18% EDF1 19945 0,15% 0,08% 174 0,05% 0,15% EDF5 19945 1,93% 1,38% 95 1,00% 2,15% Moodys PD1 14743 0,15% 0,10% 120 0,05% 0,18% Moodys PD5 14743 2,01% 1,63% 75 0,93% 2,51% Δ (EDF, Moodys-PD) 14743 0,00% -0,01% 11660 -0,11% 0,06% Correlation 19945 0,51 0,52 25 0,42 0,60 Descriptive statistics: Mean CDS-spread of ~ 50 bp, mean cumulative PD of ~ 2 %, mean correlation ~ 0.5 Descriptive statistics

  17. Median market Sharpe ratio: 37% (EDF) and 35% (Moodys) Variable N Mean Median Coeff of Variation 25th Pctl 75th Pctl Sharpe ratio market(EDF) 19945 42,46% 37,23% 76,33 21,91% 56,80% Sharpe ratio company (EDF) 19945 19,56% 19,04% 60,32 11,76% 26,71% Sharpe ratio market (EDF), after tax adjustment 19945 32,13% 27,39% 59,64 15,45% 43,12% Sharpe ratio company (EDF), after tax adjustment 19945 14,76% 13,95% 46,32 8,30% 20,27% Sharpe ratio market (Moodys) 14743 39,01% 35,25% 75,70 21,50% 53,03% Sharpe ratio company (Moodys) 14743 18,70% 17,79% 67,63 10,15% 26,51% 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) 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

  18. Implicit market Sharpe ratio fluctuates between 30 % and 50 %Volatility of market Sharpe ratio approximately 50% Downgrades of Ford and GM Subprime crisis De-coupling of spreads (increasing) and EDFs/Equity markets (decreasing volas, slightly increasing prices)

  19. Our calculations should determine an upper limit for the market Sharpe ratio / equity premium Input parameters Not considered in our calculation Effect CDS-spreads • Implicit Options (delivery) have not been considered • Part of spread may not be attributable to credit risk • Our result is upward biased • Our result is upward biased Recovery rate • 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 • Our result is upward biased • Our result is upward biased Correlations • Correlations could also be derived from historical PD-volatilities or from the Basel-II-framework • Asset correlations may be lower than equity correlations • Our result is upward biased(1) • Our result may be slightly downward biased 1. Results available on request

  20. 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

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