18th Annual Derivatives Securities and Risk Management Conference

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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
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Agenda
The equity premium Model setup Empirical
findings
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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
Remarks
Main Literature
Historical Equity Premium Estimates
(HEP) Implicit Equity Premium Estimates
(IEP) Utility-function-based estimation
(UEP) Expert estimates
EP 7-9 SR 40-50 EP 1-9 SR 5-50
EP lt 1 SR lt 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
Ibbotson (yearly) Fama/French (2001) Claus/Tho
mas (2001) Gebhardt/Lee/Swaminathan
(2001) Ohlson/Juettner-Nauroth (2005), Easton
(2004) Mehra/Prescott (1985) Welch (2000,
2001, 2008)
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 .
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Agenda
The equity premium Model setup Empirical
findings
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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 Moodys ratings, see
section Empirical findings for details
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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 ? lR
Default occurs, if assets at maturity are below
the default threshold L ? 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, ?
Correlation(Assets, Market), F cumulative
standard normal distribution
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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, ?
Correlation(Assets, Market), F cumulative
standard normal distribution
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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

• ?Q spread (1-RR), PDQ exp(-?Q 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!)

?Asset, 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
?Q 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)
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Two further model classes examined Merton style
first passage time models and a model with
incomplete information
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Merton
Model ingredients
Source
Merton (1974)
Asset value in t0
V0 ? 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
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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, ? Correlation(Assets, Market), F
cumulative standard normal distribution
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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) (T5,
First passage and Duffie/Lando (2001), s 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) s lt 10 leads to larger adjustment factors. s
lt 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
s 10 30 , Asset Sharpe ratio SR 10 40
, Risk neutral drift (after payouts) m 0 5
, Asset value uncertainty a 0 30 ,
Uncertainty time factor s 0 3 years, Default
barrier L 100, Asset Value in t0 All values
that resulted in a rating between AA and B for
any of the above combinations
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The Merton-formula is robust with respect to
noise in the input parameters
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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 ratio10 37 bp
• Company Sharpe ratio40 140 bp

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Agenda
The equity premium Model setup Empirical
findings
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Data sources Appr. 20.000 US-Investment-Grade
CDS from 2003-2007 were analyzed
Data sources
Parameter
1. See our paper for methodological details
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Descriptive statistics Mean CDS-spread of 50
bp, mean cumulative PD of 2 , mean correlation
0.5
Descriptive statistics
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Median market Sharpe ratio 37 (EDF) and 35
(Moodys)
75th Pctl
25th Pctl
Coeff of Variation
Median
Mean
N
Variable
56,80
21,91
76,33
37,23
42,46
19945
Sharpe ratio market(EDF)
26,71
11,76
60,32
19,04
19,56
19945
Sharpe ratio company (EDF)
43,12
15,45
59,64
27,39
32,13
19945
Sharpe ratio market (EDF), after tax adjustment
20,27
8,30
46,32
13,95
14,76
19945
Sharpe ratio company (EDF), after tax
adjustment
53,03
21,50
75,70
35,25
39,01
14743
Sharpe ratio market (Moodys)
26,51
10,15
67,63
17,79
18,70
14743
Sharpe ratio company (Moodys)
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
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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)
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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
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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