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Credit Risk Premia and a Link to the Equity Premium
Tobias Berg, Christoph Kaserer
Draft, this version: October 08, 2007
Abstract
While the equity premium is - both from a conceptual and
empirical perspective - awidely researched topic in finance, the
analysis of credit risk premia has only recentlyattracted wider
attention. Credit risk premia are usually measured as the
multiplebetween risk neutral (derived from bonds or CDS spreads)
and actual default proba-bilities (derived from ratings of rating
agencies or market implied rating as MoodysKMV or Altmans Z-Score),
which we will call Q-to-P-ratio. Empirical studies havethough
shown, that these Q-to-P-ratios are not simply a measure of risk
aversion, butalso depend on factors such as credit quality and
industry sector.
In this paper, we propose a different measure for extracting
risk premia out of creditvaluations which is based on structural
models. This approach is able to - qualitatively- explain the
observed variations in the Q-to-P-ratio from empirical studies and
hasseveral advantages: First, it is only based on observable
parameters; second, it is con-sistent with classical portfolio
theory; third, it is robust with respect to model changes(besides
the classical Merton model we examine the Duffie/Lando (2001) model
withunobservable asset values and deviations from the log-normal
assumption) and fourth- and most importantly - it directly yields
the market sharpe ratio and therefore allowsfor a direct comparison
with the equity premium.
Based on an CDS spreads of the 125 most liquid CDS in the U.S.
from 2003 to 2007,we show that appr. 80% of the CDS spreads can be
explained by credit risk basedon structural models with
unobservable asset values. We derive an average implicitmarket
sharpe ratio of appr. 40%, adjusting for taxes yields an average
market sharperatio of appr. 30%. This confirms research on the
equity premium, which indicates,that the historically observed
sharpe ratio of 40-50% (corresponding to an equity pre-mium of 7-8%
and a volatility of 15-20%) was partly due to one-time effects.
Keywords: credit risk premium, equity premium, credit risk,
structural models of default
Tobias Berg, Department of Financial Management and Capital
Markets, Munich University of Tech-nology
Prof. Christoph Kaserer, Head of the Department of Financial
Management and Capital Markets,Munich University of Technology
I
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1 INTRODUCTION 1
1 Introduction
Risk premia in equity markets are a widely researched topic. The
risk premium in equitymarkets is usually defined as the equity
premium, e.g. the excess return of equities over riskfree bonds.
Adjusting for different volatilities yields the sharpe ratio, i.e.
the excess returndivided by the overall market volatility.
Generally, the measurement of equity premia canbe dividend into
three main approches: Models based on historical averages, dividend
orcash-flow-discount models and models based on utility functions.
While historical averageshave long dominated theory and practical
applications, current research suggest an upwardbias, e.g. the ex
post realized equity returns do not correctly mirror the ex ante
priced equitypremium.1 Dividend-/Cash-flow-discount models have
become more popular, but are alsosubject to debate, especially for
their rather high sensitivity to
growth/dividends/earningsforecasts. While approaches based on
utility functions have been subject to intensive debatein the
academic literature2, its use in practical applications is
currently of minor importance.
Risk premia are though not limited to equity markets, the same
logic does also apply tocredit markets in theory and it has been
observed in empirical studies. For example, theaverage 5-year CDS
spread for a A-rated obligor in the CDS.NA.IG-index over the past
3years has been 36 bp, whereas the average annual expected loss
over 5 years is less than 10bp, e.g. 5 year CDS investments
(approximately) yield an average return of appr 26 bp overthe risk
free rate, see figure 1. In absolute terms, this premium increases
with decreasingcredit quality (i.e. the expected net returns
increase with increasing riskyness). Measuredrelative to the
expected loss (or the actual default probability) it decreases with
decreasingcredit quality. Over the last year, research about this
default risk premium has developed,but there has not yet emerged
consensus on the methodology for measuring this defaultpremia.
Duffie and Singleton (1999) model the default event via default
intensity processes asan inaccesible stopping time. This approach
has gained popularity especially for hedgedbased pricing
applications restricted to debt markets. Assuming a recovery of 0%,
a riskyzero coupon bond can simply be priced via a simple
additional discount factor to the riskfree interest rate, which
represents the loss in survival probability over the next instance
oftime, i.e.
B(T ) = EQ[e T0 r(s)+(s)ds],
where r captures the risk free rate dynamics whereas captures
the risk neutral defaultintensity dynamics.Fixing the risk attitude
of investors, it is though not clear if - when examinign differ-ent
obligors - the absolute default risk premium should be a fixed
percentage of the actual
1Reasons are survivorship bias, risk premium volatility,
interest rate level and state of the economy, seefor example
Claus/Thomas (2001), Illmanen (2003), Fama/French (2002).
2This debate is mainly based on the so called Equity Premium
Puzzle by Mehra/Prescott (1985),see Mehra (2003) for an overview
about different utility based approaches including alternative
preferencestructures, disaster states and survivorship bias and
borrowing constraints.
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1 INTRODUCTION 2
Figure 1: Relationship between 5-year CDS spreads (mid) and
5-year annualized expectedloss per rating grade for investment
grade ratings.
default probability (i.e. Q = c P ), a fixed amount (in bp)
independent of the ratinggrade/actual default probability (i.e. Q =
P + c) or if other (e.g. logarithmic) relation-ships are to be
preferred. Many researches opt for a fixed multiple3, i.e. they
examinethe relationship between risk neutral and real world default
probability or default intensity,e.g.
Q
P, which we will denote throughout this paper as Q-to-P-ratio4.
Berndt et.al. (2005)
come to the conclusion, that the Q-to-P-ratio is higher for high
quality firms and lower forlow quality firms. They use a linear and
a log/log relationship between CDS spreads anddefault
probabilities, resulting - especially for the linear relationship -
in an intercept ofroughly 50 bp, which is more 30 times the
standard error and does not seem to be plau-sible even accounting
for liquidity risk. Amato and Remolona (2005) principallly
confirmthese findings. In addition, Amato (2005) finds a
significantly smaller default premium forfinancial services
companies. Both papers find a positive correlation between risk
premium
3Among others Berndt et.al. (2005), Amato (2005), Driessen
(2003), Liu et.al. (2000).4Dependent on the specific rating
methods, credit risk methodology and the overall context this
ratio
is also referred to as risk neutral to actual default intensity
(see for example Berndt et.al. (2005)) or - ifadjusted for recovery
rates - CDS-to-PD or CDS-to-EDF-ratio (if KMVs expected default
frequencies areuse) by other authors.
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1 INTRODUCTION 3
and average default probability, i.e. an increasing average
default probability in the marketis accompanied by an increasing
risk premium. Driessen (2003) uses constant Q-to-P-ratiosby
modelling the risk neutral default intensity as a constant multiple
of the actual defaultintensity (over time and for different
obligors). On average, he reports a Q-to-P-ratio of 2.31(S&P)
and 2.15 (Moodys) respectively, although the standard errors are
quite large (> 1%).
The main contribution of this paper is threefold. First, we
explain the variations in theQ-to-P-ratio for different rating
grades and different industry sectors observed in the litera-ture
based on a simple Merton style model. Second, this paper is - to
our best knowledge -the first to theoretically analyze the
Q-to-P-ratio in an incomplete information setting (basedon the
model of Duffie/Lando (2001)) - which enables us to model the
effect of uncertainty onthe Q-to-P-ratio - and the effect in
deviations from the lognormal distribution assumption.We show, that
higher Q-to-P-ratios may - besides higher risk aversion - also be
explained byhigher uncertainty of the current asset value, due to a
concave relationship between defaultprobability and Q-to-P-ratio.
Third - and most importantly - we directly extract the marketsharpe
ratios out of CDS spreads, which gives us a direct link to the
equity premium. Wemeasure an upper bound for the average sharpe
ratio of 29 - 42%, which is consistent withresearch of equity
markets.
We use a simple relationship between risk neutral and real world
default probability inthe Merton framework, i.e.
Qdef (t, T ) = [1(P def (t, T )) + SRAssets
T t
](1)
The resulting relationship between risk neutral and actual
default probability5 (Q-to-P-ratio) is neither linear nor
log-linear, but involves the cumulative normal distribution.
Pleasenote, that we do not try to estimate the actual and risk
neutral default probability seper-ately; we simply assume that we
know the actual default probability from rating
information;calibrate the model accordingly and from there derive
risk neutral default probabilities; e.g.we do not have to calibrate
the asset value, default barrier or volatility separately. With
thismodel, we are able to include the correlation and volatility of
a firms assets and thereforederive a result which is consistent
with classical portfolio theory (e.g. the dependence ofreturns on
the systematic, rather than total, risk inherent in a claim). We
will then beable to explain the high intercept from Berndt et.al.
(2005) (which is due to a non-linearrelationship between real world
and risk neutral default probability), the observed high riskpremia
for high quality firms (which is a combined effect of the
non-linearity of real worldand risk neutral PD and a larger part of
ideosyncratic risk incorporated in high quality firms).
5Please note, that we there is minor a difference between the
ratio of cumulative default probabilities, i.e.Qdef (t,T )Pdef (t,T
)
and the ratio of the default intensities QP =ln(1Qdefdef(t,T
)ln(1Pdefdef(t,T ) defined by Q := ln(1Q
defdef(t,T ))T
and P := ln(1Pdefdef(t,T ))T respectively, which is in first
order equal to the ratio of cumulative default
probabilities. Throughout this paper, we will always analyze the
ratio of the default intensity for reasons oftheoretical
consistency.
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1 INTRODUCTION 4
The model also allows for an extraction of the risk premium out
of CDS spreads, whichis - in theory - not dependent on the quality
of the firm or the sector the firm operates,but merely mirrors the
risk attitude of investors and directly yields the sharpe ratio of
themarket portfolio:
SRM :=M rM
1(Qdef (t, T )) 1(P def (t, T ))
T t 1
E,M=: Merton, (2)
where E,M denotes the correlation between the market portfolio
and the equity value of thefirm.6 Since credit spreads are very
sensitive to the sharpe ratio, this is a very convenientway: for
example, a BBB-rated obligor would have a 5-year credit spread of
73 bp with asharpe ratio of 10% compared to a 5-year credit spread
of 280 bp with a sharpe ratio of 40%,see subsection 2.1 for
details. This difference shows, that the common noise in the data
willnot significantly reduce the possibility to extract the sharpe
ratio out of market prices.
Extending the simple Merton model to more advanced models (we
examine a classical firstpassage time model with observable asset
values, a model with unobservable asset valuesproposed by Duffie
et.al. (2001) and deviations from the lognormal assumption.)
showsan astonishing robustness of the results (1) and (2) derived
in the Merton framework forinvestment grade companies as long as
the asset volatility is larger than approximately 10%,which can
reasonably be assumed for all companies outside the financial
services sector.7
For smaller asset volatilities, the results of the first passage
time models deviate from thestandard Merton model. With these
models we are able to theoretically explain the lowerobserved risk
premia for financial service companies (which is due to a
significantly smallerasset volatility of financial services
companies)8 More generally speaking, by introducing a(model- and
parameter-dependent) adjustment factor by
SRM = Merton AF, (3)we show that
The adjustment factor is close to one for all models analyzed as
long as the asset volatil-ity is below 10% and the resulting actual
default probability belongs to an investmentgrade rating.
The adjustment factor can be accurately determined simply based
on knowledge of thematurity and the actual default probability
(i.e. parameters that can be observed in
6The correlation between equity value and market portfolio was
used as a proxy for the correlationbetween asset value and the
market portfolio. In the Merton framework as well as in more
general structualmodels, it can be shown that for reasonable
parameter choices (e.g. resulting in a credit quality / ratinggrade
of B or higher) these two correlations do not differ by more than
1%, see section 2.
7In our sample of 125 companies of the CDS index CDX.NA.IG appr.
90% of all non-financial companieshad an asset volatility of 10% or
larger based on data from Moodys KMV. In contrast, for fincial
servicescompanies, the volatility is 10% or smaller in appr. 75% of
all cases.
8Due to a lack of data of financial services companies with
asset volatilities above 10% and/or non-financial services
companies with an asset volatility below 10%, it is though hard to
empirically verifiy thesetheoretical results, see section 3 for
details.
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2 MODEL SETUP 5
the market) as long as > 10%, i.e. for a given combination of
default probabilityand maturity, parameters that can not be
observed easily (e.g. asset volatility, defaultbarrier, asset value
or accounting noise) do not significantly affect the
adjustmentfactor.
Furthermore, the model directly yields the sharpe ratio of the
market portfolio and thereforeallows a direct comparison with the
equity premium. Applied to all NYSE-listed companiesin the
investment grade CDS index CDX-NA.IG from 2003 to 2007 and using
EDFs fromKMV as a proxy for the actual default probability results
in an average implied marketsharpe ratio of 42% and an average
company sharpe ratio of 20%. Adjusting for tax effectsyields a
market sharpe ratio of 32% and an average company sharpe ratio of
16%. UsingMoodys ratings instead of EDFs shows similar results
(market sharpe ratio of 39% and 29%after tax adjustments). We used
CDS spreads as they are unfunded exposures, so that therates should
be less sensitive to liquidity effects.9 Based on the theoretical
models, appr.80% of the CDS spreads can be explained by credit
risk.
The remainder of the paper is structured as follows. Section 2
describes the theoreticalframework for credit risk premia based on
asset value models, including a discussion ofthe impact of
different asset models and deviations from the lognormal
distribution on thewidely used Q-to-P-ratios. We examine a
classical Merton model, a first passage time model,a model based on
unobservable asset values as proposed by Duffie/Lando (2001) and
assetdistributions based on actual S&P returns. Using a model
with unobservable asset values isfundamental in our point of view,
since only these models are able to explain credit spreadsobserved
in the markets and yield a default intensity, which constitutes the
basis of moderncredit pricing models. Section 3 describes our data
sources and shows our empirical resultsfor the risk premium.
Section 4 concludes.
2 Model setup
This section discusses the theoretical framework for extracting
risk premia out of CDS prices.The basic idea is to use structural
asset models to derive a relationship between risk neutraland
actual default probability. Most structural models show a quite
poor performance inempirical studies10. One of the main reasons is
the calibration process usually needed tospecifiy structural
models, e.g. determination of leverage, asset volatilities, etc. In
contrastto most parts of the literature11, we do though not aim to
derive actual and risk neutraldefault probabilities from structural
models, we are simply interested in the relation between
9Compare Berndt et.al. (2005) for similar arguments.10See
Schonbucher (2003) for an overview of empirical
studies.11Huang/Huang (2003), Bohn (2000) and Delianedis/Geske
(1998) use a similar approach, our approach
differs though in at last three ways: First, we work in an
incomplete information setting, whereas we explicitlyfocus on
models with information uncertainty; second, we use CDS spreads,
which should be less sensitiveto liquidity distortions; third, we
are - to our best knowledge - the first who directly aim to extract
the riskattitude out of credit prices.
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2 MODEL SETUP 6
risk neutral and actual default probabilities. We simply assume,
that there exists a struc-tural model yielding the correct actual
default probability and from there derive the riskneutral default
probability. We can therefore omit (most of) the calibration
process whichresults in stable calculation.
Subsection (2.1) discusses a simple Merton model and shows, that
most empirical resultsabout the Q-to-P-ratio can already be
explained - at least qualitatively - in this setting.Subsection
(2.2) expands the framework to a simple first passage time model.
The results donot materially differ compared to the simple Merton
framework as long as the asset volatilityis above 10 Percent. Asset
volatilities smaller than 10% (which are usually only observedfor
financial services companies) lead to a quite large deviation from
the standard Mertonmodel. We are then able to explain the impact of
volatility on the Q-to-P-ratio, especiallylower observed
Q-to-P-ratios for financial services companies.Subsection (2.3)
examines the model of Duffie/Lando (2001) and again shows the
robustnessof the simple Merton model for our purpose. Subsetion
(2.4) analyzes the impact of devia-tions from the assumption of
lognormally-distributed asset returns.
The following subsections will always be structurd in the same
way: first, the model setupincluding the main assumptions is given
and formulaes for calculating actual and risk neu-tral default
probabilities are given; second, the Q-to-P-ratio is analyzed
within the respectivemodel; third, the implications for estimating
the sharpe ratio are analyzed.
2.1 Merton
Model setup:
Structural models for the valuation of debt and the
determination of default probabili-ties were already mentioned in
the well-known Black/Scholes (1973)12 paper. The Mertonframework
presented in the subsection is based on the famous paper of Merton
(1974)13,which explicitely focusses on the pricing of corporate
debt.
In this framework, a companys debt simply consists of one
zero-bond. Default occurs,if the asset value of the company falls
below the nominal value of the zero bond at thematurity of the
bond. A company can therefore only default at one point in time,
whichobviously poses a simplification of the real world.
Before we will derive an estimator for the market price of risk
() based on the Mertonframework, the main assumptions will be
presented:
Assumption 2.1 (Assumptions Merton framework)
12See Black/Scholes (1973).13See Merton (1974).
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2 MODEL SETUP 7
1. The Assets Vt follow a geometric Brownian motion with
constant drift = V (underP) and r (under Q) and constant volatility
= V > 0
14, i.e.
under P : dVt = Vtdt + VtdBt VT = Vt e( 122)(Tt)+(BTBt)
under Q : dVt = rVtdt + VtdBt VT = Vt e(r 122)(Tt)+(BTBt).2. The
companys debt consist of one single zero-bond with nominal N and
maturity T.
3. Default occurs if VT < N (a default can therefore only
occur at t=T).
Assumptions 1 is a standard assumption in financial economics,
assumption 2 and 3 aresimplifications of the real world and will be
relaxed in the next subsections.
Under these assumptions, the real world default probability P
def (t, T ) between t and T15
can be calculated as follows:
P def (t, T ) = P [ VT < N ] = P [ Vt e ( 122)(Tt)+(WTWt)
< N ]= P
[ (WT Wt) < ln
(N
Vt
) ( 1
22) (T t)
]=
[lnN
Vt ( 1
22) (T t)
T t
]. (4)
The default probability under the risk neutral measure Q can be
calculated accordingly as
Qdef (t, T ) = Q[ VT < N ] =
[lnN
Vt (r 1
22) (T t)
T t
]. (5)
Combining (4) and (5) yields16
Qdef (t, T ) =
[1(P def (t, T )) +
r
T t]
(6)
andV rV
=1(Qdef (t, T )) 1(P def (t, T ))
T t (7)
respectively, see figure 2 for an illustration. Please note a
main advantage of this formula:it directly yields the sharpe ratio
of the assets; i.e. neither V and V nor Vt, N or r have tobe
estimated separately (and - on the other hand - can not be inferred
from (7) separately).
14For practical reasons, the index V will usually be omitted in
the following formulas. To avoid potentialconfusion, the index V
will though be used in the main formulas and results of this
section.
15Of course, in the Merton framework, a default is only possible
at time T, for reasons of consistency withthe following
subsections, we will always talk about default probabilities
between t and T in this context,too.
16See for example Duffie/Singleton (2003) for the first
equation.
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2 MODEL SETUP 8
Figure 2: Illustration of the relationship between actual and
risk neutral default probabilitiesin the Merton framework. PDcum:
actual cumulative default probability, Qcum: risk neutralcumulative
default probability, SRV : sharpe ratio of the assets, T:
maturity.
Q-to-P-ratio:
The resulting relationship between CDS spreads / risk neutral
default probabilites and ac-tual default probabilities is shown in
figure 4. This relationship is increasing and concave,as can be
seen by the first two derivatives of (6) with respect to P def
:
Q
P=
12 pi e
0.5(SRT+1(P def )2 1
P def
= e0.5(SR2T+2SRT1(P def ) > 0
and
2Q
P 2= Q
P SR
T2pie0.5(
1(P def ))2 < 0,
If one tries to make a linear regression between the risk
neutral and the actual defaultprobability, the intercept will be
significantly above zero, which is a result of the concavity of
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2 MODEL SETUP 9
Figure 3: Relationship between CDS spread and actual default
probability in the Mertonframework and results of a linear
interpolation (i.e. assumption of a credit quality indepen-dent
Q-to-P-ratio. Parameters: r=5%, SRA = 30%, RR=50%.
the relationship between the risk neutral and the actual default
probability implied by (10)and confirms the empirical research, see
for example Berndt et.al. (2005). Please note thatthe specific
parameters of the regression (slope and intercept) are rarely
arbitrarely based onthe choosen interval of actual default
probabilities and the choosen data points (which areusually
clustered at regions of investment grade default probabilities,
e.g. one-year-defaultprobabilities up to appr. 0.3% and 5-year
cumulative default probabilities up to appr. 3.5%).Plotting the
Q-to-P-ratio and the actual default probabilities (or, similarly,
the rating) basedon (10) shows, that the Q-to-P-ratio declines with
declining credit quality (see figure 4),which underlines empirical
research on this topic (See Berndt et.al. (2005) and Amato(2005)).
Again, this is a direct result of the concave funcion between the
risk neutral andthe actual default probabilities implied by
(10).
Market sharpe ratio:
If we try to extract the market sharpe ratio out of (6), we are
faced with an additionalproblem: the sharpe ratio of the assets V
r
Vwill usually differ from the market sharpe ratio,
since the assets Vt will usually not be on the efficient
frontier. In other words, the sharperatio of the assets does not
only capture the risk attitude of investors, but also depends
on
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2 MODEL SETUP 10
Figure 4: Relationship between the Q-to-P-ratio and the
rating/actual default probabilityin the Merton framework.
Parameters: r=5%, SRA = 20%.
the correlation of the assets with the market portfolio. The
market price of risk can thoughbe calculated via a straight forward
application of the CAPM:
V = r +M rM
V,M V 17 M rM
=V rV
1V,M
, (8)
where V,M denotes the correlation coefficient between the asset
return and the market re-turn.18
17Without loss in generality we assume V,M 6= 0.18An application
of the CAPM does require the respective assets to be traded or -
equally - it requires
a self-financing trading strategy resulting in a contingent
claim equal to the asset payoff. Asset values areusually not traded
on financial markets, we do though believe that this application is
justified for two reasons.First, the asset value can - in theory -
be duplicated by the equity value and a risk free bond in the
Mertonframework, so the asset value lies in the asset span of the
market. Second, most of the claims on the assetsof large
corporations are either directly traded (e.g. equity and bonds) or
market values can be inferred fromdirectly traded instruments with
a certain accuracy (e.g. bank loans via bonds). In addition,
non-tradableparts like insolvency costs in more advanced models do
have an impact on the choice of the optimal capitalstructure, given
the actual default probabilities observed in the markets (most
large companies are ratedinvestment grade), these do not have a
significant effect on the asset value and are also hedgeable in
thesemodel-setups.
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2 MODEL SETUP 11
Therefore, in addition to the sharpe ratio of the assets, we
will need an estimate of thecorrelation between the asset value and
the market portfolio. At first, this correlation (V,M)seems to be a
problem for practical applications, since it can neither be
directly measurednor implicitly inferred e.g. from option prices.
In the following applications, we will ap-proximate V,M by the
correlation between the corresponding equity return and the
marketreturn (denotet by E,M), i.e. we will assume that
V,M E,MThe error of this approximation is almost negligible,
since - within the Merton framework -the equity value of a company
equals a deep-in-the-money call option on the assets. For
allreasonable parameter choices, the approximation error is less
than 1%, see appendix A fordetails. Hence the following
approximation holds:
M rM
1(Qdef (t, T )) 1(P def (t, T ))
T t 1
E,M
and we can formulate a Merton estimator for the market price of
risk = MrM
:
Merton :=1(Qdef (t, T )) 1(P def (t, T ))
T t1
E,M; (9)
or, solved for the risk neutral default probability:
Qdef (t, T ) =
[1(P def (t, T )) + SRM
1
E,M
](10)
Please note, that we will need a sufficient sensitivity of the
risk neutral default probabilityQdef (t, T ) with respect to the
sharpe ratio in order for an empirical application. Otherwisenoise
in the data (e.g. bid-ask-spreads, inaccuracies in determining
correlations and actualdefault probailities) will result in a very
inaccurate estimation. That this sensitivity is indeedgiven can be
seen by calculating the first derivative of (10) with respect to
the sharpe ratio,i.e.
Qdef (T )
SRV=
12pi
e 12(SRV T+1(P def (T )))2
T. (11)
see figure 5 for an illustration. If we look for example at a
BBB-rated obligor with a 5-yearcumulative actual default
probability of appr 2.17%, the resulting risk neutral default
prob-ability should be either 3.6% (for an asset sharpe ratio of
10%) or 13% (for an asset sharperatio of 40%) respectively.
Assuming a recovery rate of 50% transforms this into a CDSspread of
either 73 bp or 279 bp19, a difference that is certainly above any
noise induced byliquidity effects, bid/ask-spread or inaccurate
measurement of actual default probability.
19Using the approximation CDS spread = Q LGD and by deriving the
risk neutral default intensity fromthe risk neutral cumulative
default probability by Qdef (t, T ) = 1 eQ(Tt).
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2 MODEL SETUP 12
Figure 5: Influence of the sharpe ratio on risk neutral default
probabilities in the Mertonframework for different sharpe ratios
(10%-40%). Other Parameters: T=5.
Please note again, that this relationship is independent of the
asset volatility, the assetvalue Vt and the default barrier L
(which is a direct implication of (10)). A calibration of theasset
value model will therefore not be necessary to specify the
relationship between actualand risk neutral default
probabilities.
Although we found a compelling result for an estimation of the
market price of risk, theassumptions made in the Merton framework
do not fully reflect the true world. In the fol-lowing subsections
we will relax the assumptions about the default timing (see
subsection2.2) and the assumption about complete information (see
subsection 2.3) and look at moregeneral first passage time models.
The subsections will always be structured as follows: first,the
model setup is explained and formulas for actual and risk neutral
default probabilities arederived; second, the resulting
Q-to-P-ratio is analyzed; third, the impact on the estimationof the
sharpe ratio is discussed.
Before moving on, please note that our estimator for the market
price of risk contains acertain kind of robustness against changes
in the underlying assumptions: Since both de-fault probabilities (P
def and Qdef ) are measured within the same model and
substracted
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2 MODEL SETUP 13
from each other, the effect of changes in the default modelling
(e.g. first passage time) orin the assumptions about distribution
of returns (e.g. deviation from the assumption of alog-normal
distribution of the asset returns) is - qualitatively spoken -
reduced significantly.
2.2 First passage time modell with certain default barrier
Model setup:
In first passage time models, a default occurs as soon as the
asset value20 falls below acertain barrier. The asset value and the
default barrier can both be either observable orunobservable. The
model with a certain default barrier and a certain asset value
based onis treated in this section. A model with an uncertain
default barrier and unobservable assetvalue based on Duffie/Lando
(2001) is analyzed in the following subsection.
For this subsection, the following assumptions apply:
Assumption 2.2 (Assumptions first passage time model with
certain default barrier)
1. The Assets Vt follow a geometric Brownian motion with
constant drift = V (underP) and r (under Q) and constant volatility
= V > 0, i.e.
under P : dVt = Vtdt + VtdBt VT = Vt e( 122)(Tt)+(BTBt)
and
under Q : dVt = rVtdt + VtdBt VT = Vt e(r 122)(Tt)+(BTBt)
2. Default occurs as soon as the asset value Vt falls below a
predefined and certain barrierL (i.e. L R)(a default can therefore
occur at any time in (t,T].
Under these assumptions, the real world default probability P
def (t, T ) between t and Tcan be calculated as
P def (t, T ) = P [Vs < L for any t s T] = 1 P[Vs L, t s T]=
1 P [ min
tsTVs L] = 1 P
[mintsT
[Vt e( 122)(st)+(WsWt)
] L;
]= 1 P
[mintsT
[ 1
22)(s t) + (Ws Wt)
] ln( L
Vt)
]=
(bm(T t)T t
) e 2mb2
(b+m(T t)T t
)(12)
20Some authors use an even more general version of an
ability-to-pay process, see Bluhm/Overbeck/Wagner(2003) for
example.
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2 MODEL SETUP 14
with
b = ln(L
Vt); m = 1
22; = V ,
see especially Musiala/Rutkowski (1997) for the last step. The
default probability under therisk neutral measure Q can be
calculated accordingly as
Qdef (t, T ) =
(b m(T t)T t
) e 2mb2
(b+ m(T t)T t
)(13)
with
b = b = ln(L
Vt); m = r 1
22; = V .
There is - in contrast to the Merton framework - no closed form
solution for the sharpe ratioV rV
in this model.
Unfortunately, we have two equations ((12) and (13) with three
unknown variables ( LVt, , ,
assuming we can derive r, T t, P def , Qdef from bond/CDS market
data), so we can notdirectly derive and in order to determine the
sharpe ratio r
Q-to-P-ratio. We will
though be able to determine as a function of and therefore also
the sharpe ratio of theassets (r
) as a function of . This function is nearly constant for >
10%, see figure 6, so
apart from assets with very low asset volatility we will again
be able to estimate the sharperatio without calibrating the asset
volatility , the asset value Vt or the default barrier L.Please
note, that the sharpe ratios in figure 6 do not imply, that assets
with a lower volatilityhave a higher sharpe ratio, rather they
implicitly derive sharpe ratios out of CDS spreadsbased on
empirical data and a first passage time model. A company with a low
asset volatil-ity will therefore (all other parameters being equal,
especially the actual default probabilityand the asset correlation)
trade at lower spreads than a company with a high asset
volatility.
Q-to-P-ratio:
The resulting slope of the Q-to-P-ratio as a function of the
asset volatility is shown infigure 7 for different rating classes
(identified by specific cumulative default probabilities).Again,
one can see that the Q-to-P-ratio is higher for higher rating
classes, (almost) inde-pendent of the asset volatility for < 10%
and sharply declining for asset volatilites smallerthan 10%. It
converges to 1 for all rating classes, thus, the slope is more
pronounced forhigher rating classes. The dependency on the asset
volatility can be explained by the defaulttiming: for lower asset
volatilities, the expected default time conditional on a default
untilT is lower for lower asset volatilities conditional on a fixed
cumulative default probability upto T.21 In other words, fixing the
cumulative default probability until time T, defaults willoccur
with a higher probability at the beginning of the period if the
volatility is low. Since
21Please note that - all other parameters being equal - the
default probability declines with declining assetvolatility.
Therefore, the expected value of the default time will also
decrease. In this case, the decliningasset volatility is always
balanced by a lower t0-asset-value to result in order to yield the
same rating grade.
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2 MODEL SETUP 15
Figure 6: Relationship between the asset volatility and the
implicit sharpe ratio derivedfrom CDS spreads for rating class A.
P5, mean and P95 denote observed CDS spreads forthe CDX-NA-IG-index
from 2004-2006 for all obligor rated A1/A2/A3 by Moodys at the5% (8
bp), 50% (19 bp) and 90% percentile (42 bp). Risk neutral default
probabilities werederived with a recovery rate assumption of 50%.
The slope of the graph shows quite similarpatterns for other rating
classes or recovery rate assumptions.
the Q-to-P-ratio decreases with decreasing maturity, see figure
4 and the related discussionin subsection 2.1, it also decreases
with decreasing asset volatility in the first passage
timeframework. In addition, one can easily see from (6), that the
Q-to-P-ratio converges to oneif the maturity (T-t) converges to
zero.
Estimation of the sharpe ratio:
We now want to test the robustness of the simple Merton
estimator for the sharpe ratio. Wetherefore introduce an
adjustement factor AFFP by
SRM :=M rM
=:1(Qdef (T ) 1P def (T )
T 1M,E
AFFP = Merton AFFP , (14)
i.e. the adjustment factor shows, how far the estimate of the
market sharpe ratio via thestandard Merton model deviates from the
true market sharpe ratio if a first passage modelapplies. Again, we
have assumed that V,M = E,M , i.e. that the correlation between
marketand asset returns equals the correlation between market and
equity returns. This equation
-
2 MODEL SETUP 16
Figure 7: Relationship between the Q-to-P-ratio and the asset
volatility for different ratinggrades the first passage time model.
Parameters: r=5%, SRA = 15%.
holds true for reasonable parameter choices in the first passage
time framework as well, aswe will show in the next subsection.The
adjustment factor is dependent on the volatility, the sharpe ratio
and the credit quality(interpreted as actual default probability)
of the company. The relationship is shown infigure 8 for a sharpe
ratio of 20%. It shows, that the adjustment factor increases
withdecreasing credit quality and with increasing volatility, but
for investment-grade titles anda volatility smaller than 10% the
adjustment factor is very close to 1.
2.3 A model with unobservable asset values
Model setup:
Although an improvement compared to the Merton model, credit
spreads predicted by sim-ple first passage time models are still
not able to predict the credit spreads that can beobserved on the
markets.22 Especially for short term maturities, market credit
spreads (orrisk neutral default probabilities respectively) are
higher than a simple first passage timemodel would suggest.
22See Duffie/Lando (2001), Duffie/Singleton (2003) and
Schonbucher (2003) for a detailed discussion.
-
2 MODEL SETUP 17
Figure 8: Credit quality smile: Adjustment factor in the first
passage time framework.Parameters: r=5%, SRA = 20%. Please note
that the majority of traded bonds and CDS(by volume) has an
investment grade rating.
These higher credit spreads for short term maturitites seem to
be mainly attributable tocredit risk and are not due to liquidity
effects, other risk categories or market imperfections(See for
example Schonbucher (2003).). Several methodological reasons are
discussed in theliterature:
The asset value may be uncertain due to imperfect information
structures, or - tobe more precise - the current asset value can
not be observed exactly. Therefore,the current asset value becomes
a random variable, which in turn has the effect ofincreasing short
term default probabilities.
The default barrier may be uncertain/unobservable.This is
consistent with the fact that the recovery rate is usually assumed
to be a randomvariable rather than a fixed value.23 In the simple
first passage time model presentedin the last subsection, the
barrier was certain, therefore the recovery rate should alsobe
non-random. An unobservable default barrier leads to a significant
increase in the
23See for example Moodys (2007). A random recovery rate could
though also be induced by introducingrandom insolvency costs, i.e.
costs incurred at default due to direct insolvency expenses, losses
in asset valuedue to a forced sale in an insolvency process and
revaluation of assets serving a specific purpose for therespective
company.
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2 MODEL SETUP 18
short term default probability, long term default probabilities
are less effected, sincethe asset volatility dominates uncertainty
for longer time periods.
Asset values are not lognormally distributed and may include
jumps. This increasesthe short term probability, that the asset
value will fall below the default barrier.24
The arguments all have an economic foundation and seem to be
reasonable. It is not withinthe scope of this thesis to evaluate,
which one (or which combination) best produces thetrue economic
world. Incorporating jumps in the asset value process has been
analyzed byZhou (1997). Duffie/Lando (2001) have developed a first
passage time model with uncertainasset value. The assumption of an
uncertain default barrier has been implemented in thecommercial
model CreditGrades by Finger et.al. (2002).
In this subsection, we will focus on the model of Duffie/Lando
(2001) and show that -although the default probabilities implied by
this model differ substantially from the clas-sical Merton model -
the difference between risk neutral and actual default
probabilities isalmost the same than in the Merton model as long as
the asset volatility is above 10%. Tobe more precise, we will show
that the simple Merton estimator for the market sharpe ratio(see
(9)) is - to a certain degree of accuracy - still valid in the
Duffie/Lando-framework forasset volatilities above 10%.
We choose the Duffie/Lando model for this analysis as it is the
most general frameworkfor credit risk modeling we know of. It
incorporates a sophisticated structural model ofdefault (i.e. a
strategic setting of the default barrier based on the asset value
process, taxshield and insolvency costs) and - even more important
for this setting - an unobservableasset process resulting in
realistic default term strucures with a default intensity.25
The main assumptions in the Duffie/Lando model are as
follows:26
Assumption 2.3 (Assumptions Duffie/Lando model)
1. The Assets Vt follow a geometric Brownian motion with
constant drift m = (under P) and r (under Q) - where denotes the
asset payoff rate - and constantvolatility = V > 0, i.e.
under P : dVt = mVtdt + VtdBt VT = Vt e(m 122)(Tt)+(BTBt)24Note
that for long term maturities, a higher volatility has the same
effect as adding jumps to the process,
therefore long term default probabilities will not be effected
in the same manner than short term defaultprobabilities.
25Defaults in a Merton framework can not be described via
default intensity processes, since the probabilityof a default from
t (today) until t+ t is always zero or one for a sufficient small
t. A default intensity doesalso not exist in the the Zhou (1997)
framework, since the default time cannot be represented by an
totallyinaccessible stopping time (which is a consequence of the
fact, that the default barrier may be hit/crossedvia the normal
diffusion process with positive probability), see Duffie/Lando
(2001) for details.
26See Duffie/Lando (2001) for details.
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2 MODEL SETUP 19
and
under Q : dVt = r Vtdt + VtdBt VT = Vt e(r 122)(Tt)+(BTBt)
2. Default occurs as soon as the asset value Vt falls below a
predefined and certain barrierVB (i.e. L R), which is determined by
the equity holders in t=0 as to optimize thetotal initial firm
value (equity + proceeds of the sale of debt).
3. The bond holders / CDS investors are not able to observe the
asset process directly.
Instead they receive imperfect information Y (ti) := ln(Vti
)= ln(Vti) + Uti, where
U(ti) is normally distributed and independent of B(t) and is a
parameter specifyingthe degree of noise in the information received
by the bond/CDS investors. Therefore,the information filtration
given to the bond/CDS investors is27
Ht = ({Y (ti), ..., Y (Tn), 1s : 0 s t}) (15)
We will examine the case n=1. Under these assumptions, the
conditional density of Zt :=ln(Vt), > t
28 conditional on the noisy observation Yt and with a fixed
starting pointln(V0) =: z0 can be calculated as
29
b(x, t|Yt, z0) := P [ > t Zt dx|Yt) = P [ > t Zt = x Yt =
Yt]P [Yt = Yt]
=P [ > t Zt = x Ut = Yt x]
P [Yt = Yt]=P [ > t|Zt = x] P [Zt = x] P [Ut = Yt x]
P [Yt = Yt],
where we used the equivalance of Yt() = Yt and Ut() = Yt x under
the condition thatZt() = x in the second step and Bayes rule in the
first and third step. Conditioning on > t30, this yields the
conditional density g(x|Yt, z0, t) of the log asset value Yt
conditionalon the information Yt = Yt and > t:
g(x|Yt, z0, t) = b(x|y, z0, t)P [ > t|Yt, z0]
This conditional density can be explicitly calculated, if it is
assumed, that asset values followa geometric Brownian motion and Ut
has a normal distribution, see Duffie/Lando (2001) fordetails.To
calculate cumulative default probabilities requires a weighted
application of (12) over all
27Of course, the bond/CDS investors can also obvserve, if a
default has already occured.28 denotes the stopping time
representing the default point.29In the following, we will use the
conventional informal notations P [X = x] or P [X dx].30Please note
that the investors are of course able to observe, that no default
has taken place up to time
t.
-
2 MODEL SETUP 20
possible asset values Vt, where the weight is - roughly speaking
- the probability of the assetvalue Vt
31, i.e.
P def (t, T ) =
VB
P defFP (t, T, x) PD(first passage time), if Vt=x
g(x|Yt, z0, t) Prob., that Vt=x
dx (16)
where P defFP (t, T, x) denotes the probability, that an asset
value process starting in t=t atVt = x will fall below the default
barrier up time t=T, see (12), and g is the conditionaldensity of
the asset value at t=t given the filtration Ht.
Q-to-P-ratio:
Formula (16) can now be used to calculate the risk neutral as
well as the real world de-fault probability by either using the
risk neutral drift of the asset (e.g. the risk free rate lessthe
payout rate) or the actual drift of the assets.32
Unfortunately, there is no closed form solution for (16), we
though have to draw back onnumerical solutions. Figure 9 shows the
Q-to-P-ratio for = 10% and varying actual defaultprobabilities
(identified as ratings) and asset volatilities.It can be seen, that
- for a given rating grade - the Q-to-P-ratio is again almost
independentof the asset volatility as long as > 10% and sharply
declines for < 10%.Uncertainty (measured by and T1) results in
slightly higher Q-to-P-ratios. This effect israther small but
increasing with increasing credit quality, see figure 10 for = 15%
andvarying values for ). The explanation is rather simple:
To calculate default probabilities in the Duffie/Lando model, we
simply have to calculatethem in the first passage time framework
with observable asset values and average the resultfor different
possible levels of PDcum (representing different levels of the
actual asset level Vt)with respect to the conditional density of
Vt. Since the Q-to-P-ratio is a convex function of the
default probability in the Merton framework (see figure 4), i.e.
E[Qdef (P def )P def
] Qdef (E[P def ])E[P def ]
as a result of Jensens inequality, an increasing leads to an
increase in the Q-to-P-ratio,although the effect is rather small
for the parameters analyzed. The effect is though stillquite
interesting: An increase in the Q-to-P-ratio does not necessarily
mean, that agentshave become more risk averse, it can also reflect
a higher (actual or perceived) noise in assetvalues. This effect is
larger for high quality companies, so that in states of higher
uncertainty,spreads on good quality companies should increase most.
This finding can also have effectson trading strategies which try
to use risk aversion measures depicted from credit marketsfor other
market segments (e.g. equities)33
31Of course the probability of a single value Vt will be zero
for non-degenerated parameter choices, since weoperate in a
continious setting. We will still use this informal notation to
allow for a better understanding.
32Please note, that the drift changes P defFP (t, T, x) as well
as the conditional density g(x|Yt, z0, t). Theinfluence on the
conditional density decreases with decreasing .
33There are several tactical asset allocation strategies which
use risk aversion derived from credit markets,equity markets and
money markets (e.g. TED spreads) to allocate funds to different
asset classes. Using
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2 MODEL SETUP 21
Since asset volatilites and accounting noise is usually hard to
estimate, this a very convenientresult for our purpose. If we want
to extract the risk premium out of CDS spreads, assetvolatilites
and accounting noise does merely play a role given a specific
actual default prob-ability. The basic logic behind this is quite
simple: the accounting noise does not pose anysystematic risk,
therefore the difference between risk neutral and real world
default proba-bility is merely unaffected; the same logic applies
to the payout rate, the asset value leveland the time between the
start of the asset value process and the first noise observation,
too.
Figure 9: Relationship between the Q-to-P-ratio and the asset
volatility for different ratinggrades in the Duffie/Lando model.
Parameters: = 10%, m=2.5%, SRA = 20%, T=5, s=1.
Estimation of the sharpe ratio:
simple measures as the development of average CDS spread or
Q-to-P-ratios may be misleading in somemarket environments, where a
high uncertainty about true values exists. An example is probably
thesubprime crisis in 2007, which lead market participants highly
unsecure about where risks actually turnedabout and what dimension
potential losses may have. It is not within the scope of this
paper, to empiricallyanalyze the market reactions on credit and
equity markets, but this could provide an area for further
researchon this topic.
-
2 MODEL SETUP 22
Figure 10: Relationship between the Q-to-P-ratio and the
rating/actual default probabilityin the first passage time model
for different levels of . Parameters: r=5%, SRA = 15%,=15%.
As in subsection 2.2, we now want to test the robustness of the
Merton model estimator,e.g. we again define an adjustment factor
AFDL by
M rM
= Merton AFDL (17)
This adjustment factor may depend on all parameters included in
the model, which we willgroup into two different classes: Class 1
captures all parameters that can easily be observedin the market,
i.e. the actual default probability (which is actually a combined
parameter ofall other input parameters) and the maturity. Class 2
captures all parameters that can notbe easily observed in the
markets, i.e. the asset volatility , the payout rate or the
riskneutral net asset growth rate m := r , the starting point of
the asset value process in t=0(Z0), the default barrier VB, the
noise asset value observed at t=t (Vt) and the accountingnoise . If
the adjustment factor depends on any class-2 parameter, this will
affect ourability to correctly measure the market sharpe ratio,
since these parameters will be subjectto possibly significant
calibration errors.
-
2 MODEL SETUP 23
We have evaluated (17) for different combinations of input
parameters34, the main resultscan be summarized as follows:
The adjustment factor is close to 1 for all parameter
combinations as long as theasset volatility is below 10% and the
resulting actual default probability belongs to aninvestment grade
rating, see figure 11. This can be explained by looking at the
impactof the parameters introduced in the Duffie/Lando framework:
all of them do effect theactual default probability as well as the
risk neutral default probability in the samedirection, e.g.
increasing the information uncertainty increases the actual as well
asthe risk neutral default probability. The share ratio is the only
parameter that simplyhas an effect on the actual default
probability only. This explains qualitatively, whythe adjustment
factor is close to one in many cases.
The adjustment factor can be accurately determined simply based
on knowledge ofthe class-1 parameters and the actual default
probability as long as > 10%, i.e. fora given combination of
default probability and maturity, parameters that can not
beobserved easily (e.g. asset volatility, default barrier, asset
value or accounting noise)do not significantly affect the
adjustment factor.
Please note the special role of the actual default probability:
For example, an adjustmentfactor of appr. 1.7 (i.e. significantly
above 1) occurs for an asset value of 108, =10%,T=5, SR=40% and
=0%. If this were due to any class-2 parameter, empirical
applicationswould be hardly possible due to calibration errors of
class-2 parameters. But as soon aswe change any of these parameters
so that the resulting actual default probability belongsto an
investment grade rating (e.g. increasing Vt, decreasing ,
decreasing (up to a levelof 10%)) the adjustment factor will be
close to 1. In other words, any combination ofthese parameters,
that yields a given actual default probability also yields (almost)
the sameadjustment factor. All in all, class-2 parameters do
actually have a significant influence onthe adjustment factor; but
this influence can (almost fully) be captured simply by the
ratingsmile.If we take a closer look at the parameter combinations
leading to the minimal and maximaladjustment factor in figure 11,
we see, that these actually belong to unlikely
parametercombinations. Table 6 shows the dependeny of the adjusment
factor for a maturity of 5years and a Baa-rating35 for = 0% and s =
0, e.g. for the extreme of observable asset
34Input parameters used were: : 3%30% (The 5% and 95% quantile
for the asset volatility from KMVwas 6% and 25% respectively),
sharpe ratio of the ability-to-pay process: 10% to 40% (The market
sharperatio is usually assumed to be anywhere between 20% and 50%,
due to a correlation of lower than 1, theasset sharpe ratio should
be smaller), m : 0% 5% (m < 0 would imply, that the payout rate
is larger thanthe risk free rate, m=5% was choosen as an upper
limit to reflect (almost) zero payout at a risk free interestrate
of 5%.), : 0% 30% ( = 0% reflects the classical first passage model
with observable asset values,Duffie/Lando use 10% as a standard
value, the upper limit of 30% is also based on
Duffie/Lando(2001)),Vt = Z0 and VB for all combination that
resulted in rating grades from AA to B. The case Vt > Z0 andVt
< Z0 was also analyzed, the results merely differ from the case
Vt = Z0 and are available upon request.
35We choose this as an example, since 5-year CDS are the most
liquid ones usually used in empiricalstudies, see for example
Berndt et.al. (2005) and Amato (2005) and Baa is the most common
rating amongnon-financial companies.
-
2 MODEL SETUP 24
values, table 7 for = 30% and s = 3 representing large
uncertainty about the asset value.The bold numbers depict the
minimum and the maximum values for the adjustment factor.The
absolut maximum is taken for small asset volatilites, no
uncertainty and a high riskneutral asset growth rate (i.e. a low
payout rate). Even ignoring the unrealistic default termstructure
implied by a certain asset value, one would usually expect small
asset volatilitiesto belong to a value firm whereas low payout
rates usually apply to growth companies.The absolut minimum is
taken for small asset volatilities and high payout rates (i.e. a
lowrisk neutral asset growth rate m), which would suit the usual
assumptions about value firms,but for a high uncertainty about the
current asset value, which one would usually assume forgrowth
companies. Depicting values one would usually assume for value
firms and growthfirms, we can see that the adjustment factor will
lie even closer around the mean value.
Figure 11: Adjustment factor in the Duffie/Lando model for
different rating grades. Theminimum and maximum is taken over the
parameters 10% 30%, 0 30%,0 m 5%, 10% SRV 40%, 0 s 3. Other
parameter: T=5.
2.4 Deviations from the lognormal assumption
It is a widely discussed topic in empirical finance, that log
asset returns are not normallydistributed (as a geometric Brownian
motion suggests) but rather fat tailed or leptokurtic,i.e. their
standardized third moment is larger than the third moment of a
standard normaldistribution (which has a fourth moment of 3). In
first passage time structural models, a
-
2 MODEL SETUP 25
default occurs, as soon as the asset value has fallen below a
predefined default barrier. Sincethese default probabilities are
usually small (e.g. the 5-year default probability of a
A-ratedobligor is 0.60%, the 5-year default probability of a
BBB-rated obligor is appr. 2.2%36),leptokurtis of the asset value
returns has a significant influence on default probabilities, and-
as we will show in this subsection - on the difference between risk
neutral and real worlddefault probabilities as well.
To analyze the impact of leptokurtic asset returns, we will
return to a Merton setting,where default can only occur at maturity
of the bond/CDS. The reason for this approachis mainly its
simplicity and tractability. Using fat-tailed distributions in
first-passage-timesetting usually involves quite sophisticated
mathematical framework based on discontiniousmartingales.37 In
addition, the results usually loose the simplicity and
intuition.
We will assume in the following, that the log asset return RT :=
ln(VTVt) at time T (the
maturity) has a distribution identified by its cumulative
distribution function F (x). Thedistribution is supposed to have
first and second order moments, so that we can standardizeit by RT
=
RTTT
. The expected one-year return is supposed to equal under the
realworld probability measure and r under the risk neutral
probability measure. Furthermore,we assume stationary and
independent increments of the 1-year returns, so that T = Tand T
=
T , which yields RT =
RTTT . Therefore, the real world default probability
can be calculated as
P def (t, T ) = P [ VT < N ] = P
[RT < ln(
N
Vt)
]= P
[RT