# Presentation: 241843

## Linking Credit Risk Premia to the Equity Premium(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](http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1019279)_

## 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

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

**Approach**

- Historical Equity Premium Estimates (HEP)
- Implicit Equity Premium Estimates (IEP)
- Utility-function-based estimation (UEP)
- Expert estimates

**Estimates**
**EP / SR**

- EP: 7-9 %
- SR:  40-50 %
- EP: 1-9 %
- SR: 5-50%
- EP: < 1 %
- SR: < 5 %
- EP: 4-7 %
- SR: 25-40 %

**Remarks**

- 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

**Main 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 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 %.

## Agenda

** ****The equity premium**
** ****Model setup**
** ****Empirical findings**

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

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

- 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)

**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

**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 probability

- Asset Sharpe ratio

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

- Market Sharpe ratio

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

- 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

**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

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

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

**1**

**Input parameter**

- PDQ

**Explanation**

- Risk neutral default probability

**Source**

- CDS-spreads

**Remark**

- 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 t****0****-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

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

**Model ingredients**

**Merton**

- Asset value in t=0

- Asset value process

- Default boundary

- Default mechanism

- V0 є lR

- Geometric Brownian Motion

- Exogenous

- Default only at maturity

**2**

- Source

- 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 characteristics

- First structural default model
- Simple framework

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

**2**

- Merton

- First passage

- Duffie/Lando

**Model**

- 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

**Model (and parameter) changes affect both PD****P**** and PD****Q**** in the same direction – **
**the Sharpe ratio is the only parameter that solely has an influence on PD****P**

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

**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)****)**

- Adjustment 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)**

- 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

## Agenda

** ****The equity premium**
** ****Model setup**
** ****Empirical findings**

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

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

**Data sources**

- 1. See our paper for methodological details

**Parameter**

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

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

**Descriptive statistics**

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

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

- 26,51%

- 10,15%

- 67,63

- 17,79%

- 18,70%

- 14743

- Sharpe ratio company (Moodys)

- 53,03%

- 21,50%

- 75,70

- 35,25%

- 39,01%

- 14743

- Sharpe ratio market
- (Moodys)

- 20,27%

- 8,30%

- 46,32

- 13,95%

- 14,76%

- 19945

- Sharpe ratio company
- (EDF),
- after  tax adjustment

- 43,12%

- 15,45%

- 59,64

- 27,39%

- 32,13%

- 19945

- Sharpe ratio market
- (EDF),
- after  tax adjustment

- 26,71%

- 11,76%

- 60,32

- 19,04%

- 19,56%

- 19945

- Sharpe ratio company
- (EDF)

- 56,80%

- 21,91%

- 76,33

- 37,23%

- 42,46%

- 19945

- Sharpe ratio market(EDF)

- 75th Pctl

- 25th Pctl

- Coeff of Variation

- Median

- Mean

- N

- Variable

- 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% 

**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)**

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

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

**Input parameters**

**Not considered in our calculation**

- CDS-spreads

- Recovery rate

- Correlations

- 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

**Effect**

- Our result is upward biased
- Our result is upward biased

- Our result is upward biased
- 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