Bond spread-based approaches as insolvency forecasting procedures

The basic idea of ​​the bond spread-based insolvency forecast method is to use the credit spread that a company has to pay for its capital market-traded bonds compared to “risk-free liabilities” to infer the probability of default (PD) of the company implied by this market valuation .

Simple model with risk-neutral assessment

For the sake of simplicity, we consider a bullet bond in which the entire interest payment is due on the due date together with the repayment. The following sizes are considered:

• ${\displaystyle V_{mkt}}$ - market value of the bond
• ${\displaystyle V_{nom}}$ - Nominal value of the bond
• ${\displaystyle i_{nom}}$- nominal interest rate
• ${\displaystyle i_{mkt}}$- Gross market return (interest of , if the bond debtor does not default) ${\displaystyle V_{mkt}}$
• ${\displaystyle i_{rl}}$ - risk-free interest rate,
• ${\displaystyle PD}$- probability of default (English probability of default )
• ${\displaystyle LGD}$- Loss given default .

The bondholder is likely to receive a payment equal to the nominal amount of the bond plus the agreed interest on the due date . There is a likelihood of default and the creditor only receives a fraction of this. With a risk-neutral valuation , the market value of the bond then results as follows: ${\displaystyle 1-PD}$ ${\displaystyle V_{nom}}$ ${\displaystyle i_{nom}\cdot V_{nom}}$ ${\displaystyle PD}$ ${\displaystyle 1-LGD}$

${\displaystyle V_{mkt}={\frac {(1-PD)\cdot V_{nom}\cdot (1+i_{nom})+PD\cdot (1-LGD)\cdot V_{nom}\cdot (1+i_{nom})}{1+i_{rl}}}}$

By moving

${\displaystyle V_{mkt}\cdot (1+i_{rl})=V_{nom}\cdot (1+i_{nom})\cdot (1-PD+PD-LGD\cdot PD)}$

and further transformation

${\displaystyle {\frac {V_{mkt}\cdot (1+i_{rl})}{V_{nom}\cdot (1+i_{nom})}}=1-LGD\cdot PD}$

the formula can be solved according to the probability of failure : ${\displaystyle PD}$

${\displaystyle PD={\frac {V_{nom}\cdot (1+i_{nom})-V_{mkt}\cdot (1+i_{rl})}{LGD\cdot V_{nom}\cdot (1+i_{nom})}}}$

The following applies to the (not directly observable) gross market yield of the bond : ${\displaystyle i_{mkt}}$

${\displaystyle 1+i_{mkt}={\frac {V_{nom}\cdot (1+i_{nom})}{V_{mkt}}}}$

This resolved after results ${\displaystyle V_{mkt}}$

${\displaystyle V_{mkt}=V_{nom}\cdot {\frac {1+i_{nom}}{1+i_{mkt}}}}$

Now this can be inserted into the above formula for : ${\displaystyle V_{mkt}}$ ${\displaystyle PD}$

${\displaystyle PD={\frac {V_{nom}\cdot (1+i_{nom})-V_{nom}\cdot {\frac {1+i_{nom}}{1+i_{mkt}}}\cdot (1+i_{rl})}{LGD\cdot V_{nom}\cdot (1+i_{nom})}}}$

The nominal values and are thus abbreviated to numerators and denominators, so that the estimation of the probability of default does not depend on these instrument-specific values ​​that can be freely selected by the debtor: ${\displaystyle V_{nom}}$ ${\displaystyle i_{nom}}$

${\displaystyle PD={\frac {1-{\frac {1+i_{rl}}{1+i_{mkt}}}}{LGD}}={\frac {i_{mkt}-i_{rl}}{LGD\cdot (1+i_{mkt})}}}$

The counter on the right shows the credit premium ; solving the equation for the credit add-on results ${\displaystyle i_{mkt}-i_{rl}}$

${\displaystyle {\text{Kreditaufschlag}}=PD\cdot LGD\cdot (1+i_{mkt})}$

With very low recovery rates, i. H. for LGD close to 100 percent, the credit spread roughly corresponds to the probability of default PD.

Empirical findings on the performance of the simple model with risk-neutral assessment

Real market data show that the model in this form is not suitable for determining default probabilities. The credit spreads observed in the bond markets are not compatible with a risk-neutral valuation (and presumably also not with any other rational type of valuation) in terms of level and also with regard to time variability . Particularly evident this assessment are anomalies in bonds in investment worthy credit classes (English investment grade ), the assessments made by renowned so as rating agencies have low probability of default. For example, the average credit premium on bonds with a BBB rating according to Standard & Poor’s (S&P) rose from December 1999 to December 2000 by 150 basis points to 300 basis points, see the following figure.

Average interest rate differentials for bonds of various S&P rating categories compared to US Treasuries, 1999–2002.

A recovery rate of around 50 percent was achieved in the period 1982–2005 for senior, unsecured corporate bonds of creditworthy companies. Assuming an expected recovery rate for bonds of around 50 percent for the sake of simplicity, the annual probability of default of bonds rated BBB would have to have risen by 3% to around 6% within this period - at least with a risk-neutral assessment. In fact, the realized one-year default rates of BBB bonds are much lower. The highest one-year default rate ever realized in the roughly 25-year observation period 1981–2004 was only 1.2% (in 2002, see the following figure); on average for the years 1981–2004 the corresponding failure rate was only 0.29%. A default level of 6% is only achieved with BBB-rated bonds - at least on average for the years 1981–2004 - with regard to the cumulative 10-year default rates. Similar findings result from a comparison of the realized default rates and the credit spreads for bonds rated AA or A, see the graphs on the left in the figures above and below.

Realized default rates for bonds of various credit ratings from Standard & Poor's (S&P) over time, 1981–2004.

Only a small fraction of the implausibly high credit spreads on bonds can be attributed to the tax unequal treatment of interest income on corporate and government bonds at the US state level. Factors that are more difficult to quantify - such as lower liquidity and regulatory investment restrictions for certain groups of potential bond buyers - are also cited to explain credit spreads. Overall, however, it is estimated that only around 25% of the observable credit spreads on corporate bonds can be traced back to expected losses - around half, however, to compensation for “ systematic risks ”. Empirical studies have shown that the credit spreads on bonds correlate over time with the interest rate level and various stock indices . However, it is hardly conceivable that there could be a closed formula for determining the probability of default of a company from the current characteristics of these "systematic risk factors" and the current credit spread.

Alternative modeling approaches

Despite the (apparently) unpredictable changes in bond spreads over time, studies show that the ranking of companies implied by the level of bond spreads (which must be adjusted for instrument-specific option rights , e.g. early redemption rights ) at a given point in time is a very high (ordinal ) Has the ability to forecast insolvency . By mapping the spreads, which are not meaningful in their absolute amount and which cannot be compared over time, by means of a dynamically adaptable and possibly term-specific mapping rule on an agency rating scale, default forecasts can be generated from the bond spreads with a high forecast quality across all points in time, which even the forecasting ability exceeded by agency ratings!

literature

• EI Altman: Measuring Corporate Bond Mortality and Performance . In: Journal of Finance , Volume 44 (4), 1989, pp. 909-922, 1989
• Basel Committee on Banking Supervision (Ed.): Range of Practice in Banks' Internal Ratings Systems . (PDF; 172 kB), discussion paper, Bank for International Settlements (BIS), 01/2000
• M. Bemmann: Development and validation of a stochastic simulation model for the forecast of corporate insolvencies . Approved dissertation, Technische Universität Dresden, 2007. TUDpress Verlag der Wissenschaften, Dresden, ISBN 978-3-940046-38-3
• C. Bluhm, L. Overbeck, C. Wagner: An introduction to Credit Risk Modeling . Chapman and Hall / CRC Financial Mathematics Series, Boca Raton 2003
• L. Breger, LR Goldberg, O. Cheyette: Market Implied Ratings . also published in Risk Magazine , 07/2003.
• R. Cantor, C. Mann: Measuring the Performance of Corporate Bond Ratings . Special Comment, Report # 77916, Moody's Investor's Service, 04/2003
• Deutsche Bundesbank (Ed.): Financial Stability Report 2005 . (PDF) 11/2005
• EJ Elton, MJ Gruber, D. Agrawal, C. Mann: Explaining the Rate Spread on Corporate Bonds . In: Journal of Finance , Volume 46 (1), 2001, pp. 247-277
• GM Gupton, RM Stein: LossCalc V2: Dynamic Prediction of LGD Modeling Methodology . (PDF) Moody's KMV, Working Paper, 01/2005
• DT Hamilton: Rating Transitions and Defaults Conditional on Watchlist, Outlook and Rating History . Moody's Investors Service, Special Comment, Report # 81068, 02/2004
• JC Hull, M. Predescu, A. White: The Relationship between Credit Default Swap Spreads, Bond Yields, and Credit Rating Announcements . In: Journal of Banking and Finance , Volume 28 (11), 2004, pp. 2789-2811
• Moody’s (Ed.): Default and Recovery Rates of Corporate Bond Issuers, 1920–2005 . (PDF) Moody's Investors Service, 01/2006
• Standard and Poor’s (Ed.): Ratings Performance 2002, Default, Transition, Recovery, and Spreads . Special Report 02/2003, The McGraw Hills Companies, 2003
• Standard and Poor’s (Ed.): Ratings Performance 2003 . Special Report 03/2004, The McGraw Hills Companies
• Standard and Poor’s (Ed.): Annual Global Corporate Default Study: Corporate Defaults Poised to Rise in 2005 . Global Fixed Income Research, The McGraw Hills Companies, 2005
• S&P: see Standard and Poor’s
• SM Turnbull: Unresolved Issues in Modeling Credit-Risky Assets . In: Journal of Fixed Income , Volume 15 (1), 2005, pp. 68-87

Individual evidence

1. ↑ This article is based on Bemmann (2007, section 2.3.3.2)
2. ↑ The model shown corresponds essentially to the model by Bluhm, Overbeck, Wagner (2003, p. 186 f.) With the difference that the default loss rate (LGD) relates to all receivables (including interest) at the time of default and not only on the nominal amount of the bond.
3. ↑ see S&P (2003a, p. 14)
4. ↑ See Moody’s (2006, p. 12). For a more detailed description of the influencing variables determined in empirical studies on the losses expected in the event of a failure (LGD) see Gupton, Stein (2005).
5. ↑ Strictly speaking, these are the average 10-year failure rates of the S&P BBB cohorts 1981–1995, since no failure observations over a ten-year period were available for the cohorts formed after 1995.
6. ↑ The average annual failure rate of 0.29% p. a. refers to the default rate of companies that had a BBB rating from S&P at the beginning of the year in question. Since the credit rating of the company can vary over time, conclusions cannot be drawn directly from the one-year default rates on the cumulative or average multi-year default rates. However, these can be found in the historical default statistics of the rating agencies.
7. ↑ Source: own evaluations based on S&P (2004, p. 16ff.) And S&P (2005, p. 33ff.)
8. ↑ See Altman (1989, p. 921) and Hull, Predescu, White (2004, p. 2796ff.)
9. ↑ See Basel Committee (2000c, p. 54) for a list of 14 different regulatory provisions through which various investor groups, etc. a. be discouraged or restricted from purchasing bonds of different “risk”.
10. ↑ See Elton et al. (2001, p. 249 and p. 273). The average credit spreads observed in the investigation period of the study 1987–1996 are even significantly lower than the values ​​shown in the above figure for the period 1999–2002.
11. ↑ See Turnbull (2005, p. 72ff.) And the literature cited there as well as Deutsche Bundesbank (2005, p. 141ff.). Further macroeconomic influencing factors mentioned here for explaining or forecasting credit spreads are, for example, the implied volatility of stock market indices or key figures for describing issuing activity .
12. ↑ See Breger, Goldberg, Cheyette (2003, pp. 2f.) And Cantor, Mann (2003, p. 25). Furthermore, identical expected recovery rates are assumed.
13. ↑ Cantor, Mann (2003) take into account differences in runtime, Breger, Goldberg, Cheyette (2003, pp. 2f.) Do not.
14. ↑ On the basis of an identical sample, the “[bond] market implied rating” by Cantor, Mann (2003, p. 25) achieves an accuracy ratio of 7 percentage points better than Moody's ratings over a one-year period. Over a three-year perspective, the performance is at least 1.5 percentage points (PP) better.
15. ↑ The inferiority of agency ratings compared to “bond market-implied ratings” is possibly mainly due to the efforts of the rating agencies to artificially stabilize their rating judgments. If the agency ratings are modified according to their “outlook or observation status (outlook, watchlist)”, their estimation quality, measured in the accuracy ratio over a three-year perspective, increases by approx. 6 percentage points, see Hamilton (2004, p. 12).