Fair Value and Expected Credit Loss Estimation: An Accuracy Comparison of Bond Price versus Spread Analysis Using Lehman Data

04/25/2016 05:36 AM

In both cases, the visibility of the organization’s valuation and credit

risk assessment moves from the back office or middle office, seen primarily by risk experts, to center stage under a bright spot light. Both these standards allow the use of creditworthiness assessment using approaches encompassing:

  • Probabilities of default
  • Internal or external credit ratings
  • Credit spreads

The objective of the standards is the generation of 12-month and expected lifetime loan losses based on changes to obligor creditworthiness from one observed point to the next.

It is to the last bullet point above that we focus our attention, and in this note, take a model validation approach and compare two methods of valuation and credit loss assessment from an accuracy point of view. The first approach uses market-based credit spreads to establish obligor creditworthiness, estimate values and credit losses. The second approach uses observable market prices of securities, rather than credit spreads derived from them, directly in the valuation and credit assessment process. We explain why the use of observable market prices is best practice. We also list the model validation issues that cause credit spreads (derived from market prices) to be a source of random errors in valuation and credit assessment.

Which Market? The Bond Market or the Credit Default Swap Market?

What is the best source of market data, whether it be securities prices or credit spreads derived from them, for any creditworthiness assessment, including IFRS 9 and CECL expected loss calculations? In a perfect world, the simple answer would be “all markets.” Sadly, there is a substantial imbalance in the transparency and price discovery available in two key markets that are potentially important for IFRS 9 and CECL: the corporate bond market in the United States and the market for single name credit default swaps.

The chart below presents the trading volume and most heavily traded reference names in the U.S. corporate bond market on a representative day, April 19, 2016:

The data is provided by the Financial Industry Regulatory Authority (“FINRA”) via the Trade Reporting and Compliance Engine (“TRACE”) system which records every trade in the U.S. corporate bond market and the prices at which trades took place. On April 19, there were 33,680 bond trades on 4,722 bonds issued by 1,290 bond issuers for an underlying principal amount of $11.4 billion.

The volume of single name credit default swaps traded, but not the spreads or prices, has been reported by the Depository Trust & Clearing Corporation since July, 2010. Data is reported weekly with a four calendar day lag, not daily. Trading volume for the week ended April 15 is given in the chart below:

The trading volume during the week ended April 15 was $56.2 billion for 5 business days, or $11.2 billion per day. While this is comparable to the daily volume in the U.S. corporate bond market, trading volume is dominated by trades in sovereign reference names at the top of the volume ranking, as the chart shows. The number of reference names traded in a week, at 734, is more than 500 fewer than the daily number of reference names traded in the U.S. bond market. The number of trades, at 14,000 per week or 2,800 per day, is less than one-tenth of the U.S. corporate bond market trade count. As mentioned above, the Depository Trust and Clearing Corporation does not report actual traded spreads or prices. In addition, the leading vendors of credit default swap data report quotes, not traded spreads or prices. Kamakura Corporation’s estimates, using the volume numbers above, are that less than 3% of the credit spreads reported by these data vendors could possibly be associated with real trades, and that 3% is not identified by the vendor. In other words, reliance on anonymous quotes from Wall Street is certainly not a reliable basis for any expected loss calculations because of the obvious conflict of interest that the anonymous dealers have.

A Lehman Brothers Example

Valuation and credit assessment for IFRS 9 and CECL for a riskless borrower is quite simple. The analysis becomes progressively more difficult to do accurately as the credit risk of the obligor increases. The impact of non-default probability factors on credit spreads, however, gives credit spreads a much higher volatility than the matched maturity default probability (in orange) and one year default probability (in red). This volatility differential is very visible in this graph of credit spreads (in blue), matched maturity default probabilities (in orange), and 1 year default probabilities (in red) on the Bristol-Myers Squibb bond due 2022:

Given the fact that spread modeling is even more complex for high default risk issuers, it is instructive to do our model validation using data for the now bankrupt Lehman Brothers. Lehman Brothers Holdings Inc. announced its intention to file for bankruptcy on Sunday, September 14, 2008. Bond prices and spreads on Monday, September 15, 2008, fully reflect this information. We use this data from TRACE on 22 senior non-call bonds that traded at least $5 million in daily volume on that day:

The credit spreads on these issues ranged from 16.91% to 116.96%, depending on the maturity of the bond, among other things. The credit spreads are graphed by maturity date in the figure below:

With this data in hand, we now turn to a discussion of common practice and best practice using either bond prices or credit spreads as input.

Common Practice: Model Validation Using Credit Spreads

In the graph above, the credit spreads on Lehman Brothers Holdings Inc. senior non-call bonds had an average credit spread of 46.86% with a standard deviation of 29.73% on September 15, 2008. A common practice approach to the computation of expected losses using credit spreads ascribes the variation in spreads to noise in the bond price data, and the analyst therefore “solves” this problem by fitting a smooth function to the Lehman Brothers spread data. The result is a commonly observed downward sloping spread as a function of years to maturity:

In the graph above, we have used a cubic function of years to maturity because of the common use of cubic splines in yield curve smoothing in general. At first glance, the fitting seems successful, because the three terms in the regression (years to maturity, years squared and years cubed) are all statistically significant and the adjusted r-squared in the regression is a respectable 88.49%. A red flag appears, however, in the form of a root mean squared error of the regression with a value of 10.09%, meaning the errors in the regression have a standard deviation of 10 full percentage points. Should that be worrisome?

We can address that question from a model validation perspective by answering a more basic question about the mathematical formula for the credit spread. For a bond with semi-annual payments, a semi-annual coupon of C dollars, and a principal amount of $100 due in exactly n semi-annual periods, the value of the credit spread s (expressed as a decimal) is the constant such that this formula equals the bond’s net present value V, which is the sum of price P and accrued interest A (which will be zero in this example):

We can say that the implied annualized value of the spread s is a function of the periods to maturity n, the matched maturity risk free yield on U.S. Treasuries r (expressed as a decimal), and the bond’s semi-annual dollar coupon C and value V (or alternatively price P and accrued interest A):

We have learned a lot from this exercise, and all of it is troubling. First, the cubic polynomial we have fitted above is mis-specified, because we have omitted the risk free Treasury yield level r, the bond’s semi-annual dollar coupon C, and its net present value V. Because of these omitted variables, we have caused our original Lehman credit spreads to be LESS accurate than they were before. The original credit spreads, by definition, were 100% consistent with the original bond prices, because they were derived from the equation for net present value V. The new fitted spreads no longer match observed bond prices.

Sadly, there are many more problems with the equation above, because the equation involves a large number of assumptions which we now know to be wrong in the case where default risk is not zero. Using Lehman Brothers data as an extreme example makes these false assumptions clear:

  1. The corporate bond will pay its full principal amount (this argument is false: the bond is defaulting and will pay its recovery value). In the Lehman case, the average bond price is 33.80, with a relatively small standard deviation of 1.60 over the 22 bond issues. If we say that the recovery amount is roughly 33.80, the assumption that the bond will pay 100 is grossly wrong and overstated.
  2. The full principal amount will be paid at maturity (false: the recovery amount will be paid upon resolution of bankruptcy proceedings in court. The longest maturity bond from Lehman in the chart above is 2027, but most of the recovery payments to Lehman bond holders have already been made).
  3. All interest coupons will be paid (false: only those interest payments prior to the bankruptcy filing on September 15, 2008 will be paid).
  4. Bonds of different maturities and coupons have different cash flows (false: they have identical cash flows upon default: interest payments are zero and the principal that will be paid is the recovery amount; and the payment date is the date [or series of dates] that recovery payments are made after the bankruptcy is resolved in court).
  5. Credit spreads are constant for all periods prior to maturity of bond k (false, they vary by maturity for firms that are not near bankruptcy).
  6. Credit spreads for bond k are different from bond j if they have different maturities, but these constant spreads are inconsistent from time zero to years to maturity = min(j,k). To give a specific example, the credit spread formula implies that the credit spread is 16.96% for the 2027 bond but 45.23% for the bond due in January 2012. In short, for the period from September 2008 to January 2012, the spread formula implies that the coupons for the 2012 bond have a spread that is almost 30 percentage points higher than the 16.96% spread that applies to coupons covering the same time period on the bond due in 2027. This inconsistency is nonsense.
  7. The risk free yield is constant for all periods until the risk-free bond’s maturity (false, this is a well-known problem with the yield to maturity calculation). Even for the risk free curve, the yield to maturity for bonds of different maturities implies different discount rates during the overlapping period when both bonds are outstanding.


After reviewing the model validation errors in the credit spread formula, one comes to a horrible realization: the “noise” in the credit spread graph above is not due to the noise in bond prices per se; instead the noise is due to omitted variables in the spread formula and false assumptions that are relevant whenever the default probability of the bond issuer is not zero (note that issues 6 and 7 apply even if the default probability is zero).

The result of all of these issues is simple to summarize: by “cleansing” noisy spread data that is in fact due to false model assumptions, the resulting predicted bond prices will be much more volatile than the original bond data itself. Our conclusion is inescapable: we reject the use of traditional credit spreads as an intermediate calculation in any obligor valuation and credit assessments.

Best Practice: Model Validation Using Bond Prices and No Arbitrage Assumptions

We now know that the original bond prices for Lehman have far less “noise” in them than bond prices predicted by smoothing flawed credit spreads. We plot the original bond prices here with the addition of one outlier where the trading volume is $48,000 on September 15 instead of the floor we imposed of $5 million trading volume:

As mentioned above, the average price of the bonds as they converge to the market’s perceived recovery value is 33.80 with a standard deviation of 1.60. We have four questions:

  • How do we estimate the price for the outlier bond which the price of 72 is just “bad data” due to small volume and a disadvantaged panic in execution?
  • How do we estimate the price of other Lehman bonds for which there were no trades?
  • How do we estimate all bond prices for a less distressed firm, ABC Brothers, if only some of ABC Brothers bonds are observable?
  • How do we estimate all bond prices for ABC Brothers if some of the bonds are callable?


The pricing of securities whose value depends on a number of macro factors (like the Bristol-Myers Squibb bonds above) in an environment where multiple factors drive the risk-free yield curve was described by Amin and Jarrow [1992], as modified for default risk by Jarrow [2013]. Detailed technical guides (Kamakura Corporation, 2015 and 2016) discuss the no arbitrage valuation for discrete Monte Carlo simulation of the risk free yield curve, macro factors, and default probabilities. Both risk free yields and credit spreads are smoothed while analyzing all bonds of ABC Brothers jointly. Best practice valuation has these characteristics:

  • The risk free zero coupon bonds vary by maturity and apply to all of the bonds of ABC Brothers in a consistent way.
  • The defaultable bonds of ABC Brothers comprise a series of “building block” securities that apply to all bonds in a consistent manner. Assume that there are 3 bonds outstanding which mature in 2, 4, and 6 semi-annual periods. The primitive securities consist of these 9 building blocks:I. A security (which represents a coupon payment) which pays $1 in 1 period if default has not occurred before that time
    II. A security which pays $1 in 2 periods if default has not occurred before that time
    III. A security which pays $1 in 3 periods if default has not occurred before that time
    IV. A security which pays $1 in 4 periods if default has not occurred before that time
    V. A security which pays $1 in 5 periods if default has not occurred before that time
    VI. A security which pays $1 in 6 periods if default has not occurred before that time
    VII. A security (which represents principal) and pays $1 in 2 periods if default has not occurred before then and which pays a random recovery D at the time of default if default occurs before period 2
    VIII. A security (which represents principal) and pays $1 in 4 periods if default has not occurred before then and which pays a random recovery D at the time of default if default occurs before period 4
    IX. A security (which represents principal) and pays $1 in 6 periods if default has not occurred before then and which pays a random recovery D at the time of default if default occurs before period 6
  • Let the three bonds of ABC Brothers have coupons of C[1], C[2] and C[3] dollars per semi-annual period and have a principal amount of 100. Let the value of the 9 building block securities be W[1], W[2],…W[9]. Then the net present value V[1], V[2] and V[3] of the 3 ABC Brothers bonds are the sum of the interest and principal parts:
    V[1] = C[1](W[1]+W[2])+100 W[7]
    V[2] = C[2](W[1]+W[2]+W[3]+W[4])+100 W[8]
    V[3] = C[3](W[1]+W[2]+W[3]+W[4]+W[5]+W[6])+100 W[9]


We also know the values of these building block securities if the real name of “ABC Brothers” is “Lehman Brothers.” If

W[1]= W[2]= W[3]= W[4]= W[5]= W[6]=0


W[7]= W[8]= W[9]=0.3380

then all three bonds will be valued at the average price of Lehman Brothers Holdings Inc. senior non-call debt on September 15, 2008:


Jarrow and Turnbull [1995] were the first to value these building block securities in a random interest rate environment incorporating obligor defaults. Of course, once we have the predicted bond prices (which should be very close to, if not exactly equal to, observable prices) we can then apply the (flawed) credit spread equation above.

The result of this no arbitrage valuation derived using all bond issues of ABC Brothers for which bond prices are observable minimizes the sum of squared pricing errors in a best practice no arbitrage/multi-factor economy. Even when the bonds are callable, the no arbitrage framework of Amin and Jarrow [1992], Heath Jarrow and Morton [1992], and Jarrow [2013] applies.


The common use of credit spreads, which are derived from observable bond prices but which contain false assumptions, contributes large errors to valuation and credit assessment in the obligor creditworthiness analytical process, and this includes obligor creditworthiness estimates based on credit spreads for IFRS9 or CECL expected loss computation processes. We can avoid this incremental source of error by fitting basic building block securities to observable bond prices in a no arbitrage framework. The result is a consistent and highly accurate valuation and credit estimation framework that meets best practice standards of financial theory, econometrics, and trading precision.


Amin, Kaushik and Robert A. Jarrow, “Pricing American Options on Risky Assets in a Stochastic Interest Rate Economy,” Mathematical Finance, October 1992, pp. 217-237. Heath, David, Robert A. Jarrow and Andrew Morton, “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claim Valuation,” Econometrica, 60(1), 1992, pp. 77-105.
Jarrow, Robert, “Amin and Jarrow with Defaults,” Kamakura Corporation and Cornell University working paper, March 18, 2013.
Jarrow, Robert, Jens Hilscher, and Donald R. van Deventer, “Parameter Estimation for Heath, Jarrow and Morton Term Structure Models, Technical Guide, Appendix A, U.S. Treasury Yields, January 1962 Through December 2015,” Kamakura Risk Information Services, Version 2.0, March 31, 2016.
Jarrow, Robert, Jens Hilscher, Thuy Le, Mark Mesler and Donald R. van Deventer, “Kamakura Public Firm Default Probabilities, Technical Guide, Version 6.0, Edition 7.0,” June 30, 2015.
Jarrow, Robert and Stuart Turnbull, “Pricing Derivatives on Financial Securities Subject to Credit Risk,” Journal of Finance 50 (1), 1995, pp. 53-85.
Jarrow, Robert, and Donald R. van Deventer, “Monte Carlo Simulation in a Multi‐Factor Heath, Jarrow and Morton Term Structure Model, Technical Guide, Kamakura Risk Manager and KRIS Credit Portfolio Manager, Version 4.0,” June 16, 2015.
Jarrow, Robert and Donald R. van Deventer, “Monte Carlo Amin and Jarrow Simulation of Traded Macro Factors and Securities in a Multi-Factor Heath, Jarrow and Morton Economy with Both Zero and Non-Zero Probability of Default, Technical Guide, Kamakura Risk Manager and KRIS Credit Portfolio Manager, Version 4.0,” April 13, 2016.

Copyright ©2016 Donald R. van Deventer and Suresh Sankaran