1. Managing Corporate Bond Risk: New Evidence in the Light of the Sub-prime Crisis Giovanni Barone-Adesi Nicola Carcano Hakim Dall’O

2. Agenda Motivation Model Data Results Conclusions 2

3. Motivation (I) Most commonly used techniques to hedge bond portfolios against interest rate risk (PCA, Duration Vector, Key Rate Duration) assume the following dynamics of the zero-coupon risk-free rate R(t,D k ) of maturity D k : where: – F l t represents the change in the l-th factor between time t and t+1 – c lk represents the sensitivity of the zero-coupon rate of maturity D k to this change 3

4. Motivation (II) When the number of hedging assets is not smaller than the number of risk factors, these techniques set the sensitivity of the hedged portfolio to each individual risk factor equal to zero For high-quality hedging relying on more than one risk factor, these techniques have been found both by academicians and practitioners to be unstable and to lead to contradictory results (see, for example, Falkenstein and Hanweck (1997)) 4

5. Motivation (III) For default risk-free bonds, Carcano (2009) and Carcano and Dall’O (2011) show that accounting for the modeling errors in the development of the hedging equations can make these techniques more robust, thus reducing hedging errors as well as transaction costs In this framework, the dynamics of the zero-coupon risk- free rate are described by the following equation: where  represents the model error. 5

6. Motivation (IV) The main goals of our research were: – To assess if the inclusion of modeling errors in the hedging equations can lead to better hedging also of corporate bond portfolios – To test the performance of alternative hedging instruments before and during the financial crisis of 2007-2009. In particular, we intended to test the following hedging instruments: US T-Note / T-Bond Futures S&P500 Futures CDX Contracts – To test the effectiveness of liquid credit derivatives, like CDS, to hedge credit spreads in times of financial distress 6

7. Model (I) – Dynamics of BBB-rated Bonds where: – ⍵ i (t,D k ) indicates the present value of bond i cash-flow with maturity D k as a percentage of its total present value at time t. – P indicates the bond dirty price – R BBB (t,D k ) indicates the zero-coupon market yield for BBB-rated bonds with duration D k at time t – and  i (t) indicates the idiosyncratic return provided by bond i from time t to time t+Dt – 7

8. Model (II) – Dynamics of T-Bond Futures where: – FP indicates the future price and s is the maturity of the future contract – R RF (t,D k ) is the zero-coupon yield for risk-free bonds with duration D k – CTD(y) indicates the cheapest-to-deliver bond of the futures contract y – for k > s, where cf stands for cash-flow – for k = s 8

9. Model (III) – Equations for Hedging only through T-Bond Futures If we represent the dynamics of risk-free rates as: the unexpected return of the hedged portfolio is: where A i represents the amount invested in bond i and  y represents the optimal weight to be invested in the y-th futures contract. 9

10. Model (IV) – Equations for Hedging only through T-Bond Futures In the context of PCA, if we assume that errors  BBB and  RF as well as idiosyncratic returns  i are independent from each other and from risk factors F, the partial derivatives of the variance of unexpected return relatively to the weight  y is approximated by: The optimal hedging strategy is obtained when the last equation is set equal to zero for each of the four considered T-bond futures 10

11. We define the dynamics of the S&P500 futures as: We perform the same steps presented above including the S&P500 futures in the estimation of the PCA parameters and adding one additional weight  S&P in the set of hedging equations 11 Model (V) – Equations for Hedging through T-Bond and S&P500 Futures

12. Because of lack of observations to estimate one integrated PCA including CDX, we followed a two-stage approach: – In the first stage, the optimal portfolio of T-bond futures has been identified following the procedure presented above – In the second stage, we determined the optimal weighting for the CDX contract by following a traditional approach based on linear regression: where: – e FUT represents the hedging error produced by the strategy including only the T- bond futures – R CDX represents the return to the protection buyer of the CDX contract – e CDX represents the hedging error remaining after the inclusion of the CDX in the hedging portfolio. 12 Model (VI) – Equations for Hedging through T-Bond Futures and CDX

13. In order to explain the residual error obtained by using the CDX contracts, we estimated the hedging error which would have been obtained by using CSF free of liquidity as well as counterparty risk The market rate of these CSF has been calculated as: where Y BBB (t,D k ) is the yearly-compounded par market yield. The hedging regression now is: where CSD BBB is the Credit Spread Duration of the considered spread 13 Model (VII) – Equations for Hedging through T-Bond Futures and Credit Spread Forwards

14. In order to perform realistic tests, we intended to build corporate bond portfolios displaying both a good correlation with market yields as well as a certain level of idiosyncratic risk: bond portfolios consisting of 8 bonds of different issuer and maturity displayed these characteristics Following Eom et al. (2004), we restricted our bond selection to BBB-rated industrial issuers We defined 8 time buckets with maturity equal to – respectively – 2, 4, 6, 8, 10, 16, 20, and 26 years. At the beginning of each year, we selected new, fixed-coupon bonds based on three conditions: a publicly held face value outstanding of at least 100 million US$, an already-paid first coupon and a maturity as close as possible to the one of the corresponding time bucket Relying on results reported by Carcano (2009) and Carcano and Dall’O (2011), we performed our tests on equally-weighted portfolios Bond prices and reference data have been kindly provided by Wilshire Associates 14 Data (I) – Corporate Bond Portfolios

15. Zero and par yield curves for BBB-rated bonds issued by industrial companies as well as the continuous S&P500 future series have been provided by Bloomberg We referred to the next expiring contract of US T-bond and T-note futures with denomination of – respectively – 2, 5, 10, and 30 years. For each contract and period, we identified CTD bonds following the net basis method and relying on the monthly baskets of deliverable bonds and conversion factors (CF) kindly provided by the CME. Future closing prices have been provided by Datastream. We extracted all data related to US Treasury bonds needed for estimating the sensitivity of each T-bond futures to the risk factors from the CRSP database. We calculated the present value of each individual cash-flow using US Treasury zero-coupon rates estimated through the Unsmoothed Fama-Bliss methodology described in Bliss (1997). 5-year CDX.NA.IG series from 2004 to 2010 (no. 3 to 13) have been downloaded from Bloomberg. Markit supported us on the unexpected return calculation. 15 Data (II) – Other Data

16. Sub-periods include – respectively – 119, 191, and 48 by-weekly observations Since our hedging strategies assume that idiosyncratic bond returns are independent from market returns, they should not be expected to hedge the residual idiosyncratic return provided by the bond portfolio. Accordingly, the fourth column of the table provides us with a rough estimate of the maximum variance reduction which can be expected from the tested strategies. 16 Results (I) – Summary Statistics Average Correlation Average Correlation Bond Portfolio: Explanatory BBB Systematic Risk: Explanatory Average Ratio Spread Variance / Period BBB Rates BBB Spreads Power of BBB Systematic Risk Power of Risk-Free Rates Total Variance 1995-Sep. 200075.5%30.2% N.A.87.8%27.2% Oct. 2000 - 200765.0%37.0%83.8%83.5%26.8% 2008-200976.0%94.9%75.5%43.4%57.6%

17. Sub-periods include – respectively – 100, 72, and 47 by-weekly observations Results for the period 2000-2007 compare well to the maximum variance reduction of 50% reported by previous studies hedging corporate bonds through T-bond and S&P500 futures. We attribute this improvement to the use of four futures contracts and to the consideration of modeling errors. 17 Results (II) – Variance Reduction Obtained by Alternative Hedging Strategies Hedging InstrumentsT-Bond Futures T-bond Futures and S&P500 Futures T-bond Futures and CDX T-bond Futures and Credit Spread Forward Bond PortfolioVarianceR 2 of Hedging ErrorVariance PeriodVarianceReductionon Spread ChangesReduction 2000-20040.015%82.3%0.6%76.8%N.A. 83.1% 2005-20070.008%84.7%0.0%84.0%86.7% 84.9% 2008-20090.035%23.5%49.8% (***)26.1%30.3%64.3%(**)

18. 18 Results (III) – Level of corporate bond spread and CDX spread from 2004 to 2009 The CDX Spread has been obtained by Bloomberg for the 5-year Investment Grade CDX contract. The Spread to Swap has been calculated as the difference between the average yield of the CDX basket and the 5-year swap rate. The CDX Basis has been calculated as the difference between the CDX Spread and the Spread to Swap.

19. Conclusions (I) – From 2000 to 2007 A hedging strategy based only on T-bond futures would have reduced the variance of the bond portfolio by circa 83.5%. This reduction is significantly higher than the 50% reported by previous studies attempting to hedge corporate bonds through T-bond futures. Hedging errors tend to be independent from the dynamics of the average credit spread paid on BBB-rated bonds. This suggests that they are mainly due to the idiosyncratic returns provided by the bond portfolio to be hedged. Accordingly, it is plausible to think that increasing the number of bonds in the portfolio might lead to even better hedging. 19

20. Conclusions (II) – 2008 and 2009 During this period – which was characterized by credit spreads of BBB-rated bonds in excess of 2% - properly hedging the dynamics of these spreads was of paramount importance The S&P500 future would have not helped a lot, since the weights assigned to it by our strategies are quite low. This is due to the low correlation of this future with changes in BBB yields (even though we followed Marcus and Ors (1996) in considering the S&P500 future only in periods of low consumer confidence) Surprisingly, also the CDX would have not helped a lot; the good performance of hedging through CSF indicates that the problem is not the methodology, but rather the specific dynamics of the CDX basis due to liquidity issues and/or counterparty risk 20

21. Conclusions (III) – Summing up… In periods of financial stability, corporate bond portfolios can be hedged very effectively by simply using T-bond futures However, when credit spreads exceed ordinary levels indicating financial distress, this is no longer the case and hedging instruments capturing the dynamics of credit spreads should be used Unfortunately, all currently liquid derivatives which could plausibly hedge these spreads do not seem to perform well A part of the problem is that - to use a non-financial analogy - credit derivatives subject to counterparty risk are like fire-extinguishers stored within inflammable cases! This motivates launching fully-collateralized CDS or credit spread forward contracts which should move more in-line with actual bond spreads 21

22. References Bliss, Robert R., 1997. Testing term structure estimation methods. Advances in Futures and Options Research 9, 197-231. Carcano, N., 2009. Yield curve risk management: adjusting principal component analysis for model errors. Journal of Risk 12, n.1, 3-16. Carcano, Nicola, and Hakim Dall’O, 2011, Alternative models for hedging yield curve risk: An empirical comparison, Journal of Banking and Finance, 35, November, 2991-3000. Eom, Young, Jean Helwege, and Jing-zhi Huang. 2004. Structural Models of Corporate Bond Pricing: An Empirical Analysis. Review of Financial Studies 17:499–544. Falkenstein, E., and Hanweck, J., 1997. Minimizing basis risk from non- parallel shifts in the yield curve. Part II: Principal components. Journal of Fixed Income (June), 85–90. Marcus, Alan. and Evren. Ors, 1996, Hedging Corporate Bond Portfolios Across the Business Cycle, Journal of Fixed Income, Vol. 4, No. 4 (March), pp. 56-60. 22