MS&E 348 Winter 2011

BOND PORTFOLIO MANAGEMENT: INCORPORATING CORPORATE

BOND DEFAULT

March 19, 2011

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Assignment Overview In this project, we sought to design a system for optimal bond management. Within this framework, we analyzed the impact that the default risk of corporate bonds has on the optimized bond portfolio. This assignment was comprised of several distinct objectives, which, in combination, would fully assess our research questions. These objectives were:

1) Create an appropriate utility function to model investor preferences – The utility function is the lens through which our model viewed portfolio scenarios. It was thus an essential part of our model that affected the quality and content of our results.

2) Select and calibrate an interest rate model – By definition, a stochastic model must assess actions over a range of potential scenarios. In our case, where the optimized portfolio is made up of bonds (i.e. interest rate derivatives), the range of scenarios is a myriad of yield curve possibilities. These scenarios are generated from an interest rate model. The selection and calibration of an appropriate model was our next step in creating our system.

3) Implement our two-stage stochastic optimization model – With these first two steps in place, we were ready to combine them in our initial optimization model. This initial model was able to analyze US Treasury bonds over possible future interest rate environments.

4) Develop a method of incorporating bond default risk –We then adjusted our model to include an additional dimension of uncertainty. We examined several options for incorporating this risk, and implemented the most appropriate method.

5) Conduct sensitivity analysis over key inputs – To truly understand the factors that were important in reaching our optimized results, we then analyzed the impact that changes in the utility function, interest rate model, and bond default risk methodology had in our end results. Much of this was done in an attempt to study the behavior of our model under different market assumptions.

Stochastic Model Overview Our stochastic model has two main phases – generation of bond price data, and use of this as input to optimize our initial bond selection. During the data generation phase, we first created a scenario tree, as shown in the figure below. At each node in this tree, we generated a unique interest rate curve and re-priced every bond and cash flow under that interest rate environment. These computations were done using MATLAB. In the next phase, this data was imported into GAMS, which ran an optimization of initial bond allocations. Key aspects of our optimization model include:

• Two stages – Our model included an initial bond allocation, a rebalancing of the portfolio at the end of stage 1, and an evaluation of final portfolio value at the end of stage 2.

• Objective Function – We maximized the expected utility of our portfolio value at the end of stage 2. This was based our rebalanced bond allocations from stage 1, multiplied by the final bond prices and any accumulated cash flow (coupon payments or matured bonds) from stage 2.

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• Budget constraint – An initial wealth of $1,000,000 limited the initial bond portfolio. No explicit transaction costs were considered in this initial allocation, but all bonds purchased in period 1 were purchased at the market-observed ask prices.

• Inventory balance constraint – This maintained equivalence between the initial bond allocation, the amount bought and sold, and the rebalanced portfolio.

• Cash balance constraint – This forced maintenance of the value of portfolio’s value during the rebalancing. It also instituted transaction costs.

 Figure 1 - Model Overview and Optimization Equations

Utility Function Our utility function played a central role in risk-management for the optimization, as it incorporates the risk-aversion factor of our investor. We examined several possible utility functions, including a power function and a logarithmic function. In the end, we settled on a concave piece-wise linear function, based on our ability to utilize it in conjunction with LP optimization techniques. Our utility function has four distinct pieces, as shown in the figure below. These pieces corresponded to ! returns of 3%, 5%, and 7%. A steep gradient for the first piece was used to heavily penalize downside risk. For GAMS implementation, we split our final portfolio value variable into 4 pieces, then used the following equations to value the pieces and compute the final utility:

!"#$%"&"' !"#$#% = !"#$!%

!"#$%"&"'.!" !"#$#% =  ∆!

!"#$#"% = !"#$%"&"' ∗ !"#$#%&'("#)*

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Figure 2 - Utility function

Interest Rate Model A variety of interest rate models exist that are used in the valuation of interest rate derivatives. In our implementation, we chose one of the most popular, the Cox, Ingersoll, and Ross model. This model was an extension of the Vasicek model, and similarly incorporates mean reversion for short rates to prevent a “random walk” scenario and limits deviation. The following equation is used to create a new simulated short rate:

!!!!!   = !!!   + ! ! − !!! ! + ! !!!!!! Where:

• !!!!!   = new short rate

• !!!   = old short rate

• !,  ! = CIR model parameters • ! = model time step • !!! = N(0, !)

Once these short rates are generated, each one is used to form an entire yield curve through the following equation:

! ! =! ! ∗ ! − ln!(!)

!

! ! =2ℎ  exp  [ ! + ℎ !2 ]

2ℎ + (! + ℎ)(!"# ! ∗ ℎ − 1)

!!"!!

! ! =  2 exp ! ∗ ℎ − 1

2ℎ + ! + ℎ exp ! ∗ ℎ − 1

ℎ =   !! + 2!!

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Where k, !, and ! are model parameters. We adjusted these parameters to fit our model as closely as possible to the current term structure from the initial bond prices. This was important to minimize the effect of moving from empirical, actual prices (used in our initial portfolio valuation) to hypothetical, simulated prices (used to value our portfolio in after stages 1 and 2). The figures below show our CIR yield curves under our initial parameterization (on left) and under a revised parameterization that better fit the initial term structure (on right). The thick blue line in the right hand side plot is the term structure implied by the empirical prices of the Treasuries.

 Figure 3 - CIR Curves (Hor Axis = time (half-year periods), Vert Axis = spot rate)

Bond Pricing Once we had generated our interest rate scenarios, we used these yield curves to price bonds after stages 1 and 2. Our bond valuation was simply a calculation of the net present value of the bond cash flows under continuous compounding. A small risk-free interest rate of 1% was used for accumulating cash from coupon payments and matured bonds.

Adding Corporate Bond Default In order to appropriately consider corporate bonds, we turned to developing a suitable model for incorporating bond default risk. Two main methods were considered:

1) Straight-to-default probability – Under this method, each bond had a certain probability of defaulting in every stage given its current rating and age. Each bond either defaulted or remained at its current rating.

2) Rating transition probability matrix – Here, each bond had a constant probability of changing ratings each period, as a function of its current rating and maturity. While the probability of transitioning directly to default was trivially small for investment grade bonds (such as AAA), default could happen through multiple rating downgrades over a number of periods. We concluded that this approach was more realistic because it captured not only default probabilities but also upgrade/downgrade possibilities.

Each method factored in a credit spread (see figure below) for corporate bonds to provide a risk premium to compensate for default and downgrade risk. This credit spread was in the

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form of a yield premium – the yield for bond was computed from its risk-free price, the yield premium was added, and the bond was re-priced at the higher yield.

 Figure 4 - Credit Spreads

To ensure that each method achieved an appropriate number of bond defaults, we selected a sample of bonds across the rating spectrum and ran a test over 10,000 scenarios. The charts below show the number of defaults or transitions for a typical bond of each rating. These appear to be realistic figures and indicate the adequacy of the scenario sample size of 10,000.

 Figure 5 - Bond Defaults (Straight-to-Default Method)

 

 Figure 6 - Bond Transitions

In the end, we proceeded with the rating transition matrix approach because it captured both the default and downgrade/upgrade dynamics.

Corporate Bond Model Assumptions Our corporate bond model relied on several simplifying assumptions. First, we assumed that credit ratings were issued only at the rating bucket level (“AAA”, “AA”, “A”, “BBB”) rather than at a rating notch level (e.g. A+, A, A-). This introduced additional pricing error in the first period, as bonds that were rated BBB+ and BBB-, for example, were combined into the same category. In the second period, however, all bonds were priced off of the simulated curve, so pricing complexity is reduced because fewer credit spread curves were required to price the whole range of corporate bonds.

0.0%  

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3.0%  

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5.0%  

6.0%  

7.0%  

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0   5   10   15   20   25   30  

Yield  

Years  to  Maturity  

BBB  

A  

AA  

AAA  

TSY  

AAA AA A BBBUpgrade 0 59 448 1015

Downgrade 1727 2027 1234 1054Defaults 1 8 11 134

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Our model assumed that credit spreads were deterministic over time and maturity. We used empirical data found in Duffee (1998) and Elton et al. (2001) to extrapolate “typical” or long-run average credit spread curves. These curves were derived primarily from credit spreads from 1985 to 1995. Therefore they were not entirely consistent with the empirical prices of Treasury bonds and corporate bonds used in our optimization, which were obtained from 2008. Finally, we assumed that credit spreads and default recovery rates were constant across sectors, instead of using different values for the utility, industrial, and finance sectors. Restricting analysis to one sector or rating category might allow for the most accurate calibration of the parameters.

Initial Results With 100 scenarios in each stage, totaling 10,000 overall scenarios, conducting an analysis of the optimized bond allocation in each scenario is untenable. However, to help us gain insight into the optimization, we keep track of mean returns, standard deviation of returns, and, most importantly, the correlation matrix. These statistics, which are traditionally used in mean-variance analysis, can still provide intuitive insights into the allocation even though stochastic programming aims to improve the shortcomings of the mean-variance framework.

Initial Results with US Treasury Bonds For a sample of 10 Treasury bonds with a mix of different maturities and coupon rates, the optimizer always allocates 100% of wealth into a long-term bond with the highest yield in the first stage, and 100% of wealth into a maturing (and thus risk-free) bond at the second decision stage. This concentration of allocation is relatively insensitive to adjustments in the utility function. While the lack of diversification is somewhat discouraging, a more careful analysis and economic intuition convince us that a concentrated portfolio is actually the right decision. It becomes obvious if one considers the correlation among Treasuries (see Figure 7 below). Since Treasuries were assumed to be risk-free, by the Law of One Price, the correlations among them should be very close to unity for all practical purposes. When assets are highly correlated, it makes sense not to diversify. In the mean-variance framework, extremely high correlations result in a singular covariance matrix, and there is no unique solution because every asset is just as good and as risky as any other asset. The consequence in the scenario-based framework is that the optimizer simply chooses the highest yield bond to maximize return in the first period and a risk-free bond in the second period to minimize risk.

Figure 7 - US Treasury Correlations (6 bonds of various lengths)

1 2 3 4 5 61 1 1 1 1 0.999 0.9992 1 1 1 1 1 0.9993 1 1 1 1 1 0.9994 1 1 1 1 1 15 0.999 1 1 1 1 16 0.999 0.999 0.999 1 1 1

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Initial Results with Corporate Bonds We divided corporate bonds into two segments, short-term bonds with maturities less than 5 years and long-term bonds with maturities between 25 and 30 years. For each maturity category, we ran optimizations on two sets of sample data, one with corporate bonds only and one with both corporate and Treasury bonds. The initial results were produced with two important assumptions – 1) Independent rating transition probabilities, and 2) deterministic credit spread for any given credit rating and maturity. One consistent result from both short-term sample and long-term sample is that none of the Treasury bonds receives any allocation. Additionally, when compared to the US Treasury results, there is much more diversification at the second stage. The diversification is a direct result of the lower correlations among corporate bonds (see Figure 8 below), which result from our assumption of independent rating transition probabilities.

TSY AAA AA A BBB

TSY 1.00 0.92 0.78 0.65 0.23 AAA 0.92 1.00 0.73 0.60 0.20 AA 0.78 0.73 1.00 0.48 0.15 A 0.65 0.60 0.48 1.00 0.46

BBB 0.23 0.20 0.15 0.46 1.00

Figure 8 - Corporate Bond Correlations

Initial Results: Short-term Corporate Bonds Due to the change from empirical pricing to model-based pricing in period 1, we expect the least amount of diversification at this stage. Certain bonds will outperform and others will underperform due solely to the switch. There is no way to precisely calibrate our model and all of the credit spread, due to the use of bonds from a variety of sectors and credit ratings. In the first stage, the optimizer allocates 100% of wealth to a single-A bond that matures within the first year. This is most likely due to the highly risk-averse utility function. In the second stage, the allocation is relatively diversified. For any given scenario, the wealth is allocated to between 1 to 4 bonds. Figure 9 below shows the average period 2 allocations over all 10,000 scenarios. The actual weights and the bonds chosen vary across the scenarios. Every corporate bond receives some allocation in certain scenarios. However, the optimizer seems to favor high-yield bonds (BBB) for most of the scenarios, most likely because the portfolio is already in the positive territory, thus it can afford to take on more risk.

Initial Results: Long-term Corporate Bonds A similar pattern can be observed for the optimization with only long-term bonds. In the first period, 100% of the wealth is allocated to a single-A bond with the highest coupon. The second period allocation still tends to favor high-yield bonds but the concentration is not as pronounced as in the case with short-term bonds.

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Corporate Only Treasury + Corporate

Rating Coupons Maturity Stage 1 Stage 2 Stage 1 Stage 2

Cor

pora

te

AAA 5.45 4 1% 1% AA 4.375 2 1% 0% AA 4.75 5 6% 4% AA 6.25 3m N/A N/A A 4.25 5 3% 5% A 6.339 7m 100% N/A 100% N/A A 7.25 2 0% 7%

BBB 6.45 3 17% 24% BBB 6.95 4 21% 17% BBB 6.15 5 24% 19% BBB 6.25 5 25% 23%

Trea

surie

s

TSY 3.875 5 0% TSY 4.25 5 0% TSY 4.5 6m 0% TSY 4.875 1 0% TSY 5 3 0% TSY 5 3 0% TSY 4.375 4 0%

Figure 9 - Short-term Corporate Optimization Summary

Corporate Only Treasury and Corporate

Rating Coupons Maturity Stage 1 Stage 2 Stage 1 Stage 2

Cor

pora

te

AAA 5.95 29 2% 8% AA 5.55 29 7% 5% A 5.375 27 11% 9% A 6.5 29 100% 2% 100% 3% A 6.125 30 6% 11%

BBB 6.65 28 14% 17% BBB 6.75 29 20% 18% BBB 6.8 30 19% 16% BBB 7 30 20% 14%

TSY

TSY 4.5 28 0% TSY 4.75 29 0% TSY 4.375 30 0%

Figure 10 - Long-term Corporate Optimization Summary

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Sensitivity Analysis After observing the absence of Treasury bonds in the optimized corporate/Treasury mixed portfolio, we experimented with different parameter values in the specifications of the utility function and the credit spreads. Our first hypothesis for the lack of allocation to Treasuries is that the utility function is not risk-averse enough, i.e. it does not penalize downside risk severely enough. As we increase the risk-aversion by increasing the slope of the first line segment in the piece-wise linear utility function, we still do not observe any allocation to Treasuries. Instead, the optimizer allocates to cash, even at a risk-free interest rate of as low as 0.01%. This is likely due to the relatively low Sharpe ratio of the Treasuries, as the empirical-to-simulated mismatch produces additional variance in bond returns. The second experiment involves increasing credit spreads across the board for all corporate bonds. This approach also failed to produce any allocation to Treasuries. The problem here also lies in the mismatch among empirical bond prices, empirical credit spreads, and simulated price data. In the real world, the bond prices, credit spreads, and default probabilities should all be correlated with one another. Artificially increasing the credit spreads distorts their relationship with the prices and default probabilities.

Model Alterations The assumption of deterministic credit spreads over time and across maturity skewed the optimized results towards an allocation comprised solely of corporate bonds. In the absence of bond defaults or rating transitions, the deterministic credit spreads resulted in a similar variance between corporate bonds and Treasury bonds. This caused corporate bonds to have a superior Sharpe ratio. Better diversification could be obtained by investing in other corporate bonds rather than US Treasuries. The assumption of independence of transitions and defaults also tilted the optimized portfolio toward corporate bonds because corporate bonds as an asset class have lower correlations. Our initial model failed to factor in the general assumption that default risk is generally driven by economic forces. Generally, corporate bonds have a higher likelihood of defaulting in an environment in which other corporate bonds have defaulted. In order to model a self-exciting process, we experiment with two approaches that introduce correlation in the direction of credit spreads and in the rating transition probabilities.

Correlated Credit Spread Approach With the correlated credit spread approach, we sample from an unbiased Bernoulli distribution in each period to decide whether spreads should widen or tighten in general. We then sample from another uniform distribution of a given range to determine how many basis points the bonds in each credit rating should widen or tighten. The exact range depends on the credit rating, with BBB bonds having the widest range. This approach attempts to introduce correlation in the general direction of credit spreads movement, while allowing for idiosyncratic fluctuations in the credit spreads for individual bonds.

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Correlated Transition Approach Our original rating transition approach uses an independent uniform random variable for each bond to model the probability of a rating transition. With the correlated credit spread approach, we add a single exponential random variable to all of the independent uniform random variables to model the correlation. The exponential random variable can be viewed as market risk (to model “flight-to-quality”) whereas the independent uniform can be viewed as idiosyncratic or company-specific risk. Exponential random variables have a high probability of taking on small values, which represent the low correlation among corporate bonds under normal economic conditions. In the few scenarios where the exponential random variable takes on large values, the correlation among corporate bonds increases. Treasury bonds become attractive when corporate bonds all lose value together. This is our attempt to model the “flight-to-quality” effect observed in the credit crisis.

Results with Correlation Introduced Both of the correlation approaches successfully result in allocation to Treasury bonds. The Correlated Transition approach results in a higher level of first-period diversification and a higher allocation to Treasury bonds and to cash in the second period. The parameters used in both models were based on heuristic assumptions with little empirical justification. Hence, the exact numerical results in Figure 11 should not be attached with too much significance. What we have demonstrated are merely two possible methods to model the self-exciting default process among corporate bonds. The exact model parameters, however, will still need further refinement from empirical research.

Correlated Transition Correlated Spread

Rating Coupons Maturity Stage 1 Stage 2 Stage 1 Stage 2

Cor

pora

te

AAA 5.95 29 13% 3% 4% AA 5.55 29 3% 5% A 5.375 27 63% 7% 14% A 6.5 29 24% 5% 100% 10% A 6.125 30 9% 8%

BBB 6.65 28 9% 12% BBB 6.75 29 15% 14% BBB 6.8 30 11% 12% BBB 7 30 13% 17%

TSY

TSY 4.5 28 6% 1% TSY 4.75 29 11% 1% TSY 4.375 30 2% 1%

Cas

h

6%

1%

Figure 11 - Results of Correlation Approaches

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Conclusions and Recommendations Our optimized portfolio proved to be highly sensitivity bond pricing and default probability assumptions. These in turn were influenced by interest rate models and credit spreads. A key issue we encountered was the mismatch between empirical and simulated prices. This turned out to be a major hurdle in producing a sensibly optimized portfolio. Further, diversification is beneficial from a risk perspective, but it is highly dependent on the risk-aversion factor used, the market credit spreads (which essentially convey the risk-aversion factor of the market), and the correlations among bonds. Finally, generating interest rate scenarios and optimizing bond performance came at high computational cost. Even after narrowing down our bond choices to only 10-15 bonds, running an full optimization over 10,000 scenarios took over 30 minutes (including scenario generation and bond pricing). While implementation of an appropriate solution algorithm (Benders decomposition, perhaps) could speed the actual optimization up significantly, much of the time was spent generating interest rate scenarios and pricing bonds, for which no quicker methods are immediately apparent. There are thousands of US corporate bonds available for purchase. Optimizing over the entire realm of portfolio possibilities is simply untenable. This requires strict selection criterion to narrow the bond list to a smaller sample before performing the optimization.

Questions for Future Research We have produced two frameworks for modeling the self-exciting process of corporate bond defaults into the stochastic optimization model. Further research will be needed to properly calibrate the model parameters for each of these approaches. There may also be value in combining the two approaches to allow for correlations in both credit spreads and rating transitions. Another potential research topic is the implementation of a more sophisticated interest rate models in order to reduce the model risk inherent in our approach. The CIR model was initially selected due to its simplicity and ability to produce inverted yield curves. A two-factor model might allow for additional interest rate shocks that better reflect the actual market environment. However, a key to understanding the usefulness of a different model will be assessing the impact it actually has on the optimized portfolio and its performance. An additional question is how the improvement in curve fitting affects optimized outcomes within a particular interest rate model. This will determine how much time and effort should be put into correct parameterization and the returns that can be expected from it. Finally, our return results are, as of yet, only theoretical. A true test of the value of bond portfolio optimization will be a historical back-test against the performance of bond index funds. For example, we could select a variety of bond index funds, pick a subset of bonds in each fund to optimize, and then compare the performance of our portfolio against that of the bond fund as a whole. The performance improvement, if any, will provide further insight into the value of the stochastic programming approach to bond portfolio optimization.

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Bibliography J.C. Cox, J.E. Ingersoll, and S.A. Ross. “A Theory of the Term Structure of Interest Rates." Econometrica. Vol. 53, Issue 2, 1985, pp. 385-407. A.J. Diaco. Fixed-Income Portfolio Construction Via Simulation and Stochastic Programming. PhD. Dissertation, 2008. Stanford University. G.R. Duffee. “The Relation Between Treasury Yields and Corporate Bond Yield Spreads.” The Journal of Finance. Vol. 53, Issue 6, 1998, pp. 2225-2241. E.J. Elton, M.J. Gruber, D. Agrawal, and C. Mann. “Explaining the Rate Spread on Corporate Bonds.” The Journal of Finance. Vol. 56, Issue 1, 2001, pp. 247-277. G. Infanger. Planning Under Uncertainty, Solving Large-Scale Stochastic Linear Programs. Boyd & Fraser Publishing Co., 1994. O. Korn, and C. Koziol. “Bond Portfolio Optimization: A Risk-Return Approach.” Centre for Financial Research, University of Cologne. Working Paper No. 06-03, 2006. Moody’s Global Credit Research. “Corporate Default and Recovery Rates, 1920-2007.” February 2008. Moody’s Global Credit Research. “Corporate Default and Recovery Rates, 1920-2009.” February 2010. Wharton Research Data Services (WRDS) was used in preparing this piece. Market prices of Treasury Bonds and Corporate Bonds were obtained from two of its constituent databases, the CRSP Daily US Treasury Database and the FINRA TRACE service. Credit ratings and other security-specific data were obtained from Moody’s Investors Services (www.moodys.com).