Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and Optimisation Modelling *sponsored by EPSRC and OptiRisk

Outline 1. Motivation & insights 2. Problem setting 3. Novel features 4. Asset model 5. Computational study

1. Motivation & Insights Credit risk – risk that a counterparty’s creditworthiness changes, in particular that of default on financial obligations. An important consideration for investors. An important distinction Rogers (1999) notes “the first and most important thing to realise about modelling credit risk is that we may be trying to answer questions of two different types”; Pricing credit risky assets and quantifying their risk exposure. Appropriate integration of significant risks. - Market risk in particular, interest rate risk. - Credit risk.

1. Motivation & Insights Two main categories credit risk model - Structural models. Merton (1974), Option theoretic, Debt contingent claim on firm’s assets. - Reduced form models. Jump processes.

2. Problem Setting - Jarrow, Lando & Turnbull (1997) (JLT) Credit migration process described by Markov chain. Fractional recovery on default; η of par value. Interest rate process. - Any process can be used to represent the default-free rate. Connecting risk neutral measure, P with physical world measure, leading to… Arbitrage-free pricing. - Prices of defaultable bonds are the expected values under P, discounted at the default-free rate.

2. Problem Setting - Thomas, Allen & Morkel-Kingsbury (2002) Adapt JLT framework. Markov chain which represents the economy; “Regime-switching model”. Price defaultable bonds.

3. Problem Setting – Our Work Pricing. - Price a portfolio of defaultable bonds. - Similar model to Thomas et al. Risk quantification. - Extend the model to the physical measure to simulate the bond portfolio value. - Calculate Value at Risk (VaR) and Conditional Value at Risk (CVaR) one year ahead.

3. Novel Features Yield Curve Modelling; no arbitrage across interest rates. Bond stripping: Discovering of yield and credit spreads using quadratic programming (QP1), - constraints remove price anomalies. Calibration of the risk neutral credit migration process using quadratic programming (QP2), - constraints remove negative probabilities.

4. Asset Model Economy Process C t E Rating Migration Process C t R Interest Rate Process C t I 0 - Treasury 1 – Highest rated bonds M-1 Worst rated bonds M Default - absorbing state.

4. Asset Models – Interest Rate Process Thomas et al. model spot rate without restrictions, on process for arbitrage. Whole yield curve should be modelled, – Heath, Jarrow & Morton (1992). Interest Rates are functions of two time dimensions, - current time t, - time to maturity T.

4. Asset Models – Interest Rate Process

Model an auxiliary binomial process to describe underlying interest rate process. Figure 1: Possible evolution of state space tree over t = {0,1,2,3} S 0 ={0} S 1 ={u, d}S 2 ={uu, ud, du, dd} S 3 ={uuu, …}

4. Asset Models – Interest Rate Process Forward Rate Process

4. Asset Models – Interest Rate Process Treasury zero coupon bond prices process and forward rate process are equivalent. Zero coupon bond price process Absence of arbitrage can be understood as

4. Asset Models – Interest Rate Process Continuously compounding forward rate Continuous time process described by SDE Defining, the discrete process converges to the continuous.

4. Asset Models – Interest Rate Process Assuming No arbitrage condition becomes Drift of process under the risk neutral measure is well defined, given the observed volatility.

4. Asset Models – Risky bond price The price of a credit risky bond is, discounted expected value under the risk neutral measure, where,

5. Computational Study Part 1 – Bond stripping; Discovering of yield and credit spreads. Part 2 – Model calibration. Part 3 – Simulation of bond portfolio value so determining its distribution.

5. Computational Study Bond Stripping - Motivation Bonds priced using zero coupon bond prices. - Any bond’s price is determined using zero coupon bonds prices as discount factors. - Models for term structure describe zero coupon bond prices. Market Data Available. - Mainly coupon bonds available in market. - Model calibration involves stripping coupons from coupon bonds to derive underlying zero coupon bond prices. Jarrow et al. - Bucket bonds by rating and maturity and find average prices and coupons for each bucket. - Solve system triangular equations to get zero coupon prices. - Can lead to mispricing.

5. Computational Study Bond Stripping - Motivation Allen, Thomas and Zheng - Use linear programming to find zero coupon bond prices. - Minimise absolute pricing errors. - Constraints are used to remove anomalies =>mispricing. Our Work - Use same formulation but with quadratic programming so minimising squared pricing errors. - Equivalent to constrained regression.

5. Computational Study Bond Stripping - Formulation Credit ratings {0, 1,…, M-1} Time set { 0,…,T} N coupon bonds - where bond b, 1  b  N, - current market price v b, - rating r(b), - pays cash flow c b (t) at time t. Zero coupon bond prices - z k (t) price of a zero coupon bond, with rating k, which pays 1 at time t.

5. Computational Study Bond Stripping - Formulation for bond b let o b be the pricing error ‘over’ let u b be the pricing error ‘under’ Minimise  b (o b +u b ) 2 (1) subject to v b + o b =  t c b (t) z r(b) (t) + u b  b  {1,…,N}(2) z 0 (t)  z 0 (t+1) (1+ m(t))  t  {0,…,T}(3) z k (t+1) - z k+1 (t+1)  z k (t) - z k+1 (t)  t  {0,…,T}  k  {0,…,M-1} (4) m(t) – minimum discount rate over (t,t+1].

5. Computational Study Bond Stripping

5. Computational Study Bond Stripping – Allowing Mispricing If we adapt (4) we allow some types of mispricing of zero coupon bonds. Minimise  b (o b +u b ) 2 (1) subject to v b + o b =  t c b (t) z r(b) (t) + u b  b  {1,…,N}(2) z 0 (t)  z 0 (t+1) (1+ m(t))  t  {0,…,T}  k  {0,…,M-1} (3) z 0 (t+1) - z k+1 (t+1)  z 0 (t) - z k+1 (t)  t  {0,…,T}  k  {0,…,M-1} (4)

5. Computational Study Bond Stripping – Allowing Mispricing

5. Computational Study - Model Calibration Each chain -Determine relevant states. -Determine physical world and risk neutral measures. Economy - Classify years as good or bad, using observed transitions to lower ratings and default. - Physical probabilities as observed frequencies of transitions. - Risk neutral measure assumed to be same as real world.

5. Computational Study - Model Calibration Interest Rate Process - From observations of daily yield curve, we derive observations of volatility. - Fit to functional form, to give our volatility function. Vasicek model. - Risk neutral drift is then well defined by no arbitrage condition.

5. Computational Study - Model Calibration Interest Rate Process - The states of the processes are known, given Forward rates/ zero coupon bonds processes. - Risk neutral probabilities are ½ by construction. - Physical probabilities are derived from the observed drift of the yield curve.

5. Computational Study – Model Calibration Credit Rating Migrations - States are well defined; ratings classes and default state. - Historical Standard & Poors default frequency data for physical world measure. - Risk neutral measure backed out from zero coupon bond prices.

5. Computational Study – Model Calibration Credit Rating Migrations - physical world measure - risk neutral / pricing measure - Similar expressions for when the economy is in state B

5. Computational Study – Model Calibration Assuming the economy does not change, pricing equation is. So given the zero coupon bond prices we are able to derive the implied probabilities of default under the risk neutral measure,, that is risk neutral probabilities over (0, T] are known. Credit Rating Migrations

5. Computational Study – Model Calibration Risk neutral probabilities over (0, T 0 ] can be found from zero coupon bond prices for bond maturing at time T 0. Risk neutral probabilities over (0, T n ] can be found from zero coupon bond prices for bond maturing at time T n. As process is Markov, if probabilities over (0, T n -1 ] and probabilities over (0, T n ] are known we can derive probabilities over (T n -1, T n ]. We are able to derive risk neutral probabilities over all timeperiods, recursively. Credit Rating Migrations

5. Computational Study – Model Calibration Negative values for the probabilities possible. -Nothing in structure of recursive equations solved that ensure probabilities take sensible values. Solve set of recursive QPs with restrictions on values of probabilities. - Initial Credit Fit – derives probabilities over (0,T 0 ] given zero coupon bond prices maturing at T 0 - Recursive Credit Fit – derives probabilities over (T n-1,T n ], given zero coupon bond prices maturing at T n and probabilities over (0,T n-1 ] Use prices at more than one date. - Incorporates more information into risk neutral measure that is derived. Credit Rating Migrations

5. Computational Study – Model Calibration Initial Credit Fit – derives probabilities over (0,T 0 ]

5. Computational Study – Model Calibration Recursive Credit Fit – derives probabilities over (T n-1,T n ]

5. Computational Study- Simulation of VaR & CVaR

5. Computational Study- Simulation of VaR & CVaR RatingMaturity Expected Return 95% VaR Loss 95% CVaR Loss Treasury7 years4.29%1.00%2.77% AA7 years4.52%2.12%4.31% BBB7 years6.95%14.16%27.77%

Questions ?

5. QP vs LP Efficient Frontier Allen, Thomas and Zheng – LP Formulation - Objective: Minimise  b (o b +u b )