Book 1. Market Risk

FRM Part 2

MR 12. Arbitrage Pricing with Term Structure Models

Presented by: Sudhanshu

Module 1. Interest-Rate Trees and Risk-Neutral Pricing

Module 2. Binomial Trees

Module 3. Option-Adjusted Spread

Module 1.  Regression Hedging And Principal Component Analysis

Topic 1. DV01-Neutral Hedge

Topic 2. Regression Hedge

Topic 3. Hedge Adjustment Factor

Topic 4. Face amount of Offsetting Position

Topic 5. Two-Variable Regression Hedge

Topic 6. Level and Change Regressions

Topic 7. Reverse Regressions

Topic 8. Principal Component Analysis

Topic 1. DV01-Neutral Hedge

  • Definition: A standard DV01-neutral hedge assumes that the yield on a bond and the yield on a hedging instrument will rise and fall by the exact same number of basis points.

  • Drawbacks:

    • Unrealistic Assumption: A one-to-one relationship between yield changes does not always exist in practice.

    • Imprecision: For example, hedging a T-bond (nominal yield) with a Treasury Inflation-Protected Security (TIPS) (real yield) will likely be imprecise due to differing yield behaviors.

    • Empirical Observation: Empirically, nominal yields often adjust by more than one basis point for every basis point adjustment in real yields, indicating a dispersion.

    • Limited Scope: DV01-style metrics and hedges primarily focus on how rates change relative to one another, which may not capture all real-world complexities.

Practice Questions: Q1

Q1. If a trader is creating a fixed-income hedge, which hedging methodology would be least effective if the trader is concerned about the dispersion of the change in the nominal yield for a particular change in the real yield?

A. One-variable regression hedge.

B. DV01 hedge.

C. Two-variable regression hedge.

D. Principal components hedge.

Practice Questions: Q1 Answer

Explanation:  B is correct.

The DV01 hedge assumes that the yield on the bond and the assumed hedging instruments rises and falls by the same number of basis points; so with a DV01 hedge, there is not much the trader can do to allow for dispersion between nominal and real yields.

Topic 2.  Regression Hedge 

  • Concept: Regression hedging enhances DV01-style hedges by examining and adjusting for historical yield changes over time.

  • Improvement over DV01-Neutral Hedge:

    • It accounts for projected nominal yield changes compared to projected real yield changes.

    • Utilizes least squares regression analysis to analyze the historical relationship between real and nominal yields

    • Provides an estimate of a hedged portfolio's volatility, allowing investors to gauge expected gains and assess attractiveness.

Practice Questions: Q3

Q3. What is a key advantage of using a regression hedge to fine tune a DV01 hedge?

A. It assumes that term structure changes are driven by one factor.

B. The proper hedge amount may be computed for any assumed change in the term structure.

C. Bond price changes and returns can be estimated with proper measures of price sensitivity.

D. It gives an estimate of the hedged portfolio’s volatility over time.

Practice Questions: Q3 Answer

Explanation:  D is correct.

A key advantage of using a regression approach in setting up a hedge is that it automatically gives an estimate of the hedged portfolio’s volatility.

Topic 3. Hedge Adjustment Factor

  • Purpose: To profit from a hedge, variability in the spread between real and nominal yields over time must be assumed.

  • Methodology: Least squares regression is used to analyze these changes and determine the alpha and beta coefficients.

  • Regression Equation: 

    •           : Changes in the nominal yield

    •           : Changes in the real yield

    •           : Intercept term

    •           : Slope of the data plot (hedge adjustment factor)

  • Interpretation of Beta: If the yield beta is, for example, 1.0198, it means the nominal yield increases by 1.0198 basis points for every basis point increase in real yields over the sample period.

\Delta y_t^N=\alpha+\beta \Delta y_t^R+\epsilon_t
\Delta y_t^N
\beta
\alpha
\Delta y_t^R

Topic 4.  Face Amount of Offsetting Position

  • Adjusted DV01 Hedge Formula: The DV01 hedge is adjusted by the hedge adjustment factor (beta) to determine the face amount of the real yield bond (FR) needed to hedge a nominal bond (FN).

​

  •               : Face amount of the real yield bond

  •               : Face amount of the nominal bond

  •               : DV01 of the real yield bond

  •               : DV01 of the nominal bond

  • Considerations: This accounts for both the size of the underlying instrument and differences between nominal and real yields over time.

    • The regression hedge assumes the slope coefficient (beta) is constant over time, which may not always be true, so it's advisable to estimate beta over different periods.

    • Other factors to consider include the R-squared (coefficient of determination) and the standard error of the regression (SER).

F^R=F^N \times\left(\frac{D V 01^N}{D V 01^R}\right) \times \beta
F^R
F^N
D V 01^R
D V 01^N

Practice Questions: Q2

Q2. Assume that a trader is making a relative value trade, selling a U.S. Treasury bond and correspondingly purchasing a U.S. TIPS. Based on the current spread between the two securities, the trader shorts $100 million of the nominal bond and purchases $89.8 million of TIPS. The trader then starts to question the amount of the hedge due to changes in yields on TIPS in relation to nominal bonds. He runs a regression and determines from the output that the nominal yield changes by 1.0274 basis points per basis point change in the real yield. Would the trader adjust the hedge, and if so, by how much?

A. No.

B. Yes, by $2.46 million (purchase additional TIPS).

C. Yes, by $2.5 million (sell a portion of the TIPS).

D. Yes, by $2.11 million (purchase additional TIPS).

Practice Questions: Q2 Answer

Explanation:  B is correct.

The trader would need to adjust the hedge as follows:
$89.8 million × 1.0274 = $92.26 million
Thus, the trader needs to purchase additional TIPS worth $2.46 million.

\alpha

Practice Questions: Q4

Q4. What does the regression hedge assume about the slope coefficient?

A. It moves in lockstep with real rates.

B. It stays constant over time.

C. It generally tracks nominal rates over time.

D. It is volatile over time, similar to both real and nominal rates.

Practice Questions: Q4 Answer

Explanation:  B is correct.

It should be pointed out that while it is true that the regression hedge assumes a constant beta, this is not a realistic assumption; thus, it is best to estimate beta over several time periods and compare accordingly.

Topic 5.  Two-Variable Regression Hedge

  • Application: Regression hedging can be extended to include two independent variables.

  • Scenario: Hedging an illiquid 20-year swap position by selling a combination of more liquid 10- and 30-year swaps.

  • Regression Equation: Approximates the relationship between changes in the target swap rate and changes in the hedging instrument swap rates.

     

    •                   : Change in 20-year swap rate

    •                   : Change in 10-year swap rate

    •                   : Change in 30-year swap rate

    •                   : Risk weights (beta coefficients)

  • Hedge Application: The beta coefficients indicate the proportion of the target DV01 to be hedged with each independent variable. For example, a beta of 0.2221 for a 10-year swap means hedging 22.21% of the 20-year swap DV01 with the 10-year swap.

  • Advantages: Provides a better hedge (higher R-squared) compared to a single-variable approach.

\Delta y_t^{20}=\alpha+\beta^{10} \Delta y_t^{10}+\beta^{30} \Delta y_t^{30}+\epsilon_t
\Delta y_t^{20}
\Delta y_t^{10}
\Delta y_t^{30}
\beta^{10},\beta^{30}

Topic 6. Level and Change Regressions

  • Two Schools of Thought:

    • Change-on-Change Regression: Regressing changes in yields on changes in yields.

      • Δyt​=yt​−yt−1​ 

      • Δxt​=xt​−xt−1​ 

    • Level-on-Level Regression: Regressing yields on yields.  yt​=α+βxt​+ϵt​

  • Comparison of Error Terms:

    • In both approaches, the estimated regression coefficients are unbiased and consistent.

    • However, the error terms (ϵt​) are unlikely to be independent of each other (i.e., they are serially correlated).

    • This serial correlation means the estimated regression coefficients are not efficient.

  • Alternative Model for Error Terms: ϵt​=ρϵt−1​+vt​ 

    • This model assumes today's error term is a function of yesterday's error term plus a new random fluctuation.

\Delta y_t=\alpha+\beta \Delta x_t+\Delta \epsilon_t

Practice Questions: Q5

Q5. Traci York, FRM, is setting up a regression-based hedge and is trying to decide between a changes- in-yields-on-changes-in-yields approach versus a yields-on-yields approach. Which of the following is a correct statement concerning error terms in these two approaches?

A. In both cases, the error terms are completely uncorrelated.
B. With change-on-change, there is no correlation in error terms, while yield-on-yield error terms are completely correlated.

C. Error terms are correlated over time with both approaches.

D. With yield-on-yield, there is no correlation in error terms, while change-on-change error terms are completely correlated.

Practice Questions: Q5 Answer

Explanation: C is correct.

With the level-on-level approach, error terms are somewhat correlated over time, while with the change-on-change approach, the error terms are completely correlated. Thus, error terms are correlated over time with both approaches.

Topic 7. Reverse Regressions

  • Definition: In a reverse regression, the dependent and independent variables are switched compared to a standard regression hedge.

  • Example:

    • Regression Hedge: Regresses the yield change of a target bond (e.g., XYZ corporate bond) on the yield change of a hedging instrument (e.g., T-bond).

    • Reverse Regression: Regresses changes in the yield of the T-bond on changes in the yield of the XYZ corporate bond.

  • Key Differences: Hedge Ratio and Risk Weight: Lead to different calculated hedge amounts.

    • Volatility of Hedged Position: The standard deviation of the hedged position will differ between the two approaches.

  • Choice of Hedge: The selection depends on the trader's objectives.

    • Regression Hedge: Ideal for minimizing the variance of hedging the target bond position.

    • Reverse Regression: Preferred if the priority is managing Treasury exposure while maintaining a fixed Treasury position.

  • Importance: Defining clear goals is crucial before selecting a hedging strategy to align with risk management and portfolio objectives

Topic 8. Principal Component Analysis

  • Purpose: Provides a single empirical description of term structure behavior applicable across all bonds, contrasting with regression analysis's focus on a small number of bonds.

  • Core Idea: Attempts to explain all factor exposures using a small number of uncorrelated exposures that effectively capture risk.

  • How it Works (Example with Swap Rates):

    • For 30 annual swap rates, PCA creates 30 interest rate factors or components, each describing a change in one of the rates.

    • Unlike regression, PCA does not directly look at pairwise correlations but rather focuses on deriving uncorrelated components.

  • Properties of Principal Components (PCs):

    1. The sum of the variances of the PCs equals the sum of the variances of the individual rates, capturing the overall volatility.

    2. The PCs are not correlated with each other (uncorrelated).

    3. Each PC is chosen to contain the highest possible variance, given the previously chosen PCs.

  • Advantage: A small number of PCs (e.g., the first three) can adequately approximate the sum of variances of all rates, significantly simplifying the description of term structure movements.

    • Changes in 30 rates can be expressed with changes in just three factors, leading to a much simpler approach for constructing hedging portfolios.