Book 1. Market Risk

FRM Part 2

MR 5. VaR Mapping

Presented by: Sudhanshu

Module 1.  VaR Mapping

Module 2.  Mapping Fixed-Income Securities

Module 3. Stress Testing, Perfomance Benchmarks, And Mapping Derivatives

Module 1.  VaR Mapping

Topic 1. Introduction to VaR Mapping

Topic 2. Principles of VaR Mapping

Topic 3. Mapping Process

Topic 4. Capturing General and Specific Risk in Mapping Process

Topic 5. Calculating General and Specific Risk from Primitive Risk Factors

Topic 1. Introduction to VaR Mapping

• VaR mapping involves replacing the current values of a portfolio with risk factor exposures.

• Purpose: It simplifies risk management for complex, multi-asset portfolios by separating the risk into identifiable risk factors.

• Process Overview:

  • Measure all current positions within a portfolio.

  • Map these positions to common risk factors (e.g., changes in interest rates, equity prices).

  • This reduction in variables simplifies the risk management process, especially for large portfolios where managing individual positions is difficult and time-consuming.

• Benefits:

  • Assists risk managers in evaluating positions with changing characteristics, such as fixed-income securities.

  • Provides an effective way to manage risk when sufficient historical data for an investment (e.g., an IPO) is unavailable. In such cases, the focus shifts to evaluating risk factors likely to impact the portfolio's risk profile.

Topic 2. Principles for VaR Mapping 

• The principles for VaR risk mapping are summarized as follows:

• Aggregates Risk Exposure: VaR mapping aggregates risk exposure when considering each position separately is impractical due to the high number of computations needed.

• Simplifies Risk Exposures: It simplifies risk exposures into primitive risk factors. For instance, thousands of positions linked to a specific exchange rate can be summarized with one aggregate risk factor.

• Differs from Pricing Methods: VaR risk measurements can differ from pricing methods where prices cannot be aggregated; however, aggregating a number of positions to one risk factor is acceptable for risk measurement purposes.

• Measures Changes Over Time: VaR mapping is useful for measuring changes over time, such as with bonds or options, where risk exposure can be mapped to spot yields reflecting the current position as bonds mature.

• Useful When Historical Data is Unavailable: It is useful when historical data is not available.

Topic 3. Mapping Process

• The VaR mapping process involves several steps:

• Identify Common Risk Factors: The first step is to identify common risk factors for different investment positions.

  • Illustration (Figure 5.1 - Conceptual): Market values (MVs) of each position or investment are matched to identified common risk factors.

• Construct Risk Factor Distributions: The risk manager constructs risk factor distributions and inputs all data into the risk model.

  • Illustration (Figure 5.2 - Conceptual): Market value of each position (MVi​) is allocated to specific risk exposures (xij​) related to different risk factors.

• Aggregate Risk Exposures: Summing the risk factors in each column creates a vector consisting of aggregated risk exposures.

Topic 4. Capturing General and Specific Risk in Mapping Process

• General Risk Factors (Primitive Risk Factors): These are broad market risks that affect a wide range of assets. The number of general risk factors chosen impacts the model's complexity and accuracy.

• Specific Risks (Residual or Asset-Specific Risks): These arise from unsystematic or asset-specific risks of various positions in the portfolio.

• Relationship between General and Specific Risk: The more precisely general risk is defined (i.e., by adding more risk factors), the smaller the specific risk becomes.

  • Example (Bonds): If duration is the only risk factor for a bond portfolio, there will be significant specific risk due to differences in ratings, terms, and currencies. Adding credit risk and currency risk as additional factors would reduce specific risk.

Topic 5. Calculating General and Specific Risk from Primitive Risk Factors

• For an equity portfolio with N stocks mapped to a market index (primitive risk factor):

• Risk Exposure (βi​): Computed by regressing the return of stock 'i' on the market index return:

  • ϵi​ represents specific risk and is assumed to be uncorrelated with other stocks or the market portfolio.

• Portfolio Return (Rp​):

   

• Aggregated Risk Exposure (βp​):

   

• Decomposition of Portfolio Return Variance (V(Rp​)):

​

• General Market Risk:

• Specific Risk: ​ 

Practice Questions: Q1

Q1. Which of the following could be considered a general risk factor?

I. Exchange rates.

II. Zero-coupon bonds.

A. I only.

B. II only.

C. Both I and II.

D. Neither I nor II.

Practice Questions: Q1 Answer

Explanation:  B is correct.

Bootstrapping from historical simulation involves repeated sampling with replacement. The 5% VaR is recorded from each sample draw. The average of the VaRs from all the draws is the VaR estimate. The bootstrapping procedure does not involve filtering the data or weighting observations. Note that the VaR from the original data set is not used in the analysis.

Module 2.  Mapping Fixed-Income Securities

Topic 1. Three Methods for Mapping Portfolios of Fixed-Income Securities

Topic 2. Mapping Portfolios of Fixed-Income Securities: An Example

Topic 3. Principle Mapping

Topic 4.  Duration Mapping

Topic 5. Cash Flow Mapping

Topic 1. Three Methods for Mapping Portfolios of Fixed-Income Securities

• Once general risk factors are selected, the portfolio is mapped onto these factors using one of three methods for fixed-income securities:

• Principal Mapping: Concept: Considers only the risk of repayment of principal amounts.

  • Process: Uses the average maturity of the portfolio. VaR is calculated using the risk level from a zero-coupon bond that matches this average maturity.

  • Simplicity: It is the simplest of the three approaches.

• Duration Mapping: Concept: The bond's risk is mapped to a zero-coupon bond with the same duration.

  • Process: VaR is calculated using the risk level of the zero-coupon bond that equals the portfolio's duration.

  • Challenge: It may be difficult to find a zero-coupon bond that exactly matches the portfolio's duration.

• Cash Flow Mapping: Concept: The bond's risk is decomposed into the risk of each of its cash flows.

  • Precision: This is the most precise method.

Practice Questions: Q1

Q1. Which of the following methods is not one of the three approaches for mapping a portfolio of fifixed-income securities onto risk factors?

A. Principal mapping.

B. Duration mapping.

C. Cash flow mapping.

D. Present value mapping.

Practice Questions: Q1 Answer

Explanation:  D is correct.

Present value mapping is not one of the approaches

Topic 2.  Mapping Portfolios of Fixed-Income Securities: An Example

Let's illustrate the three mapping methods using a portfolio of two par value bonds:

• Bond 1: One-year $100 million bond with a 3.5% coupon rate.

• Bond 2: Five-year $100 million bond with a 5% coupon rate.

• Total Portfolio Value: $200 million.

VaR Percentages for Zero-Coupon Bonds (95% Confidence Level):

The primary difference among these techniques lies in how they consider the timing and amount of cash flows.

Topic 3.  Principle Mapping 

• Simplest Method: Only considers the timing of redemption or maturity payments, ignoring coupon payments.

• Calculation:

  • Weights: Both bonds have a 50% weight ($100 million / $200 million).

  • Weighted Average Life: $\left[0.50\right(1)+0.50(5)]=3$ years.

  • Assumption: The entire portfolio value of $200 million occurs at the average life of three years.

  • VaR Percentage: For a three-year zero-coupon bond, the VaR percentage is 1.4841%.

  • Principal Mapping VaR: $200 million $\times$ 1.4841% = $2.968 million.

Practice Questions: Q2

Q2. If portfolio assets are perfectly correlated, portfolio VaR will equal:

A. marginal VaR.

B. component VaR.

C. undiversified VaR.

D. diversified VaR.

Practice Questions: Q2 Answer

Explanation:  C is correct.

If we assume perfect correlation among assets, VaR would be equal to undiversified VaR.

Practice Questions: Q3

Q3. The VaR percentages at the 95% confidence level for a bond with maturities ranging from one year to five years are as follows:

A bond portfolio consists of a $100 million bond maturing in two years and a $100 million bond maturing in four years. What is the VaR of this bond portfolio using the principal VaR mapping method?

A. $1.484 million.

B. $1.974 million.

C. $2.769 million.

D. $2.968 million.

Practice Questions: Q3 Answer

Explanation: D is correct.

The VaR percentage is 1.4841 for a three-year zero-coupon bond [(2 + 4) / 2 = 3]. We compute the VaR under the principal method by multiplying the VaR
percentage times the market value of the average life of the bond: principal
mapping VaR = $200 million × 1.4841% = $2.968 million.

Topic 4.  Duration Mapping

• Concept: Replaces the portfolio with a zero-coupon bond having the same maturity as the portfolio's duration.

• Calculation (Macaulay Duration):

  • Numerator: Sum of (time $\times$ present value of cash flows)

  • Denominator: Present value of all cash flows.

  • Example Calculation: For the two-bond portfolio, Macaulay Duration = $553.69 million / $200 million = 2.768 years.

• Interpolating VaR:

  • VaR for a two-year zero-coupon bond: 0.9868%.

  • VaR for a three-year zero-coupon bond: 1.4841%.

  • Interpolated VaR for 2.768 years:

    $0.9868+(1.4841-0.9868)\times (2.768-2)=0.9868+(0.4973\times 0.768)=1.3687\%$.

• Duration Mapping VaR: $200 million $\times$ 1.3687% = $2.737 million.

Topic 5. Cash Flow Mapping

• Concept: Decomposes the bond's risk into the risk of each cash flow, mapping present values to zero-coupon bonds of the same maturities and including inter-maturity correlations.

• Calculations: Present value of cash flows (PV(CF)) are determined.

  • PV(CF) are multiplied by the corresponding zero-coupon VaR percentages.

• Undiversified VaR: If all five zero-coupon bonds were perfectly correlated, the undiversified VaR is the sum of the absolute values of (present value of cash flows $\times$ VaR percentage).

  • For the example, Undiversified VaR = 2.674 (sum of the '$x\times v$' column in Figure 5.5).

• Diversified VaR: Requires incorporating the correlations between the zero-coupon bonds using a correlation matrix (R).

  • Computed using matrix algebra:
    ​ where 'x' is the present value of cash flows vector, 'V' is the vector of VaR for zero-coupon bond returns, and 'R' is the correlation matrix.

  • For the example, Diversified VaR (square root of 6.840) =

    2.615.

Module 3. Stress Testing, Perfomance Benchmarks, And Mapping Derivatives

Topic 1. Supporting Stress Testing with Risk Factors Mapping

Topic 2. VaR Application Relative to a Performance Benchmark

Topic 3. Benchmarking Process: Matching Duration with ZCBs

Topic 4. Benchmarking Process: Absolute VaR

Topic 5. Benchmarking Process: Tracking Error

Topic 6. Mapping Approaches for Linear Derivatives

Topic 7. Mapping Approach for Forward Contracts: Delta-Normal Method

Topic 8. Mapping Approach for Forward Contracts: Example Portfolio

Topic 9. Mapping Approach for Forward Contracts: Diversified and Undiversified VaR

Topic 10. Forward Rate Agreements (FRAs): Example Portfolio

Topic 11. Forward Rate Agreements (FRAs): Diversified and Undiversified VaR

Topic 12. Interest Rate Swaps

Topic 13. Mapping Approaches for Nonlinear Derivatives

Topic 1. Supporting Stress Testing with Risk Factors Mapping

• Perfect Correlation Scenario: If perfect correlation among maturities of zero-coupon bonds is assumed, portfolio VaR equals undiversified VaR (sum of individual VaRs).

• Stress Testing Approach:

  • Instead of directly calculating undiversified VaR, each zero-coupon value can be reduced by its respective VaR, and the portfolio revalued.

  • The difference between the revalued portfolio and the original portfolio value should equal the undiversified VaR.

  • This is a simpler approach than incorporating correlations but is only viable if correlations are perfect (i.e., 1).

• Example (Two-Bond Portfolio):

  • Stress testing involves calculating VaR-adjusted present value factors for each zero-coupon bond based on its VaR percentage.

  • The portfolio's cash flows are then valued using these adjusted factors.

  • The change in portfolio value from stress testing (e.g., $2.67 for the example portfolio) is equivalent to the previously computed undiversified VaR.

Topic 2. VaR Application Relative to a Performance Benchmark

• Benchmarking a Portfolio: This involves measuring VaR relative to a benchmark portfolio.

• Purpose: Portfolios can be constructed to match the risk factors of a benchmark but have a higher or lower VaR.

• Tracking Error VaR:

  • The VaR of the deviation between the target portfolio and the benchmark portfolio.

  • It measures the difference between the VaR of the target portfolio and the benchmark portfolio.

Topic 3. Benchmarking Process: Matching Duration with ZCBs

• First Step: Match the duration of the portfolio with two zero-coupon bonds.

• Adjustment: The market value weights of the zero-coupon bonds are adjusted to match the benchmark portfolio's duration.

• Example: Creating five two-bond portfolios (e.g., Portfolio A: 23% in 4-year ZCB, 77% in 5-year ZCB) that each have a duration of 4.77, equivalent to the benchmark duration.

Topic 4. Benchmarking Process: Absolute VaR

• First Step: Match the duration of the portfolio with two zero-coupon bonds.
  

• Adjustment: The market value weights of the zero-coupon bonds are adjusted to match the benchmark portfolio's duration.
  

• Example: Creating five two-bond portfolios (e.g., Portfolio A: 23% in 4-year ZCB, 77% in 5-year ZCB) that each have a duration of 4.77, equivalent to the benchmark duration.​
  

Topic 5. Benchmarking Process: Tracking Error

• Definition: The tracking error, or difference between the VaR for the benchmark and zero-bond portfolios, is primarily due to nonparallel shifts in the term structure of interest rates.

• Calculation (Conceptual):                                                             ​                  where 'x' is the vector of market value positions for each zero-coupon bond portfolio and ' x0​' is the vector of market value positions of the benchmark.

• Minimizing Tracking Error: The smallest tracking error often occurs when the cash flows of the zero-coupon bond portfolio are most closely aligned with the benchmark's largest market weights.

• Absolute VaR vs. Tracking Error: Minimizing absolute VaR is not the same as minimizing tracking error. A portfolio with the lowest absolute VaR can still have the highest tracking error to the index.

• Variance Improvement: Tracking error can be used to compute variance reduction:

• Example: For Portfolio C, variance improvement =  

Topic 6. Mapping Approaches for Linear Derivatives

• Delta-Normal Method: Provides accurate estimates of VaR for portfolios and assets that can be expressed as linear combinations of normally distributed risk factors.

  Process: Once expressed as a linear combination of risk factors, a covariance (correlation) matrix can be generated, and VaR can be measured using matrix multiplication.

  Applicability: This method is suitable for instruments like forwards, forward rate agreements, and interest rate swaps, as their values are linear combinations of a few general risk factors with readily available volatility and correlation data.

Topic 7. Mapping Approach for Forward Contracts: Delta-Normal Method

• Current Value of a Forward Contract: Equal to the present value of the difference between the current forward rate (Ft​) and the locked-in delivery rate (K):  

   

• Analogous Risk Positions: A forward position, such as purchasing euros with U.S. dollars, is analogous to three separate risk positions:

  1. A short position in a U.S. Treasury bill.

  2. A long position in a one-year euro bill.

  3. A long position in the euro spot market.

Topic 8. Mapping Approach for Forward Contracts: Example Portfolio

• Example: Computing the diversified VaR of a forward contract used to purchase $100 million euros with $126.5 million U.S. dollars one year from now.

• Positions: A long position in a EUR contract ($122.911 million today) and a short position in a one-year U.S. T-bill ($122.911 million today).

• Calculation Inputs: Requires pricing information for the forward contract and the correlation matrix between the positions. The absolute present value of cash flows is multiplied by the VaR percentage.

Topic 9. Mapping Approach for Forward Contracts: Diversified and Undiversified VaR

• Undiversified VaR: For the forward contract example, the undiversified VaR is $6.01 million. This is computed as the sum of the absolute present values of cash flows multiplied by their respective VaR percentages.

• Diversified VaR: For the forward contract example, the diversified VaR is $5.588 million. This is computed using matrix algebra, multiplying the vector of absolute values by the correlation matrix.

Topic 10. Forward Rate Agreements (FRAs): Example Portfolio

• Concept: The general procedure for forwards also applies to FRAs.

• Example: Selling a 6x12 FRA on $100 million. This is equivalent to borrowing $100 million for 6 months (180 days) and investing at the 12-month rate (360 days).

• Present Values of Cash Flows: If the 360-day spot rate is 4.5% and the 180-day spot rate is 4.1%, the present value of the notional $100 million contract is x = $100 / 1.0205 = $97.991 million.

• Forward Rate: Computed as (1+F1,2​/2)=[1.045/(1+0.041/2)]=[(1.045/1.0205)−1]×2=4.8%

Topic 11. Forward Rate Agreements (FRAs): Diversified and Undiversified VaR

• Undiversified VaR: Computed by multiplying the VaR percentages by the absolute value of the present values of cash flows and summing them. For the FRA example, the undiversified VaR is  $0.62 million at the 95% confidence level.

• Diversified VaR: Matrix algebra is used to multiply the vector (from the undiversified VaR calculation) by the correlation matrix. For the FRA example, the diversified VaR is $0.348 million.

Topic 12. Interest Rate Swaps

• Decomposition: Interest rate swaps can be broken down into fixed and floating parts.

  • The fixed part is priced with a coupon-paying bond.

  • The floating part is priced as a floating-rate note.

• VaR Calculation Steps (Undiversified and Diversified):

  1. Create Present Value of Cash Flows: Show a short position for the fixed portion (paying fixed interest rates and fixed bond maturity) and a long present value for the variable rate bond.

  2. Compute Undiversified VaR: Multiply the vector representing the absolute present values of cash flows by the VaR percentages (e.g., at 95% confidence) and sum the values.

  3. Compute Diversified VaR: Use matrix algebra to multiply the correlation matrix by the absolute values.

Topic 13. Mapping Approaches for Nonlinear Derivatives

• Challenge: The delta-normal VaR method relies on linear relationships between variables. Options, however, exhibit nonlinear relationships between movements in the underlying instrument's value and the option's value.

• Applicability: In many cases, the delta-normal method can still be applied because an option's value can be expressed linearly as the product of the option's delta.

• Limitations: Delta-normal VaR may not provide accurate VaR measures over long risk horizons where deltas are unstable.

Practice Questions: Q1

Q1. Suppose you are calculating the tracking error VaR for two zero-coupon bonds using a $100 million benchmark bond portfolio with the following maturities and market value weights. Which of the following combinations of two zero-coupon bonds would most likely have the smallest tracking error?

A. 1 year and 7 year.
B. 2 year and 4 year.
C. 3 year and 5 year.
D. 4 year and 7 year.

Practice Questions: Q1 Answer

Explanation:  C is correct.

The three-year and five-year cash flows are highest for the benchmark portfolio at $24 million and $18 million, respectively. Thus, tracking error VaR will likely be the lowest for the portfolio where the cash flows of the benchmark and zero- coupon bond portfolios are most closely matched.