Book 5. Risk and Investment Management

FRM Part 2

IM 5. Portfolio Risk: Analytical Methods

Presented by: Sudhanshu

Module 1. VaR Measures

Module 2. Managing Portfolios with VaR

Module 1. VaR Measures

Topic 1. Diversified VaR, Individual VaR and Undiversified VaR

Topic 2. Portfolio Variance and Standard Deviation

Topic 3. Portfolio VaR: Examples

Topic 4. Marginal VaR

Topic 5. Incremental VaR: Overview

Topic 6. Incremental VaR: Example

Topic 7. Component VaR: Overview

Topic 8. Component VaR: Examples

Topic 9. VaR for Non-Elliptical Distibutions

Topic 1. Diversified VaR, Individual VaR and Undiversified VaR

Diversified VaR: The VaR of a portfolio that accounts for the diversification effects among the assets.
Individual VaR: It is the VaR of an individual position in isolation.
Undiversified VaR: If the positions in the portfolio are perfectly correlated, then there is no benefit of diversification and the resultant VaR is known as undiversified VaR. It is equal to the sum of VaR of individual positions.
Bounds on Portfolio VaR
- Upper bound on Portfolio VaR = Portfolio VaR with Correlated Positions
- Lower bound on Portfolio VaR = Portfolio VaR with Uncorrelated Positions

\text{VaR}_{\mathrm{i}}=\mathrm{Z}_{\mathrm{c}} \times \sigma_{\mathrm{i}} \times\left|\mathrm{P}_{\mathrm{i}}\right|=\mathrm{Z}_{\mathrm{c}} \times \sigma_{\mathrm{i}} \times\left|\mathrm{w}_{\mathrm{i}}\right| \times \mathrm{P}

\mathrm{VaR}_{\mathrm{p}}=\mathrm{Z}_{\mathrm{c}} \times \sigma_{\mathrm{p}} \times \mathrm{P}

\text{Undiversified VaR}_{\mathrm{p}}=\Sigma{VaR}_i

Topic 2. Portfolio Variance and Standard Deviation

Portfolio Variance
Standard Deviation
Consider an equally weighted portfolio in which all the individual positions have the same standard deviation of returns and the correlations between each pair of returns are the same. In this case:

\sigma_P^2=\sum_{i=1}^N w_i^2 \sigma_i^2+2 \sum_{i=1}^{N} \sum_{j\le i}^N w_i w_j \rho_{ij} \sigma_{i}\sigma_{j}

\sigma_P=\sqrt{\sigma_P^2}=\sqrt{\sum_{i=1}^N w_i^2 \sigma_i^2+2 \sum_{i=1}^N \sum_{j\le i}^N w_i w_j \rho_{i j} \sigma_i \sigma_j}

\sigma_{\mathrm{p}}=\sigma \sqrt{\frac{1}{\mathrm{~N}}+\left(1-\frac{1}{\mathrm{~N}}\right) \rho}

Topic 3. Portfolio VaR: Examples

Example 1: An analyst computes the VaR for two positions in her portfolio as following: Compute portfolio VaR if the returns of the two assets are: (1) uncorrelated and (2) perfectly correlated
Solution:
- Case 1: When returns of two assets are uncorrelated
- Case 2: When returns of two assets are perfectly correlated
Example 2: A portfolio has five positions of $2 million each. The standard deviation of the returns is 30% for each position. The correlations between each pair of returns is 0.2. Calculate the VaR using a z-score of 2.33.
- Solution: The standard deviation of the portfolio returns is:
- The VaR in nominal terms is:

\begin{aligned} \mathrm{VaR}_{\mathrm{p}} & =\sqrt{\mathrm{VaR}_1^2+\mathrm{VaR}_2^2} =\sqrt{\left(2.4^2+1.6^2\right)(\$ \text { millions })^2}\\ & =\sqrt{8.32(\$ \text { millions })^2} =\$ 2.8844 \text { million }\end{aligned}

VaR_1=\$ 2.4 \text{ million and } VaR_2 =\$ 1.6 \text{ million.}

\mathrm{VaR}_{\mathrm{P}}=\mathrm{VaR}_1+\mathrm{VaR}_2=\$ 2.4 \text{ million } +\$ 1.6 \text{ million } =\$ 4 \text{ million }

\sigma_P=30 \% \sqrt{\frac{1}{5}+\left(1-\frac{1}{5}\right) 0.2} =30 \% \sqrt{0.36} =18 \%

\mathrm{VaR}_{\mathrm{p}}=\mathrm{Z}_{\mathrm{c}} \times \sigma_{\mathrm{p}} \times \mathrm{V}=(2.33)(18 \%)(\$ 10 \text{ million }) = \$4,194,000

Practice Questions: Q1

Q1. Which of the following is the best synonym for diversified VaR?

A. Vector VaR.

B. Position VaR.

C. Portfolio VaR.

D. Incidental VaR.

Practice Questions: Q1 Answer

Explanation: C is correct.

Portfolio VaR should include the effects of diversification. None of the other answers are types of VaRs.

Practice Questions: Q2

Q2. When computing individual VaR, it is proper to:

A. use the absolute value of the portfolio weight.

B. use only positive weights.

C. use only negative weights.

D. compute VaR for each asset within the portfolio.

Practice Questions: Q2 Answer

Explanation: A is correct.

The expression for individual VaR is: . The

Absolute value signs indicate that we need to measure the risk of both positive and negative positions, and risk cannot be negative.

VaR_i=Z_c \times \sigma \times |P_i|=Z \times \sigma_i \times |w_i| \times P

Practice Questions: Q3

Q3. A portfolio consists of two positions. The VaR of the two positions are $10 million and $ 20 million. If the returns of the two positions are not correlated, the VaR of the portfolio would be closest to:

A. $5.48 million.

B. $ 15.00 million

C. $ 22.36 million.

D. $ 25.00 million.

Practice Questions: Q3 Answer

Explanation: C is correct.

For uncorrelated positions, the answer is the square root of the sum of the squared VaRs:

VaR_P=\sqrt{(10^2+20^2)} \times (\$ \text{ million})=\$22.36 \text{ million}

Topic 4. Marginal VaR

Definition: The per-unit change in a portfolio's VaR resulting from an additional investment in a specific position.
Mathematical Expression:
Using Beta
Example: Assume Portfolio X has a VaR of €400,000. The portfolio is made up of four assets: Asset A, Asset B, Asset C, and Asset D. These assets are equally weighted within the portfolio and are each valued at €1,000,000. Asset A has a beta of 1.2. Calculate the marginal VaR of Asset A.
Solution:
- Thus, portfolio VaR will change by 0.12 for each euro change in Asset A.

M V a R_i=\frac{\partial V a R_P}{\partial(\text {monetary investment in } \mathrm{i})}

M V a R_i= \mathrm{Z}_{\mathrm{c}} \frac{\partial \sigma_{\mathrm{p}}}{\partial \mathrm{w}_{\mathrm{i}}} =Z_c \frac{\operatorname{cov}\left(R_i, R_P\right)}{\sigma_P}

M V a R_i=\frac{V a R_P}{\text { portfolio value }} \times \beta_i

\begin{aligned} & \operatorname{Marginal} \operatorname{VaR}_A=\left(\operatorname{VaR}_P / \text { portfolio value }\right) \times \beta_A \\ & \operatorname{Marginal} \operatorname{VaR}_A=(400,000 / 4,000,000) \times 1.2=0.12\end{aligned}

Topic 5. Incremental VaR: Overview

Definition: The change in a portfolio's VaR from adding a new position.
Component VaR is generally larger than marginal VaR as it applies to entire positions and includes nonlinear relationships that marginal VaR typically assumes away.
Full Revaluation Method: Incremental VaR is calculated by full revaluation method which requires a complete recalculation of the portfolio VaR after adding the new position.
- Incremental VaR = $Va R_{n e w p or t f o l i o} - Va R_{or i g ina l p or t f o l i o}$
- Note: This is the most accurate method but it is time-consuming for large portfolios due to a large covariance matrix.
Approximation: For small additions to a portfolio, incremental VaR can be approximated with the following steps
- Step 1: Estimate the risk factors of the new position and include them in a vector [η].
- Step 2: Estimate the vector of marginal VaRs for the risk factors
- Step 3: Take the cross product.
This probably requires less work and is faster to implement because it is likely the managers already have estimates of the vector of values in Step 2.

[MVaR_j].

MVaR_j

Topic 6. Incremental VaR: Example

Example: A portfolio consists of Assets A and B. The volatilities are 6% and 14%, respectively. There are $4 million and $2 million invested in them respectively. If we assume they are uncorrelated with each other, compute the incremental VaR for an increase of $10,000 in Asset A. Assume a z-score of 1.65.
Solution: To find incremental VaR, we compute the per dollar covariances of each risk factor:
- These per dollar covariances represent the covariance of a given risk factor with
  
  the portfolio. Thus, we can substitute these values into the marginal VaR equations for the risk factors as follows. The marginal VaRs of the two risk factors are:
- Since the two assets are uncorrelated, the incremental VaR of an additional $10,000 investment in Position A would simply be $10,000*0.064428 = $644.28.

\left[\begin{array}{l}\operatorname{cov}\left(\mathrm{R}_{\mathrm{A}}, \mathrm{R}_{\mathrm{p}}\right) \\ \operatorname{cov}\left(\mathrm{R}_{\mathrm{B}}, \mathrm{R}_{\mathrm{p}}\right)\end{array}\right]=\left[\begin{array}{cc}0.06^2 & 0 \\ 0 & 0.14^2\end{array}\right]\left[\begin{array}{l}\$ 4 \\ \$ 2\end{array}\right]=\left[\begin{array}{l}0.0144 \\ 0.0392\end{array}\right]

\begin{aligned} & \mathrm{MVaR}_{\mathrm{A}}=\mathrm{Z}_e \times \frac{\operatorname{cov}\left(\mathrm{R}_{\mathrm{A}}, \mathrm{R}_{\mathrm{p}}\right)}{\sigma_{\mathrm{p}}}=1.65 \times \frac{0.0144}{\sqrt{0.136}}=0.064428 \\ & \mathrm{MVaR}_{\mathrm{B}}=\mathrm{Z}_c \times \frac{\operatorname{cov}\left(\mathrm{R}_{\mathrm{B}}, \mathrm{R}_{\mathrm{p}}\right)}{\sigma_{\mathrm{p}}}=1.65 \times \frac{0.0392}{\sqrt{0.136}}=0.175388\end{aligned}

Practice Questions: Q4

Q4. Which of the following is true with respect to computing incremental VaR? Compared to using marginal VaRs, computing with full revaluation is:

A. more costly, but less accurate.

B. less costly, but more accurate.

C. less costly, but also less accurate.

D. more costly, but also more accurate.

Practice Questions: Q4 Answer

Explanation: D is correct.

Full revaluation means recalculating the VaR of the entire portfolio. The marginal VaRs are probably already known, so using them is probably less costly, but will not be as accurate.

Topic 7. Component VaR: Overview

Definition: The amount of risk a particular fund contributes to a portfolio of funds.
In a large portfolio with many positions, the approximation is simply the marginal VaR multiplied by the dollar weight in position i:
Total Portfolio VaR: The sum of all component VaRs equals the total portfolio VaR.
Example: Asset A, Asset B, Asset C, and Asset D. These assets are equally weighted within the portfolio and are each valued at €1,000,000. Asset A has a beta of 1.2. Calculate the component VaR of Asset A.
Solution:
- Thus, portfolio VaR will decrease by €120,000 if Asset A is removed.

VaR_P=\sum_{i=1}^N CVaR_i

\begin{aligned} \mathrm{CVaR}_{\mathrm{i}} & =\left(\mathrm{MVaR}_{\mathrm{i}}\right) \times\left(\mathrm{w}_{\mathrm{i}} \times \mathrm{P}\right)=\operatorname{VaR} \times \beta_{\mathrm{i}} \times \mathrm{w}_{\mathrm{i}} \text{ (because } \beta_{\mathrm{i}}=\left(\rho_{\mathrm{i}} \times \sigma_{\mathrm{i}}\right) / \sigma_{\mathrm{p}}) \\ & =\left(\alpha \times \sigma_{\mathrm{p}} \times \mathrm{P}\right) \times \beta_{\mathrm{i}} \times \mathrm{w}_{\mathrm{i}}=\left(\alpha \times \sigma_{\mathrm{i}} \times \mathrm{w}_{\mathrm{i}} \times \mathrm{P}\right) \times \rho_{\mathrm{i}}=\operatorname{VaR}_{\mathrm{i}} \times \rho_{\mathrm{i}}\end{aligned}

\begin{aligned} \text { Component } \operatorname{VaR}_A & =\operatorname{VaR}_p \times \beta_A \times \text { asset weight } \\ \text { Component } \operatorname{VaR}_A & =400,000 \times 1.2 \times(1,000,000 / 4,000,000) =€ 120,000\end{aligned}

Topic 8. Component VaR: Examples

Example 1: Consider a portfolio in which $4 million is invested in asset A and $2 million is invested in asset B. Using the respective marginal VaRs of A and B as 0.064428 and 0.175388, compute the component VaRs.
Solution: The component VaRs for A and B are calculated as:
Example 2: Using the results from the previous example, compute the percentage of contribution to VaR of each component.
Solution: The answer is the sum of the component VaRs divided into each individual component VaR:

\begin{aligned} & \mathrm{CVaR}_{\mathrm{A}}=\left(\mathrm{MVaR}_{\mathrm{A}}\right) \times\left(\mathrm{w}_{\mathrm{A}} \times \mathrm{P}\right)=(0.064428) \times(\$ 4 \text { million })=\$ 257,713 \\ & \mathrm{CVaR}_{\mathrm{B}}=\left(\mathrm{MVaR}_{\mathrm{B}}\right) \times\left(\mathrm{w}_{\mathrm{B}} \times \mathrm{P}\right)=(0.175388) \times(\$ 2 \text { million })=\$ 350,777\end{aligned}

\begin{aligned} & \% \text { Contribution to VaR from } A=\frac{\$ 257,713}{(\$ 257,713+\$ 350,777)}=42.35 \% \\ & \% \text { Contribution to VaR from } B=\frac{\$ 350,777}{(\$ 257,713+\$ 350,777)}=57.65 \%\end{aligned}

Topic 9. VaR for Non-Elliptical Distibutions

Normal distributions are a subset of elliptical distributions, which have fewer assumptions and are commonly assumed in risk management for estimating component VaRs.
If returns don't follow an elliptical distribution but are homogeneous of degree one, Euler's theorem can be used to estimate component VaRs. For a constant k,
We can find component VaRs for a non-elliptical distribution using historical returns using following steps:
- Step 1: Sort the portfolio's historical returns from lowest to highest.
- Step 2: Identify the portfolio return corresponding to the chosen VaR level.
- Step 3: Find the individual position returns that occurred when occurred.
- Step 4: Use each position's return associated with as that position's component VaR.
To improve estimates, obtain position returns for portfolio returns slightly above and below , then compute averages to better approximate each position's component VaR.

k \times R_P=\sum_{i=1}^N k \times w_i \times R_i

\mathrm{R}_{\mathrm{P}(\mathrm{VaR})}

Module 2. Managing Portfolios With VaR

Topic 1. Portfolio Management using Marginal VaR

Topic 2. Risk Management Vs Portfolio Management

Topic 1. Portfolio Management using Marginal VaR

A manager can lower portfolio VaR by reducing allocations to positions with the highest marginal VaR while maintaining constant total invested capital.
This strategy involves increasing allocations to positions with lower marginal VaR to reduce overall portfolio risk.
Portfolio risk reaches a global minimum when all marginal VaRs are equal across all positions:
Example: Consider a portfolio of $6 million with two assets A and B. The volatilities are 6% and 14% respectively. We'll consider two asset allocation cases.
- Case 1: Asset A: $4 million and Asset B: $2 million
  - In this case,
- Case 2: Asset A: $5 million and Asset B: $1 million
  - In this case,
We can note that as we increase weight of Asset A (in Case 2), which has lower volatility, the resultant marginal VaRs of A and B get closer to each other.

MVaR_i=MVaR_j

MVaR_A= 0.064428\text{ and } MVaR_B=0.175388

MVaR_A= 0.08971 \text{ and } MVaR_B=0.09769

Topic 2. Risk Management Vs Portfolio Management

Risk management focuses on risk and ways to reduce risk. However, minimizing risk may not produce the optimal portfolio.
Portfolio management requires assessing both risk measures and return measures to choose the optimal portfolio.
Conditions for global minimum, lowest portfolio VaR and optimal portfolio:
- Global Minimum: Portfolio risk will be at a global minimum where all the marginal VaRs are equal for all i and j:
- Lowest Portfolio VaR: Equating the MVaRs will obtain the portfolio with the lowest portfolio VaR.
- Optimal Portfolio: Equating the excess return/MVaR ratios will obtain the optimal portfolio.

\frac{\text { (Position i return - Risk-free rate) }}{\left(M V a R_i\right)}=\frac{\text { (Position j return - Risk-free rate) }}{\left(M V a R_j\right)}

Practice Questions: Q1

Q1. A portfolio has an equal amount invested in two positions, X and Y. The expected excess return of X is 9 % and that of Y is 12 %. Their marginal VaRs are 0.06 and 0.075, respectively. To move toward the optimal portfolio, the manager will probably:

A. increase the allocation in Y and/or lower that in X.

B. increase the allocation in X and/or lower that in Y.

C. do nothing because the information is insufficient.

D. not change the portfolio because it is already optimal.

Practice Questions: Q1 Answer

Explanation: A is correct.

The expected excess-return-to-MVaR ratio for X is 0.09/0.06=1.5 and for Y is 0.12/0.075=1.6. Therefore, the portfolio weight in Y should increase to move the portfolio toward the optimal portfolio.