Portfolio VaR

• A portfolio can be characterized by positions on a certain number of constituent assets, expressed in the base currency, say, dollars. If the positions are fixed over the selected horizon, the portfolio rate of return is a linear combination of the returns on underlying assets, where the weights are given by the relative amounts invested at the beginning of the period

\(R_{P,t+1} = \sum_{i=1}^{N} w_i R_{i,t+1}\)

The rate of return is defined as the change in the dollar value, or dollar return, scaled by the initial investment. This is a unitless measure.

In matrix notation

\(R_P = w_1 R_1 + w_2 R_2 + \cdots + w_N R_N = \begin{bmatrix} w_1 & w_2 & \cdots & w_N \end{bmatrix} \begin{bmatrix} R_1 \\ R_2 \\ \vdots \\ R_N \end{bmatrix} = w’R \)

• The portfolio expected return is

\(E(R_P) = \mu_P = \sum_{i=1}^{N} w_i \mu_i \)

and the Variance is

As the number of assets increases, it becomes difficult to keep track of all covariance terms, which is why it is more convenient to use matrix notation. The variance can be written as

Defining ∑ as the covariance matrix, the variance of the portfolio rate of return can be written more compactly as

This also can be written in terms of dollar exposures 𝑥 as

• Diversified𝑽𝒂𝑹 is the portfolio 𝑉𝑎𝑅, taking into account diversification benefits between components

\( \text{Portfolio VaR} = \text{VaR}_P = \alpha \sigma_P W = \alpha \sqrt{x’ \Sigma x} \)

• Individual𝑽𝒂𝑹 is the 𝑉𝑎𝑅 of one component taken in isolation. At this point, the individual risk of each component can be defined as

\( \text{VaR}_i = \alpha \sigma_i \left| x_i \right| = \alpha \sigma_i \left| w_i \right| W \)

The correlation coefficient is a more convenient, scale-free measure of linear dependence:

\( \rho_{12} = \frac{\sigma_{12}}{\sigma_1 \sigma_2} \)

• Risk tends asymptotically to zero. More generally, portfolio risk is

\( \sigma_P = \sigma \sqrt{\frac{1}{N} + \left(1 – \frac{1}{N}\right) \rho } \)

which tends to \( \sigma \sqrt{\rho} \) as 𝑁 increases.

• For a two asset portfolio, the “diversified” portfolio variance is

\( \sigma_P^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2 \)

The portfolio 𝑉𝑎𝑅 is then

\( VaR_P = \alpha \sigma_P W = \alpha \sqrt{ w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2 } W \)

• 𝑉𝑎𝑅 is a coherent risk measure for normal and, more generally, elliptical distributions.
• When the correlation is exactly unity and 𝑤₁ and 𝑤₂ are both positive,
• Undiversified𝑽𝒂𝑹 The sum of individual 𝑉𝑎𝑅s, or the portfolio 𝑉𝑎𝑅 when there is no short position and all correlations are unity.
• This interpretation differs when short sales are allowed. Suppose that the portfolio is long asset 1 but short asset 2 (𝑤₁ is positive, and 𝑤₂ is negative). This could represent a hedge fund that has $1 in capital and a $1 billion long position in corporate bonds and a $1 billion short position in Treasury bonds, the rationale for the position being that corporate yields are slightly higher than Treasury yields. If the correlation is exactly unity, the fund has no risk because any loss in one asset will be offset by a matching gain in the other. The portfolio 𝑉𝑎𝑅 then is zero.
• Instead, the risk will be greatest if the correlation is -1, in which case losses in one asset will be amplified by the other.

Portfolio VaR Example

• Consider a portfolio with two foreign currencies, the Canadian dollar (𝐶𝐴𝐷) and the euro (𝐸𝑈𝑅). Assume that these two currencies are uncorrelated and have a volatility against the dollar of 5 and 12 percent, respectively. The first step is to mark to market the positions in the base currency. The portfolio has 𝑈𝑆 $2 million invested in the 𝐶𝐴𝐷 and 𝑈𝑆 $1 million in the 𝐸𝑈𝑅. Find the portfolio 𝑉𝑎𝑅 at the 95 percent confidence level.

VaR Tools – Marginal VaR

• Marginal 𝑉𝑎𝑅 is the change in portfolio 𝑉𝑎𝑅 resulting from taking an additional dollar of exposure to a given component. It is also the partial (or linear) derivative with respect to the component position.

\( \Delta VaR_i = \frac{\partial VaR_P}{\partial x_i} = \frac{\partial VaR_P}{\partial w_i W} = \alpha \frac{\partial \sigma_P}{\partial w_i} = \alpha \frac{cov(R_i, R_P)}{\sigma_P} \)

• This marginal 𝑉𝑎𝑅 is closely related to the beta,defined as

\( \beta_i = \frac{cov(R_i, R_P)}{\sigma_P^2} = \rho_{iP} \frac{\sigma_i}{\sigma_P} \)

• Using matrix notation, the vector 𝛽, including all assets, can be written as

\( \beta = \frac{\Sigma w}{w’ \Sigma w} \)

The relationship between the Δ𝑉𝑎𝑅 and 𝛽 is

\( \Delta VaR_i = \alpha (\beta_i \times \sigma_P) = \frac{VaR_P}{W} \times \beta_i \)

VaR Tools – Incremental

• Incremental 𝑽𝒂𝑹 is the change in the 𝑉𝑎𝑅 of a portfolio from the addition of a new position in a portfolio. The incremental 𝑉𝑎𝑅 then as described in this Figure, as

𝐼𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡𝑎𝑙 𝑉𝑎𝑅 =  𝑉𝑎𝑅_P+a — 𝑉𝑎𝑅_P

• Incremental 𝑉𝑎𝑅 differs from the marginal 𝑉𝑎𝑅 in that the amount added or subtracted can be large, in which case 𝑉𝑎𝑅 changes in a nonlinear fashion.

• The main drawback of this approach is that it requires a full revaluation of the portfolio 𝑉𝑎𝑅 with the new trade. This can be quite time-consuming for large portfolios.

• A shortcut can be taken to get an approximation

\( Incremental \, VaR \approx (\Delta VaR)’ \times a \)
This measure is much faster to implement because the Δ𝑉𝑎𝑅 vector is a by-product of the initial 𝑉𝑎𝑅_P computation. The new process is described in this Figure.

• Going back to the previous two-currency example, now consider increasing the 𝐶𝐴𝐷 position by 𝑈𝑆 $10,000.

VaR Tools – Component VaR

• Component𝑽𝒂𝑹 is a partition of the portfolio 𝑉𝑎𝑅 that indicates how much the portfolio 𝑉𝑎𝑅 would change approximately if the given component was deleted. By construction, component 𝑉𝑎𝑅s sum to the portfolio 𝑉𝑎𝑅.

\( Component \, VaR_i = (\Delta VaR_i) \times w_i W = \frac{VaR \beta_i}{w_i W} = VaR \beta_i w_i \)

The quality of this linear approximation improves when the 𝑉𝑎𝑅 components are small. Hence this decomposition is more useful with large portfolios, which tend to have many small positions.

\( CVaR_1 + CVaR_2 + \cdots + CVaR_N = VaR_P \left( \sum_{i=1}^{N} w_i \beta_i \right) = VaR_P \)

because the term between parentheses is simply the beta of the portfolio with itself, which is unity.

• The component 𝑉𝑎𝑅 can be simplified further.

\( CVaR_i = VaR_P w_i \beta_i = (\alpha \sigma_P W) w_i \beta_i = (\alpha \sigma_i w_i W) \rho_i = VaR_i \rho_i \)

This conveniently transforms the individual 𝑉𝑎𝑅 into its contribution to the total portfolio simply by multiplying it by the correlation coefficient.

• Percent contribution to 𝑉𝑎𝑅 of component

\( i = \frac{CVaR_i}{VaR} = w_i \beta_i \)

VaR Tools – Example

• Continuing with the previous two-currency example, find the component 𝑉𝑎𝑅 for the portfolio.

VaR Tools -Summary

VaR And Portfolio Management

• Marginal 𝑉𝑎𝑅 and component 𝑉𝑎𝑅 are useful tools, best suited to small changes in the portfolio. This can help the portfolio manager to decrease the risk of the portfolio. Positions should be cut first where the marginal 𝑉𝑎𝑅 is the greatest, keeping portfolio constraints satisfied. For example, if the portfolio needs to be fully invested, some other position, with the lowest marginal 𝑉𝑎𝑅, should be added to make up for the first change.
• This process can be repeated up to the point where the portfolio risk has reached a global minimum. At this point, all the marginal 𝑉𝑎𝑅s, or the portfolio betas, must be equal:

\( \Delta VaR_1 = \frac{VaR}{W} \times \beta_i = constant \)

• This table illustrates this process with the previous two currency portfolio.

• The next step is to consider the portfolio expected return as well as its risk. Indeed, the role of the portfolio manager is to choose a portfolio that represents the best combination of expected return and risk.
• For simplicity, all returns are defined in excess of the risk- free rate. In the figure, this translates all the points down by the same amount so that the risk-free asset is at the origin.

• Suppose now that the objective function is to maximize the ratio of expected return to risk. This Sharpe ratio is

\( SR_P = \frac{E_P}{\sigma_P} \)

• This can be modified and written with 𝑉𝑎𝑅 in the denominator instead of standard deviation.

• At the optimum,

\( \frac{E_i}{\Delta VaR_i} = \frac{E_i}{\beta_i} = \text{constant} \)