LO 67.3: Describe the challenges associated with VaR measurement as portfolio

LO 67.3: Describe the challenges associated with VaR measurement as portfolio size increases.
Incremental VaR is the change in VaR from the addition of a new position in a portfolio. Since it applies to an entire position, it is generally larger than marginal VaR and may include nonlinear relationships, which marginal VaR generally assumes away. The problem with measuring incremental VaR is that, in order to be accurate, a full revaluation of the portfolio after the addition of the new position would be necessary. The incremental VaR is the difference between the new VaR from the revaluation minus the VaR before the addition. The revaluation requires not only measuring the risk of the position itself, but it also requires measuring the change in the risk of the other positions that are already in the portfolio. For a portfolio with hundreds or thousands of positions, this would be time consuming. Clearly, VaR measurement becomes more difficult as portfolio size increases given the expansion of the covariance matrix. Using a shortcut approach for computing incremental VaR would be beneficial.
For small additions to a portfolio, we can approximate the incremental VaR with the following steps: Step 1: Estimate the risk factors of the new position and include them in a vector [r|]. Step 2: For the portfolio, estimate the vector of marginal VaRs for the risk factors [MVaR]. 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 MVaR- values in Step 2 .
Before we take a look at how to calculate incremental VaR, lets review the calculation of delta-normal VaR using matrix notation (i.e., using a covariance matrix).
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
Example: Computing VaR using matrix notation
A portfolio consists of assets A and B. These assets are the risk factors in the portfolio. 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 VaR of the portfolio using a confidence parameter, Z, of 1.65.
We can use matrix notation to derive the dollar variance of the portfolio:
a 2V 2 = [\$4 \$2 ] 0.062
0
0
\$4
0.142 \$2 = 0.0576 + 0.0784 = 0.136
This value is in (\$ millions)2. VaR is then the square root of the portfolio variance times 1.65:
VaR = (1.65)(\$368,782) = \$608,490
Professors Note: Matrix multiplication consists o f multiplying each row by each column. For example: (4 x 0.062) + (2 x 0) = 0.0144; 0.0144 x 4 = 0.0576. Had the positions been positively correlated, some positive value would replace the zeros in the covariance matrix.
Example: Computing incremental VaR
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.
To find incremental VaR, we compute the per dollar covariances of each risk factor:
cov(RA,RP) cov(RB,RP)
0.062 0
0 \$4 0.142 \$2
0.0144 0.0392
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.
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
The marginal VaRs of the two risk factors are:
MVaRA =
c ov(Ra ,Rp)
CTp
= 1.65 x
MVaRB = ^ c 1.65 x ,, c ov(Rb ,RP) _ = 1.65 x
CTp
0.0144 Jo. 136
0.0392 Jo. 136
= 0.064428
= 0.175388
Since the two assets are uncorrelated, the incremental VaR of an additional \$10,000 investment in Position A would simply be \$10,000 times 0.064428, or \$644.28.
C o m p o n e n t Va R Component VaR for position i, denoted CVaRp is the amount of risk a particular fund contributes to a portfolio of funds. It will generally be less than the VaR of the fund by itself (i.e., stand alone VaR) because of diversification benefits at the portfolio level. In a large portfolio with many positions, the approximation is simply the marginal VaR multiplied by the dollar weight in position i:
CVaR; = (MVaRj) x (w- x P) = VaR x pi x w. (a x o x P) x pj x Wj = (a x aj x Wj x P) x p; = VaRj x pi = (a x o x P) x pj x Wj = (a x aj x Wj x P) x p; = VaRj x pi
The last two components consider the fact that beta; = (pj x cq) / crp.
Using CVaRp we can express the total VaR of the portfolio as:
/
N
i=i V
\
/
i
\ /
N S wi x f3i i=i
/ \
VaR = J 2 CVaR; = VaR
N
i=l
Given the way the betas were computed we know:
Example: Computing component VaR (Example 1)
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 component VaR of Asset A.
Component VaRA = VaRp x (3A x asset weight
Component VaRA = 400,000 x 1.2 x (1,000,000 / 4,000,000) = 120,000
Thus, portfolio VaR will decrease by 120,000 if Asset A is removed.
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
Example: Computing component VaR (Example 2, Part 1)
Recall our previous incremental VaR example of a portfolio invested \$4 million in A and \$2 million in B. Using their respective marginal VaRs, 0.064428 and 0.175388, compute the component VaRs.
CVaRA = (MVaRA) x (wA x P) = (0.064428) x (\$4 million) = \$257,713
CVaRB = (MVaRB) x (wB x P) = (0.175388) x (\$2 million) = \$350,777
Professor’s Note: The values have been adjusted for rounding.
Example: Computing component VaR (Example 2, Part 2)
Using the results from the previous example, compute the percent of contribution to VaR of each component.
The answer is the sum of the component VaRs divided into each individual component VaR
% contribution to VaR from A
% contribution to VaR from B
\$257,713
(\$257,713 + \$350,777)
\$350,777
(\$257,713 + \$350,777)
42.35%
57.65%
Normal distributions are a subset of the class of distributions called elliptical distributions. As a class, elliptical distributions have fewer assumptions than normal distributions. Risk management often assumes elliptical distributions, and the procedures to estimate component VaRs up to this point have applied to elliptical distributions.
If the returns do not follow an elliptical distribution, we can employ other procedures to compute component VaR. If the distribution is homogeneous of degree one, for example, then we can use Eulers theorem to estimate the component VaRs. The return of a portfolio of assets is homogeneous of degree one because, for some constant, k, we can write:
kxRp = y > x w; x R|
N
i=l
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
The following steps can help us find component VaRs for a non-elliptical distribution using historical returns: Step 1: Sort the historical returns of the portfolio. Step 2: Find the return of the portfolio, which we will designate Rp(yaRp that corresponds
to a return that would be associated with the chosen VaR.
Step 3: Find the returns of the individual positions that occurred when Rp(VaR) occurred. Step 4: Use each of the position returns associated with Rp(yaR) for component VaR for
that position.
To improve the estimates of the component VaRs, an analyst should probably obtain returns for each individual position for returns of the portfolio slightly above and below Rp(yaRy For each set of returns for each position, the analyst would compute an average to better approximate the component VaR of the position.
M a n a g in g P o r t f o l io s U s in g Va R