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LO 68.2: Describe the impact of horizon, turnover, and leverage on the risk management process in the investment management industry.
The sell side of the investment industry largely consists of banks that have developed VaR techniques and have used them for many years. Investors make up the buy side of the investment industry. Investors are now using VaR techniques, but they have to adapt them to the different nature of that side of the business. To understand why the needs are different, we should compare the characteristics of the two sides. Figure 1 makes direct comparisons.
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Topic 68 Cross Reference to GARP Assigned Reading – Jorion, Chapter 17
Figure 1: Sell Side and Buy Side Characteristics
Characteristic
Horizon Turnover Leverage Risk measures
Risk controls
Sell Side
Buy Side
Short-term (days)
Long-term (month or more)
Fast High VaR
Stress tests
Position limits VaR limits Stop-loss rules
Slow Low
Asset allocation Tracking error Diversification Benchmarking
Investment guidelines
Banks trade rapidly, which is why they cannot rely on traditional measures of risk that are based on historical data. For banks, yesterdays risk may not have anything to do with todays positions. Investors usually try to hold positions for longer periods of time (e.g., years).
Having a more dynamic method for measuring risk such as VaR is also important for banks because of their high leverage. Institutional investors often have much stronger constraints with respect to leverage; therefore, they have a much lower need to control downside risk.
Th e In v e s t m e n t P r o c e s s

LO 68.1: Define risk budgeting.
Risk budgeting is a top-down process that involves choosing and managing exposures to risk. The main idea is that the risk manager establishes a risk budget for the entire portfolio and then allocates risk to individual positions based on a predetermined fund risk level. The risk budgeting process differs from market value allocation since it involves the allocation of risk.
M a n a g in g Ris k W it h Va R

LO 67.6: Explain the difference between risk management and portfolio management, and describe how to use marginal VaR in portfolio management.
As the name implies, 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. Traditional efficient frontier analysis tells us that the minimum variance portfolio is not optimal. We should note that the efficient frontier is the plot of portfolios that have the lowest standard deviation for each expected return (or highest return for each standard deviation) when plotted on a plane with the vertical axis measuring return and the horizontal axis measuring the standard deviation. The optimal portfolio is represented by the point where a ray from the risk-free rate is just tangent to the efficient frontier. That optimal portfolio has the highest Sharpe ratio:
Sharpe ratio = —————- 77—————-7———— (standard deviation of portfolio return)
(portfolio return risk-free rate)
We can modify this formula by replacing the standard deviation with VaR so that the focus then becomes the excess return of the portfolio over VaR:
(portfolio return risk-free rate)
(VaR of portfolio)
This ratio is maximized when the excess return in each position divided by its respective marginal VaR equals a constant. In other words, at the optimum:
(Position i return risk-free rate)
(Position j return risk-free rate)
(MVaRi) for all positions i andy
~
(MVaRj)
Professors Note: Equating the excess return/MVaR ratios will obtain the optimal portfolio. This differs from equating just the MVaRs, which obtains the portfolio with the lowest portfolio VaR.
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
Assuming that the returns follow elliptical distributions, we can represent the condition in a more concise fashion by employing betas, (3p which are obtained from regressing each positions return on the portfolio return:
(Position i return risk-free rate) _ (Position j return risk-free rate)
Pi
for all positions i andy
3j
The portfolio weights that make these ratios equal will be the optimal portfolio. We now turn our attention to determining the optimal portfolio for our example portfolio of A and B. We will assume the expected excess return of A is 6% and that of B is 11%. Even without this information, we should know that the optimal portfolio will have an allocation in A less than $5 million and in B greater than $1 million. This is because the marginal VaRs were almost equal with those allocations. Thus, the resulting portfolio would be close to the minimum variance, which will not be optimal. We might want to find out how to adjust the allocation with respect to the original values of $4 million in A and $2 million in B. By comparing the ratios of the two assets we find:
Excess return of A
———————–= ————- = 0 9313
0.06
MVaRA
0.064428
^ ,___ Excess return of B ———————- = ————-= 0.6272
0.11
MVaRB
0.173388
We see that there is too much allocated in B. Before we adjust the portfolio, we compute the excess-return-to-VaR ratio for the entire portfolio. The return is:
. .. % excess return on portfolio = 7.67% = ————-(6%) H————— (11%)
^
r $4 million .
. $2 million. $4 million . $6 million $2 million. $6 million
The return to VaR (scaled by the size of the portfolio) is:
0.7339
0.0767 $608,490
x $6 million
Now, because the return to MVaR ratio was greater for A, we will increase the allocation in A to $4.5 million and decrease that in B to $1.5 million. With those changes, the portfolio variance is:
p 2 V 2
[$4.5 $1.5] 0.062 0
0 $4.5 0.142 $1.5

0.0729 + 0.0441 = 0.1170
(cid:0)
This value is in ($ millions)2. VaR is then the square root of the portfolio variance times 1.65 (95% confidence level):
VaR = (1.65)($342,053) = $564,387
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In this case, the marginal VaRs are found by:
Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
cov(RA,RP) cov(RB,RP)
0.062 0
0 $4.5 0.142 $ i-5
0.0162 0.0294
(cid:0)
(cid:0)
The marginal VaRs of the two positions are then:
MVaRA = Z c x —- v A 17 = 1.65x

c o v ( R a ,R p )
Op Vo.1170 0.0162 ,
Vo. 1170
^
= = 0.0781
_____ MVaRB = Zc x —- – b- = 1.65 x
cov(Rr>,Rp)
__
_
Op
0.0294 , VO. 1170
= = 0.1418
We see the expected excess-return-to-marginal VaR ratios are much closer:
0.06 0.0781
0.11 0.1418
0.7678
0.7756
The portfolio return is now:
. % excess return on portfolio = 7.25% = ————— (6%) H——————(11%) .. _ ..
$4.5 million $6 million
$1.5 million $6 million
The portfolio return divided by the portfolio VaR has risen. The return to VaR (scaled by the size of the portfolio) is:
0.7707 = ————x $6 million
….
0.0725 $564,387
This is greater than the 0.7559 value associated with the original $4 million and $2 million allocations. The result is a more optimal portfolio allocation.
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
Ke y Co n c e pt s
LO 67.1 Diversified VaR is simply the VaR of the portfolio where the calculation takes into account the diversification effects.
Individual VaR is the VaR of an individual position in isolation.
Diversified VaR is simply the VaR of the portfolio where the calculation takes into account the diversification effects. The basic formula is:
VaR = Z x a x P
e p
p
where: Z = the .z-score associated with the level of confidence c COp = the standard deviation of the portfolio return P = the nominal value invested in the portfolio
Individual VaR is the VaR of an individual position in isolation. If the proportion or weight in the position is iv{, then we can define the individual VaR as:
VaR = Z x o
1
C
P:1 = Z x O- x
C
1
W -l
p
1
where: P = the portfolio value Pj = the nominal amount invested in position i
Marginal VaR is the change in a portfolio VaR that occurs from an additional one unit investment in a given position. Useful representations are:
Marginal VaR = MVaRj = Z c covtRi>Rp)
T
Marginal VaR = MVaRj
Incremental VaR is the change in VaR from the addition of a new position in a portfolio. It can be calculated precisely from a total revaluation of the portfolio, but this can be costly. A less costly approximation is found by (1) breaking down the new position into risk factors, (2) multiplying each new risk factor times the corresponding partial derivative of the portfolio with respect to the risk factor, and then (3) adding up all the values.
Component VaR for position z, denoted CVaRj, is the amount a portfolio VaR would change from deleting that position in a portfolio. In a large portfolio with many positions, the approximation is simply the marginal VaR multiplied by the dollar weight in position i:
CVaR; = (MVaR;) x (Wj x P) = VaR x
x
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
There is a method for computing component VaRs for distributions that are not elliptical. The procedure is to sort the historical returns of the portfolio and designate a portfolio return that corresponds to the loss associated with the VaR and then find the returns of each of the components associated with that portfolio loss. Those position returns can be used to compute component VaRs.
LO 67.2 For a two-asset portfolio, two special cases are:
1. VaR for uncorrelated positions:
VaRp = -y/VaRj2 + VaR22
2. VaR for perfectly correlated positions:
Undiversified VaR = VaRp = ^VaRj2 + VaR22 + 2VaR! VaR2 = VaRj + VaR2
LO 67.3 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.
LO 67.4 Portfolio risk will be at a global minimum where all the marginal VaRs are equal for all i andy:
MVaRj = MVaRj
LO 67.5 Equating the MVaRs will obtain the portfolio with the lowest portfolio VaR. Equating the excess return/MVaR ratios will obtain the optimal portfolio.
LO 67.6 The optimal portfolio is the one for which all excess-return-to-marginal VaR ratios are equal:
(Position i return – risk-free rate)
(Position j return – risk-free rate)
(MVaR^
(MVaRj)
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
Co n c e pt Ch e c k e r s
1.
2.
3.
Which of the following is the best synonym for diversified VaR? A. Vector VaR. B. Position VaR. C. Portfolio VaR. D. Incidental VaR.
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.
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. $25.00 million.
Which of the following is true with respect to computing incremental VaR? Compared to using marginal VaRs, computing with full revaluation is: A. more costly, but less accurate. less costly, but more accurate. B. C. less costly, but also less accurate. D. more costly, but also more accurate.
A portfolio has an equal amount invested in two positions, X and Y. The expected excess return of X is 9% and that of Y is 12%. Their marginal VaRs are 0.06 and 0.075, respectively. To move toward the optimal portfolio, the manager will probably: A. B. C. D.
increase the allocation in Y and/or lower that in X. increase the allocation in X and/or lower that in Y. do nothing because the information is insufficient, not change the portfolio because it is already optimal.
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
Co n c e pt Ch e c k e r An s w e r s
1. C Portfolio VaR should include the effects of diversification. None of the other answers are
types of VaRs.
2. A The expression for individual VaR is VaRj = Zc x a x | Pj | = Z x Gj x |wj x P. The absolute
value signs indicate that we need to measure the risk of both positive and negative positions, and risk cannot be negative.
3. C For uncorrelated positions, the answer is the square root of the sum of the squared VaRs:
VaRp = yj(102 + 2 0 2) x ($ million) = $22.36 million.
4. D 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.
5. A The expected excess-return-to-MVaR ratios for X and Y are 1.5 and 1.6, respectively. Therefore, the portfolio weight in Y should increase to move the portfolio toward the optimal portfolio.
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The following is a review of the Risk Management and Investment Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in:
Va R a n d Ri s k Bu d g e t i n g i n In v e s t m e n t M a n a g e m e n t
Topic 68
Ex a m Fo c u s
Banks on the sell side of the investment industry have long used risk budgeting and value at risk (VaR). There is a trend for the buy side investment firms to increasingly use VaR. One reason for increased demand for risk budgeting is the increased complexity, dynamics, and globalization of the investment industry. Use of VaR can help set better guidelines than more traditional limits. By measuring marginal and incremental VaRs, a manager can make better decisions concerning portfolio weights. For the exam, be comfortable with the concept of surplus at risk (SaR). Also, understand how to budget risk across asset classes and active managers.
Ris k Bu d g e t in g

LO 67.5: Explain the risk-minimizing position and the risk and return-optimizing position of a portfolio.
A manager can lower a portfolio VaR by lowering allocations to the positions with the highest marginal VaR. If the manager keeps the total invested capital constant, this would mean increasing allocations to positions with lower marginal VaR. Portfolio risk will be at a global minimum where all the marginal VaRs are equal for all i andy:
MVaRj = MVaR
We can use our earlier example to see how we can use marginal VaRs to make decisions to lower the risk of the entire portfolio. In the earlier example, Position A has the smaller MVaR; therefore, we will compute the marginal VaRs and total VaR for a portfolio which has $5 million invested in A and $1 million in B. The portfolio variance is:
0.062 0
0 $5′ 0.142 $1

*
0.0900 + 0.0196 = 0.1096
This value is in ($ millions)2. VaR is then the square root of the portfolio variance times 1.65 (95% confidence level):
VaR = (1.65)($331,059) = $546,247
The VaR of $546,247 is less than the VaR of $608,490, which was produced when Portfolio A had a lower weight. We can see that the marginal VaRs are now much closer in value:
cov(RA,RP) cov(RB,RP)
0.062 0
0 0.142

(cid:0)
$5 $1
0.0180 0.0196
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
The marginal VaRs of the two positions are:
MVaRA Z , x ^ A . R p ) = 1 .6 5 x
CTp
= 0.08971
j o .1096
s
MVaRB = Zc x C0V(RB>r p )
CTp
‘196 = 0.09769 VO. 1096

LO 67.4: Apply the concept of marginal VaR to guide decisions about portfolio VaR.

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.
Answer:
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.
Answer:
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.
Answer:
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.
Answer:
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.
Answer:
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

LO 67.2: Explain the role of correlation on portfolio risk.
In a two-asset portfolio, the equation for the standard deviation is:
P = VW12(J12 + w 22(J22 + 2 wlw2Pl,2CTl2
and the VaR is:
VaRp Z cP Jw ^aj2 + w 2 2 c t 2 2 + 2 w1w2p12a 1a 2
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
We can square Z and P and put them under the square-root sign. This allows us to express VaRp as a function of the VaRs of the individual positions, which we express as VaRj for each position i. For a two-asset portfolio we will have VaRj and VaR2. If the correlation is zero, p 2 = 0, then the third term under the radical is zero and:
VaR for uncorrelated positions: VaRp = ^VaRj2 + VaR22
The other extreme is when the correlation is equal to unity, p 1 2 = 1. With perfect correlation, there is no benefit from diversification. For the two-asset portfolio, we find:
Undiversified VaR = VaRp = ^VaR,2 + VaR22 + 2 VaR,VaR2 = VaR! + VaR2
In general, undiversified VaR is the sum of all the VaRs of the individual positions in the portfolio when none of those positions are short positions. Notice how evaluating VaR using both uncorrelated positions and perfectly correlated positions will place a lower and upper bound on the total (or portfolio) VaR. Total VaR will be less if the positions are uncorrelated and greater if the positions are correlated. The greatest risk is a correlation o f1 where one asset magnifies the loss of the other asset. The following examples illustrate this point.
Example: Computing portfolio VaR (part 1)
An analyst computes the VaR for the two positions in her portfolio. The VaRs: VaRj = $2.4 million and VaR2 = $1.6 million. Compute VaRp if the returns of the two assets are uncorrelated. Answer:
For uncorrelated assets:
VaRp = ^VaRj2 +VaR22 = ^ 2 .4 2 + 1.62 j($millions)2 = ^8.32($millions)2
VaRp = $2.8844 million
Example: Computing portfolio VaR (part 2)
An analyst computes the VaR for the two positions in her portfolio. The VaRs: VaRj = $2.4 million and VaR2 = $1.6 million. Compute VaRp if the returns of the two assets are perfectly correlated.
Answer:
For perfectly correlated assets:
VaRp = VaRj + VaR2 = $2.4 million + $1.6 million = $4 million
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
Under certain assumptions, the portfolio standard deviation of returns for a portfolio with more than two assets has a very concise formula. The assumptions are: The portfolio is equally weighted. All the individual positions have the same standard deviation of returns. The correlations between each pair of returns are the same. The formula is then:
where: N = the number of positions a = the standard deviation that is equal for all Appositions p = the correlation between the returns of each pair of positions
Professors Note: This formula greatly simplifies the process o f having to calculate portfolio standard deviation with a covariance matrix.
To demonstrate the benefits of diversification, we can simply set up a 2 x 2 table where there is a small and large correlation (p) column and a small and large sample size {N) row. Assuming that the standard deviation of returns is 20% for both assets, we see how the portfolio variance is affected by the different inputs.
Figure 1: Portfolio Standard Deviation
Sample size/correlation N = 4 N = 10
p = 0.1
o p = 11.40% cjp = 8.72%
ia
i
.
Op = 15.81% Op = 14.83%
Example: Computing portfolio VaR (part 3)
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-value of 2.33.
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
Answer:
The standard deviation of the portfolio returns is:
ii 1 – + 1- – 5 5
0.2
Op = 30%V036
aP = 18%
The VaR in nominal terms is:
VaRp = Zc x a p x V = (2.33)(18%)($10 million)
VaRp = $4,194,000
M a r g in a l Va R
Marginal VaR applies to a particular position in a portfolio, and it is the per unit change in a portfolio VaR that occurs from an additional investment in that position. Mathematically speaking, it is the partial derivative of the portfolio VaR with respect to the position:
Marginal VaR = MVaRj
________ <9 VaRp________ ^(monetary investment in i) cov(Rj,Rp) Op Using CAPM methodology, we know a regression of the returns of a single asset i in a portfolio on the returns of the entire portfolio gives a beta, denoted pj, which is a concise measure that includes the covariance of the positions returns with the total portfolio: cov(Ri5 Rp) Using the concept of beta gives another expression for marginal VaR: Marginal VaR = MVaR; =------------------x ft portfolio value 2018 Kaplan, Inc. Page 77 Topic 67 Cross Reference to GARP Assigned Reading - Jorion, Chapter 7 Example: Computing marginal VaR 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. Answer Marginal VaRA = (VaRp / portfolio value) x (3A Marginal VaRA = (400,000 / 4,000,000) x 1.2 = 0.12 Thus, portfolio VaR will change by 0.12 for each euro change in Asset A. In c r e m e n t a l Va R

LO 67.1: Define, calculate, and distinguish between the following portfolio VaR measures: individual VaR, incremental VaR, marginal VaR, component VaR, undiversified portfolio VaR, and diversified portfolio VaR.
Professor’s Note: LO 67.1 is addressed throughout this topic.
D iv e r s if ie d Po r t f o l io Va R
Diversified VaR is simply the VaR of the portfolio where the calculation takes into account the diversification effects. The basic formula is:
VaR = Z x a x P
e p
p
where: Z = the ^-score associated with the level of confidence c Op = the standard deviation of the portfolio return P = the nominal value invested in the portfolio
C
2018 Kaplan, Inc.
Page 73
Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
Examining the formula for the variance of the portfolio returns is important because it reveals how the correlations of the returns of the assets in the portfolio affect volatility. The variance formula is:
Op2 =
N
i=l
N N
+ 2 y ^ y ^ w iw ipiia ia i
i=l jja iCTj
\ i=l
i=l j