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