Portfolio Risk: Analytical Methods -...

© Pristine FRM – II © Pristine – www.edupristine.com

Portfolio Risk: Analytical Methods

© Pristine FRM – II

Individual VaR: is the Var of an isolated position.

VaR = Z value * σ * Portfolio value * Position weight age

Diversified Portfolio VaR: is the VaR of the portfolio.

Portfolio VaR = Z value * σP * Portfolio value

For uncorrelated assets:

VaR of the portfolio = Sqrt(VaR12 + VaR2

2)

For perfectly correlated assets, we have;

VaR of the portfolio = VaR1 + VaR2

Standard deviation of an equally-weighted portfolio with equal standard deviations and correlations (ρ):

σP = σSqrt(1/N + (1-1/N)ρ)

1

VaR Concepts for Portfolio


VaR Concepts for Portfolio

Marginal VaR

Marginal VaR is the change in the portfolio VaR for an additional investment in an position.

Marginal VaR = ZC*cov(Ri, Rp )/σP

Marginal VaR = VaRP * βi / Portfolio Value

Portfolio VaR can be reduced by reducing allocation to those positions which have a high Marginal VaR.

Incremental VaR

Incremental VaR is the increase in VaR from the addition of a new posotion in a portfolio.

Component VaR

Component VaR is the amount of risk a position contributes to a Portfolio

2


Individual VaR

The VaR of an individual position in isolation

The proportion or weights in the position is wi

Absolute weights can be used as both long and short positions pose risk

• P: Portfolio value

• Pi: Nominal amount invested in position i

3

PwZPZVARiiciici

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Diversified Portfolio VaR

The VaR of the portfolio where the calculation takes into diversification effects

• Zc: The z-score associated with the level of confidence c

• σp: std. dev. Of portfolio returns

• P: Nominal value invested in the portfolio

The standard deviation

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N

i

N

i

N

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,

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4


Role Correlation has on Portfolio Risk

The VAR for uncorrelated portfolio is

The VAR for undiversified portfolio when the correlation is one

For a two asset portfolio the general equation is

2

2

2

1)( VARVARpositionseduncorrelatVAR

P

2121

2

2

2

1**2)( VARVARVARVARVARVARpositionsiedundiversifVAR

P

212,121

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212,121

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5


Portfolio Standard Deviation of Returns

The following formula has the following assumptions

• Portfolio is equally weighted

• Individual positions have the same standard deviation of returns

• Correlation between each pair of returns is the same

Where:

• N: Number of positions

• σ: std. dev. That is equal for all N positions

• ρ: correlation between the returns of each pair of positions

6

*1

11

*

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Example

A portfolio has 10 positions of 3 million USD each with standard deviation/volatility for each position being 20%. The correlation between each pair is 0.3, and we need to calculate VaR using a z value of 2.5.

Solution:

σP = σSqrt(1/N + (1-1/N)ρ)

= 0.2 Sqrt(1/10 + (1- 1/10)0.3)

= 12.16%

VaR of the portfolio = 2.5 x12.16 x 10 x 3 million USD

= 9.12 million USD

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Marginal VaR

The per unit change in a portfolio VaR that occurs for an additional investment in that position

The beta for the entire portfolio is

Using concept of beta (CAPM) we can calculate marginal VaR as follows

P

pi

ci

RRZMVARVaRinalM

),cov(*arg

2

),cov(

P

PA

i

RR

i

P

i

ValuePortfolio

VARMVAR *

8


Incremental VaR

The change in VaR whenever a new position is added to the portfolio

Incremental VaR is the new VaR after the revaluation minus the VaR before the addition

VaR measurement becomes more complicated as the portfolio size increases given the expansion of covariance matrix

The steps we take to approximate incremental VaR are:

• Step 1: Estimate the risk factors of the new position that include them in a vector *ƞ+

• Step 2: For the portfolio, estimate the vector of marginal VARs for the risk factors [MVARi]

• Step 3: Take the cross product

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Component VaR

The risk contributed by each fund to a portfolio of funds

Generally it could be less than the VaR of the fund by itself because of diversification benefit

10

iiiiiwVARPwMVARCVAR **)*(*)(

N

i

iCVARVAR

1


Component VaR for a Non-Elliptical Distribution

The above calculations is assuming a normal distribution which is a form of elliptical distribution

For non-elliptical distribution we use the following steps

• Step 1: Sort the historical returns of the portfolio

• Step 2: Find the returns of the portfolio, which we designate RP(VAR), that corresponds to a return that would be associated with the chosen VAR

• Step 3: Returns of the individual positions that occurred when RP(VAR) occurred

• Step 4: Use each of the position returns associated with RP(VAR) for component VAR for that position

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Managing Portfolios Using VaR

A manager can lower the portfolio VAR by lowering the allocation to the positions with the highest marginal VAR

Portfolio risk is at a global minimum where all the marginal VARs are equal for all the i and j

MVARi = MVARj

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Difference Between Risk Management and Portfolio Management

The efficient frontier is the plot of portfolio that have the lowest standard deviation.

The optimal portfolio also has the highest Sharpe ratio

The std. dev. Can also be replaced by the VAR of the portfolio

The ratio is maximized when the excess return in each position divided by its respective marginal VaR equals a constant.

turnsPortfolioofStdDev

RateFreeRiskturnPortfolioRatioSharpe

Re

Re

PortfolioofVaR

RateFreeRiskturnPortfolio Re

jiMVaR

RateFreeRiskturnjPortfolio

MVaR

RateFreeRiskturniPortfolio

ReRe

ji

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13


Concept Checkers

1. Given the following information calculate the portfolio VaR. Calculate it at the 97.5% confidence interval

A. 48.26

B. 54.23

C. 17.25

D. None of the above

Portfolio Portfolio Value Std. Dev.

A 40mn 25%

B 50mn 45%

Correlation of A & B 0

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Concept Checkers - Solution

1. A

Var (port.) = sqrt(Var(1) + Var(2))

Var (1) = Zc*P*w(i)*σ(i) = 1.96*(40/90)*90*(0.25) = 19.6

Var (2) = Zc*P*w(i)*σ(i) = 1.96*(50/90)*90*(0.45) = 44.1

Var (port.) = sqrt ( 19.62 + 44.12 ) = 48.26 mn

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