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© 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)
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VaR Concepts for Portfolio
© Pristine FRM – II
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
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© Pristine FRM – II
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
*****
© Pristine FRM – II
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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i
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4
© Pristine FRM – II
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
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P
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5
© Pristine FRM – II
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
*
NNP
© Pristine FRM – II
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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© Pristine FRM – II
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
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pi
ci
RRZMVARVaRinalM
),cov(*arg
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i
RR
i
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i
ValuePortfolio
VARMVAR *
8
© Pristine FRM – II
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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© Pristine FRM – II
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
© Pristine FRM – II
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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© Pristine FRM – II
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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© Pristine FRM – II
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
RateFreeRiskturnjPortfolioRateFreeRiskturniPortfolio
ReRe
13
© Pristine FRM – II
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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© Pristine FRM – II
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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