Portfolio Risk - Analytical Tools

Question 1

What is Diversified Portfolio VaR

Answer 1

VaR = Z X Sigma of protfolio X Nominal value invested in portfolio

Sigma will be for the entire portfolio, based on the formula of multiple securities (the variance one)

Variance and SD will be lower when correlations are lower

Question 2

How to calculate individual VaR?

Answer 2

Individual VaR is the VaR of the individual position in isolation. If the proportion in position is w, then we can define the individual VaR as

VaR = Z X Sigma X Weight Individual X Portfolio Value

Question 3

Formula for VaR based on VaR of constituting assets (in a 2 asset portfolio)

Answer 3

VaR (Portfolio) = sqrt (VaR (1)^2 + VaR (2)^2 + 2VaR(1)VaR(2)*Correlation between 1 and 2)

Obviously total VaR will be least, when correlation is zero, and will be highest, when correlation is one

Question 4

What is undiversified VaR

Answer 4

It is the sum of all the VaRs of the individual positions in the portfolio, when none of those positions are short positions.

Question 5

Formula for portfolio SD, when there are more than two assets, portfolio is equally weighted, all the securities have same SD, and the correlations are the same

Answer 5

Sigma(P)=Sigma(security) X Sqrt(1/N + (1-1/N)*correlation)

Question 6

What is marginal VaR or mVaR

Answer 6

mVaR is the per unit change in a portfolio VaR that occurs from an additional investment in that position.

(partial derivative of the portfolio VaR with respect to the position)

mVaR=Z X change in portfolio SD/ change in weight
=Z X cov (R(individual), R(portfolio)/SD(portfolio)

By the concept of beta,

MVaR = VaR (Portfolio)/portfolio value X beta (individual)

Question 7

What is incremental VaR?

Answer 7

Incremental VaR is the change in VaR from the addition of a new position in a portfolio.

Incremental Var<= Var of the individual asset

Question 8

What is component VaR

Answer 8

CVaR is the amount of risk a particular fund contributes to a portfolio of funds

CVaR(i) = MVAR(i) X Weight of security (i) X P
=VaR (P) X beta (i) X Weight (i)
= Var (i) X Correlation (i)

Question 9

Expression of VaR using CVAR

Answer 9

VaR = Summation of CVaRs of individual securities

Question 10

When will portfolio risk be at a global minimum

Answer 10

When all the marginal VaRs are equal for all i and j

MVaR (i) = MVar (j)

Question 11

Sharpe Ratio

Answer 11

(R(p)-R(f))/Sigma of portfolio

Question 12

A metric of return upon risk is (R(p)-R(f))/VAR(p). When will this be maximised?

Answer 12

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

For all i, (R(i)-R(f)/mVaR(i)=Constant

If all the returns follow elliptical distributions,

(R(i)-R(f))/beta(i) = Constant for all i

Question 13

Two assets A and B make a portfolio. Return/ mVAR of A< return/mVaR of B. How should we optimize this?

Answer 13

Increase allocation of B, and/ or reduce allocation of A

Basically we need to reduce the mVAR of A, and mVaR is dependent on weight