1. TK 6413 / TK 5413 : ISLAMIC RISK MANAGEMENT TOPIC 6A: VALUE AT RISK (VaR) (EXTENSION) 1

2. (I) Properties of Risk Measures A risk measure for specifying capital requirements can be thought of as the amount of cash (or capital) that must be added to a position to make its risk acceptable to regulators. Artzner et al. (1999) provide an interesting approach to the choice of risk measures by postulating four desirable properties for capital adequacy purposes: i.Monotonicity: If a portfolio (W 1 ) has lower returns than another portfolio (W 2 ), i.e. W 1 ≤ W 2, for every state of the world, its risk measure should be greater, i.e. ρ(W 1 ) ≥ ρ(W 2 ) ii.Translation invariance: If we add an amount of cash K to a portfolio, its risk measure should go down by K, i.e. ρ(W+K)=ρ(W)-K 2

3. iii.Homogeneity: Changing the size of a portfolio by a factor λ while keeping the relative amounts of different items in the portfolio the same should result in the risk measure being multiplied by λ. i.e., ρ(λW)=λρ(W) iv.Subadditivity: The risk measure for two portfolios after they have been merged should be no greater than the sum of their risk measures before they are merged, i.e. ρ(W 1 +W 2 ) ≤ ρ(W 1 )+ρ(W 2 ) A risk measure that satisfies the above four properties is said to be coherent. VaR satisfy the first three conditions, however, it does not always satisfy the fourth one i.e. subadditivity; as such, VaR cannot be considered as a coherent measure. 3

4. (II) Value-at-Risk Versus Expected Shortfall In essence VaR asks the question: “How bad can things get?”; whereas Expected Shortfall (ES) asks the question: “If things do get bad, what is the expected loss?”. In other words, the expected shortfall will tell us how much we could lose if we are “hit” beyond the VaR. In the literature, expected shortfall is also known as conditional loss or Expected Tail Loss (ETL). ES has better properties than VaR in that it encourages diversification. One disadvantage is that it does not have the simplicity of VaR and, as a result, is slightly more difficult to understand. 4

5. (III) Portfolio VaR The VaR of a portfolio can be constructed from a combination of the risks of underlying securities. If we define the portfolio rate of return from (t) to (t+1) as where N=number of assets R i, t+1 =rate of return on assets i W i =weight 5

6. Putting it in a matrix form, R p =W 1 R 1 +W 2 R 2 + ……. +W N R N =[W 1,W 2 …. W N ] = W I R 6 R1R2...RNR1R2...RN

7. The variance can be written as, σ 1 2 ………..σ 1N 2. σ N1 2 ……….σ N 2 = W I ΣW Since W are weightswhich have no units, the above expression can be written in terms of dollars exposures as 7 W1W2...WNW1W2...WN

8. Portfolio VaR = VaR p = and individual VaR = V a R i = = If the portfolio consists of two assets then the diversified portfolio variance is, 8

9. Therefore, The portfolio risk must be lower than the sum of the individual VaRs:VaR p <(VaR 1 +VaR 2 ) 9 Assuming ρ=0

10. When correlation is exactly unity and W 1, W 2 are positive, This is a case of an undiversified VaR. 10

11. Example: Consider a portfolio with two foreign currencies, the Canadian dollar (CAD) and the Euro (€UR). Assume that these two currencies are uncorrelated and have a volatility against the U.S$ of 5 and 12 percent respectively. The portfolio has $U.S2million invested in CAD and $U.S1million in the €UR. Find the portfolio VaR at the 95% confidence level. If x is the dollar amount allocated to each risk factor in millions, Σx = 0.05 2 02 = $0.0050 0 0.12 2 1$0.0144 σ p 2 W 2 = x I Σx = [$2 $1] $0.0050 = 0.0100 + 0.0144 = 0.0244 $0.0144 σ p W = = $0.156205 million = $156,205 Therefore VaR p = ασ p W = 1.65x156,205 = $257,738 11

12. The individual undiversified VaR i = ασ i xi, = VaR 1 = 1.65 x 0.05 x $2million= $165,000 VaR 2 1.65 x 0.12 x $1million $198,000 therefore the sum of the undiversified VaR= $165,000 +$198,000 = $363,000 which is greater than the portfolio VaR of $257,738 12

13. (IV) VaR Tools Initially VaR was developed as a methodology to measure portfolio risk. However, over time risk managers have discovered that they could use the VaR process for active risk management. A typical question may be, “which position should I alter to modify my VaR most effectively?” Such information is quite useful because portfolios typically are traded incrementally owing to transaction costs. This is the purpose of VaR tools which include marginal, incremental and component VaR. 13

14. a)Marginal VaR Marginal VaR is defined as the change in portfolio VaR resulting from taking an additional dollar of exposure to a given component. It is also the partial (or linear) derivative with respect to the component position. Given a portfolio of N securities; a new portfolio is obtained by adding one unit of security i – To assess the impact of this trade we measure its “marginal” contribution to risk by increasing w by a small amount or differentiating σ p 2 i.e. however marginal VaR or ∆VaR can be written as 14

15. On the other hand marginal VaR closely related to β i (systematic risk of security i); since Therefore The marginal VaR can be used for a variety of risk management purposes. Suppose that an investor wants to lower the portfolio VaR and has the choice to reduce all positions by a fixed amount say, $100,000. the investor should rank all marginal VaR numbers and pick the asset with the largest ∆VaR because it will have the greatest hedging effect 15

16. b)Incremental VaR Incremental VaR is defined as the change in VaR owing to a new position. It differs from the marginal VaR in that the amount added or subtracted can be large, in which case VaR changes in a nonlinear fashion. Suppose the portfolio VaR at the initial position is VaR p and at the new position is VaR p+a, then the incremental VaR is =VaR p+a -VaR p This “before and after” comparison is quite informative. If VaR is decreased, the new trade is risk-reducing or is a hedge; otherwise, the new trade is risk-increasing. The main drawback of this approach is that it requires a full revaluation of the portfolio VaR with the new trade. This can be quite time consuming for large portfolios. 16

17. However, an approximation method can be used to estimate the incremental VaR Example: Taking the previous example, let say we are considering the CAD position by $10,000 since therefore β = 3 x $0.0050 ($0.0156 2 ) $0.0144 = 3 x 0.205 =0.615 0.5901.770 Marginal VaR∆VaR = = 1.65 x 0.0050 /(0.0156) 0.0144 = 0.0528 0.1521 17

18. As we increase the first position by $10,000, the incremental VaR is, (∆VaR) I x a = [0.0528, 0.1521] 10,000 0 =$528 Using the full revaluation method, adding $0.01 million to the first position we find, VaR p+a = = [$20.1, $1] 0.05 2 0 $20.1 0 0.12 2 $1 = $258,267 Since the initial VaR p = $257,738 Incremental VaR= VaR p+a – VaR p = $258,267 - $257,738 = $529 18

19. c)Component VaR By definition component VaR is a partition of the portfolio VaR that indicates how much the portfolio VaR would change approximately if the given component was deleted. By construction, component VaRs sum to the portfolio VaR. Component VaR = Since VaR = ασ p W, therefore CVaR i = = 19

20. Example: Using the previous example, CVaR i =∆VaR i X i CVaR 1 = 0.0528x $2million =$105,630 = VaR x 41.0% CVaR 2 = 0.1521 x $1million =$152,108 59.0% We verify that these two components indeed sum to the total VaR of $257,738. CVaR 1 /VaR= W 1 β 1 =0.667 x 0.615 =41.0% CVaR 2 /VaR W 2 β 2 =0.333 x 1.77059.0% 20