Solution of the Financial Risk Management Examination

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1 Solution of the Financial Risk Management Examination Thierry Roncalli January 8 th 014 Remark 1 The first five questions are corrected in TR-GDR 1 and in the document of exercise solutions, which is available in my web page. 1 The BCBS regulation Market risk 3 Credit risk 4 Counterparty credit risk 5 Operational risk 6 Value-at-risk of an equity portfolio The main reference for this exercise is TR-GDR pages 61-63). 1. We note P A t) resp. P B t)) the value of the stock A resp. B) at the date t. The portfolio value is: P t) x A P A t) + x B P B t) with x A and x B the number of stocks A and B. We deduce that the PnL between t and t + 1 is: PnL P t + 1) P t) x A P A t + 1) P A t)) + x B P B t + 1) P B t)) x A P A t) R A + x B P B t) R B W A R A + W B R B where R A and R B are the asset returns of A and B between the dates t and t + 1. We have also W A x A P A t) and W B x B P B t). Because x A + and x B +1, we have W A 300 and W B 00. It follows that: PnL 300 R A + 00 R B We deduce that the yearly volatility of the PnL is: σ PnL) Thierry Roncalli, La Gestion des Risques Financiers, Economica, deuxième édition, 009. The direct link is 1

2 We obtain σ PnL) euros. It follows that the Gaussian VaR for a one-week time horizon and a 99% confidence level: VaR Φ 1 α) σ PnL) The VaR is equal to euros or 8.1% of the portfolio value 3 ).. The PnL becomes: PnL W A R A + W B R B W I R I a) We have: We obtain σ PnL) euros. horizon and a 99% confidence level: σ PnL) It follows that the Gaussian VaR for a one-week time VaR Φ 1 α) σ PnL) The VaR is equal to euros or 3.95% of the portfolio value). b) We have: σ PnL) We obtain σ PnL) 60 euros. It follows that the Gaussian VaR for a one-week time horizon and a 99% confidence level: VaR Φ 1 α) σ PnL) The VaR is equal to euros or 3.88% of the portfolio value). c) W I 500 corresponds to a fully hedged portfolio in terms of notional. Indeed, we have: W I W A + W B However, we observe that W I 560 is better to hedge the portfolio in terms of value-at-risk. This means that hedging the portfolio by considering the sum of nominal exposures is not optimal. 3 The portfolio value is equal to 500 euros.

3 3. We deduce that the PnL of the hedged portfolio is: PnL W A R A + W B R B W I R I W A β A R I + ε A ) + W B β B R I + ε B ) W I R I W A β A + W B β B W I ) R I + W A ε A + W B ε B W R I + W A ε A + W B ε B where W W A β A + W B β B W I is the residual exposure on the index. a) If W I 500, we have: If W I 560, we have: W W A β A + W B β B W I W W A β A + W B β B W I We observe that the residual exposure is equal to zero if W I 560. This is not the case if W I 500. The reason is that the second asset has a high beta β B 1.6), meaning that the sensitivity of the portfolio P 0 is higher than its current value 500). b) We have: ε A R A β A R I σ ε A ) σ A + β A σ I β Aρ A,I σ A σ I % Because R A β A R I + ε A, we have also: σa βaσ I + σ ε A ) It follows that: σ ε A ) σ A β A σ I In a same way, we have σ ε B ) 4%. c) If W I 560, W 0. It follows that: PnL W A ε A + W B ε B σ PnL) We retrieve the result obtained in Question.b): VaR

4 d) We have: It follows that: σ PnL) In the case of Portfolio P 0, we have: In the case of Portfolio P 1, we have: PnL W R I + W A ε A + W B ε B W σ I + W A σ ε A ) + W B σ ε B ) W P 0 ) W A β A + W B β B 560 W P 1 ) W A β A + W B β B W I 560 W I The VaR of the hedged portfolio P 1 is lower than the VaR of the original portfolio P 0 if 4 : σ PnL P 1 )) σ PnL P 0 )) W P 0 )) W P 1 )) W I ) W I Risk contribution in the Basle II model 1. We assume that TR-GDR, page 179): a) The loss given default LGD i is independent of the default time τ i ; b) the default times depend on common factors X 1,..., X m ; c) the portfolio is infinitely granular: there is no concentration on a specific credit, meaning that: EAD i / EAD i 0. Let R be the risk measure. The risk contribution of the credit i is the product of the exposure-atdefault EAD i and the marginal risk TR-GDR, page 497): In the case of a convex risk measure, we have: RC i EAD i R EAD i R RC i The risk measure is then equal to the sum of the different risk contributions. 3. We have: EL E [L] UL F 1 α) EL where F is the cumulative distribution function of L. If the default times are independent, we obtain: E [L X 1,..., X m ] EAD i E [LGD i ] PD i EL L is then not random and we have Pr {L EL} 1. It follows that F 1 α) EL and UL 0. 4 By construction, we have W I > 0. 4

5 4. We set TR-GDR, page 180): By definition, we have: g x) EAD i E [LGD i ] PD i x) E [L X x] g x) F l) Pr {L l} Pr {g X) l} Because EAD i 0 and E [LGD i ] 0, g x) is an increasing resp. decreasing) function if PD i x) is an increasing resp. decreasing) function of x. In the case where g x) is an increasing function, we have: F 1 α) Pr {g X) l} α Pr { X g 1 l) } α H g 1 l) ) α l g H 1 α) ) EAD i E [LGD i ] PD i H 1 α) ) In the case where g x) is a decreasing function, we have: F 1 α) Pr {g X) l} α Pr { X g 1 l) } α H g 1 l) ) 1 α l g H 1 1 α) ) EAD i E [LGD i ] PD i H 1 1 α) ) If at least one exposure EAD i is negative, the function g x) is not monotonic. We don t check anymore g X) l X g 1 l) increasing case) or g X) l X g 1 l) decreasing case), implying that the expression F 1 α) is not valid. We can not use the Basle II model if the credit portfolio has one or more) negative exposure. For the management of the credit portfolio, this implies that buying a CDS protection on the i th credit can only be done to reduce the exposureat-default on the counterparty, but not to be short on the counterparty. 5. We have TR-GDR, pages ): and B i Φ 1 PD i ). We have also: PD i Pr {τ i M i } Pr {Z i B i } Φ B i ) Pr {τ i M i X x} Pr {Z i B i X x} { ρx } Pr + 1 ρεi B i X x { Pr ε i B i } ρx X x 1 ρ Φ 1 PD i ) ) ρx Φ 1 ρ 5

6 6. Because the conditional default probability is a decreasing function with respect to x, we have: F 1 α) EAD i E [LGD i ] PD i Φ 1 1 α) ) Φ 1 PD i ) ρφ 1 ) 1 α) EAD i E [LGD i ] Φ 1 ρ Φ 1 PD i ) + ρφ 1 ) α) EAD i E [LGD i ] Φ 1 ρ 7. ρ is the constant correlation between assets: cor Z i, Z j ) E [Z i Z j ] E [ρx + ] ρ 1 ρ)x ε i + ε j ) + 1 ρ) ε i ε j ρ 8. The second pillar concerns the non-respect of the assumptions H). In particular, we have to understand the impact on the credit risk measure if the portfolio is not granular or if the asset correlation is not constant. 8 Correlation and log-normal random variables 1. a) The density of X N µ, σ ) is: f x) 1 σ π exp 1 ) ) x µ σ Let Y LN µ, σ ). We have: It comes that: with x ln y. g y) Y e X g y) f x) dx dy 1 σ π exp 1 ) ) x µ 1 σ y 1 yσ π exp 1 ) ) ln y µ σ b) Let m 1. We have: E [Y m ] 0 0 y m g y) dy y m yσ π exp 1 ) ) ln y µ dy σ 6

7 By considering the change of variable z σ 1 ln y µ), we have: E [Y m e µ+σz ) m ] exp 1 ) π z dz e mµ 1 exp 1 ) π z + mσz dz e mµ 1 exp 1 z π mσ) + 1 ) m σ dz e mµ+ 1 m σ 1 exp 1 ) z π mσ) dz By considering the change of variable t z mσ, we obtain: E [Y m ] e mµ+ 1 m σ 1 exp 1 ) π t dt e mµ+ 1 m σ Φ ) Φ )) e mµ+ 1 m σ c) We have shown that: E [Y ] exp µ + 1 ) σ E [Y m ] exp mµ + 1 ) m σ d) It follows that: var Y ) E [ Y ] E [Y ] e µ+σ e µ+σ ) e µ+σ e σ 1 e) See TR-GDR on page 39.. a) X 1 + X is a Gaussian random variable because it is a linear combination of the Gaussian vector X 1, X ). We have: E [X 1 + X ] µ 1 + µ and: b) We have: var X 1 + X ) σ 1 + ρσ 1 σ + σ X 1 + X N µ 1 + µ, σ1 + ρσ 1 σ + σ ) cov Y 1, Y ) E [Y 1 Y ] E [Y ] E [Y ] E [ e X1+X] E [Y ] E [Y ] Using Question.a, we know that e X1+X is a log-normal random variables. E [ e X1+X] e µ1+µ+ 1 σ 1 +ρσ1σ+σ ) e µ1+ 1 σ 1 e µ + 1 σ e ρσ 1σ We finally obtain: cov Y 1, Y ) e µ1+ 1 σ 1 e µ + 1 σ e ρσ 1σ 1) 7

8 c) We have: ρ Y 1, Y ) e µ1+ 1 σ 1 e µ + 1 σ e ρσ 1σ 1) e µ1+σ 1 e σ1 1 ) e µ+σ e σ 1 ) e ρσ1σ 1 e σ 1 1 e σ 1 ρ Y 1, Y ) is an increasing function with respect to ρ. { ρ 1 ρ Y 1, Y ) 1 σ 1 σ The lower bound is reached if ρ 1. In this case, we have: It follows that ρ Y 1, Y ) 1. e σ1σ 1 ρ Y 1, Y ) e σ 1 1 e σ 1 > 1 d) A concordance measure reached the lower bound 1 resp. the upper bound +1) if the random variables are countermonotonic resp. comonotonic). It is not the case with the linear correlation. For instance, if ρ 1, the dependence function is C and ρ Y 1, Y ) 1. if ρ 1, the dependence function is C +. In this case, ρ Y 1, Y ) 1 if and only if ln Y 1 and ln Y have the same variance. 9 Credit spreads 1. We have TR-GDR, page 47) : Let U S τ). We have U [0, 1] and: F t) 1 e λt S t) e λt f t) λe λt Pr {U u} Pr {S τ) u} Pr { τ S 1 u) } S S 1 u) ) u We deduce that S τ) U [0,1] TR-GDR, page 48). It comes that τ S 1 U) with U U [0,1]. Let u be a uniform random variate. Simulating τ is equivalent to transform u into t: t 1 λ ln u. We have TR-GDR, pages ) : P 1 4 s N P + 1 R) N with s the spread, N the notional, P the premium leg and P + the protection leg. The quarterly payment of the premium leg explains the factor 1/4 in the formula of P. We deduce the flow chart given in Figure 1. 8

9 Figure 1: Flow chart from the viewpoint of the protection buyer 1 4sN euros every three months if the default does not occur {}}{ 3M 6M 9M 1Y 15M 18M 1M Y τ 1 R) N euros at the default time τ) t 3. The ATM margin or spread) is the value of s such that the CDS price is zero TR-GDR, page 410): [ ) ] P t) E P P + 0 We have the following triangle relationship TR-GDR, page 410): s λ 1 R) 4. Let PD be the annual default probability. We have PD 1 S 1) 1 e λ 1 1 λ) because λ is generally small λ 10%). We have: PD λ s 1 R PD % 67 bps 1 5% 10 Extreme value theory and stress-testing 1. See TR-GDR, page We have TR-GDR, pages ): 3. See TR-GDR, page 139. G n x) Pr {max X 1,..., X n ) x} Pr {X 1 x,..., X n x} n Pr {X i x} x µ Φ σ ) n 9

10 4. a) An extreme value EV) copula C satisfies the following relationship: for all t > 0. C u t 1, u t ) C t u 1, u ) b) The product copula is an EV copula because: C u t 1, u t ) u t 1u t u 1 u ) t [ C u 1, u ) ] t c) We have: C u t 1, u t ) [ ) exp ln u t θ ) ] ) 1 + ln u t θ 1/θ exp [ t ln u 1 ) θ + t ln u ) θ] ) 1/θ exp t [ ln u 1 ) θ + ln u ) θ] ) 1/θ [e [ ln u1)θ + ln u ) θ ] 1/θ ] t C t u 1, u ) d) The upper tail dependence λ is defined as follows: λ lim u u + C u 1, u ) 1 u It indicates the probability to have an extreme in one direction knowing that we have already an extreme in the other direction. If λ 0, extremes are independent and the copula of extreme values is the product copula C. If λ 1, extremes are comonotonic and the copula of extreme values is the upper Fréchet copula C +. Moreover, the upper tail dependence of the copula between the random variables is equal to the upper tail dependence of the copula between the extremes. e) We obtain using L Hospital s rule: 1 u + e [ ln u) θ + ln u) θ ] 1/θ λ lim u u 1 u + e [ ln u) θ ] 1/θ lim u u 1 u + u 1/θ lim u u 0 + 1/θ u 1/θ 1 lim u lim 1/θ u 1/θ 1 u 1 + 1/θ f) If θ 1, λ 0. It comes that the copula of the extremes is the product copula. Extremes are then not correlated. This result is not surprising because the Gumbel-Houggard copula is equal to the product copula: e [ ln u1)1 + ln u ) 1 ] 1 u 1 u C u 1, u ) 10

11 5. a) i. We have: G x 1, x ) C G x 1 ), G x )) Λ x 1 ) Ψ 1 x 1) ii. We have: G x 1, x ) C G x 1 ), G x )) Λ x 1 ) Φ α 1 + x ) α b) We know that the upper tail dependence is equal to zero for the Gaussian copula if ρ < 1. We then obtain exactly the same results as previously: i. ii. G x 1, x ) exp e x1 + x 1 ) G x 1, x ) exp e x1 1 + x ) ) α α c) If ρ 1, the Gaussian copula is the upper Fréchet copula C +, which is an EV copula. We deduce that: i. G x 1, x ) min Λ x 1 ), Ψ 1 x 1)) ii. G x 1, x ) min Λ x 1 ), Φ α 1 + x )) α d) We have shown that the Gumbel-Houggard copula is an EV copula. i. G x 1, x ) e [ ln Λx1))θ + ln Ψ 1x 1)) θ ] 1/θ exp [e θx1 + 1 x ) θ] ) 1/θ ii. [ G x 1, x ) e ln Λx 1)) θ + ln Φ α1+ x α )) θ] 1/θ ) ] ) αθ 1/θ exp [ e θx x α 11