Value at Risk: Quantifying Overall Net Market Risk

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1 Overview Chapter 14 : Quantifying Overall Net Market Risk

2 Overview Overview a Factor-based, Linear Approach Why work with factors not assets Linking Bonds to Interest-rate Factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A Zero-Drift ( Martingale ) Process A Constant-Variance Process Constant Correlations Between Factors. Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing CDOs/CDSs are hard to price Moral Hazard type risks

3 Overview Overview a Factor-based, Linear Approach Why work with factors not assets Linking Bonds to Interest-rate Factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A Zero-Drift ( Martingale ) Process A Constant-Variance Process Constant Correlations Between Factors. Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing CDOs/CDSs are hard to price Moral Hazard type risks

4 Overview Overview a Factor-based, Linear Approach Why work with factors not assets Linking Bonds to Interest-rate Factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A Zero-Drift ( Martingale ) Process A Constant-Variance Process Constant Correlations Between Factors. Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing CDOs/CDSs are hard to price Moral Hazard type risks

5 Overview Overview a Factor-based, Linear Approach Why work with factors not assets Linking Bonds to Interest-rate Factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A Zero-Drift ( Martingale ) Process A Constant-Variance Process Constant Correlations Between Factors. Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing CDOs/CDSs are hard to price Moral Hazard type risks

6 What are we after? (in this chapter) VaR = maximal loss that could be suffered on the current portfolio with 99% confidence VaR: losses worse than VaR should occur only one day in 100 basis for a bank s Capital Requirement calculations ( Market Risk Charge ), which is set at three times VaR or more can be computed... from a normality model if σ p is known with normal factors, linearly related to returns (Riskmetrics) from a reconstructed history of portfolio returns (backtesting) from a simulated series of possible future events with... resampled actual returns (bootstrapping), or normal or non-normal computer-generated factors, mapped into returns (Monte Carlo)

7 What are we after? (in this chapter) VaR = maximal loss that could be suffered on the current portfolio with 99% confidence VaR: losses worse than VaR should occur only one day in 100 basis for a bank s Capital Requirement calculations ( Market Risk Charge ), which is set at three times VaR or more can be computed... from a normality model if σ p is known with normal factors, linearly related to returns (Riskmetrics) from a reconstructed history of portfolio returns (backtesting) from a simulated series of possible future events with... resampled actual returns (bootstrapping), or normal or non-normal computer-generated factors, mapped into returns (Monte Carlo)

8 What are we after? (in this chapter) VaR = maximal loss that could be suffered on the current portfolio with 99% confidence VaR: losses worse than VaR should occur only one day in 100 basis for a bank s Capital Requirement calculations ( Market Risk Charge ), which is set at three times VaR or more can be computed... from a normality model if σ p is known with normal factors, linearly related to returns (Riskmetrics) from a reconstructed history of portfolio returns (backtesting) from a simulated series of possible future events with... resampled actual returns (bootstrapping), or normal or non-normal computer-generated factors, mapped into returns (Monte Carlo)

9 Outline Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget a Factor-based, Linear Approach Why work with factors not assets Linking Bonds to Interest-rate Factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A Zero-Drift ( Martingale ) Process A Constant-Variance Process Constant Correlations Between Factors. Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing CDOs/CDSs are hard to price Moral Hazard type risks

10 Why? Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Suppose we know std( r p ): VaR follows easily: Example data: current value 100m, expected return 10%, stdev 30%, both p.a. all assets are liquid horizon is 1 trading day, i.e. 1/260 year Computations expected value tomorrow: 100m = m 260 variance is linear in time, so the 1-day std is How to get std? q 100m = 1.9m maximal loss below (with 99% confidence): 1.9m 2.33 = 4.13m over one day portfolio theory: need full varcov matrix of all assets Need Nobs N of assets, otherwise linear dependencies factor-covariance/normality solution: map return as functions of far fewer underlying factors

11 Why? Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Suppose we know std( r p ): VaR follows easily: Example data: current value 100m, expected return 10%, stdev 30%, both p.a. all assets are liquid horizon is 1 trading day, i.e. 1/260 year Computations expected value tomorrow: 100m = m 260 variance is linear in time, so the 1-day std is How to get std? q 100m = 1.9m maximal loss below (with 99% confidence): 1.9m 2.33 = 4.13m over one day portfolio theory: need full varcov matrix of all assets Need Nobs N of assets, otherwise linear dependencies factor-covariance/normality solution: map return as functions of far fewer underlying factors

12 More on sources of risk and factors) Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Sources of uncertainty: (in standard software) exchange risks stock price risks (by country, in local currency) interest rate risk (by currency and time to maturity; say 13 rates) commodity price risk A factor: the unexpected percentage change in the source-of-risk variable. J.P. Morgan s Riskmetrics (C) provides a covariance matrix for hundreds of these factors. Mapping: link between factor and asset return E.g. a 5-year FC bond: V = V S so V V with V V V V a function of the foreign 5-year interest rate + S S

13 More on sources of risk and factors) Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Sources of uncertainty: (in standard software) exchange risks stock price risks (by country, in local currency) interest rate risk (by currency and time to maturity; say 13 rates) commodity price risk A factor: the unexpected percentage change in the source-of-risk variable. J.P. Morgan s Riskmetrics (C) provides a covariance matrix for hundreds of these factors. Mapping: link between factor and asset return E.g. a 5-year FC bond: V = V S so V V with V V V V a function of the foreign 5-year interest rate + S S

14 More on sources of risk and factors) Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Sources of uncertainty: (in standard software) exchange risks stock price risks (by country, in local currency) interest rate risk (by currency and time to maturity; say 13 rates) commodity price risk A factor: the unexpected percentage change in the source-of-risk variable. J.P. Morgan s Riskmetrics (C) provides a covariance matrix for hundreds of these factors. Mapping: link between factor and asset return E.g. a 5-year FC bond: V = V S so V V with V V V V a function of the foreign 5-year interest rate + S S

15 Portfolio return as a function of asset returns Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Portfolio return: dv p = = NX n i,t (P i,t+1 P i,t ) i=1 NX i=1 n i,t P i,t {z } initial capital invested in i P i,t+1 P i,t P i,t {z } return on i. Example (a) (b) (c)=(a) (b) (d) (e)=(a) (d) (e) (c) (e) asset n i,t P i,t initial capital P i,t+1 final capital return cap Nikkei 1,000 13,000 13,000,000 13,650 13,650, ,000 Copper ,000 18,000, ,800, ,800,000 Total 31,000,000 33,450,000 2,450,000

16 Asset return as a function of factors Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Individual asset i as a function of factors f j, j = 1,..., M: fdv i = = = MX V i X j f dxj + j=1 MX V i V i X j {z} X j=1 j V i {z } elasticity w.r.t.x j inv. {z } pseudo-inv in j via i, E i,j MX E i,j fj +... j=1 f dx j X j {z} factor V i dt + higher-order terms t + V i t dt {z } (not risky) +..., Portfolio then behaves as series of pseudo-investments E p,j in factors j: fdv p = NX MX NX MX n idvi f +... = n i E i,j fj +... = E p,j fj +... i=1 j=1 i=1 {z } j=1 =E p,j

17 Asset return as a function of factors Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Individual asset i as a function of factors f j, j = 1,..., M: fdv i = = = MX V i X j f dxj + j=1 MX V i V i X j {z} X j=1 j V i {z } elasticity w.r.t.x j inv. {z } pseudo-inv in j via i, E i,j MX E i,j fj +... j=1 f dx j X j {z} factor V i dt + higher-order terms t + V i t dt {z } (not risky) +..., Portfolio then behaves as series of pseudo-investments E p,j in factors j: fdv p = NX MX NX MX n idvi f +... = n i E i,j fj +... = E p,j fj +... i=1 j=1 i=1 {z } j=1 =E p,j

18 Linking Bonds to Interest-rate factors 1 Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Step 1: Strip the bonds decompose every individual bond or loan into a replicating package of promissory notes. Example A 3-year 6% bond paying out 1m with first coupon date in 8 months boils down to one promissory note (PN) ad 60,000, 8 month one PN ad 60,000, 20 month one PN ad 1060,000, 32 month

19 Linking Bonds to Interest-rate factors 1 Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Step 1: Strip the bonds decompose every individual bond or loan into a replicating package of promissory notes. Example A 3-year 6% bond paying out 1m with first coupon date in 8 months boils down to one promissory note (PN) ad 60,000, 8 month one PN ad 60,000, 20 month one PN ad 1060,000, 32 month

20 Linking Bonds to Interest-rate Factors 2 Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Step 2: track dv t to dr/r V t = V T (1 + R t,t) (T t), dv t = (T t) V T (1 + R t,t) (T t) 1 dr t,t, Modif z Dur } { = T t 1 + R t,t R t,t {z } elasticity V T (1 + R t,t) T t {z } = V t dr t,t R t,t Example: 8-mo PN; V T = 100K, R 8mo = 0.03 p.a. Value V t = 100, 000/(1.03) 2/3 = Duration = (2/3)/1.03 = Elasticity = = dv = ( ) {z } dr 8mo R 8mo

21 Linking Bonds to Interest-rate Factors 2 Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Step 2: track dv t to dr/r V t = V T (1 + R t,t) (T t), dv t = (T t) V T (1 + R t,t) (T t) 1 dr t,t, Modif z Dur } { = T t 1 + R t,t R t,t {z } elasticity V T (1 + R t,t) T t {z } = V t dr t,t R t,t Example: 8-mo PN; V T = 100K, R 8mo = 0.03 p.a. Value V t = 100, 000/(1.03) 2/3 = Duration = (2/3)/1.03 = Elasticity = = dv = ( ) {z } dr 8mo R 8mo

22 Linking Bonds to Interest-rate Factors 2 Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Step 2: track dv t to dr/r V t = V T (1 + R t,t) (T t), dv t = (T t) V T (1 + R t,t) (T t) 1 dr t,t, Modif z Dur } { = T t 1 + R t,t R t,t {z } elasticity V T (1 + R t,t) T t {z } = V t dr t,t R t,t Example: 8-mo PN; V T = 100K, R 8mo = 0.03 p.a. Value V t = 100, 000/(1.03) 2/3 = Duration = (2/3)/1.03 = Elasticity = = dv = ( ) {z } dr 8mo R 8mo

23 Linking Bonds to Interest-rate Factors 3 Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Step 3: Link dr t,t /R t,t to official factors: The 8-mo rate is not in the riskmetrics database. So link it, via (non?)linear interpolation, to those that are. Example: 8-mo rate linked to 6- and 9-mo ones Simplest solution: geometric interpolation: R 8mo = R x 6moR 1 x 9mo find x if R 8mo is observed, set x = 1/3 if R 8mo must be calculated. Implication: dr 8mo R 8mo = x dr 6mo R 8mo + (1 x) dr 9mo R 9mo Value impact in previous example dv = x dr6mo R 8mo «+ (1 x) dr9mo R 9mo dr 6mo = x (1 x) {z } R 8mo {z } E to 6mo rate E to 9mo rate dr 9mo R 9mo

24 Linking Bonds to Interest-rate Factors 3 Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Step 3: Link dr t,t /R t,t to official factors: The 8-mo rate is not in the riskmetrics database. So link it, via (non?)linear interpolation, to those that are. Example: 8-mo rate linked to 6- and 9-mo ones Simplest solution: geometric interpolation: R 8mo = R x 6moR 1 x 9mo find x if R 8mo is observed, set x = 1/3 if R 8mo must be calculated. Implication: dr 8mo R 8mo = x dr 6mo R 8mo + (1 x) dr 9mo R 9mo Value impact in previous example dv = x dr6mo R 8mo «+ (1 x) dr9mo R 9mo dr 6mo = x (1 x) {z } R 8mo {z } E to 6mo rate E to 9mo rate dr 9mo R 9mo

25 Linking Bonds to Interest-rate Factors 3 Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Step 3: Link dr t,t /R t,t to official factors: The 8-mo rate is not in the riskmetrics database. So link it, via (non?)linear interpolation, to those that are. Example: 8-mo rate linked to 6- and 9-mo ones Simplest solution: geometric interpolation: R 8mo = R x 6moR 1 x 9mo find x if R 8mo is observed, set x = 1/3 if R 8mo must be calculated. Implication: dr 8mo R 8mo = x dr 6mo R 8mo + (1 x) dr 9mo R 9mo Value impact in previous example dv = x dr6mo R 8mo «+ (1 x) dr9mo R 9mo dr 6mo = x (1 x) {z } R 8mo {z } E to 6mo rate E to 9mo rate dr 9mo R 9mo

26 Stock-market Risk Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget General: Top-down approach: the investment in country-a stocks is assumed to be a position in the country s market index, M. So elasticity (or beta) 1: no correction to mkt values in HC is needed Domestic Stocks dv = V dv V and dv V dm M V dm M Foreign Stocks: exposed to both M and S: dv V V = V S dv V = dv = V dv V V dm M + ds S dm M + ds S + V ds S V

27 Stock-market Risk Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget General: Top-down approach: the investment in country-a stocks is assumed to be a position in the country s market index, M. So elasticity (or beta) 1: no correction to mkt values in HC is needed Domestic Stocks dv = V dv V and dv V dm M V dm M Foreign Stocks: exposed to both M and S: dv V V = V S dv V = dv = V dv V V dm M + ds S dm M + ds S + V ds S V

28 Stock-market Risk Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget General: Top-down approach: the investment in country-a stocks is assumed to be a position in the country s market index, M. So elasticity (or beta) 1: no correction to mkt values in HC is needed Domestic Stocks dv = V dv V and dv V dm M V dm M Foreign Stocks: exposed to both M and S: dv V V = V S dv V = dv = V dv V V dm M + ds S dm M + ds S + V ds S V

29 Stock-market Risk Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget General: Top-down approach: the investment in country-a stocks is assumed to be a position in the country s market index, M. So elasticity (or beta) 1: no correction to mkt values in HC is needed Domestic Stocks dv = V dv V and dv V dm M V dm M Foreign Stocks: exposed to both M and S: dv V V = V S dv V = dv = V dv V V dm M + ds S dm M + ds S + V ds S V

30 Currency Forwards Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Decompose into three first-pass factors: a forward purchase of ISK 100m at for delivery at time T boils down to a portfolio of two PNs: Asset: your claim on the bank = the bank s PN ad 100m Kronar, whose market value depends on the ISK risk-free rate for T, and the exchange rate Liability: you wrote a PN ad HC 1.25m, whose value depends on the HC risk-free rate for date T Bottom line: five exposures two adjacent domestic interest rates two adjacent foreign interest rates exchange rate ISK

31 Currency Forwards Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Decompose into three first-pass factors: a forward purchase of ISK 100m at for delivery at time T boils down to a portfolio of two PNs: Asset: your claim on the bank = the bank s PN ad 100m Kronar, whose market value depends on the ISK risk-free rate for T, and the exchange rate Liability: you wrote a PN ad HC 1.25m, whose value depends on the HC risk-free rate for date T Bottom line: five exposures two adjacent domestic interest rates two adjacent foreign interest rates exchange rate ISK

32 Currency Forwards Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Decompose into three first-pass factors: a forward purchase of ISK 100m at for delivery at time T boils down to a portfolio of two PNs: Asset: your claim on the bank = the bank s PN ad 100m Kronar, whose market value depends on the ISK risk-free rate for T, and the exchange rate Liability: you wrote a PN ad HC 1.25m, whose value depends on the HC risk-free rate for date T Bottom line: five exposures two adjacent domestic interest rates two adjacent foreign interest rates exchange rate ISK

33 Currency Forwards Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Decompose into three first-pass factors: a forward purchase of ISK 100m at for delivery at time T boils down to a portfolio of two PNs: Asset: your claim on the bank = the bank s PN ad 100m Kronar, whose market value depends on the ISK risk-free rate for T, and the exchange rate Liability: you wrote a PN ad HC 1.25m, whose value depends on the HC risk-free rate for date T Bottom line: five exposures two adjacent domestic interest rates two adjacent foreign interest rates exchange rate ISK

34 Options on Stocks or Forex Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Replication: in the short run, a currency option behaves like a portfolio of forex PN domestic PN C = S t (1 + R t,t )T t } {{ } HC price FC PN # of FC PN s you hold { }} { N(d 1 ) X (1 + R t,t ) T t } {{ } price HC PN face X # of HC PN s you hold { }} { N(d 2 ) Similar to forward purchase except for N(d i ) 1. Bottom line: five exposures two adjacent domestic interest rates two adjacent foreign interest rates exchange rate ISK

35 Options on Stocks or Forex Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Replication: in the short run, a currency option behaves like a portfolio of forex PN domestic PN C = S t (1 + R t,t )T t } {{ } HC price FC PN # of FC PN s you hold { }} { N(d 1 ) X (1 + R t,t ) T t } {{ } price HC PN face X # of HC PN s you hold { }} { N(d 2 ) Similar to forward purchase except for N(d i ) 1. Bottom line: five exposures two adjacent domestic interest rates two adjacent foreign interest rates exchange rate ISK

36 Swaps Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Replication: a swap is like an exchange of two bonds that differ in terms of interest rate: fixed, floating currency of denomination both FRN bond: behaves like a PN expiring at first coupon date Fixed-rate bond: behaves like a portfolio of PNs Possibly Xrisk

37 Swaps Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Replication: a swap is like an exchange of two bonds that differ in terms of interest rate: fixed, floating currency of denomination both FRN bond: behaves like a PN expiring at first coupon date Fixed-rate bond: behaves like a portfolio of PNs Possibly Xrisk

38 The Risk Budget Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Portfolio s sensitivity to factors is obtained by summing: dv p = E 1,1f 1 + E 1,2f E 1,Mf M + E 2,1f 1 + E 2,2f E 2,Mf M + E 3,1f 1 + E 3,2f E 3,Mf M... + E n,1f 1 + E n,2f E n,mf M " nx # " nx # " nx # = E i,1 f 1 + E i,2 f E i,m i=1 i=1 = E p,1f 1 + E p,2f E p,mf M Portfolio variance follows the usual formula except that we use factors not returns, and elasticity-corrected amounts not amounts: var(dv p) = MX j=1 E j,i M X i 1 E i,j cov(f i, f j) i=1 f M

39 The Risk Budget Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Portfolio s sensitivity to factors is obtained by summing: dv p = E 1,1f 1 + E 1,2f E 1,Mf M + E 2,1f 1 + E 2,2f E 2,Mf M + E 3,1f 1 + E 3,2f E 3,Mf M... + E n,1f 1 + E n,2f E n,mf M " nx # " nx # " nx # = E i,1 f 1 + E i,2 f E i,m i=1 i=1 = E p,1f 1 + E p,2f E p,mf M Portfolio variance follows the usual formula except that we use factors not returns, and elasticity-corrected amounts not amounts: var(dv p) = MX j=1 E j,i M X i 1 E i,j cov(f i, f j) i=1 f M

40 The Risk Budget example Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Example: Three positions: (i) domestic stock worth HC 150; (ii) foreign stock worth HC 200; and (iii) a 10-year forward sale worth, in PV, HC 100 each leg. Let R 10 = 5% and R 10 = 4%. stock stock Xrate R 10 R 10 elasticities home stock foreign stock forward sale pseudo capital home stock foreign stock forward sale portfolio dv p 150 r stock r stock ds S dr dr 10. R 10 R 10

41 The Risk Budget example Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Example: Three positions: (i) domestic stock worth HC 150; (ii) foreign stock worth HC 200; and (iii) a 10-year forward sale worth, in PV, HC 100 each leg. Let R 10 = 5% and R 10 = 4%. stock stock Xrate R 10 R 10 elasticities home stock foreign stock forward sale pseudo capital home stock foreign stock forward sale portfolio dv p 150 r stock r stock ds S dr dr 10. R 10 R 10

42 The Risk Budget example Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget Example: Three positions: (i) domestic stock worth HC 150; (ii) foreign stock worth HC 200; and (iii) a 10-year forward sale worth, in PV, HC 100 each leg. Let R 10 = 5% and R 10 = 4%. stock stock Xrate R 10 R 10 elasticities home stock foreign stock forward sale pseudo capital home stock foreign stock forward sale portfolio dv p 150 r stock r stock ds S dr dr 10. R 10 R 10

43 VaR: the arrows Portfolio & the risk: boxes summary 2. Sources of uncertainty, and Why factors not assets? The interest-rate factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget back office information systems, www riskmetrics, or statistical department varcov matrix number, and characteristics per asset current prices, rates, etc middle office financial models current values elasticities pseudo-investments E portfolio risk

44 Outline Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day a Factor-based, Linear Approach Why work with factors not assets Linking Bonds to Interest-rate Factors Stock-market Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A Zero-Drift ( Martingale ) Process A Constant-Variance Process Constant Correlations Between Factors. Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing CDOs/CDSs are hard to price Moral Hazard type risks

45 Potential Flaws & Corrections Overview Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day 1. Intertemporal independence of changes in the levels of prices or interest rates 2. Constant distribution of percentage changes in the levels of prices or interest rates 3. Constant linear relationships between the changes in the levels of prices or interest rates 4. Linearizations of links between underlying variables and between asset prices and factors 5. Choice of factors: in some respects too many, in other respects too few factors 6. Normality of the portfolio value 7. Liquidity.

46 Zero-Drift ( Martingale ) Processes Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Assumption 1: f i fully unpredictable, apart from (tiny) drift Counterexamples mean reversion in R; in S; in goods prices; even in stocks dr = κ(µ R R) ɛ Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day error-correction-typelinks between vars, e.g. various Rs Evaluation dr 1 = λ(r 2 R 1 ν 1,2) ɛ not a serious problem at one-day or -week horizon

47 Zero-Drift ( Martingale ) Processes Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Assumption 1: f i fully unpredictable, apart from (tiny) drift Counterexamples mean reversion in R; in S; in goods prices; even in stocks dr = κ(µ R R) ɛ Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day error-correction-typelinks between vars, e.g. various Rs Evaluation dr 1 = λ(r 2 R 1 ν 1,2) ɛ not a serious problem at one-day or -week horizon

48 Zero-Drift ( Martingale ) Processes Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Assumption 1: f i fully unpredictable, apart from (tiny) drift Counterexamples mean reversion in R; in S; in goods prices; even in stocks dr = κ(µ R R) ɛ Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day error-correction-typelinks between vars, e.g. various Rs Evaluation dr 1 = λ(r 2 R 1 ν 1,2) ɛ not a serious problem at one-day or -week horizon

49 Zero-Drift ( Martingale ) Processes Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Assumption 1: f i fully unpredictable, apart from (tiny) drift Counterexamples mean reversion in R; in S; in goods prices; even in stocks dr = κ(µ R R) ɛ Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day error-correction-typelinks between vars, e.g. various Rs Evaluation dr 1 = λ(r 2 R 1 ν 1,2) ɛ not a serious problem at one-day or -week horizon

50 Constant-Variance Processes Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Assumption 2: var t (f i,t+1 ) is constant so that variance is easily estimated, and risk increases linearly in length of horizon T t Counterexamples managed Xrates: changes are a mixture of intra-band-changes (whose distributions depend on the position in the band), and jumps (re-alignments) whose chances are time-varying stock and bond prices are also subject to jump risk (e.g. crashes) with time-varying probabilties. asset markets experience waves of nervousness/sleepiness: variance has many possible levels instead of just two??

51 Constant-Variance Processes Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Assumption 2: var t (f i,t+1 ) is constant so that variance is easily estimated, and risk increases linearly in length of horizon T t Counterexamples managed Xrates: changes are a mixture of intra-band-changes (whose distributions depend on the position in the band), and jumps (re-alignments) whose chances are time-varying stock and bond prices are also subject to jump risk (e.g. crashes) with time-varying probabilties. asset markets experience waves of nervousness/sleepiness: variance has many possible levels instead of just two??

52 The GARCH correction Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Correction Step 1 option 1: GARCH models: the risk we expect for today depends on... [AR:] how nervous we felt yesterday morning v average day: [MA:] to what extent we were surprised last night v i/t morning var t = var + χ [var t 1 var] + ψ [ɛ 2 t 1 var t 1] {z } {z } morning evening = (1 χ) var + (χ ψ) var t 1 + ψ ɛ 2 t 1 = (1 φ ψ) var + φ var t 1 + ψ ɛ 2 t 1, φ = χ ψ. {z } {z } AR MA Generalisation to GARCH(p,q): px qx var t = var + φ i var t i + ψ j ɛ 2 t j i 1 j=1

53 The GARCH correction Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Correction Step 1 option 1: GARCH models: the risk we expect for today depends on... [AR:] how nervous we felt yesterday morning v average day: [MA:] to what extent we were surprised last night v i/t morning var t = var + χ [var t 1 var] + ψ [ɛ 2 t 1 var t 1] {z } {z } morning evening = (1 χ) var + (χ ψ) var t 1 + ψ ɛ 2 t 1 = (1 φ ψ) var + φ var t 1 + ψ ɛ 2 t 1, φ = χ ψ. {z } {z } AR MA Generalisation to GARCH(p,q): px qx var t = var + φ i var t i + ψ j ɛ 2 t j i 1 j=1

54 RiskMetrics Choice: GARCH(0,1) Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day standard statistics: constant weight, 1/Nobs GARCH(0,1): exponentially decaying weight, ψ(1 ψ) n i, i = 1,..., n: var t = (1 ψ) var t 1 + ψ ɛ 2 t 1 = (1 ψ){(1 ψ) var t 2 + ψ ɛ 2 t 2} + ψ ɛ 2 t 1 = (1 ψ){(1 ψ)[(1 ψ) var t 3 + ψ ɛ 2 t 3] + ψ ɛ 2 t 2} + ψ ɛ 2 t 1 =... = ψ [ɛ 2 t 1 + (1 ψ)ɛ 2 t 2 + (1 ψ) 2 ɛ 2 t (1 ψ) n 1 ɛ 2 t n] + (1 ψ) n {z } 0 var t n, RiskMetrics: n i daily, n i monthly:

55 RiskMetrics Choice: the Movie Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day weight for observation weights in a exponential-decay model standard (weight is 1/59= or 0) Exponential, 1-psi=0.97 exponential. 1-psi=0.94 age of observation

56 Other Corrections for Non-constant Variance Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Correction Step 1 option 2: ISD instead of GARCH smirks: as many ISDs are there are T t s and X/S s Deep ITM/OTM: sensitive to bid-ask Correction Step 2: Covariances GARCH-like models for covs: require many parameters or heavy restrictions CCC: constant conditional correlation: cov t(f i, f j) = σ i,tσ j,tρ(f i, f j) Correction Step 3: add risk as a factor variance is needed not just for varcov(f i, f j) but also as a pricing factor in derivatives options: vega bonds: convexity

57 Other Corrections for Non-constant Variance Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Correction Step 1 option 2: ISD instead of GARCH smirks: as many ISDs are there are T t s and X/S s Deep ITM/OTM: sensitive to bid-ask Correction Step 2: Covariances GARCH-like models for covs: require many parameters or heavy restrictions CCC: constant conditional correlation: cov t(f i, f j) = σ i,tσ j,tρ(f i, f j) Correction Step 3: add risk as a factor variance is needed not just for varcov(f i, f j) but also as a pricing factor in derivatives options: vega bonds: convexity

58 Other Corrections for Non-constant Variance Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Correction Step 1 option 2: ISD instead of GARCH smirks: as many ISDs are there are T t s and X/S s Deep ITM/OTM: sensitive to bid-ask Correction Step 2: Covariances GARCH-like models for covs: require many parameters or heavy restrictions CCC: constant conditional correlation: cov t(f i, f j) = σ i,tσ j,tρ(f i, f j) Correction Step 3: add risk as a factor variance is needed not just for varcov(f i, f j) but also as a pricing factor in derivatives options: vega bonds: convexity

59 T-Garch(1,1) plots (MY, RS, AR); GARCH(1,1) (AR) Zero Drift Processes A Constant-Variance Process Constant Correlations Between Factors Linearizations in the Mapping f dv Choice of the factors Normality of dv Assets can be Liquidated in one Day Conditional variance

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