Value at Risk: Quantifying Overall Net Market Risk


 Marcus Bennett
 2 years ago
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1 Overview Chapter 14 : Quantifying Overall Net Market Risk
2 Overview Overview a Factorbased, Linear Approach Why work with factors not assets Linking Bonds to Interestrate Factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A ZeroDrift ( Martingale ) Process A ConstantVariance 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 Factorbased, Linear Approach Why work with factors not assets Linking Bonds to Interestrate Factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A ZeroDrift ( Martingale ) Process A ConstantVariance 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 Factorbased, Linear Approach Why work with factors not assets Linking Bonds to Interestrate Factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A ZeroDrift ( Martingale ) Process A ConstantVariance 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 Factorbased, Linear Approach Why work with factors not assets Linking Bonds to Interestrate Factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A ZeroDrift ( Martingale ) Process A ConstantVariance 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 nonnormal computergenerated 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 nonnormal computergenerated 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 nonnormal computergenerated factors, mapped into returns (Monte Carlo)
9 Outline Why factors not assets? The interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget a Factorbased, Linear Approach Why work with factors not assets Linking Bonds to Interestrate Factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A ZeroDrift ( Martingale ) Process A ConstantVariance 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 interestrate factors Stockmarket 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 1day 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 factorcovariance/normality solution: map return as functions of far fewer underlying factors
11 Why? Why factors not assets? The interestrate factors Stockmarket 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 1day 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 factorcovariance/normality solution: map return as functions of far fewer underlying factors
12 More on sources of risk and factors) Why factors not assets? The interestrate factors Stockmarket 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 sourceofrisk 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 5year FC bond: V = V S so V V with V V V V a function of the foreign 5year interest rate + S S
13 More on sources of risk and factors) Why factors not assets? The interestrate factors Stockmarket 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 sourceofrisk 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 5year FC bond: V = V S so V V with V V V V a function of the foreign 5year interest rate + S S
14 More on sources of risk and factors) Why factors not assets? The interestrate factors Stockmarket 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 sourceofrisk 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 5year FC bond: V = V S so V V with V V V V a function of the foreign 5year interest rate + S S
15 Portfolio return as a function of asset returns Why factors not assets? The interestrate factors Stockmarket 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 interestrate factors Stockmarket 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 } pseudoinv in j via i, E i,j MX E i,j fj +... j=1 f dx j X j {z} factor V i dt + higherorder terms t + V i t dt {z } (not risky) +..., Portfolio then behaves as series of pseudoinvestments 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 interestrate factors Stockmarket 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 } pseudoinv in j via i, E i,j MX E i,j fj +... j=1 f dx j X j {z} factor V i dt + higherorder terms t + V i t dt {z } (not risky) +..., Portfolio then behaves as series of pseudoinvestments 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 Interestrate factors 1 Why factors not assets? The interestrate factors Stockmarket 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 3year 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 Interestrate factors 1 Why factors not assets? The interestrate factors Stockmarket 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 3year 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 Interestrate Factors 2 Why factors not assets? The interestrate factors Stockmarket 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: 8mo 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 Interestrate Factors 2 Why factors not assets? The interestrate factors Stockmarket 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: 8mo 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 Interestrate Factors 2 Why factors not assets? The interestrate factors Stockmarket 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: 8mo 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 Interestrate Factors 3 Why factors not assets? The interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget Step 3: Link dr t,t /R t,t to official factors: The 8mo rate is not in the riskmetrics database. So link it, via (non?)linear interpolation, to those that are. Example: 8mo rate linked to 6 and 9mo 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 Interestrate Factors 3 Why factors not assets? The interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget Step 3: Link dr t,t /R t,t to official factors: The 8mo rate is not in the riskmetrics database. So link it, via (non?)linear interpolation, to those that are. Example: 8mo rate linked to 6 and 9mo 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 Interestrate Factors 3 Why factors not assets? The interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget Step 3: Link dr t,t /R t,t to official factors: The 8mo rate is not in the riskmetrics database. So link it, via (non?)linear interpolation, to those that are. Example: 8mo rate linked to 6 and 9mo 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 Stockmarket Risk Why factors not assets? The interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget General: Topdown approach: the investment in countrya 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 Stockmarket Risk Why factors not assets? The interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget General: Topdown approach: the investment in countrya 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 Stockmarket Risk Why factors not assets? The interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget General: Topdown approach: the investment in countrya 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 Stockmarket Risk Why factors not assets? The interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget General: Topdown approach: the investment in countrya 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 interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget Decompose into three firstpass 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 riskfree rate for T, and the exchange rate Liability: you wrote a PN ad HC 1.25m, whose value depends on the HC riskfree 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 interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget Decompose into three firstpass 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 riskfree rate for T, and the exchange rate Liability: you wrote a PN ad HC 1.25m, whose value depends on the HC riskfree 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 interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget Decompose into three firstpass 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 riskfree rate for T, and the exchange rate Liability: you wrote a PN ad HC 1.25m, whose value depends on the HC riskfree 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 interestrate factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget Decompose into three firstpass 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 riskfree rate for T, and the exchange rate Liability: you wrote a PN ad HC 1.25m, whose value depends on the HC riskfree 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 interestrate factors Stockmarket 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 interestrate factors Stockmarket 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 interestrate factors Stockmarket 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 Fixedrate bond: behaves like a portfolio of PNs Possibly Xrisk
37 Swaps Why factors not assets? The interestrate factors Stockmarket 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 Fixedrate bond: behaves like a portfolio of PNs Possibly Xrisk
38 The Risk Budget Why factors not assets? The interestrate factors Stockmarket 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 elasticitycorrected 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 interestrate factors Stockmarket 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 elasticitycorrected 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 interestrate factors Stockmarket 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 10year 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 interestrate factors Stockmarket 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 10year 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 interestrate factors Stockmarket 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 10year 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 interestrate factors Stockmarket 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 pseudoinvestments E portfolio risk
44 Outline Zero Drift Processes A ConstantVariance 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 Factorbased, Linear Approach Why work with factors not assets Linking Bonds to Interestrate Factors Stockmarket Risk Currency Forwards Options Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A ZeroDrift ( Martingale ) Process A ConstantVariance 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 ConstantVariance 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 ZeroDrift ( Martingale ) Processes Zero Drift Processes A ConstantVariance 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 errorcorrectiontypelinks between vars, e.g. various Rs Evaluation dr 1 = λ(r 2 R 1 ν 1,2) ɛ not a serious problem at oneday or week horizon
47 ZeroDrift ( Martingale ) Processes Zero Drift Processes A ConstantVariance 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 errorcorrectiontypelinks between vars, e.g. various Rs Evaluation dr 1 = λ(r 2 R 1 ν 1,2) ɛ not a serious problem at oneday or week horizon
48 ZeroDrift ( Martingale ) Processes Zero Drift Processes A ConstantVariance 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 errorcorrectiontypelinks between vars, e.g. various Rs Evaluation dr 1 = λ(r 2 R 1 ν 1,2) ɛ not a serious problem at oneday or week horizon
49 ZeroDrift ( Martingale ) Processes Zero Drift Processes A ConstantVariance 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 errorcorrectiontypelinks between vars, e.g. various Rs Evaluation dr 1 = λ(r 2 R 1 ν 1,2) ɛ not a serious problem at oneday or week horizon
50 ConstantVariance Processes Zero Drift Processes A ConstantVariance 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 intrabandchanges (whose distributions depend on the position in the band), and jumps (realignments) whose chances are timevarying stock and bond prices are also subject to jump risk (e.g. crashes) with timevarying probabilties. asset markets experience waves of nervousness/sleepiness: variance has many possible levels instead of just two??
51 ConstantVariance Processes Zero Drift Processes A ConstantVariance 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 intrabandchanges (whose distributions depend on the position in the band), and jumps (realignments) whose chances are timevarying stock and bond prices are also subject to jump risk (e.g. crashes) with timevarying 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 ConstantVariance 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 ConstantVariance 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 ConstantVariance 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 ConstantVariance 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 exponentialdecay model standard (weight is 1/59= or 0) Exponential, 1psi=0.97 exponential. 1psi=0.94 age of observation
56 Other Corrections for Nonconstant Variance Zero Drift Processes A ConstantVariance 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 bidask Correction Step 2: Covariances GARCHlike 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 Nonconstant Variance Zero Drift Processes A ConstantVariance 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 bidask Correction Step 2: Covariances GARCHlike 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 Nonconstant Variance Zero Drift Processes A ConstantVariance 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 bidask Correction Step 2: Covariances GARCHlike 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 TGarch(1,1) plots (MY, RS, AR); GARCH(1,1) (AR) Zero Drift Processes A ConstantVariance 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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