Risk Management and Financial Institutions

Transcription

1 Rsk Management and Fnancal Insttutons By John C. Hull Chapter 3 How Traders manage Ther Exposures... Chapter 4 Interest Rate Rsk...3 Chapter 5 Volatlty...5 Chapter 6 Correlatons and Copulas...7 Chapter 7 Bank Regulaton and Basel II...9 Chapter 8 The VaR Measure... Chapter 9 Market Rsk VaR: Hstorcal Smulaton Approach...4 Chapter 0 Chapter Market Rsk VaR: Model-Buldng Approach...6 Credt Rsk: Estmatng Default Probabltes...7 Chapter Credt Rsk Losses and Credt VaR...0 Chapter 3 Credt Dervatves... Chapter 4 Operatonal Rsk...4 Chapter 5 Model Rsk and Lqudty Rsk...5 Chapter 7 Weather, Energy, and Insurance Dervatves...7 Chapter 8 Bg Losses and What We Can Learn From Them...8 T Bootstrap...30 T T3 Prncpal Component Analyss...30 Monte Carlo Smulaton Methods

2 Chapter 3 How Traders manage Ther Exposures. Lnear products: a product whose value s lnearly dependent on the value of the underlyng asset prce. Forward, futures, and swaps are lnear products; optons are not. E.g. Goldman Sachs have entered nto a forward wth a gold mnng frm. Goldman Sachs borrows gold from a central bank and sell t at the current market prce. At the end of the lfe of the forward, Goldman Sachs buys gold from the gold mnng frm and uses t to repay the central bank.. Delta neutralty s more feasble for a large portfolo of dervatves dependent on a sngle asset. Only one trade n the underlyng asset s necessary to zero out delta for the whole portfolo. 3. Gamma: f t s small, delta changes slowly and adjustments to keep a portfolo delta neutral only need to be made relatvely nfrequently. Gamma = Π S - Gamma s postve for a long poston n an opton (call or put). - A lnear product has zero Gamma and cannot be used to change the gamma of a portfolo. 4. Vega - Spot postons, forwards, and swaps do not depend on the volatlty of the underlyng market varable, but optons and most exotcs do. Π - ν = σ - Vega s postve for long call and put; - The volatltes of short-dated optons tend to be more varable than the volatltes of long-dated optons. 5. Theta: tme decay of the portfolo. - Theta s usually negatve for an opton. An excepton could be an n-the-money European put opton on a non-dvdend-payng stock or an n-the-money European call opton on a currency wth a very hgh nterest rate. - It makes sense to hedge aganst changes n the prce of the underlyng asset, but t does not make sense to hedge aganst the effect of the passage of tme on an opton portfolo. In spte of ths, many traders regard theta as a useful statstc. In a delta neutral portfolo, when theta s large and postve, gamma tends to be large and negatve, and vce versa. 6. Rho: the rate of change of a portfolo wth respect to the level of nterest rates. Currency optons have two rhos, one for the domestc nterest rate and one for the foregn nterest rate. 7. Taylor expansons: - f gnorng terms of hgher order dt and assumng delta neutral and volatlty and nterest rates are constant, then - -

3 ΔΠ = ΘΔ t+ ΓΔ S - Optons traders make themselves delta neutral or close to delta neutral at the end of each day. Gamma and vega are montored, but are not usually managed on a daly bass. - There s one aspect of an optons portfolo that mtgates problems of managng gamma and vega somewhat. Optons are often close to the money when they are frst sold so that they have relatvely hgh gammas and vegas. However, as tme elapses, the underlyng asset prce has often changed suffcently for them to become deep out of the money or deep n the money. Ther gammas and vegas are then very small and of lttle consequence. The nghtmare scenaro for a trader s where wrtten optons reman very close to the money as the maturty date s approached. Chapter 4 Interest Rate Rsk. LIBID, the London Interbank Bd Rate. Ths s the rate at whch a bank s prepared to accept deposts from another bank. The LIBOR quote s slghtly hgher than the LIBID quote. - Large banks quote, 3, 6 and month LIBOR n all major currences. A bank must have an AA ratng to qualfy for recevng LIBOR deposts. - How LIBOR yeld curve be extended beyond one year? Usually, create a yeld curve to represent the future short-term borrowng rates for AA-rated companes. - The LIBOR yeld curve s also called the swap yeld curve or the LIBOR/swap yeld curve. - Practtoners usually assume that the LIBOR/swap yeld curve provdes the rsk-free rate. T-rates are regarded as too low to be used as rsk-free rates because: a. T-blls and T-bonds must be purchased by fnancal nsttutons to fulfll a varety of regulatory requrements. Ths ncrease demand for these Ts drvng ther prces up and yelds down. b. The amount of captal a bank s requred to hold to support an nvestment n Ts s substantally smaller than the captal requred to support a smlar nvestment n other very low-rsk nstruments. c. In USA, Ts are gven a favorable tax treatment because they are not taxed at the state level.. Duraton - Contnuous compoundng. n yt db ce D= = t ( ), Δ B= BDΔ y. Bdy B Ths s the weghted average of the tmes when payments are made, wth the =

4 weght appled to tme t beng equal to the proporton of the bond s total present value provded by the cash flow at tme t. - Compoundng m tmes per year, then modfed duraton. 3. Convexty. - n yt ct e d B = C = = Bdy B to the recept of cash flows. Δ B = BDΔ y+ BC( Δ y) D * D = + y/ m. Ths s the weghted average of the square of the tme - The duraton (convexty) of a portfolo s the weghted average of the duratons (convexty) of the ndvdual assets comprsng the portfolo wth the weght assgned to an asset beng proportonal to the value of the asset. - The convexty of a bond portfolo tends to be greatest when the portfolo provdes payments evenly over a long perod of tme. It s least when the payments are concentrated around one partcular pont n tme. - Duraton zero protect small parallel sht n the yeld curve. - Both duraton and convexty zero protect large parallel shfts n the yeld curve. - Duraton and convexty are analogous to the delta and gamma. 4. Nonparallel yeld curve shfts - Defnton: D p dp =, where dy s the sze of the small change made to the P dy th pont on the yeld curve and dp s the resultant change n the portfolo value. - The sum of all the partal duraton measures equals the usual duraton measure. 5. Interest rate deltas - DV0: defne the delta of a portfolo as the change n value for a one-bass-pont parallel shft n the zero curve. It s the same as the duraton of the portfolo multpled by the value of the portfolo multpled by 0.0%. - In practce, traders lke to calculate several deltas to reflect ther exposures to all the dfferent ways n whch the yeld curve can move. One approach corresponds to the partal duraton approach. the sum of the deltas for all the ponts n the yeld curve equals the DV0. 6. Prncpal components analyss - The nterest rate move for a partcular factor s factor loadng. The sum of the squares of factor loadngs s. - Factors are chosen so that the factor scores are uncorrelated. - 0 rates and 0 factors solve the smultaneous equatons: the quantty of a partcular factor n the nterest rate changes on a partcular day s known as the - 4 -

5 factor score for that day. - The mportance of a factor s measured by the standard devaton of ts factor score. - The sum of the varances of the factor scores equal the total varance of the data. the mportance of st factor = the varance of st factor s factor score/total varance of the factor scores. - Use prncpal component analyss to calculate delta exposures to factors: dp/df = dp/dy *dy/df. Chapter 5 Volatlty. What causes volatlty? - It s natural to assume that the volatlty of a stock prce s caused by new nformaton reachng the market. Fama (965), French (980), and French and Roll (986) show that the varance of stock prce returns between Frday and Monday s only %, 9% and 0.7% hgher than the varance of stock prce return between two consecutve tradng days (not 3 tmes). - Roll (984) looked at the prces of orange juce futures. By far the most mportant news for orange juce futures prces s news about the weather and news about the weather s equally lkely to arrve at any tme. He found that the Frday-to-Monday varance s only.54 tmes the frst varance. - The only reasonable concluson from all ths s that volatlty s to a large extent caused by tradng tself.. Varance rate: rsk managers often focus on the varance rate rather than the volatlty. It s defned as the square of the volatlty. 3. Impled volatltes are used extensvely by traders. However, rsk management s largely based on hstorcal volatltes. 4. Suppose that most market nvestors thnk that exchange rates are log normally dstrbuted. They wll be comfortable usng the same volatlty to value all optons on a partcular exchange rate. But you know that the lognormal assumpton s not a good one for exchange rates. What should you do? You should buy deep-out-of-the-money call and put optons on a varety of dfferent currences and wat. These optons wll be relatvely nexpensve and more of them wll close n the money than the lognormal model predcts. The present value of your payoffs wll on average be much greater than the cost of the optons. In the md-980s, the few traders who were well nformed followed the strategy and made lots of money. By the late 980s everyone realzed that out-of-the-money optons should have a hgher mpled volatlty than at the money optons and the tradng opportuntes dsappeared. 5. An alternatve to normal dstrbutons: the power law- has been found to be a good descrptons of the tals of many dstrbutons n practce. - The power law: for many varables, t s approxmately true that the value of v of - 5 -

6 the varable has the property that, when x s large, Pr ob( v > x) = Kx α, where K and alpha are constants. 6. Montorng volatlty - The exponentally weghted movng average model (EWMA). σ λσ λ n = n + ( ) u - The GARCH(,) MODEL σ n γvl αun βσ n n, RskMetrcs use λ =0.94. = + +, Where γ + α + β =, f γ =0, then GARCH model s EWMA - ML method to estmate GARCH (,) 7. How good s the model: - The assumpton underlyng a GARCH model s that volatlty changes wth the passage of tme. If a GARCH model s workng well, t should remove the autocorrelaton ofu. We can consder the autocorrelaton of the varables u / σ. If these show very lttle autocorrelaton, the model for volatlty has succeed n explanng autocorrelatons n the u. We can use Ljung-Box statstc. If ths statstc s greater than 5, zero autocorrelaton can be rejected wth 95% confdence. 8. Usng GARCH to forecast future volatlty. σ α β σ n+ t VL= ( un+ t VL) + ( n+ t VL), snce the expected value of u n + t s σ n + t, hence: E[ σ V ] = ( α + β) E[ σ V ] = ( α + β) ( σ V t n+ t L n+ t L n L 9. Volatlty term structures the relatonshp between the mpled volatltes of the optons and ther maturtes. - Defne - Vt ( ) = Eσ ( n t ) and a=ln α + β + at T e Vtdt () VL [ V(0) VL] T = + 0 at, or n per year at e - σ ( T) = 5 VL + [ V(0) V L ] can be used to estmate a at volatlty term structure based on the GARCH (,) model. 0. Impact of volatlty changes. - at e σ (0) σ ( T) = 5 VL + [ V L]. at 5 ) - 6 -

7 When σ (0) change by d σ (0), σ ( T ) changes by at e σ (0) Δ σ (0) at σ ( T ) - Many banks use analyses such as ths when determnng the exposure of ther books to volatlty changes. Chapter 6 Correlatons and Copulas. Correlaton measures lnear dependence. There are many other ways n whch two varables can be related. E.g. for normal values, two varables may be unrelated. However, ther extreme values may be related. Durng a crss the correlatons all go to one.. Montorng correlaton - Usng EWMA: cov = λ cov + ( λ) x y n n n n - Usng GARCH cov = ω + αcov + βx y n n n n 3. Consstency condton for covarances - Varance-covarance matrx should be postve-semdefnte. That s, T ω Ωω 0 for any vector omega. - To ensure that a postve-semdefnte matrx s produced, varances and convarances should be calculated consstently. For example, f varance rates are updated usng an EWMA model wth λ=0.94, the same should be done for covarance rates. 4. Multvarate normal dstrbuton - E[Y x] = μ Y + ρ(σ Y /σ X )(x - μ X ), and - The condtonal mean of Y s lnearly dependent on X and the condtonal standard devaton of Y s constant. 5. Factor models U = a F + a F a F + a a... a Z. The M M M factors have uncorrelated standard normal dstrbutons and the Z are uncorrelated both wth each other and wth the factors. In ths case the correlaton between U and Uj s M m= a a m jm - 7 -

8 6. Copulas - The margnal dstrbuton of X (uncondtonal dstrbuton) s ts dstrbuton assumng we know nothng about Y. But often there s no natural way of defnng a correlaton structure between two margnal dstrbutons. - Gaussan copula approach. Suppose that F and F are the cumulatve margnal probablty dstrbutons of V and V. We map V = v to U = u and V = v to U = u, where F ( v ) = N( u ) and F ( v ) = N( u ), and N s the cumulatve normal dstrbuton functon. - The key property of a copula model s that t preserves the margnal dstrbutons of V and V whle defnng correlaton structure between them. - Student t-copula U and U are assumed to have a bvarate Student t-dstrbuton. Tal correlaton s hgher n a bvarate t-dstrbuton than n a bvarate normal dstrbuton. - A factor copula model: analysts often assume a factor model for the correlaton structure between the U. 7. Applcaton to loan portfolos - Defne T (<= <= N) as the tme when company default and cumulatve probablty dstrbuton of T by Q. To defne the correlaton structure between the T usng the one-factor Gaussan copula model, we map, for each, T to a varable U that has a standard normal on a percentle-to-percentle bass. U af - We assume: U = af + a Z Pr ob( U < U F) = N[ ] a - The mappngs between the U and T mply Prob(U<U) = Prob(T<T) when N [ Q( T)] af U = N [ Q( T)] Pr ob( T < T F) = N[ ] a - Assumng Q of tme to default s the same for all and equal to Q and the copula correlaton between any two names s the same and equals rho a = ρ for all. N [ Q( T)] ρ F - Pr ob( T < T F) = N[ ]. For a large portfolo of loans, ths ρ equaton provdes a good estmate of the proporton of loans n the portfolo that default by tme T. We refer to ths as the default rate. - As F decreases, the default rate ncreases. Snce F s standard normal, the probablty that F wll be less than N - (Y) s Y. There s therefore a probablty of Y that the default rate wll be greater than N [ Q( T)] ρ N ( Y) N[ ] ρ - Defne V(T,X) as the default rate that wll not be exceeded wth probablty X, so - 8 -

9 that we are X% certan that the default rate wll not exceed V(T,X). The value of V(T,X) s determned by substtutng Y = X nto the above expresson: V(T, X) = N Q T N X [ ( )] + ρ ( ) N[ ] ρ Chapter 7 Bank Regulaton and Basel II. The captal a fnancal nsttuton requres should cover the dfference between expected losses over some tme horzon and worst-case losses over the same tme horzon. The dea s that expected losses are usually covered by the way a fnancal nsttuton prces ts products. For example, the nterest charged by a bank s desgned to recover expected loan losses. Captal s a cushon to protect the bank from an extremely unfavorable outcome.. The 988 BIS Accord (Basel I) - Assets-to-captal <+0 - The Cooke Rato calculate rsk-weghted assets. 3. Nettng: If a counterparty defaults on one contract f has wth a fnancal nsttuton then t must default on all outstandng contracts wth that fnancal nsttuton. Wthout nettng the fnancal nsttuton s exposure n the event of a default today s N = max( V,0) N max( V,0) = (N contracts wth the defaulted party). Wth nettng, t s 4. Basel II s based on three pllars: - Mnmum captal requrements - Supervsory revew: allow regulators n dfferent countres some dscreton n how rules are appled but seeks to acheve overall consstency n the applcaton of the rules. It places more emphass on early nterventon when problems arse. Part of ther role s to encourage banks to develop and use better rsk management technques and to evaluate these technques. - Market dscplne: requre banks to dsclose more nformaton about the way they allocate captal and the rsks they take. 5. Mnmum captal requrements: Total captal = 0.08*(credt rsk RWA + Market rsk RWA + Operatonal rsk RWA). 6. Market rsk (996 amendment to Basel I, contnue to be used under Basel II): - It requres fnancal nsttutons to hold captal to cover ther exposures to market - 9 -

10 rsks as well as credt rsks. It dstngushes between a bank s tradng book (normally marked to market daly) and ts bankng book. - The market rsk captal requrement: k * VaR + SRC, where SRC s a specfc rsk charge. The VaR s the greater of the pervous day s VaR and the average VaR over the last 60 days. The mnmum value for k s 3. - SRC s a captal charge for the dosyncratc rsks related to ndvdual companes. E.g. a corporate bond has two rsks: nterest rate rsk and credt rsk. The nterest rate rsk s captured by the bank s market VaR measure; the credt rsk s specfc rsk. 7. Credt rsk captal (NEW FOR BASEL II) - For an on-balance-sheet tem a rsk weght s appled to the prncpal to calculate rsk-weghted assets reflectng the credtworthness of the counterparty. For off-balance-sheet tems the rsk weght s appled to a credt equvalent amount. Ths s calculated usng ether credt converson factors or add-on amount. - Standardzed approach (for small banks. In USA, Basel II wll apply only to the largest banks and these banks must use the foundaton nternal ratngs based (IRB) approach). rsk weghts for exposures to country, banks, and corporatons as a functon of ther ratngs. - IRB approach one-factor Gaussan copula model of tme to default. WCDR: the worst-case default rate durng the next year that we are 99.9% certan wll not be exceeded PD: the probablty of default for each loan n one year EAD: The exposure at default on each loan (n dollars) LGD: the loss gven default. Ths s the proporton of the exposure that s lost n the event of a default. Suppose that the copula correlaton between each par of oblgors s rho. Then WCDR = N [ PD] + ρ N (0.999) N[ ] ρ It follows that there s a 99.9% chance that the loss on the portfolo wll be less than N tmes EAD*LGD*WCDR. - For corporate, soveregn, and bank exposures, Basel II assumes the relatonshp between rho and PD s: ρ 50 PD 0.( + e ). As PD ncreases, rho decreases. As a frm becomes less credtworthy, ts PD ncreases and ts probablty of default becomes more dosyncratc and less affected by overall market condtons. - The formula for the captal requred s: EAD*LGD*(WCDR PD)*MA. We use WCDR PD nstead of WCDR because we are nterested n provdng captal for the excess of the 99% worst-case loss over the expected loss. The MA s the maturty adjustment. - The rsk-weghted assets (RWA) are calculated as.5 tmes the captal requred: RWA =.5 * EAD*LGD*(WCDR PD)*MA - 0 -

11 8. Operatonal rsk: Banks have to keep captal for operatonal rsk. Three approaches. - Basc ndcator approach: operatonal rsk captal = the bank s average annual gross ncome (=net nterest ncome + nonnterest ncome) over the last three years multpled by Standardzed approach: smlar to basc approach, except that a dfferent factor s appled to the gross ncome from dfferent busness lnes. - Advanced measurement approach: the bank uses ts own nternal models to calculate the operatonal rsk loss that t s 99.9% certan wll not be exceeded n one year. 9. The lmtaton of Basel II: - The total requred captal under Basel II s the sum of the captal for three dfferent rsks (credt, market and operatonal). Ths mplctly assumes that the rsks are perfectly correlated. It wll be desrable f banks can assume less than perfect correlaton between losses from dfferent types of rsk. - Basel II does not allow a bank to use ts own credt rsk dversfcaton calculatons when settng captal requrements for credt rsk wthn the bankng book. The prescrbed rho must be used. In theory a bank wth $ bllon of lendng to BBB-rated frms n a sngle ndustry s lable to be asked to keep the same captal as a bank that has $ bllon of lendng to a much more dverse group of BBB-rated corporatons. Chapter 8 The VaR Measure. Choce of parameters for VaR - VaR = σ N ( X ), Assumng normal dstrbuton and the mean change n the portfolo value s zero, where X s the confdence level, sgma s the standard devaton (n dollars) of the portfolo change over the tme horzon. - The tme horzon: N-day VaR = -day VaR * square root N. - autocorrelaton the above formula a rough one. - The confdence level: f the daly portfolo changes are assumed to be normally dstrbuted wth zero mean convert a VaR to another VaR wth dfferent confdence level. N ( X*) ( ) VaR ( X *) = VaR ( X ) N X - -

12 . Margnal VaR: ( VaR), where x s the th component of a portfolo. For an x nvestment portfolo, margnal VaR s closely related to the CAPM s beta. If an asset s beta s hgh, ts margnal VaR wll tend to be hgh. 3. Incremental VaR s the ncremental effect on VaR of a new trade or the ncremental effect of closng out an exstng trade. It asks the queston: what s the dfference between VaR wth and wthout the trade. If a component s small relatve to the sze of a portfolo, t may be reasonable to assume that the margnal VaR remans constant as x s reduced to zero ( VaR) x x 4. Component VaR s the part of the VaR of the portfolo that can be attrbuted to ths component: - The th component VaR for a large portfolo should be approxmately equal to the ncremental VaR. - The sum of all the component VaRs should equal to the portfolo VaR. - VaR = N = ( VaR) x x 5. Back testng: - The percentage of tmes the actual loss exceeds VaR. Let p = -X, where x s confdence level. m = the number of tmes that the VaR lmts s exceeded, n the total number of days. Two hypotheses: a. The probablty of an excepton on any gven day s p. b. The probablty of an excepton on any gven day s greater than p. The probablty (bnomnal dstrbuton) of the VaR lmt beng exceeded on m or n n! k n k more days s: p ( p). We usually use confdence level as 5%. k!( n k)! k= m If the probablty of the VaR lmt beng exceeded on m or more days s less than 5%, we reject the frst hypothess that the probablty of an excepton s p. The above test s one-taled test. Kupec has proposed a two-taled test (Frequency-of-tal-losses or Kupec test). If the probablty of an excepton under the VaR model s p and m exceptons are observed n n trals, then n m m n m m ln[( p) p ] + ln[( m/ n) ( m/ n ) ] should have a ch-square dstrbuton wth one degree of freedom. The Kupec test s a large sample test Kupec test focuses solely on the frequency of tal losses. It throws away potentally valuable nformaton about the szes of tal losses. Ths suggests that the Kupec test should be relatvely neffcent, compared to a sutable test that took account of the szes as well as the frequency of tal losses. - -

13 - Szes-of-tal-losses test: compare the dstrbuton of emprcal tal losses aganst the tal-loss dstrbuton predcted by model Kolmogorov-Smrnov test (t s the maxmum value of the absolute dfference between the two dstrbuton functons). Another backtest s Crnkovc and Drachman (CD) test. The test s to evaluate a market model by testng the dfference between the emprcal P/L dstrbuton and the predcted P/L dstrbuton, across ther whole range of values. - The extent to whch exceptons are bunched: n practce, exceptons are often bunched together, suggestng that losses on successve days are not ndependent. 6. Stress testng: nvolves estmatng how the portfolo would have performed under extreme market moves. Two Man approaches: a. Scenaro analyses, n whch we evaluate the mpact of specfed scenaros (e.g., such as a partcular fall n the stock market) on our portfolo. The emphass s on specfyng the scenaro and workng out ts ramfcatons. b. Mechancal stress tests, n whch we evaluate a number (often a large number) of mathematcally or statstcally defned possbltes (e.g., such as ncreases or decreases of market rsk factors by a certan number of standard devatons) to determne the most damagng combnaton of events and the loss t would produce. 7. stress testng scenaro analyses: - Stylzed scenaros: Some stress tests focus on partcular market varables. a. Shftng a yeld curve by 00 bass ponts b. Changng mpled volatltes for an asset by 0% of current values. c. Changng an equty ndex by 0%. d. Changng an exchange rate for a major currency by 6% or changng the exchange rate for a mnor currency by 0%. - Actual hstorcal events: stress tests more often nvolve makng changes to several market varables use hstorcal scenaros. E.g. set the percentage changes n all market varables equal to those on October 9, If movements n only a few varables are specfed n a stress test, one approach s to set changes n all other varables to zero. Another approach s to regress the nonstressed varables on the varables that are beng stressed to obtan forecasts for them, condtonal on the changes beng made to the stressed varables (condtonal stress testng) 8. Stress testng mechancal stress testng - Factor push analyss: we push the prce of each ndvdual securty or (preferably) the relevant underlyng rsk factor n the most dsadvantageous drecton and work out the combned effect of all such changes on the value of the portfolo. a. We start by specfy a level of confdence, whch gves us a confdence level parameter alpha. b. We then consder each rsk factor on ts own, push t by alpha tmes ts standard devaton, and revalue the portfolo at the new rsk factor value; c. We do the same for all rsk factors, and select that set of rsk factor - 3 -

14 movements that has the worst effect on the portfolo value. d. Collectng these worst prce movements for each nstrument n out portfolo gves us our worst-case scenaro, and the maxmum loss (ML) s equal to the current value of our portfolo mnus the portfolo value under ths worst-case scenaro. e. Factor push test s only approprate for certan relatvely smple types of portfolo n whch the poston value s a monotonc functon of a rsk factor. - Maxmum loss optmzaton: search over the losses that occur for ntermedate as well as extreme values of the rsk varables Chapter 9 Market Rsk VaR: Hstorcal Smulaton Approach. Methodology Suppose that we calculate VaR for a portfolo usng a one-day tme horzon, a 99% confdence level, and 500 days of data. - Identfy the market varables affectng the portfolo. They typcally are exchange rates, equty prces, nterest rates, and so on. - Collect data on the movements n these varables over the most recent 500 days. v - The value of the market varable tomorrow under th scenaro s v (n v today, say n=500, = 0,, 499) - Get change n portfolo value under each scenaro accordng to the calculated market varables. - VaR = 5 th worst number of the change n portfolo value.. The confdence nterval: - Kendall and Stuart calculate a confdence nterval for the quantle of a dstrbuton when t s estmated from sample data. - The standard errors of the estmate s: q ( q), where f(x) s the f( x) n probablty densty functon of the loss evaluated at x. 3. Weghtng of observatons: the basc hstorcal smulaton approach assumes that each day n the past s gven equal weght. Boudoukh et alpha. Suggest that more recent observatons should be gven more weght the weght gven to the change between day n- and n-+ s λ ( λ) n λ. VaR s calculated by rankng the observatons from the worst outcome to the best. Startng at the worst outcome, n - 4 -

15 weghts are summed untl the requred quantle of the dstrbuton s reached. 4. Incorporatng volatlty updatng (Hull and Whte (998)). - Get daly volatlty for each market varable. - Suppose σ n + (current estmate of the volatlty of the market varable between today and tomorrow) s twceσ. we expect to see changes between today and tomorrow that are twce as bg as changes between day - and day. the value of a market varable under the th scenaro becomes: v n v + ( v v ) σ n+ / σ v 5. Extreme value theory s a way of smoothng the tals of the probablty dstrbuton of portfolo daly changes calculated usng hstorcal smulaton. It leads to estmates of VaR that reflect the whole shape of the tal of the dstrbuton, not just the postons of a few losses n the tals. It can also be used to estmate VaR when the VaR confdence level s very hgh. E.g., even f we have only 500 days of data, t could be used to come up an estmate of VaR for a VaR confdence level of 99.9%: - F(x) s CDF, F u (y) as the probablty that x les between u and u+y condtonal on x>u. F ( y) = u Fu ( + y) Fu ( ) Fu ( ) - Gnedenko states that for a wde class of dstrbutons F(x), the dstrbuton of F u (y) converges to a generalzed Pareto dstrbuton as the threshold u s ncreased. y Gξβ, ( y) = ( + ξ ) β / ξ, where ξ s the shape parameter and determnes the heavness of the tal of the dstrbuton. Beta s a scale parameter. - Estmatng ξ and beta MLE. The PDF of G(y) s g ξβ, y ( y) = ( + ξ ) β β / ξ - Frst, choose a value of u, say the 95 percentle. Then focus attenton on those observatons for whch x > u. Suppose there are n u such observatons and they are x (<=n u ). lng(x) = n u x u ξ ξ / ln[ ( + ) ] = β β - Prob(x>u+y x>u) = G(y), prob(x>u) = F(u) ~ n u /n prob(x>u+y) = nu x u / ξ [-F(u)][-G(y)] F( x) = ( + ξ ), ths s the estmator of the n β tal of the CDF of x when x s large. Ths s reduced to the Power Law f we set - 5 -

16 u = β / ξ. - Calculaton of VaR wth a confdence level of q F(VaR) = q ξ β n VaR = u + ( q) ξ n u Chapter 0 Market Rsk VaR: Model-Buldng Approach. Basc methodology two-asset case dversfcaton. Note: VaR does not always reflect the benefts of dversfcaton. n - The lnear model: Δ P = αδx =, where dx s the return on asset n one day. Alpha s the dollar value beng nvested n asset. get mean and standard devaton of dp then we are done under multvarate normal dstrbuton.. Handlng nterest rates- cash flow mappng. The cash flows from nstruments n the portfolo are mapped nto cash flows occurrng on the standard maturty dates. Snce t s easy to calculate zero T-blls or T-bonds volatltes and correlatons, after mappng, t s easy to calculate the portfolo s VaR n terms of cash flows of zero T-blls. 3. Prncpal components analyss: A PCA can be used (n conjuncton wth cash flow mappng) to handle nterest rates when VaR s calculated usng the model-buldng approach. 4. Applcatons of Lnear model: - A portfolo wth no dervatves. - Forward contracts on foregn exchange (to be treated as a long poston n the foregn bond combned wth a short poston n the domestc bond). - Interest rate swaps 5. The lnear model and optons: Δ P = δδ S = SδΔ x= Sδ Δ x = α Δx n = = n, where dx = ds/s s the return on a stock n one day. Ds s the dollar change n the stock prce n one day. - The weakness of the model: when gamma s postve (negatve), the pdf of the value of the portfolo tends to be postvely (negatvely) skewed have a less heavy (heaver) left tal than the normal dstrbuton. f we assume the dstrbuton s normal, we wll tend to calculate a VaR that s too hgh (low). 6. The Quadratc model: - 6 -

17 - Δ P = δδ S+ γ( Δ S) = SδΔ x+ S γ( Δx ) (gnorng theta term). - For a portfolo wth n underlyng market varables, wth each nstrument n the portfolo beng dependent on only one of the market varables, Δ P = S Δ x + S Δx n n δ γ( ) = =, where S s the value of the th market varable. - When some of the ndvdual nstruments are dependent on more than one market varable, ths equaton takes the more general form: n n n Δ P = Sδ Δ x + SS γ ΔxΔx, where gamma j s a cross gamma. j j j = = j= - Cornsh Fsher expanson: estmate percentles of a probablty dstrbuton from ts moments that can take account of the skewness of the probablty dstrbuton. Usng the frst three moments of dp, the Cornsh-Fsher expanson estmates the q-percentle of the dstrbuton of dp as μp + ωσ q P, where ( q q q ) ω = z + z ξp and Z s q-percentle of the standard normal dstrbuton 6 and ξ P s the skewness of dp. 7. The model-buldng approach s frequently used for nvestment portfolos. It s less popular for the tradng portfolos of fnancal nsttutons because t does not work well when deltas are low and portfolos are nonlnear. Chapter Credt Rsk: Estmatng Default Probabltes. Credt-rsk arses from the possblty that borrowers, bond ssuers, and counter-partes n dervatves transactons may default. In theory, a credt ratng s an attrbute of a bond ssue, not a company. However, n most cases all bonds ssued by a company have the same ratng. A ratng s therefore often referred to as an attrbute of a company. - Ratngs changes relatvely nfrequently. One of ratng agences objectves s ratngs stablty. They want to avod ratngs reversals where a frm s downgraded and then upgraded a few weeks later. Ratngs therefore change only when there s reason to beleve that a long-term change n the frm s credtworthness has taken place. The reason for ths s that bond traders are major users of ratngs. Often they are subject to rules governng what the credt ratngs of the bonds they hold - 7 -

18 must be. If these ratngs changed frequently they mght have to do a large amount of tradng just to satsfy the rules. - Ratng agences try to rate through the cycle. Suppose that an economc downturn ncreases the probablty of a frm defaultng n the next 6 months, but makes very lttle dfference to the frm s cumulatve probablty of defaultng over the next three to fve years. A ratng agency would not change the frm s ratng.. Internal credt ratngs: most banks have procedures for ratng the credtworthness of ther corporate and retal clents. The nternal ratngs based (IRB) approach n Basel II allows bank to use ther nternal ratngs n determnng the probablty of default, PD. Under the advanced IRB approach, they are also allowed to estmate the loss gven default, LGD, the exposure at default, EAD, and the maturty, M. 3. Altman s Z-score Usng dscrmnant analyss, Altman attempted to predct defaults from fve accountng ratos. Z =.*workng captal +.4*Retaned earnngs + 3.3* EBIT + 0.6*Market value of equty *sales, all varables are scaled by assets, except for market equty, whch s scaled by book value of total labltes. If Z-score > 3, the frm s unlkely to default. If t s between.7 and 3.0, we should be on alert. If t s between.8 and.7, there s a good chance of default. If t s less than.8, the probablty of a fnancal embarrassment s very hgh. 4. Hstorcal default probabltes - For nvestment-grade bonds, the probablty of default n a year tends to be an ncreasng functon of tme. - For bonds wth a poor credt ratng, the probablty of default s often a decreasng functon of tme. The reason s that, for a bond lke ths, the next year or two may be crtcal. If the ssuer survves ths perod, ts fnancal health s lkely to have mproved. - Default ntenstes (hazard rate): λ(t) at t s defned so thatλ(t)δt s the probablty of default between tme t and t +Δt condtonal on no default between tme 0 and tme t. Let V(t) s the cumulatve probablty of the frm survvng to tme t (no default by tme t), then V(t +Δt) V(t) = -λ(t) V(t)Δt dv () t dt t - λτ ( ) dτ 0 λt = λ() tv() t V(t)=e = e, where λ s the average default ntensty between tme zero and t. Defne Q(t) as the probablty of default by tme t Q(t) = V(t) = - e t - λ ( τ) dτ 0 = - e λt λ () t = ln[ Q()] t, where Q(t) t comes from hstorcal data. 5. Recovery rates: the bond s market value mmedately after a default as a percent of ts face value = LGD. - Recovery rates are sgnfcantly negatvely correlated wth default rates

19 - Average recovery rate = *average default rate. a bad year for the default rate s usually doubly bad because t s accompaned by a low recovery rate. 6. Estmatng default probabltes from bond prces - An approxmate calculaton: PD(T) = e R [ y Rf ] T [ PD]00e + PD00Re = 00e ~ spread/(-r) RfT RfT yt - Rsk-free rate: Traders usually use LIBOR/swap rates as proxes for rsk-free rates when calculatng default probabltes. - Credt default swaps can be used to mply the rsk-free rate assumed by traders. The rate used appears to be approxmately equal to the LIBOR/swap rate mnus 0 bass ponts on average. (Credt rsk n a swap rate s the credt rsk from makng a seres of 6-month loans to AA-rated counterpartes and 0 bass ponts s a reasonable default rsk premum for an AA-rated 6-month nstrument. 7. Asset swap: traders often use asset swap spreads as a way of extractng default probabltes from bond prces. Ths s because asset swap spreads provde a drect estmate of the spread of bond yelds over the LIBOR/swap curve. The present value of the asset swap spread s the amount by whch the prce of the corporate bond s exceeded by the prce of a smlar rsk-free bond. 8. Real-world vs. Rsk-neutral probabltes - The default probabltes mpled from bond yelds are rsk-neutral default probabltes. - The default probabltes mpled from hstorcal data are real-world default probabltes. If there was no expected excess return, the real-world and rsk-neutral default probabltes would be the same, and vce versa. - Reasons for the dfference a. Corporate bonds are relatvely llqud and bond traders demand an extra return to compensate for ths. b. The subjectve default probabltes of traders may be much hgher than those gven n table.. Traders may be allowng for depresson scenaros much worse than anythng seen durng the 970 to 003 perod. c. Bonds do not default ndependently of each other. Ths gves rse to systematc rsk and traders should requre an expected excess return for bearng the rsk. The varaton n default rates from year to year may be because of overall economc condtons or because a default by one company has a rpple effect resultng n defaults by other companes (the latter s referred as credt contagon). d. Bond returns are hghly skewed wth lmted upsde. As a result t s much more dffcult to dversfy rsks n a bond portfolo than n an equty portfolo. - When valung credt dervatves or estmatng the mpact of default rsk on the prcng of nstruments, we should use rsk-neutral default probabltes. - When carryng out scenaro analyses to calculate potental future losses from - 9 -

20 defaults, we should use real-world default probabltes. The PD used to calculate regulatory captal s a real-world default probabltes. 9. Estmatng default probabltes from equty prces: - Equty s an opton on the assets of the company. the value of equty today s: E = VNd ( ) De Nd ( ) rt 0 0 -, where - d ln( V0 / D) + ( r+ σv / ) T and d = = d σv σ T V - The rsk-neutral probablty that a frm wll default on the debt s N(-d ). We need T V0 and σ V, whch are not drectly observable. E σ E = σ V = N( d ) σvv V - From Ito s lemma E 0 V 0 - The above two equatons the values of V0 and σv N(-d ). 0. The default probablty can be estmated from hstorcal data (real-world probabltes), bond prces (rsk-neutral probabltes), or equty prces (n theory rsk-neutral probabltes. But, the output from the model can be calbrated so that ether rsk-neutral or real-world default probabltes are produced). 0 Chapter Credt Rsk Losses and Credt VaR. Estmatng credt losses: - Credt losses on a loan depend prmarly on the probablty of default and the recovery rate. - The credt rsk n a dervatve transacton s more complcated because the exposure at the tme of default s uncertan. Some dervatves transactons (e.g., wrtten optons) are always labltes and gve rse to no credt rsk. Some (e.g., long postons n optons) are always assets and ental sgnfcant credt rsks. Some may become ether assets or labltes durng ther lfe (e.g., forward and swaps).. Adjustng dervatves valuatons for counterparty default rsk - Assume the value of the dervatve to the fnancal nsttuton at tme t s f, =0,, n, and q s the rsk-neutral probablty of default at tme t, the expected recovery rate s R. the rsk-neutral expected loss from default at tme t s: [ ] q ( R) E max( f, 0) costs of defaults. n = q ( R), takng present values and sum them up over v, where v s the value today of an nstrument that pays off the exposure on the dervatve under consderaton at tme t

21 - The mpact of default rsk on nterest rate swaps s consderably less than that on currency swaps, largely because prncpals are exchanged at the end of the lfe of a currency swap and there s uncertanty about the exchange rate at that tme. - Two sded default rsk (when contracts can become ether assets or labltes). Human nature beng what t s, most frms consder that there s no chance that they themselves wll default but want to make an adjustment to contract terms for a possble default by ther counterparty. Ths can make t very dffcult for the two frms to agree on terms and explans why t s dffcult for fnancal nsttutons that are not hghly credtworthy to be actve n the dervatves market. 3. Credt rsk mtgaton - Nettng. Because of nettng, the ncremental effect of a new contract on expected default losses can be negatve. Ths tends to happen when the value of the new contract s hghly negatvely correlated wth the value of exstng contracts. A frm may get dfferent quotes n a well-functonng captal market. The company s lkely to get the most favorable quote from a fnancal nsttuton t has done busness wth n the past partcularly f that busness gves rse to exposures for the fnancal nsttuton that are opposte to the exposure generated by the new transacton. - Collateralzaton - Downgrade trggers. Ths s a clause statng that f the credt ratng of the counterparty falls below a certan level, say Baa, then the fnancal nsttuton has the opton to close out a dervatves contract at ts market value. 4. Credt VaR. Whereas the tme horzon for market rsk s usually between one day and one month that for credt rsk s usually much longer often one year. 5. Vascek s modela: - We are X% certan that the default rate wll not exceed V(T,X). V(T, X) = N[ N Q T N X [ ( )] + ρ ( ) ρ ], where Q(T) s the cumulatve probablty of each loan defaultng by tme T. 6. Credt rsk plus: the probablty of n defaults follows the Posson dstrbuton and ths s combned wth a probablty dstrbuton for the losses experenced on a sngle counterparty default to obtan a probablty dstrbuton for the total default losses from the counterpartes. 7. CredtMetrcs. It s based on an analyss of credt mgraton. Ths s the probablty of a frm movng from one ratng category to another durng a certan perod of tme. Calculatng a one-year VaR for the portfolo usng CredtMetrcs nvolves carryng out Monte Carlo smulaton of ratngs transtons for bonds n the portfolo over a one-year perod. On each smulaton tral the fnal credt ratng of all bonds s calculated and the bonds are revalued to determne total credt losses for the year. The 99% worst result s the one-year 99% VaR for the portfolo. - The credt ratng changes for dfferent counterpartes should not be assumed to be ndependent. use Gaussan copula model The copula correlaton between the - -

22 ratng transtons for two companes s typcally set equal to the correlaton between ther equty returns usng a factor model. Chapter 3 Credt Dervatves. Credt dervatves are contracts where the payoff depends on the credtworthness of one or more companes or countres. Banks have been the largest buyers of CDS credt protecton and nsurance companes have been the largest sellers. - The n-year CDS spread should be approxmately equal to the excess of the par yeld on an n-year corporate bond over the par yeld on an n-year rsk-free bond. If t s markedly less than ths, an nvestor can earn more than the rsk-free rate by buyng the corporate bond and buyng protecton. If t s markedly great than ths, an nvestor can borrow at less than the rsk-free rate by shortng the corporate bond and sellng CDS protecton. - The payoff from a CDS n a credt event s notonal prncpal*(-recovery rate). Usually a CDS specfes that a number of dfferent bonds can be delvered n the credt event. Ths gves the holder of CDS a cheapest-to-delver bond opton. Therefore, recovery rate should be the lowest recovery rate applcable to a delverable bond.. Valuaton of credt default swaps - CDS can be analyzed by calculatng the present value of the expected payments (ncludng accrued nterests) and the present value of the expected payoff. - The default probabltes used to value a CDS should be rsk-neutral default probabltes, whch can be estmated from bond prces or asset swaps. An alternatve s to mply them from CDS quotes. - Bnary CDS: t s smlar to a regular CDS except that the payoff s a fxed dollar amount. - Is the recovery rate mportant? whether we use CDS spreads or bond prces to estmate default probabltes, we need an estmate of the recovery rate. However, provded that we use the same recovery rate for (a) estmatng rsk-neutral default probabltes and (b) valung a CDS, the value of the CDS s not very senstve to the recovery rate. Ths s because the mpled probabltes of default are approxmately proportonal to /(-R) and the payoffs from a CDS are proportonal to -R. 3. A total return swaps s an agreement to exchange the total return on a bond (or any portfolo of assets) for LIBOR plus a spread. The total return ncludes coupons, nterest, and the gan or loss on the asset over the lfe of the swap. The spread over LIBOR receved by the payer s compensaton for bearng the rsk that the recever wll default. The payer wll lose money f the recever defaults at a tme when the reference bond s prce has declne. The spread therefore depends on the credt qualty of the recever and of the bond - -

23 ssuer, and on the default correlaton between the two. 4. Basket CDS - Add-up CDS provdes a payoff when any of the reference enttes default. - An n th -to-default CDS provdes a payoff only when the nth default occurs. 5. CDOs - Cash CDO (based on bonds) - Synthetc CDO: the creator sells a portfolo of CDSs to thrd partes. It then passes the default rsk on to the synthetc CDO s tranche holders. The frst tranche may be responsble for the payoffs on the CDS untl they have reached 5% of the total notonal prncpal; ; The ncome from the CDS s dstrbuted to the tranches n a way that reflects the rsk they are bearng. 6. Valuaton of a basket CDS and CDO - The spread for an n th -to-default CDS and the tranche of a CDO s crtcally dependent on default correlaton. As the default correlaton ncreases the probablty of one or more defaults declnes and the probablty of ten or more defaults ncreases. - The one-factor Gaussan copula model of tme to default has become the standard market model for valung an n th -to-default CDS or a tranche of a CDO. - Consder a portfolo of N frms, each havng a probablty Q(T) of defaultng by tme T. the probablty of default, condtonal on the level of the factor F, s N [ Q( T)] ρ F QT ( F) = N[ ] ρ - The trck here s to calculate expected cash flows condtonal on F and then ntegrate over F. The advantage of ths s that, condtonal on F, defaults are ndependent. The probablty of exactly k defaults by tme T, condtonal on F, s n! k QT ( F) ( QT ( F)) k!( n k)! - Dervates dealers calculate the mpled copula correlaton rho from the spreads quoted n the market for tranches of CDOs and tend to quote these rather than the spreads themselves. Ths s smlar to the practce n optons markets of quotng B-S mpled volatltes rather than dollar prces. - Correlaton smles: the compound correlaton s the correlaton that prces a partcular tranche correctly. The base correlaton s the correlaton that prces all tranches up to a certan level of senorty correctly. In practce, we fnd that compound correlatons exhbt a smle wth the correlatons for the most junor (equty) and senor tranches hgher than those for ntermedate tranches. The base correlatons exhbt a skew where the correlaton ncreases wth the level of senorty consdered. n k - 3 -

24 Chapter 4 Operatonal Rsk. Operatonal rsk: the rsk of loss resultng from nadequate or faled nternal processes, people and systems or from external events. - It ncludes legal rsk, but does not nclude reputaton rsk or the rsk resultng from strategc decsons.. 7 Categores of operatonal rsk - Internal fraud (Barngs), - External fraud (Republc NY corp. Lost $6 mllon because of fraud commtted by a custodal clent, - Employment practces and workplace safety (Merrll Lynch lost $50 mllon n a gender dscrmnaton lawsut), - Clents, products, & busness practces (Household Internatonal lost $484 mllon from mproper lendng practces), - Damage to physcal assets (9 attacks), - Busness dsrupton and system falures (Solomon Brothers lost $303 mllon from a change n computng technology). - Executon, delvery, and process management: faled transacton processng or process management, and relatons wth trade counter-partes and vendors. E.g., Bank of Amerca and Wells Fargo Bank lost $5 and $50 mllon, respectvely, from systems ntegraton falures and transactons processng falures. 3. Loss severty and loss frequency - Loss frequency dstrbuton the dstrbuton of the number of losses observed durng the tme horzon (a Posson dstrbuton s usually used). - Loss severty dstrbuton the dstrbuton of the sze of a loss, gven that a loss occurs. (usually assume the two are ndependent). (a lognormal probablty dstrbuton s often used) - The two dstrbutons must be used for each loss type and busness lne to determne a total loss dstrbuton. Monte Carlo smulaton can be used here: a. Sample from the frequency dstrbuton to determne the number of loss events (=n). b. Sample n tmes from the loss severty dstrbuton to determne the loss experenced for each loss event (L, L, Ln) c. Determne the total loss experenced ( = L + L + + Ln). d. Repeat ths many tmes. - Data: the frequency dstrbuton should be estmated from the bank s own data as far as possble. For the loss severty dstrbuton, regulators encourage banks to use ther own data n conjuncton wth external data (through sharng arrangements between banks or from data vendors. Both nternal and external data must be adjusted for nflaton. A scale adjustment should be made to external data). 4. Operatonal rsk captal should be allocated to busness unts n a way that encourages them to mprove ther operatonal rsk management. The overall - 4 -

25 result of operatonal rsk assessment and operatonal rsk captal allocaton should be that busness unts become more senstve to the need for managng operatonal rsk. 5. The power law holds well for the large losses experenced by banks. Ths makes the calculaton of VaR wth hgh degree of confdence such as 99.9% possble. When loss dstrbutons are aggregated, the dstrbuton wth the heavest tals tends to domnate t may only be necessary to consder one or two busness-lne/loss-type combnatons. 6. Insurance two problems (moral hazard [deductbles, consurance provsons, and polcy lmts can be employed] and adverse selecton). Some nsurance companes do offer rogue trader nsurance polces. These companes tend to specfy carefully how tradng lmts are mplemented. They may also requre that the exstence of the nsurance polcy not be revealed to anyone on the tradng floor. They are lkely to want to retan the rght to nvestgate the crcumstances underlyng any loss. 7. Sarbanes-Oxley It requres boards of drectors to become much more nvolved wth day-to-day operatons. They must montor nternal controls to ensure rsks are beng assessed and handled well. 8. Two sorts of operatonal rsk: hgh-frequency low-severty rsks and low-frequency hgh-severty rsks. The former are relatvely easy to quantfy, but operatonal rsk VaR s largely drven by the latter. Chapter 5 Model Rsk and Lqudty Rsk. Model rsk s the rsk related to the models a fnancal nsttuton uses to value dervatves. Lqudty rsk s the rsk that there may not be enough buyers/sellers n the market for a fnancal nsttuton to execute the trades t desres. The two rsks are related. Sophstcated models are only necessary to prce products that are relatvely llqud. When there s an actve market for a product, prces can be observed n the market and models play a less mportant role.. Explotng the weakness of a compettor s model: a LIBOR-n-arrears swap s an nterest rate swap where the floatng nterest rate s pad on the day t s observed, not one accrual perod later. Whereas a plan vanlla swap s correctly valued by assumng that future rates wll be today s forward rates, a LIBOR-n-arrears swap should be valued on the assumpton that the future rate s today s forward nterest rate plus a convexty adjustment. 3. Models for actvely traded products. - Traders frequently quote mpled volatltes rather than the dollar prces of optons, because mpled volatlty s more stable than the prce. - Volatlty smles: the volatlty mpled by B-S as a functon of the strke prce for a partcular opton maturty s known as a volatlty smle. The relatonshp between strke prce and mpled volatlty should be exactly the same for calls - 5 -