FOUNDATIONS OF RISK MANAGEMENT

FOUNDATIONS OF RISK MANAGEMENT Types of Risk Key classes of risk include marker risk, credir risk, liquidity risk, operarional risk, legal and regulatory risk, business risk, srraregic risk, and repuracion risk. Market risk includes interest race risk, equity price risk, foreign exchange risk, and commodity price risk. Credit risk inc ludes default risk, bankruptcy risk, downgrade risk, and sctdcmcnt risk. Liquidity risk includes fundin g liquidiry risk and crading liquidity risk. • • • Enterprise Risk Management (ERM) Comprehensive and integraced framework for managing firm risks in order co meec business objeccives, minimize unexpecred earnings volacility, and maximize firm value. Benefits include (I) increased organizarional effecciveness, (2) beccer risk reporting, and (3) improved business performance. Determining Optimal Risk Exposure Target certain default probability or specific credit rating-. high credit racing may have opporcunity coses (e.g., forego risky/proficable projeccs). i Sensitivity or scenario analys s: examine adverse impaccs on value from specific shocks. Diversifiable and Systematic Risk The pare of the volacility of a single security's recurns chac is uncorrelaced wich che volatility of the markec porcfolio is chat securicy's diversifiable risk. The pare of an individual securicy's risk char arises because of the posirive covariance of thac securicy's recurns with overall marker recurns is called its systematic risk. A standardized measure of systematic risk is beta: beta·= I Cov(R;.RM) 2 OM Capital Asset Pricing Model (CAPM) In equilibrium, all investors hold a porcfolio of risky assecs thac has the same weigh rs as rhe market porcfolio. The CAPM is expressed in che equacion of the security market line (SML). For any single security or portfolio of securicies i, the expected return in equilibrium, is: E(R;) = Ri= + b eca ; [E(RM )- RF) CAPM Assumptions Investors seek to maximize the expected utility of thei r wealth at the end of the period, and all investors have the same inv estment horizon. Investors are risk averse. Investors o nly consider the mean and standard deviation of returns (which impli cic ly assumes the asset returns are normally distrib uted . Inv estors can borrow and lend at the same risk-free • • • • • rate. ) Investors have the same expectations con c ernin g ret urns . • are neither raxes nor transactions costs, and are infinitely divisible. This is often referred to as "perfect markets." There asse ts Arbitrage Pricing Theory (APT) The APT describes expecced recurns as a linear function of exposures to common risk factors: E(R) R,. + G;iRP, + G;iRP l + ... + 0,kRP k where: 0, = /' fac tor beta for stock i i RP = risk premium associated with risk factor j i The APT defines the scruccure of rerurns but does noc define which faccors should be used in the model. The CAPM is a special case of APT with only one factor exposure-che market risk premium. The Fama-French three-factor model describes recurns as a linear funccion of che markec index recurn, firm size, and book-co-markec faccors. = Measures of Performance The Treynor measure is equal co che risk premium divided by beta, or systemacic risk: Treynor measure -[ E(Rp) - RF (3 p ] The Sharpe measure is equal co che risk premium divided by che standard deviation, or coral risk: Sharpe measure - [E(Rp)-RF] Op The Jensen measure (a.k.a. Jensen's alpha or jusc alpha), is the asset's excess return over the return predicred by the CAPM: Jensen measure o.p = E(Rp)-{Ri= + 13p[E(RM)- RF)} The information ratio is essentially the alpha of the managed porcfolio relative co its benchmark divided by che cracking error. IR = [ ] E(Rp)-E(Rs) crackmg error The Sortino ratio is similar co the Sharpe ratio excepc we replace the risk-free race wich a minimum acceptable return, denoted Rm,.• and we replace the scandard deviarion wich a cype of semi-srandard deviation. Sortino racio 1 ir ,_ "' m _..,_ p_)_-_·R R _,_ E_(_ _ - _ -­ semi- standard deviation Financial Disasters Drysdale Securities: borrowed $300 million in unsecured funds from Chase Manhaccan by exploiting a Raw in che syscem for compucing che value of collateral. Kit.Ukr Peabody: Joseph Jett reporced subscancial arcificial profits; afcer the fake profics were dececced, $350 million in previously reporced gains had co be reversed. Barinf(s: rogue crader, Nick Leeson, cook speculative derivative posicions (Nikkei 225 fucures) in an actempc co cover crading losses; Leeson had dual responsibilicies of crading and supervising settlement operacions, allowing him co hide crading losses; lessons include separacion of ducies and managemenc oversighc. Allied Irish Bank: currency crader, John Rusnak, hid $691 million in losses; Rusnak bullied back­ office workers inco not following-up on crade confirmations for fake trades. UBS: equicy derivacives business lose millions due co incorrecc modeling of long-daced opcions and ics srake in Long-Term Capical Managemenc. Sociite Genemle: junior crader, Jerome Kerviel, parcicipaced in unauthorized crading accivicy and hid accivicy with fake ofsT eccing cransaccions; fraud resulred in losses of $7. I billion. Metal!gesellscha.ft: shorc-cerm futures concracts used co hedge long-cerm exposure in che pecroleum markecs; scack-and-roll hedging scrategy; marking co markec on fucures caused huge cash Row problems. Long-Term Capital Management: hedge fund that used relative value stracegies with enormous amouncs of leverage; when Russia defaulced on ics debt in 1998, the increase in yield spreads caused huge losses and enormous cash Row problems from realizing marking co market losses; lessons include lack of diversificacion, model risk, leverage, and funding and crading liquidity risks. Banker's Trust: developed derivacive scruccures that were incencionally complex; in caped phone conversations, staff bragged abouc how badly chey fooled clients. JPMorgan and Citigroup: main councerparcies in Enron's derivatives transaccions; agreed to pay a $286 million fine for assiscing wich fraud against Enron investors. Role of Risk Management I. Assess all risks faced by che firm. 2. Communicace these risks co risk-caking decision makers. 3. Monicor and manage these risks. Objeccive of risk managemenc is co recognize chat large losses are possible and co develop conti ngenc y plans that de al with such losses if they should occur. Risk Data Aggregation Defining, gathering, and processing risk daca for measuring performance againsc risk colerance. Benefics of effeccive risk daca aggregacion and reporcing systems: Incre ases abiliry to anticipate problems. Ide ntifies rouces to financial he alth. Impr oves resolvabilicy in event of bank stress. I ncreases efficiency, reduces chance of loss, and • • • • increases profitability. GARP Code of Conduct Secs forth principles relaced co echical behavior wirhin che risk managemenc profession. It scresses ethical behavior in che following areas: Principles • • • Professional integrity and cchical con duct Con A ices of interest Confidentiality Kurtosis is a measure of the degree to which Professional Standards • Fundamental responsibilities distribution with mean µand variance equal to a distribution is more or less "peaked" than a • Adherence to best practices Violations of the Code of Conduct may result in tempor:iry <n<pen<ion or permanent removal normal distribution. Excesskurtosis = kurtosis-3. • Leptolwnic describes a distribution chat is more peaked than a normal di<trihution. from GARP membership. In addition, violations • FRM designation. Desirable Properties of an Estimator could lead to a revocation of the right to use the • QUANTITATIVE ANALYSIS Probabilities Unconditional probability (marginal probability) is the probability of an event occurring. Bayes' Theorem • P(IIO)= P(O I) J xP(I) P(O) expected value of the estimator is equal to the parameter you are trying to estimate. all the ocher unbiased estimators of the parameter you are trying to estimate. • response to the arrival of new information. An unbiased estimator is one for which the • An unbiased estimator is also efficient if the variance of its sampling distribution is smaller than Gmditiona/ probability, P( A J B), is the probability of an event A occurringgiven that event B has occur.red Updates the prior probability for an event in Platykunic refers to a distribution chat is less peaked, or flatter, than a normal distribution. A consistent estimator is one for which the accuracy of the parameter estimate increases as the sample size increases. A point estimate should be a linear estimator when it can be used as a linear function of sample data. Continuous Uniform Distribution Distribution where the probability of X occurring in a possible range is the length of the range relative to the total of all possible values. Letting Expected Value a and b be the lower and upper limits of the Weighted average of the possible outcomes of uniform distribution, respectively, then for a random variable, where the weights are the a� probabilities that the outcomes will occur. Variance Provides a measure of the extent of the dispersion in the values of the random variable around the mean. The square root of the variance is called of two random variables from their respective expected values. Cov(Ri,Rj) = E{[R i -E(Ri)] x [R j - E(R j )]) Correlation Measures the strength of the linear relationship between two random variables. It ranges from-1 Cov (Ri,R j) p(x) = (number of ways to choose x from n ) ( Sums of Random Variables Poisson random variable X refers to the number of successes per unit. The parameter lambda are equal to the parameter, }. Axe- X. Skewness and Kurtosis Skewness, or skew, refers to the extent to which a distribution is not symmetrical. The skewness of a normal distribution is equal to zero. • A positively skewed distribution is characterized by many outliers in the upper region, or right tail. A negatively skewed distribution has a disproportionately large amount of outliers that fall within its lower (left) tail. population standard deviation is the square root of the population variance. N E(xi -µ) 2 c? = �i=�l�-- N s2 = � i -X)2 L--(X i=� l � _ ___ n-1 Sample Covariance n (X·1 . i=l - X)(Y1 -Y) · n-1 The standard error of the sample mean is the standard deviation of the distribution of the sample means. When the standard deviation of CJx = P(X=x)=-­ x! CJ as: Fa_ Confidence Interval Normal Distribution If the population has a normal distribution with its mean and variance. 68% of observations fall within ± ls. population mean is: • z 99% of observations fallwithin ± 2.58s. standard deviation of 1. z-scort: represents number of standard deviations a given observation is from a population mean. z= observation -population mean standard deviation x -µ = -CJ Central Limit Theorem When selecting simple random samples of size n from a population with mean µ and finite variance CJ2, the sampling distribution of sample means approaches the normal probability a known variance, a confidence interval for the -± Zo./2 X 90% of observations fall within ± l.65s. 95% of observations fallwithin ± l.96s. If X and Y are NOT independent: • (X) A standardi:ud random variable is normalized Cov(X,Y) population variance is defined as the average of the squared deviations from the mean. The the population, CJ, is known, the standard error of so that it has a mean of zero and a x The the sample mean is calculated Standardized Random Variables Var(X + Y) = Var(X) + Var(Y) + 2 Population and Sample Variance For the distribution, both its mean and variance • Y) = Var(X) + Var(Y) It is used to make informces about the population mean. refers to the average number of successes per unit. If X and Y are independent random variables: Var(X + a sample of a population, EX, divided by the number of observations in the sample, n. Standard Error Poisson Distribution • E(Y) N The sample mean is the sum of all values in np variance= np(l- p) • If X and Y are any random variables: Exi µ= i=l covariance = E Normal distrihurion i< complere ly de...crihed hy o(Ri)o Rj) N n For a binomial random variable: Expected value of the product of the deviations + (b-a) of"success" on each trial equals: = observations in the population, N. n - 1 instead of n in the denominator improves the statistical properties of i2 as an estimator of CJ2• outcomes over a series of n trials. The probability expected value sums all observed values sample of n observations from a population. Using - xi) � -� Evaluates a random variable with two possible Covariance E(X + Y) = E(X) = m ean in the population and divides by the number of dispersion that applies when we are evaluating a (x2 p'(l- p)n-• variance(X) = EHX -µ)2 ] Corr (Ri,Rj)- b: Population and Sample Mean The population The sample variance, r, is the measure of Binomial Distribution the standard deviation. to +l. <is� <X<x P (x1 - 2) E(X)= EP(xi)Xi = P(x1)x1 +P(x2 )x2 + .. . + P(x0)x0 x1 CJ2/n as the sample size becomes large. z CJ Fa_ = 1.65 for 90% confidence intervals <>ll a12 (significance level 10%, 5% in each = 1.96 for 95% confidence intervals z<>ll= tail) (significance level 5%, 2.5% in each tail) 2.58 for 99% confidence intervals (significance level 1%, 0.5% in each Hypothesis Testing Null hypothesis (HJ: hypothesis tail) the researcher wants to reject; hypothesis that is actually tested; the basis for selection of the test statistics. Al.ternatiVt: hypothesis (H A): what is concluded if there is significant evidence to reject the null hypothesis. One-tailed test: tests whether value is greater than or less than another value. For example: H0: µ� 0 versus HA: 11>0 Two-tailed test: tests whether value is different from another value. For example: H0: µ= 0 versus HA: µ � 0 T-Distribution The t-distribution is a bell-shaped probability distribution that is symmetrical about its mean. It is the appropriate distribution to use when constructing confidence intervals based on small samples from populations with unknown variance and a normal, or approximately normal, distribution. t-test: t= x - µ. st ..In Chi-Square Distribution The chi-square test is used for hypothesis tests concerning the variance of a normally distributed population. . 2 (n -l)s 2 chi-square test: X = � F-Distribution The F-test is used for hypotheses tests concerning the equality of the variances of two populations. s2 F-test: F= 1s2 SimpleLinear Regression Yi= B0 + B1 x X i + Ei where: Y i = dependent or explained variable � independent or explanatory variable B0 intercept coefficient B1 =slope cocfficicnc Ei = error term = = TotalSum of Squares For the dependent variable in a regression model, there is a total sum of squares (TSS) around the sample mean. total sum of squares = explained sum of squares + sum of squared residuals TSS = ESS + SSR Coefficient of Detennination Represented by R 2, it is a measure of the "goodness of fit" of the regression. ESS = l SSR R2 = _ TSS TSS In a simple two-variable regression, the square root of R 2 is the correlation coefficient (r) between X' and Y, If the relationship is positive, then: r= JR2 Standard Error of theRegression (SER) Measures the degree of variability of the actual Y-values relative to the estimated Y-values from a regression equation. The SER gauges the "fit" of the regression line. The smaller the standard error, the better the fit. Linear RegressionAssumptions A linear relationship exists between the dependent and the independent variable. • The independent variable is uncorrelated with the error terms. • The expected value of the error term is zero. • The variance of the error term is constant for all independent variables. No serial correlation of the error terms. • The model is correctly specified (does not omit variables). • • RegressionAssumptionV iolations Heteroskedasticity occurs when the variance of the residuals is not the same across all observations in the sample. MulticoOinearity refers to the condition when two or more of the independent variables, or linear combinations of the independent variables, in a multiple regression are highly correlated with each other. Serial cornlation refers to the situation in which the residual terms are correlated with one another. Multiple Linear Regression A simple "gression is the two-variable regression with one dependent variable, Yi, and one independent variable, X.· A multivariate regression has more than one independent variable. Yi=Bo +B 1 xX1i +B 2 xX2i +ei Adjusted. R-Squared Adjusted R 2 is used to analyze the imporrance of an added independent variable to a regression. n-1 adjusted R2 = 1- (1 - R2 ) x --n - k-l TheF-Statistic The F-stat is used to test whether at least one of the independent variables explains a significant portion of the variation of the dependent variable. The homoskedasticity-only F-stat can only be clerivecl from R2 when the error rerms clisplay homoskedasticity. ForecastingModelSelection Model selection criteria takes the form of penalty factor times mean squa"d error (MSE). MSE is computed as: T E e;/T t=l Penalty factors for unbiased MSE (s2), Akaike information criterion (AIC), and Schwan information criterion (SIC) are: (T IT - k), e<2 kl11, and T(IUI), respectively. SIC has the largest penalty factor and is the most consistent selection criteria. CovarianceStationary A time series is covariance stationary if its mean, variance, and covariances with lagged and leading values do not change over time. Covariance stationarity is a requirement for using autoregressive (AR) models. Models that lack covariance stationarity are unstable and do not lend themselves to meaningfulforecasting. Autoregressive (AR) Process The first-order autoregressive process [AR(l)] is specified as a variable regressed against itself in lagged form. It has a mean of zero and a constant variance. Yt =�1-1 +et EWMAModel The exponentially weighted moving average (EWMA) model assumes weights decline exponentially back through time. This assumption results in a specific relationship for variance in the model: � =(1- >..)r;_, + )..cr�-1 where: )..= weight on previous volatility estimate (between zero and one) High values of>. will minimize the effect of daily percentage returns, whereas low values of).. will tend to increase the effect of daily percentage returns on the current volatility estimate. GARCHModd A GARCH(l,1) model incorporates the most recent estimates of variance and squared return, and also includes a variable that accounts for a long-run average level of variance. er� =w+nr;_, +0cr�-l where: =weighting on previous period's return 0 = weighting on previous volatility estimate w = weighted long-run variance Ct VL = long-run average variance = Ct+ 0 < 1 for stability w l-et-0 The EWMA is nothing other than a special case of a GARCH(l,1) volatility process, with w = 0, o. = 1 ->., and 0 = >.. The sum Ct + 0 is called the persistence, and if the model is to be stationary over time (with reversion to the mean), the sum must be less than one. SimulationMethods Monte Carlo simulations can model complex problems or estimate variables when there are small sample sizes. Basic steps are: (1) specify data generating process, (2) estimate unknown variable, (3) save estimate from step 2, and (4) go back to step 1 and repeat process N times. Bootstrapping simulations repeatedly draw data from historical data sets and replace data so it can be re-drawn. Requires no assumptions with respect to the true distribution of parameter estimates. However, it is ineffective when there are outliers or when data is non-independent. FINANCIAL MARKETS AND PRODUCTS Option andForwardContract Payoffs The payoff on a calloption to the option buyer is calculated as follows: CT=max(O, ST- X) The price paid for the call option, C0, is referred to as the callpremium. Thus, the profit to the option buyer is calculated as follows: profit= CT- C0 The payoff on a put option is calculated as follows: PT= max(O, X- ST ) The payoff to a long position in a forward contract is calculated as follows: payoff= ST - K where: ST = spot price at maturity K = delivery price Futures Market Participants Hedgers: lock-in a fixed price in advance. Speculators: accept the price risk that hedgers are unwilling to bear. Arbitrageurs: in te rested in marker inefficiencies co obtain riskless profic. Basis The basis in a hedge is defined as che difference between che spoc price on a hedged assec and che futures price of che hedging inscrument {e.g., furures concracc). When che hedged asset and che asset underlying che hedging inscrument are che same, che basis willbe zero ac maruricy . Minimum Variance Hedge Ratio The hedge ratio minimizes che variance of che combined hedge position. This is also che beca of spoc prices wich respecc co furures concracc prices. HR = Ps,F� crp HedgingWith Stock Index Futures # of cont racts =i3r x porcfolio value fucures price x concracc multiplier AdjustingPortfolio Beta If che beta of che capital asset pricing model is used as che systematic risk measure, chen hedging boils down co a reduction of che porcfolio beta. # of contracts= folio value (target beta-portfioIio beta) pon underlying asset ForwardInterest Rates Forward rates are interest rates implied by che spot curve for a speci fie d furure period. The forward rate between T1 and T2 can be calculated as: R R forward - 1T2-R1T1 T2 - TI =R 1 + (R 2 - R 1 ) x (_Ii_) T1 -T1 Forward RateAgreement (FRA) CashFlows a forward ooncract obligacing two parries to agree chat a certain interest rate will apply to a principal amount during a specified fucure rime. The T2 cash Bow of an FRA chat promises che receipt or payment of RK is: cash flow (if receiving R!<) = Lx(RK-R)x(T2 - T1J An FRA is cash flow (if paying RK ) = T. x (R - RK)x (Tz - Ti) where : L = princi pal RK = annualized rate on L R = annualized actual rate Ti = time i expressed in years Cost-of- CarryModel Forward price when underlying asset does not have cash flows: Fo = SoerT Forward price when underlying asset has cash Bows,/: lb = (S0 - I)er T Forward price wich continuous Fo = Soe (r-q )T dividend yield, q: Forward price wich storage co sts , u: T u)T lb =(So + U )er or lb = Soe(r+ Forward price wich convenience yield, F. o - Soe (r -c)T c: Forward foreign exchange rate using interest rate paricy ORP): i:;� -S e <i:.i-rr )T •o - o Arbitrage. Remember to buy low, sell high. If Fo > S0erT ,borrow, buy spot, sell forward today; deliver asset, repay loan at end. If lb < S0e rT , shon spot, invest, buy forward • • Backwardation and Contango Backwardation re!Crs to a situation where the futures price is below the spot price. For this to occur, there must be a significant benefit to holding the asset. • Contango refers to a situation where the fucures price is above the spot price. If there are no benefits to holding the asset (e.g., dividends, coupons, or convenience yield), cont ango will occur because the furures price will be greater than the spot price. • Treasury BondFutures In a T-bond futures concracc, any government bond with more chan 15 years to maruricy on che fuse of che delivery monch {and noc callable wichin 15 years) is d eliverable on che concracc. The procedu re to determine which bond is che cheapest-to-deliver (CID) is as follows: cash received by che s hore = {QFP x CF)+AI cost to purchase bo nd=QB P+AI where: Duration-Based Hedge Ratio T he obje ccive of a duration-based hedge is to create a combined position char does not change in value when yields change by a small amounc. ponfolio value x durarionp #of contracts= fucures value x durationp Interest RateSwaps Plain vanilla interest rat e swap: exchanges fixed for float ing -race payments over che life of the swap At inception, the value of che swap is zero. After inception, the value of the swap is the difference between che present value of che remaining fixed­ and floating-rate payments: Vswap to pay rlXcd = Bfloat - Brix Vswap to n:ccive fixed = B rix - Bfloat = (PMTfixcd,t, x e-re, ) B rixcd x e -rc2 ) + ... + (PMT . fixcd,t2 + [{notional + PMTfixcd t x Currency Swaps ) xe-n" J ��)J x e -n, Exchanges payments in two different currencies; payments can be fixed or Boating. If a swap has a positive value to one oounterparcy, chat parry is Vswap(DC) =Boe -(S0 x Bpc ) where: So = spot rate in DC per FC Lower bound European put on non-dividend­ paying stock: T p � max(Xe-r -S o ,O) Exercising AmericanOptions • It is never optimal to exercise an American callon a non-dividend-paying stock before ics expiration date. • American puts can be optimally exercised early if the y are sufficiently in-the-money. • An American call on a dividend-paying stock may be exercised early if the dividend exceedsthe amount of forgone interest. Put-CallParity p = c - S +Xe-rT c= p+S-Xe-rT Covered Call andProtective Put call. Protective pur. Long stock plus long put. Also called portfolio insurance. Covered call: Long scock plus short price and subsidize the purchase with sale of a call option with a higher exercise pri ce Bear sprrad: Purchase call with high strike price and shon callwich low strike price. Investor keeps difference in price of che options if stock price falls. B e ar spread wich puts involves buying puc wich high exercise price and selling put wich low exercise pr ce. Buttnft.yspmui: Threedifferent options: buy one callwith low exercise price, buy another with a high exercise price, and shon two callswith an exercise price in between. Butterfly buyer is hecring the scock price will stay near the price of the written calls. . T he CTD is che bond that minimizes che foll owing: Q B P- (QFP x CF). This formula calculates the cost of delivering che bond. exposed to credit risk. Lower bound European call on non-dividend­ paying stock: Bull sprrad: Purchase call option wich low exercise = ( notional p :$ Xe-rT; p :$ x OptionSpread Strategies QFP =quoted futures price CF = conversion factor QB P =quoted bond price AI accrued interest [ Upper bound European/American put: c � max(S0 -Xe-rT ,0) today; collecc loan, buy asset under fucures concracc, deliver to cover shon sale. Bfloating = notional + OptionPricing Bounds Upper bound European/American call: c :$ S0; C :$ S0 i Calendar sprrad: Two options with different expirations. Sell a shore-dated option and buy a long-dated option. Investor profits if stock price stays in a n arrow range. co a calendar spread except chat the options can have different strike prices in addition to different expirations. Diagonal sprrad: Similar Box spread: Combination of bull call spread and bear put spre ad on che same assec. This strategy chat is equal to che high exercise price minus che low exercise price. will produce a constant payoff Option Combination Strategies calland a put wich the same exercise price and expiration date. Profit is earned if scock price has a large change in either direction. Short straddlr. Sell a put and a callwith the same exercise price and ex.pirarion date. If stock price remains unchanged, seller keeps option premiums. Unlimited potential losses. Stranglr. Similar to straddle except purchased option is out-of-the-money; so it is cheaper to implement. Stock price has to move more to be profitable. Long straddle. Bee on volarilicy. Buy a Add an additional put (strip) or call(strap) to a straddle strategy. Strips and straps: Exotic Options Gap optWn: payoff is increased or decreased by the difference between two strike prices. Compound optron: option on another option. Chooser option: owner chooses whether option is a call or a put after initiation. Barrier option: payoff and existence depend on price reaching a certain barrier level. Binary option: pay either nothing or a fixed amount. Lookback optron: payoff depends on the maximum (call) or minimum (put) value of the underlying asset over the life of the option. This can be fixed or floating depending on the specification of a strike price. Shout option: owner receives intrinsic value of option at shout date or expiration, whichever is greater. Asian option: payoff depends on average of the underlying asset price over the life of the option; less volatile than standard option. Basket options: options to purchase or sell baskets of securities. These baskets may be defined specifically for the individual investor and may be composed of specific stocks, indices, or currencies. Any exotic options that involve several different assets are more generally referred to as rainbow optWns. Foreign Currency Risk A net long (short) currency position means a bank faces the risk that the FX rate will fall (rise) versus the domestic currency. net currency exposure (assets - liabilities) + (bought - sold) On-balance shut hedging. matched maturity and currency foreign asset-liability book. Off-balance sheet hedging. enter into a position in a forward contract. = Central Counterparties (CCPs) When trades are centrallycleared, a CCP becomes the seller to a buyer and the buyer to a seller. Advantages ofCCPs: transparency, offsetting, loss mutualizacion, legal and operational efficiency, liquidity, and default management. Disadvantages ofCCPs: moral hazard, adverse selection, separation of cleared and non-cleared products, and margin procyclicality. Risks faced by CCPs: default risk, model risk, liquidity risk, operational risk, and legal risk. Default of a clearing member and its flow through effects is the most significant risk for a CCP. MBSPrepay ment Risk Factors that affect prepayments: Prevailing mortgage rates, including (l) spread of current versus o riginal mortgage rates, (2) mortgage rate path (refinancing burnout), and (3) level of mortgage rates. • Underlying mortgage characteristics. • Seasonal f.ictors. • General economic activity. • Conditional Prepay ment Rate (CPR) rate at which a mortgage pool balance is assumed to be prepaid during the life of the pool. The single monthly mortality (SMM) rate is derived from CPR and used to estimate monthly prepayments for a mortgage pool: SMM l -(l -CPR) 1112 Annual = Option-Adjusted Spre ad (OAS) Spread after the "optionality" of the cash flows is taken into account. Expresses the difference between price and • • • • theoretic:al value. When comparing two MBSs of similar credit quality, buy the bond with the biggest OAS. OAS zero-volatility spread-option cost. = ''4'll!:ii''':''';11i1ti:1r''',jf1 . . Step 3: Discount to today using risk-free rate. can be altered so that the binomial model can price options on stocks with dividends, stock indices, currencies, and futures. Stocks with dividends and stock indices: replace e'T with tf.r-<i'JT, where q is the dividend yield of a stock -rr"P or stock index. Currencies: replace t'T with tf.r--r�T, where rr is the foreign risk-free rate of interest. Futurts: replace t'T with 1 since futures are considered zero growth instruments. Black-Scholes-MertonModel x c =So N(d1 )- Xe-rTN(d2) p = Xe-rT N(-d2)-S0N(-d1) Value at Risk (VaR) Minimum amount one could expect to lose with a given probability over a specific period of time. V aR(Xo/o) =zx% x cr where: In Use the square root of time to change daily to monthly or annual VaR Expected Shortfall (FS) Average or expected value of all losses greater than the VaR: E[4 I I,. > VaR]. • Popular measure to report along with VaR. • ES is also known as conditional VaR or expected tail loss. • Unlike VaR, ES has the ability to satisfy the coherent risk measure property of subadditivity. • Binomial Option PricingModel A one-step binomial model is best described within a two-state world where the price of a stock will either go up once or down once, and the change will occur one step ahead at the end of the holding period. In the two-period binomial model and multi­ period models, the tree is expanded to provide for a greater number of potential outcomes. Step 1: Calculate option payoffs at the end of all states. Step 2: Calculate option values using risk-neutral probabilities. f size of up move= U = ecr J size of down move = D= _!._ u e'1- D ; 'ITdown = 1- 'rrup 'ITup = U D _ 2 [ ] + r +0.5 xcr xT axJf = d1 = rime to maturity = asset price = exercise price = risk-free rate cr = stock return volatility N(•) =cumulative normal probability V aR(Xo/o)J-days = VaR(X%)1-day� VaRMethods The delta-normal method (a.le.a. the variance­ covariance method) for estimating VaR requires the assumption of a normal distribution. The method utilizes the expected return and standard deviation of returns. The historical simulation method for estimating VaR uses historical data. For example, to calculate the 5% daily VaR, you accumulate a number of past daily returns, rank the returns from highest to lowest, and then identify the lowest 5% of returns. The Monte Carlo simulation method refers to computer software that generates many possible outcomes from the distributions of inputs specified by the user. All of the examined portfolio returns will form a distribution, which will approximate the normal distribution. VaR is then calculated in the same way as with the delta­ normal method. (�) d2 T So X r -(ox.ff) Greeks estimates the change in value for an option for a one-unit change in stock price. • Calldelta between 0 and + 1; increases as stock price increases. • Calldelta close to 0 for far out-of-the-money calls; close to 1 for deep in-the-money calls. • P ut delta between -1 and O; increases from -1 to 0 as stock price increases. • P ut delta close to 0 for far out-of-the-money puts; close to -1 for deep in-the-money puts. • The delta of a forward contract i s equal to 1. The delta of a futures contract is equal to /T. • When the underlying asset pays a dividend, q, the delta must be adjusted. If a di vidend yield exists, delta of call equals riT N(d1), delta of put equals riT x [N(d,)-1], delta of forward equals riT, and delta of futures equals 1-�T. Theta: rime decay; change in value of an option for a one-unit change in rime; more negative when option is at-the-money and close to expiration. Gamma: rate of change in delta as underlying stock price changes; largest when option is at-the -money. Vega: change in value of an option for a one-unit change in volatility; largest when option is at-the­ money; close to 0 when option is deep in- or out­ of-the-money. Rho: sensitivity of option's price to changes in the risk-free rate; largest for in-the-money options. Delta: x Delta-Neutral Hedging • To completely hedge a long stock/short call position, purchase shares of stock equal to delta x number of options sold. Only appropriate for small changes in the valu e of the underlying asset. • Gammacan correct hedging error by protecting against large movements in asset price. • Gamma-neutral positions are created by matching portfolio gammawith an offsetting option position. • BondValuation There are three steps in the bond valuation process: Step 1: Estimate the cash flows. For a bond, there are two types of cash flows: (1) the annual cash flows associated with the instrument to its the recovery of principal at maturity, or will be reinvested at the YfM and assumes that or semiannual coupon payments and (2) when the bond is retired. iscount rate. The Step 2: Determine the appropriate d approximate discount rate can be either the bond's yield to maturity (YrM) or a series of spot rates. Step 3: Calculate the PVofthe estimated cash flows. The PY is determined by discounting the bond's cash fl.ow stream by the appropriate discount rate(s). Sources ofcountry risk-. (1) where the country is in the bond will be held until maturity. (3) Relationship Among Coupon, YfM, and Price If coupon rate > YTM, bond price willbe greater than par value: prmzium bond. If coupon rate < YTM, bond price willbe less iscount bond. than par value: d If coupon rate = YTM, bond price will be equal to par value: par bond. Clean and Dirty Bond Prices When a bond is purchased, the buyer must pay any accrued interest (AI) earned through the settlement date. Dollar Value of a Basis Point The DVO 1 is the change in a fixed income security's value for every one basis point change in interest rates. DVOl = Effective Duration and Convexity the seller of the bond must be paid to give up relationship; most widely used measure of bond ( �)mxn estimates of bond price changes. FVn = PV0 1 + effective duration = where: r = annual rate m = compounding periods per year 11 = y ears Continuous compounding: (second derivative) of the price/yield relationship; accounts for error in price change estimates from duration. Positive convexity always has a favorable convexity Spot Rates to maturity on a zero-coupon bond that matures in t-years. It can be calculated using a financial calculator or by using the following formula (assuming periods are semiannual), where d(t) is a discount factor: 1 121 d(t) -1 percentage bond price change :::::duration effect+ convexity effect �B 2 Callable bond: issuer has the right to buy back the bond in the future at a set price; as yields fall, Forward rates are interest rates that span future )1 = -duration x �y + .!. x convexity x �y2 Bonds With Embedded Options Forward Rates rorward rate (1 + ,. = BV_�y + BV+�y - 2 x BV0 BV0 x �y2 Bond Price Changes With Duration and Convexity B periods. 2 x BV0 x�y Convexity: measure of the degree of curvature . A t-period spot rate, denoted as z(t), is the yield (-) BV_�Y - BV+�Y impact on bond price. rx n FVn = PVoe 2 price volatility; the longer (shoner) the duration, changes in interest rates; can be used for linear Discrete compounding: z(t) = Duration: firsc derivative of the price-yield the more (less) sensitive the bond's price is to Compounding ic yield)'+! + period (I __:. _ ____:. _;.__ = _ _ _ _ bond is likely to be called; prices will rise at a decreasing rate-negative convexity. Putable bond: bondholder has the right to sell at a set price. i bond back to the ssuer (1 + periodic yield)1 Rc-1,c _ - BV, + C, - BV,_1 country's level of indebtedness, (2) obligations such as pension and social service commitments, (3) a country's level of and stability of tax receipts, (4) political risks, and (5) backing from other countries or entities. Internal Credit Ratings At-the-point approach: goal is to predict the credit quality over a relatively short horizon ofa few time horizon and includes the effects of forecasted cycles. Expected Loss The expected loss(EL) represents the decrease in value of an asset (ponfolio) with a given exposure subject to a positive probability of default. expected loss = exposure amount (EA) x x loss rate (LR) probability ofdefault (PD) Unexpected Loss Unexpected loss represents the variability of potential losses and can be modeled using the definition of standard deviation. � UL = EA x PDxcr[R + LR2 x cr�0 Operational Risk Operational risk is defined as: The risk ofdirr:ct and indirect loss mu/ting.from inadequate or failed internal processes, people, and systems or from external events. Operational Risk Capital Requirements • Basic indicator approach: capical charge measured on a 6rmwide basis as a percentage of annual gross income. • Standardized approach: banks divide activities among business lines; capical charge = sum for each business ine. Capical for each business line l determined with beta factors and annual gross income. • Advanced measurement approach: banks use their own methodologies for assessing operational risk. Capital allocation is based on the bank's operational VaR. Loss Frequency and Loss Severity Operational risk losses are independent dimensions: classified along two Loss severity. value of financial loss suffered. Often modeled with the lognormal distribution (distribution is asymmetrical and has fat tails). Stress Testing VaR tells the probability of exceeding a given loss realized return minus per period financing costs. but fails to incorporate the possible amount of a Yield to Maturity (YTM) discount rate that equates the present value of all Factors influencing sovereign default risk-. (1) a models random events). PPN: 32007227 ISBN-13: 9781475438192 The net realized return for a bond is its gross to its internal rate of return. The YTM is the the disproportionate reliance of a country time period (typically one year). Often modeled BV1-l The YfM of a fixed-income security is equivalent (4) on one commodity or service. with the Poisson distribution (a distribution that The gross realized return for a bond is its end-of­ value divided by its beginning-of-period value. the structure and the efficiency of legal systems, and Lossfrequency. the number of losses over a specific Realized Return period total value minus its beginning-of-period (2) political risks, the legal systems of a country, including both Through-the-cycle approach: focuses on a longer 10,000x�y Clean price. bond price without accrued interest. ownership. the economic growth life cycle, months or, more generally, a year. �BV DVOl = duration x 0.0001 x bond value Dirty price. includes accrued interest; price Country Risk price. The yield to maturity assumes cash flows loss that results from an extreme amount. 9 7 8 1 4 7 5 438 1 9 2 U.S. $29.00 <Cl 2015 Kaplan, Inc. All Rights Reserved. Stress testing complements VaR by providing information about the magnitude of losses that may occur in extreme market conditions.