Reading 37: Measuring and Managing Market Risk

Page No. 23Date / /Reading 31:,Measuring And ManagingMarket RiskValue at Risk (VAR)VAR measures the downside risk of a portfolio H has threeComponents : the loss size, the probability of loss , & a timeframeCan be referredto as a 95%Can be expressed as a YouExample confidencea e $ amtA5% probability that a portfolio will exp loss of $25,000or more in any given monthSpecificied loss is the minless over the periodIt is the est of min loss that is expected over a certaintime periods :Can be expressed In currency units or y. of folio valueVar Can be daily t weekly t monthly or any other period.5% monthly VAR of $1mln means that these is a 5% prob that theCompany will expenide a loss of at least $ min in one monthVaR 5% ofdistributionS% Vor Page No. 24Date/Note that $25,000 is a min loss amt so we can state "5%the time the min monthly loss that the company will expis $25,000 "To est a VaR, we must specific the time period and the sizeof the loss, so there is significant judgment involved inVaR est.if we choose the size of the loss we will est the probof losses of that size or largerif we choose the prob of the loss, we will est the minsize of the losses that will occur with that probEst VaRStep 1 : Identify risk factors leg market 7 int rate currency)Step 2 : Choose from diff approach to est VaR:Parametric (variance - (ovariance) approach IMPHistorical simulationMonte Carlo simulationIVaR using Parametric MethodCAKA Variance - covariance) Detta - Normal Var use mean, )Each risk factor is assigned a distribution, often assumedto be normal to avoid Complications of Shew & Rur tosis Page No. 25Date//Only mean and variance of normal distribution are neededParameters are based on data Over a lookback period.Uses est variances and Covariances of port/olio secVaR is est using expected return and SD for each risk factorVaR Can then be calculated from the normal distribution-Eg - t 57. VaR represents 1-65 Standard deviations below themain point.up = J (W.01)2 + (W202)2 + 2Wic 5 W2O2 X rt 2WA WB90%tAdvantage : Simple to use57.5%ExampleI-1-650 +1.650Imagine that we are provided the following90%4info about two asset + Sec A & BCI 4 + (2xo)u + (1.65 xo)SecStd dev of daily returnsMean daily returnCovariance ofA0.01580.0004daily retainB0.01120.00030.000106How would we use this info to est the 5% annulaVaR for a portfoliothat is 60% invested in sec A & 40%invested in secB?Soln -> Mean daily folio return = 10.6x(00004)+0.4x(00003)E 0.00036Variance of folio return := (0.6)2 (0.0158) + + = 0.00016 b Conversion Rule:-weekly - 5daily siymuWeeklyMonthly - 22Page No. 26sigmaWeeklyYrly - 252Date/2%5HadingdaysNStandard deviation of folio return = 0.000161 = 0.0126825% daily VaR = 0.00036 1.65 X(0.012682) = -0.0206MeanStdTevFor 250 day per yr (trading), annual mean return = 250(0.00036)Annual Stanard deviation = 250 X (D.012682) = 0.200525% annual VaR = 0.09 - 1.65 x (0.20052) C-0-24045y. daily VaR =$10 mh X (0.0206) = $ 206,000Zx5% annual VaR = $ 10mln X 6.2409) = $2,404 000WeaknessIf We want to go to - daily var then206,000 X 2.331.651.1 2scoreNot useful when folio contains optionsEst are only as good as inputs (mean, variance, covariance)The length of the lookback period will affect theparameter estIncases where normality cannot be reasonably assumedsuch as when the folio contains options, the parametricmethod has limited usefulnessIIVaR using Historical simulation Methodduring the look back periodBased on the actual changes in risk factors experiencedPeriodic return over a lookback period ale compiledThe 5% VaR is the value beyound the 5th percentileThe return are then ordered from smallest to largestFor 200 observation this would be approx the 10thsmallest return Page No. 27Date / /AdvantageNo need to assume distributionHistorical simulation method Can be used to est theVaR for portfolio that include optionsBased on actual eventsWeaknessVaR based on an unsually volatile (stable) lookback period willyield overestimates (underestimates) of VaRNo certainty that historical event will re-occurIIIVaR using Monte Carlo SimulationAprobability distribution of and corrections bet" each riskfactor is assumed (Key step 1)Based on users assumptions of distribution for each risk factorSimulation must also take correlation of asset into accountThis method will produce same result as the parametric method,if sample size is exteremely largeSoftware generates random values for each risk factor based onabove and computer periodic return based on these valuesAs in historical simulation VaR est from sorted outcomesThe Monte Carlo and parametric methods should produceidentical result if the distribution and parameters are the sameAdvantageWeaknessFlexible to handle more ComplexComplex Calculationdistributions No assumption ofTime consuming processnormality Page No. 28Date//Lookback period :- Time period used to est expected return &SD for each risk factorAdvantageVaRDisadvantagesEasy to understandSubjective methodEasy to CommunicateUsing normal distribution leads toUseful in comparing risk across asset Clasunder statement of downside riskHelps in Capital allocation decisionVaR could be understated if someCan be used for performatic evaluationassets in a folio are illiquidReliability can be verified by backtestingCorrelation risk can understateWidely accepted by regulatorspotential lossesVerified via backtestingIf the lookback period has low volatilityVaR will apprear to be lowNeeds to be supported byadditional risk measuresIgnores right tail events l downside RiskfacusAvg VAR Extensions bliven that losses of OF the VaR extent of VAR has encrededinside legionConditional VaR ( CVaR ) use by bankAka expected tail loss of expected shortfalib culculate anyloss beyound the VaR suteffInsid Hell(VaR is the expected loss, given that the loss is equal to orgreater than VaR (VaR is also referred to as theexpected tail loss or expected ShortfullAvg of HellWhen VaR is est using the historical simulation methodorMonte Carlo Simulation , all values equal to or less than the Page No. 29Date//VaR loss are Known, so it is straightforward to avg theseto get the CVaRWith the parametric method the magnitude of losses greaterthan the VaR are not Known + So calculating the expectedloss in the left-hand tail is mathematically complexIncremental VaR ( IVaR)lvaR is the change in VaR from a change in the folioallocation to a Sec (change in Var to change in sizeExampleA 2% increase in the weight of a sec In the folio increasethe folio VaR from $1,345,600 to $ 1,562,400WaR for the 2% increase in the folio weight of thesecis 1562,400 - 1,345,000 = $216,800Marginal VaR (-MVaR)MVaR is the slope of the tangent at a point in the VaRvs sec weight curveMVaR is based on Calculus (eg applicable for a small change insec weight)Similar to IVaR expect that MVaR uses formula S derived fromCalcalus to reflect the change in VaR for a given changein the size of postfolio positionolt is inaccurate to interpret it as the change in VaR for a% increase in the sec weight its sensitivity (Approx)small change in asset positive Page No. 30Date//oA reasonable approx of the sensitivity of VaR to a both V.increase Change in folio weight of a sec; thus 1the MVaR and WaR are similarEx ante tracking error (or relative VaR)It measures the VaR of the diff bet" the return onafolio and the return on its benchmark.A5% monthly relative VaR of 2.5% implies that 5% of timethe folio relative underperformace will be at least 2.5%The relative VaR Can be calculated as the VaR of aCombination of the subject folio and a short position in thebenchmark folioSensitivity AnalysisSensitivity analysts focuses on the impact on value of a givensmall change in one risk factorASensitivity analysis complements VaR in understanding folio riskbut f unlike VaR it does not involve any prediction ofthe probability of losses of any specific amtFor equity - BetaFor Bond - Duration & ConvexityDFor options - Delta f Gamma & Vega7' Page No. 31Date//*Scenario AnalysisScenario analysis provides an est of the impact (usuallynegative) on value of a set of changes of significantmagnitude in multiple risk factors. A historical Scenario approachuses a set of Changes in risk factors that actually occuredCeg during the sub-prime mortguge crisis of 2008) c as opposedto a hypothetical Scenario approachStress tests examine the effect on value (or solvency) of aScenario of extreme risk-factor changesD Used to est the changes in falle value for changes inmultiple risk factor0Hypothetical scenario: Extereme movements that have not occuredpreviously Can be usedHistorical Scenario : Based on movements that have actuallyoccured in the pastMeasures of Risk for Different Asset ClassesEquitiesThe risk factor most often used for equities is betaFixed IncomeDuration provides an est of how A instruments are affectedby changes in int rates C YTM) For larger changes inint rate , including the effective of convexity on fixedincome sec values improves sensitivity estChange in Price = -duration (D4) t 1/2 convexity (sy)2 First-order - - delta duration VegaPage No. 32Date/Second order - gamma & ConvexityOptionsSeveral risk factors affect the values of options positionDelta is an est of the sensitivity of option values asset. tochange in the various value of the underlying $1(eg a call data of 0.6 means that for every increasein the price of the underlying asset the call valueby $0.60) Index - \ deltaLong call - otolShort call - oto -1Gamma is an est of how delta changes as the price of theunderlying asset changes and is catted Catculated as theratio of the changes in delta to a calculate in theprice of the underlying asset.Vega is a measures of the sensitivity of options valuesto changes in the expected volutility of the price of theunderlying asset. We can incorporate all three of theseoptions risk measure in the following eq"Change in call price = delta (AS) + 1/2 gamma (As)2 t Vega (AV) IDSensitivity & Scenario Risk measureSensitivity risk measures and scenario risk measures allowa risk manager to hedge applipriately Page No. 33Date / /Sensitivity Risk measures can inform a purtfolio managerabout a portfolio exposure to various Risk factors tofacilitate risk mangement Managers may choose to hedge riskspercelved as being excessiveWhen using Scenaria analysis for a portfolio that containsoptions or fixed Income sec with embedded options theindividual options and bonds must be valued with apricing model using scenario values for the risk factorsFactors sensitivity can be used to est the effects of smallchanges in risk factors for these sec but for laigerrisk factor changes, pricing models for folio SecEven Combining first - order and second - order effects, such asduration and convexity, + only provides an approx of the changein value that would se sult from a relatively large changein a Q risk factorSenario analysis can be performed as if the Scenariochanges are instantaneous f or as incremental change whereapostfolio manager can respond by reducing dosingorhedging positionsAnalyzing instantaneous changes in considered more conservativeand more realistic in circum stances where canter parties areunable / unwilling to provide additional collateral req orwhere lack of liquidity makes changing poilfolio positionvery & costly or impositible Page No. 34Date//In reverse Stress testing the first step is to identify aportfolio largest risk exposures and determine anunacceptable outcome (usually one that would threatern scenario thesurvival of the organizational) and to identify resultof changes in risk factors that would bein Such an outcome.The Key que is then how likely such scenarios areScegario analysis Can be seen as the final step in therisk assessement and management process t after performingsensitivity analysisFor a firm that has limited its risk through amax VaR limits on position Sizes, limits on specific riskexposures + and So on + Scenario analysis Can provide additionalinfo on a postfolios vulnerability to a set of events orchange in correlations that would significantly reducethe value of the purtfolioFirms that use Leverage e especially banks & hedge funds + oftensuse stress tests involving a single risk factor todetermine the size of Change in that factor thatcould cause such losses that the firms's sustainabilityisComproprised Page No. 35Date//VaR, sentivity analysis and scenario analysis complement eachother and a risk manager should not reply one thismeasure- VaR provides a probability of loss-Sensitivity analysis provides est of the relative exposuresto diff risk factors. but provides DO est of the prob ofany specific changes in risk factors-Scenario analysis will provide info about exposure to simute-aneous change in several risk factors or changes in riskcorrelations but there is no probability associated with aspecific scenario (other than the empirical prob of a historyScenatio over the lookback period)Example: A bond manager controlls two spirate bond portfolioboth with a duration equal to 3.2'With equal duration , the predicted change in value for a changein yield will be equal for each portfolioHowever , the yield volatility of the two portfolio leg govt bondsVs high-yield bend) may be significantly different leading to adiff likelihood of that change in valueThe Same limitations applies to opption delta ( or gamma)Option delta /gamma may be an appropriate measure of the riskfor small changes in the price of the underlying t butthe volatility of the prices of the underlying may bequite diff for diff options Page No. 36Date/Sensitivity risk measures & Scenario risk measures allowa risk manager to hedge appropriately.Stress Trading :- Using exterme changes to examine the expectedeffects on a portfolio or the organizetII Reverse Stress:- Treading Targating most significant exposure andtesting assessing their behavior in various environmentAdvantage & limitations of Sensitivity risk measures &Scenario risk measures1MPVaR, sensitivity analysis, and Sconasia analysis complement each otherSensitivity and Scenasio risk measure provide est of changein value of portfolio, but no est of the probability ofchanges in risk factors.Application of Risk MeasuresRisk measures used by an organization are dicated by :- The type of risks it is exposed to- The regulations that govern it- Whether the organiz' uses leverage.These factors can be used to identify risk measures used by :-- Banks- Asset managers- Pension funds- Insures Page No. 37Date/Bank,Typically use Sensitivity measures (duration of held to- maturitysecurity & foreign exch risk exposure)Scenario analysis and Stress testing (for their full Bls)Leverage risk measures and VaR (especially for trading Sec)Banks also est risk from asset - liability mismatch , est VaR foreco capital & disaggreate risk by both geographic locationand biz unit typeTraditional (Tony - Way) Asset ManagersTypically focus on relative risk measures unless their goalis an absoute return targetTypically risk measures used include the following :- The size of positions- Sensitivity measures of int rate & market risk- Historical & hypothetical Scenario analysis- options riskPoth that will make your- LiquidityGlide path- Redeemptions risk , Probability of lossesrisk measures more specific to asset mang" ifactive Share the diff bet the weight of a Sec in theportfolio and its weight in the benchmarkEx-post tracking error(backward looking) over a lookback period istypically used for performance attribution Page No. 38/23th Dec'2022 FridayDatemeEx - ante tracking error (forward looking) focuses on the potentialunderperformance Managers with an absoute return targetmay use VaR insteadHedge fundRisk measures used depends to some extent on the Strategy1employed for hedge funds in general the riskmeasures used includes sensitivity analysis leverage measuresScenario analysis 7 & strees testsFunds with both long and short positions will est riskmeasure for long positions short positions as well as forVaR the overall portfolio (gross exposure). - Medge fund, that useShort Kell periodsfocus on VaR measures of less than 177. forHedge funds with signifantly non - normal returns distributionsuse a risk measure referred to as maxdraw downthe larget decrease in value over periods of aspecific lengthSensitity measures based on standard deviation of Bmay be misleading for large changes in risk factorswhen returns are non-normal40t20g- Eco Capital : losses Capital needed to overcom eSeversein The bizPage No 39Date//Defined Behefit Pension fundsBenefit pension funds are concerned with the diff betthe value of assets (MV) & the pv of estuture liability ( pay" to retirees & hiers)AKey risk measures used by pension funds is surplus at RiskaVaR for plan assets - liability e A -ve sulplus must bemade up by the firm if higher - than - expected assetfunded Stutsretuins do not reduce it g significantly over timeA glide path refers to a multi-yr plan for adj pensionfund Contributions to reverseasignificant overfundedorunderfunded statusTo reducesurplus uncertainty a pension fund may matchits assets to it liabilities A related risk measure is anest of the bedged exposure and unhedged (returns generating)exposule of the fundInsurance CompaniesProperty and Casualty insurance, sell akto, home boat liability&health insurance The insurance risk of a P&C Companyhave low correlation with the market risk of their investLife insurance primarily sell life insurance policies 4 annuities,deathsome of which make pay" until the annuity owner's Page No. 40Date//Insurance risks are reduced by purchasing reinsurance (fromanother insurance company) & by geographical diversificationProperty & Casualty InsurersP&C insurance use sen of their exposures tomarket risk factors in their invest portfolio for risk mangPrem income is expected to cover the cost of insuranceclaims in a typical yr t with the invest portfolioavailable to cover extraordinary claims lossesVaR and Capital at risk are used as measures of riskexposure in invest" accounts . Scenario analysis isalsoused, often combining portfolio risk factors & insurancelisk factors in a Scenario.policies insued ) & regulatory disc the value of riskierRegulations may req specific amt of reserves (based onassets helds as reserves in determining their edequancyLife InsurersThe insurances risk of life insurers is more highlyCorrelated with the market risk exposules oftheir invest portfolio then it is for P&C insuresAnnuities pay over relatively long period into the futureand present values are Sensitive to disc factorused and also have significant mortalility risk. Page No. 41Date//Life insurance both therefore est sensitivities to market risk factorfor inventing purtfolio and for annuity liabilities.Because life insurances are able to Somewhat matchthe market risk of their (@ portfolis assets to theirliabilities t they bet" must consider the risk of the remainingmismatch assets & liabilities life insurance alsouseScenario analysis that includes both non. market (insurance)risk factor and market risk factorContraintsRisk measurements alone cannot effectively manage risk Theymust be combined with constrains If constraints are too loose,the firm may experience flauncial streess , corportuge reorgunization105 bankruptyof Constrains are too restrictive + profitability is impaired - Imposingrestrictions at the biz until lul may be too restrictiveto the extent diversification benefits or offsetting positionsacross biz units are not taken into account - Common constiaintsincludes : :-Risk Budgeting : Allocation of total risk to diff activitiesStrategies or asset classesScenario Limits : Limits on est loss for a given ScenarioStop loss Limit : Req reduction in Size of a portfolio if lossesexceed a specified amt in a specified period Page No 42Date/Position Limits: Limit on allocations to scenarities within anasset classRisk BudgetingA risk management process that first determine the acceptabletotal risk for an organization and the allocates that riskato diff activities, strategies or asset classes as appropriateFor example , determing the max allowable 5% VaR amtand then allocating that VaR across various biz unitsAportfolio manager may set a limit for total risk relativeto a benchmark and then allocate that risk to deviationsfrom benchmask weight in specific in $ industries anddeviations from benchmark weight for firms in aspecific industriesPosition LimitsLimits risky by ensuring some min lul of diversificationExample : limits on allocations to individual sec within anasset class, or the diff beth long & short positionsfor a hedge. fund managerPosition % limits Can be expressed as Currency amt of asof a partfolio's value. Page No 43Date//Positionlimits can also be based on a liquidity measures, suchas avy daily or weekly trading volume.Stop - Loss LimitsThese req that a Fisk exposure be reduced if losses exceed asperfied amt over a certain period of time.A simple stop - loss limit reduces the portfolio allocation to a stock0 asset class (by a given amt) if it declines in value bymore than a - specified % (or $)of slightly more complex type of stop-loss limit is a reqthat a risk exposure be hedge as the value of asecurity or index falls This is referred to as postfolioinsurance when the value of a postfolio is hedged byindex putsScenario limits are limits on expected loss for a given scenarioRisk Measures in Capital AllocationCapital allocation refers to how the capital of a firm isused to funds its various biz units or activities(analogous to asset allocation for a postfdia manager)Risk management req that the risk exposure for each use ofiim capital be considered. Page No. 49Date //One way to introduce fisk exposures to various activitiesinto the Capital allocation decisions is to limit theoverall risk of all the activities.By calculating a VaR for each activity or biz unit themax acceptable VaR can be allocated across the activitiesor biz units in a process similar to risk budgetinyfor a portfolio Manager