FRM二级备考公式表.pdf

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1、MARKET RISK MEASUREMENT ANDMAN E NT Value at Risk(VaR)VaR for a given confidence level occurs at the cutoff point that separates the tail losses from the remaining distribution.Historical simulation approach:order return observations and find the observation chat corresponds co the VaR loss level.Pa

2、rametric estimation approach:assumes a distribution for the underlying observations.Normal distribution assumption:VaR=(-r+a r X ZCt.)Lognormal distribution assumption:VaR=(l-ellr-OrXZ()Expected Shortfall Provides an escimace of tail loss by averaging rhe VaRs for increasing confidence levels in the

3、 cail.Weighted Historical Simulation Approaches Age-weighted:adj uses the most recent(distant)observations co be more(less)heavily weighted.Volatility-weighted:replaces historic returns with volatility-adjusted returns;actual procedure of estimating VaR is unchanged.Correlation-weighted:updates the

4、variancecovariance matrix between assets in me portfolio.Filtered historical simulation:relies on bootstrapping of standardized returns based on volatility forecasts;able co capture conditional volatility,volatility clustering,and/or data asymmetry.Peaks-Over-Threshold(POT)Application of extreme val

5、ue theory(EVT)co the distribution of excess losses over a high threshold.One of the goals of using the POT approach is to compute VaR.From estimates of VaR,we can derive the expected shortfall(ES).Backtesting VaR Compares the number of instances when losses exceed the VaR level(exceptions)with the n

6、umber predicted by the model at tl1e chosen level of confidence.Failure rate:number of exceptions/nu1nber of observations.The Basel Committee requires backcescing at che 99o/o confidence level over one year;establishes zones for the number of exceptions wim corresponding penalties(increases in me ca

7、pital multiplier).Mapping Mapping involves finding common risk factors among positions in a given portfolio.It may be difficult and time consuming co manage che risk of each individual position.One can evaluate the value of portfolio positions by mapping chem onto common risk faccors.Pearson Correla

8、tion Coefficient Commonly used co measure the linear relationship between two variables:covxv PXY=axay Spearmans Rank Correlation Step 1:Order the set pairs of variables X and Y with respect co sec X.Step 2:Determine the ranks of X and Y for each I I time period i.Step 3:Calculate the difference of

9、the variable rankings and square the difference.n 6I:dr Ps=1-i=I n(n2-1)Where n is the number of observations for each variable and d is the difference between the I ranking for period i.Kendalls T T=_n-c-n_,d,_ n(n-1)I 2 Where the number of concordant pairs is represented as nc(pair rankings in agr

10、eement),and the number of discordant pairs is represented as nd(pair rankings not in agreement).Mean Reversion In1plies char over time variables or returns regress back co the 1nean or average return.Mean reversion race,a,is expressed as:Sc-Sc-I=a(-Sc-I)The coefficient of a regression is equal co me

11、 negative of the mean reversion race.Autocorrelation Measures the degree mac a variables current value is correlated co past values.Has che exact opposite properties of mean.reversion.The sum of the mean reversion rate and the oneperiod autocorrelation race will always equal one.Correlation Swap Use

12、d to trade a fixed correlation between cwo or 1nore assets with a realized correlation.Realized correlation for a portfolio of n assets:Prealizcd=2 2 LPi,j n-n.1 J Payoff for correlation swap buyer:notional amount x(P,.i;-pfixed)Gaussian Copula Indireccly defines a correlation relationship becween c

13、wo variables.Maps the marginal distribution of each variable to a standard normal distribution(done on percencile-co-percencile basis).The neV joint distribution is a multivariate standard normal distribution.A Gaussian default rime copula can be used for measuring the joint probability of default b

14、etween cwo assets.Regression-Based Hedge pR=FN x OVOlN x ovo1R where:fR=face amount of hedging instrument fN=face amount of initial position Bond Valuation Using Binomial Tree Using backward induction,the value of a bond at a given node in a bino1nial tree is the average of the present values of the

15、 two possible values from the next period.The appropriate discount rate is the forward rare associated with the node under analysis.There are three basic steps co valuing an option on a fixed-income instrument using a binomial tree:Step 1:Price the bond value at each node using the projected interes

16、t rates.Step 2:Calculate the intrinsic value of che derivative at each node at maturity.Step 3:Calcltlace the expected discounted value of the derivative at each node using the riskneutral probabilities and work backward through the tree.Interest Rate Expectations Expectations play an important role

17、 in determining the shape of the yield curve and can be illustrated by examining yield curves that are Rae,upward-sloping,and downward-sloping.If expected 1-year spot rates for the next three years are r1,r2,and r3,tl1en the 2-year and 3-year spot rares are computed as:r(2)=J(l+r1)(I+rz)-l Convexity

18、 Effect All else held equal,che value of convexity increases wi ch maturity and volatility.Term Structure Models Model 1:assumes no drift and that interest rares are normally discribuced:dr=adw Model 2:adds a positive drift cerm co Model l char can be interpreted as a positive risk premium associate

19、d with longer rime horizons:dr=A.de+crdw where:A=inceresc race drift Ho-Lee Model:generalizes drifc co incorporate rime-dependency:dr=A.(c)dc+adw Vasicek Model:assumes a mean-reverting process for short-term interest rates:dr=k(9-r)dt+crdw where:k=a parameter that measures the speed of reversion adj

20、usonent e=long-run value of the short-term rate assuming risk neutrality r=current interest rate level Model 3:assigns a specific parameterization of time-dependent volatility:dr=A(t)dt+cre.o.dw where:a=volatility at t=0,which decreases exponentially to 0 for a 0 Cox-Ingersoll-Ross(CIR)model mean-re

21、verting model with constant volatility,cr,and basis-point volatility,crJ;,that increases at a decreasing rate:dr=k(0-r)dt+crJ;dw Model 4(lognormal model):yield volatility,a,is constant,but basis-point volatility,crr,increases with the level of the short-term rate.There are two lognormal models of im

22、portance:(1)lognormal with deterministic drift and(2)lognormal with mean reversion.Overnight Indexed Swaps(OIS)Interest rate swap in which a fixed interest rate is swapped for a floating interest rate.The OIS rate is the best proxy for the risk-free rate in the valuation of collateralized derivative

23、s portfolios.Put-Call Parity c-p=S-Xe-rT where:c=price of a call p=price of a put S=price of the underlying security r=risk-free rate T=time left to expiration expressed in years Volatility Smiles Currency options:implied volatility is lower for at-the-money options than it is for away-fromthe-money

24、 options.If the implied volatilities for actual currency options are greater for away-fromthe-money than at-the-money options,currency traders must think there is a greater chance of extreme price movements than predicted by a lognormal distribution.Equity options:higher implied volatility for low s

25、trike price options.The volatility smirk(halfsmile)exhibited by equity options translates into a left-skewed implied distribution of equity price changes.This indicates that traders believe the probability of large down movements in price is greater than large up movements in price,as compared with

26、a lognormal distribution.CREDIT RISK MEASUREMENT AND MANAGEMENT Credit Risk Credit risk is either the risk of economic loss from default,or changes in credit events or credit ratings.Types of credit risky securities include:corporate and sovereign debt,credit derivatives,and structured credit produc

27、ts.Their interest rates include a credit spread above credit risk-free secur1t1es.Expected Loss(EL)Expected value of a credit loss:EL=PD x(1-RR)x exposure=PD x LGD Probability of default(PD):likelihood that a borrower will default within a specified time horizon.Loss gi.ven default(LGD):amount of cr

28、editor loss in the event of a default.In percent terms,it is equal to 1 minus the recovery rate(i.e.,1-RR).Exposure at default:amount of money the lender can lose in the event of a borrowers default.The Merton Model A value-based model where the value of the firms outstanding debt(D)plus equity(E)is

29、 equal to the value of the firm CV).The value of the debt can serve as an indicator of firm default risk.Since E and D are contingent claims,option pricing can be used to determine their values as follows:payment to shareholders:max(V M-DM,0)payment to debtholders:DM-max(DM-V M 0)Equity is similar t

30、o a long call option on the value of a firms assets where face value of debt is the strike price of the option.Debt is similar to a risk-free bond and short put option on the value of a firms assets where face value of debt is the strike price of the option.The KMV Model Built on the Merton model an

31、d tries to adjust for some of its shortcomings.Assumes there are only two debt issues.The default threshold is a linear combination of these values and is equal to the par value of the firms liabilities.A rule for determining the default threshold is:short-term liabilities+0.5 x long-term liabilitie

32、s The distance to default(DD)calculates the number of standard deviations between the mean of the asset distribution and the default threshold.expected asset value-default threshold DD=-standard deviation of expected asset value Once DD is computed,the probability of default can be found by evaluati

33、ng the DD of other firms that have defaulted.Credit Scoring Models Fisher linear discriminant analysis:segregates a larger group into homogeneous subgroups.Parametric discrimination:uses a score function to determine the members of the subgroups.The score determines which subgroup the observation is

34、 placed in(likely-to-default or not-likely-to-default group).K-nearest neighbor:categorizes a new entrant by how closely it resembles members already in groups.Support vector machines:divides larger group into subgroups using hyperplanes.Credit Spread Difference between the yield on a risky bond(e.g

35、.,corporate bond)and the yield on a riskfree bond(e.g.,T-bond)given that the two instruments have the same maturity.CS=-l x1n(D)-Rp(T-t)F where:D=current value of debt F=face value of debt Credit Risk Portfolio Models These models attempt to estimate a portfolios credit value at risk.Credit VaR diff

36、ers from market VaR in that it measures losses that are due specifically to default risk and credit deterioration risk.Credit Risk+:determines default probability correlations and default probabilities by using a set of common risk factors for each obligor.CreditMetrics:uses historical data to estim

37、ate the probability of a bond being upgraded or downgraded using historical transition matrices.KMY Portfolio Manager:default probability is a function of firm asset growth and the level of debt.The higher the growth and lower the debt level,the lower the default probability.CreditPortfolio View:mul

38、tifactor model for simulating joint conditional distributions of credit migration and default probabilities that incorporates macroeconomic factors.Credit Derivatives A credit derivative is a contract with payoffs contingent on a specified credit event.Credit events include:Failure to make required

39、payments.Restructuring that harms the creditor.Invocation of cross-default clause.Bankruptcy.Credit default swap(CDS):like insurance;party selling the protection receives a fee,pays based on swaps notional amount in the case of default.First-to-default put:CDS variation where a party pays an insuran

40、ce premium in exchange for being made whole for the first default from a basket of assets.More cost effective option than CDS if assets have uncorrelated default risks.Total return swap:total return on an asset bond)is exchanged for a fixed or variable)payment;total return receiver gets any apprecia

41、tion(capital gains and cash flows),pays any depreciation;payments take place whether or not a credit event occurs.Buyer exchanges credit risk of issuer defaulting for the combined risk of the issuer and the derivative counterparty.Vulnerable option:option with default risk;holder receives promised p

42、ayment only if seller of the option is able to make the payment.Asset-backed credit-linked note:embeds a default swap into a debt issuance.It is a debt instrument with its coupon and principal risk tied to an underlying debt instrument(e.g.,bond or loan).Spread Conventions Yield spread:ITM risky bon

43、d-ITM benchmark government bond i-spread:ITM risky bond-linearly interpolated ITM on benchmark government bond z-spread:basis points added co each spot rate on a benchmark curve CDS spread:market premium of CDS of issuer bond Hazard Rates The hazard rate(default intensity)is represented by the(const

44、ant)parameter A and the probability of default over the next,small time interval,dt,is.dt.Cumulative PD If the time of the default event is denoted t*,the cumulative default time distribution F(t)represents the probability of default over(O,t):P(t*-t)=1-F(t)=e-c Probability of Default The probabilit

45、y of default(PD)of a debt security can be calculated by using the following equation:PD=CS LGD CS represents the credit spread,which is the difference between the yield on risky debt and the risk-free rate.Loss given default(LGD)is equal to one minus the recovery rate.Single-Factor Model Examines th

46、e impact of varying default correlations based on a credit positions beta.Each individual firm or credit,i,has a beta correlation,/3.,with the market,m.Firm is individual asset return is defined as:a=A.m+Ji A7E.I 1-1 1-1 I where:Ji f3r=firms standard deviation of idiosyncratic risk E;=firms idiosync

47、ratic shock Originate-to-Distribute Model Enables lenders to originate a loan based on risk/reward pricing and then outsource the risk through various channels.This provides better access to capital for less creditworthy borrowers and more diversification options for investors.Collateralized Debt Ob

48、ligations General term for an asset-backed security that issues securities that pay principal and interest from a collateral pool of debt instruments.In order to create a CDO,the issuer packages a series of debt instruments and splits the package into several classes of securities called tranches.Th

49、e largest part of a CDO is typically the senior tranche,which usually carries an AA or AAA credit rating,regardless of the quality of the underlying assets in the pool.Synthetic CDO:originator retains reference assets on balance sheet but transfers credit risk to an SPY,which then creates the tradab

50、le synthetic CDO.This product bets on the default of a pool of assets,not on the assets themselves.Securitization T ransforms the illiquid assets of a financial institution into a package of asset-backed securities(ABSs)or mortgage-backed securities(MBSs).A third party uses credit enhancements,liqui