FRM-ZhiMap思维导图

FRM
进入思维导图模式
- Part 1
  - foundation of risk management
    - overview
      - what is risk
        expected loss
        unexpected loss
    - risk manager's job
      - uncover the source of risk and measure its impact: help make risk management decision
      - balance risk and reward
      - make risk transparent to key decision makers and stakeholders
      - find the right relationship between business leaders and the specialist risk managment functions
    - topology of risk
      - market risk
        equity price risk
        interest rate risk
        tranding risk
        gap risk
        foreign exchange risk
        commodity price risk
      - credit risk
        default risk
        bankruptcy risk
        downgrade risk
        settlement risk
      - liquidity risk
        funding liquidity risk
        trading liquidity risk
      - operational risk
        potential losses resulting from operational weaknesses including inadquate sysetms, management failure, faulty controls, fraud and human errors
        human factor risk
        technology risk
      - legal and regulatory risk
      - business risk
      - strategic risk
      - reputation risk
      - systematic risk
    - putting risk management into practice
      - determining the objective
      - mapping the risks
      - instruments for risk management
      - constructing and implementing a strategy
      - performance evaluation
    - ERM
      - comprehensive and integrated framework for managing key risks in order to achieve business objectives, minimise unexpected earnings volatility and maximise firm value
    - Markowitz portfolio theory
      - effective frontier
      - CML
    - CAPM
      - SML
      - Sharpe ratio $\frac{E[R_p] - R_f}{\sigma(R_p)}$
      - Treynor ratio $\frac{E[R_p] - R_f}{\beta_p}$
      - Sortino ratio $\frac{E[R_p] - MAR}{\sqrt{\frac{1}{N-1}\sum_{t=1}^N \left( R_{p_t} - MAR\right)^2 } }$
        MAR: minimum acceptable return
      - Jensen's alpha
      - information ratio $\frac{E[R_p] - E[R_B]}{\sigma(R_p-R_B) }$
        $\sigma(R_p-R_B)$ is called tracking error
    - APT
      - factor model
      - Fama-French Three-Factor model
        SMB: small minus big
        HML: high minus low
        market
    - examples of financial disasters
  - quantitative analysis
    - probability
    - statistics
      - moment, central moment, expected value, variance, skewness, kurtosis, covariance, correlation
      - sample mean, variance $s^2 = \sum_{i=1}^n \frac{(X_i-\overline{X})^2}{n-1}$
      - best linear unbiased estimator(BLUE)
        linearity
        unbiased
        has minimum variance
      - Chebyshev's inequality
        $P(|X-\mu|\leq k\sigma) \geq 1 - \frac{1}{k^2}$
        $P(|X-\mu|\leq k\sigma) \geq 1 - \frac{1}{k^2}$
      - distribution
        binomial
        poisson
        uniform
        normal
        $1 \sigma: 68\% 1.65 \sigma: 90\%, 1.96\sigma:95\%, 2.58\sigma,99\%$ $1 \sigma: 68\%, 1.65 \sigma: 90\%, 1.96\sigma:95\%, 2.58\sigma: 99\%$
        logmormal
        Chi-Square distribution
        t-distribution
        F-distribution
      - hypothesis test
        sampling and estimation
        CLT: central limit theorem
        large sample $n \geq 30la$ , assume approximately normal
        standard error of mean: $\overline{X}$ $SE(X) = \frac{s}{\sqrt{n}}$
        steps
        state null and alternative hypothesis
        identify the test statistic
        select a level of significance
        formulate a decision rule
        take a sample, arrive at decision
        do not reject / reject the hypothesis
        selecting test statistics
        mean
        normally distributed, known variance
        $z = \frac{\tilde{X}-\mu_0}{\sigma/\sqrt{n}} \sim N(0,1)$
        normally distributed, unknown variance
        $t = \frac{\tilde{X}-\mu_0}{s/\sqrt{n}} \sim t(n-1)$
        variance
        normally distributed
        $\chi^2 = \frac{(n-1)s^2}{\sigma_0^2} \sim \chi^2(n-1)$
        two independent normally distributed
        $F = \frac{s_1^2}{s_2^2} \sim F(n_1-1,n_2-1)$
        errors
        type 1
        reject the null hypothesis when it is actually true
        significance level
        the probability of making type 1 error
        = P(type 1 error)
        type 2
        fail to reject the null hypothesis when it is actually false
        power of a test
        the probability of correctly rejecting the null hypothesis when it is false
        = 1 - P(type 2 error)
      - linear regression
        standard error of regression
        measure the fit of the regression line
        $SER = \sqrt{\frac{\sum (Y_i-\hat{Y}_i)^2}{n-2}}$
        analysis of variance(ANOVA) table
        $R^2$ the coefficient of determination $R^2 = \frac{ESS}{TSS} = 1 - \frac{SSR}{TSS}$
        $R^2 = \frac{ESS}{TSS} = 1 - \frac{SSR}{TSS}$
        ESS: explained sum of squares    $= \sum (\hat{Y}_i-\overline{Y})^2$
        SSR: sum of square residuals   $= \sum (Y_i - \hat{Y}_i)^2$
        TSS: total sum of squares
        confidence interval for the regression coefficient
        $\hat{b} \pm (t_c \times s_{\hat{b}})$
        $s_{\hat{b}}$ : standard error of the regression coefficient
        regression coefficient hypothesis testing
        test if the true slop is b
        use t-test   $t = \frac{\hat{b} - b}{s_{\hat{b}}} \sim t(n-2)$
      - multiple regression
        adjusted $R^2 = 1 - \frac{SSR/n-k-1}{TSS/n-1}$
        $= 1 - \frac{SSR/n-k-1}{TSS/n-1}$
        does not increase when a new independent variable is added
        homoskedasticity and heteroskedasticity
        the standard errors are usually not reliable estimate
        serial correlation
        multicolinearity
        omitted variable bias
        bias arises when one or more regressors are correlated with an omitted variable
      - forecasting trends
        measure of model fitness
        mean squared error(MSE)
        adjusted MSE
        $S^2 = \frac{\sum e_t^2}{T-k}$
        deduct the degree of freedom to reduce MSE bias
        Aksikr information criterion(AIC)
        deal with trade-off between the goodness of fit of the model and the complexity of the model
        AIC = $e^{\frac{2k}{T}} \times MSE$
        Schwarz information criterion(SIC)
        lower SIC implies either fewer explanatory variables, better fit, or both
        the only consistency criterion
        cycle
        Wold's representation
      - estimating volatilities and correlation
        ARCH model
        EWMA model
        GARCH(1,1) model
      - copula
        nonlinear relationship
        tail dependence
      - simulation methods
        Monte Carlo simulation
        price increments are assumed to have a normal distribution
        select other distributions
        bootstrapping
        draw from historical scenarios
        ineffective situations
        outliers in the data
        non-independent data
        find a distribution using historical data
        variance reduction techniques
        antithetic variables
        control variates
        random number generation
  - financial market and products
    - Bond markets
      - interest rate
        treasury rates, LIBOR, repo rates
        overnight indexed swap(OIS) rates, riskfree rate
        spot rates, forward rates
        simple interest, compounding interest
        major theories of term structure
        expection theory
        market segmentation theory
        liquidity preference theory
      - bond valuation
        = sum of discounted cash flow
        risk metrics
        Maculay duration
        average period of cash flow returning weighted by discounted cash flow
        modified duration
        $= -\frac{\Delta P}{\Delta y} \times \frac{1}{P} = \frac{D_{Macaulay}}{1+y}$
        DV01
        = modified duration * bond value * 0.0001
        convexity
        $P = P_0 - DP_0\Delta y + \frac{1}{2}CP_0(\Delta y)^2$
        reinvestment risk
      - treasury market
        treasury bills
        maturity of one year or less
        cash price = 100 $\left(1- \text{discount rate} \times \frac{n}{360} \right)$
        treasury notes and treasury bonds
        make interest payments semi-annually
        clean price
        not including any accrued interest
        dirty price
        = clean price + accrued interest
        day convention
        treasury bond: actual / actual
        corporate and municipal bonds: 30 / 360
        money market instruments: actual / 360
      - corporate bond
        credit risk
        credit default risk
        credit spread risk
        loss resulting from changes in the level of credit spreads
      - MBS
    - derivatives markets
      - forward and futures
        futures
        traded on an exchange
        standardized contracts
        settled daily
        high liquidity
        guaranteed by clearinghours
        margin required and adjusted
        pricing
        hedging strategies using forward
      - swap
        main types
        interest rate swap
        currency swap
      - option
        trading strategies involving options
        bull spread
        bear spread
        butterfly
        canlendar
        straddle
        strangle
        strip
        strap
        call option
        without dividend, European = American
        $D > K(1-e^{-r(T-t)})$ shuld exercise at time when dividend is paid
        greeks
    - MBS(mortgage-backed securities)
      - prepayment of mortgage loans
        SMM(single monthly mortality rate)
        CPR(conditional prepayment rate)
        $CPR_n = 1 - (1-SMM_n)^{12}$
    - central conterparties
      - exchanges
      - OTC
        main risk
        counterparty risk
        systemic risk
    - investment banks
      - IPO
    - insurance companies
    - mutual funds and hedge funds
  - valuation and risk model
    - fixed income
      - valuation
        sum of discounted cash flow
        bond replication
        law of one price
      - risk metrics
        key rate shifts
    - option
      - binomial trees
      - Black-Scholes-Merton model
        Greek letters
    - risk models
      - market risk
        types of risk measures
        mean-variance framework
        value at risk(VaR)
        conditional VaR / expected shortfall
        spectral risk measures
        Putting VaR to work
      - credit risk
        external and internal ratings
        capital structure in banks
        country risk
        loss distribution approach
      - operational risk
        operational risk management
        risk control and self-assessment(RCSA)
        key risk indicators(KRIs)
        regulatory capital requirement
        capital requirement
        basic indicator approach(BIA)
        standardized approach(SA)
        advanced measurement approach(AMA)
        data requirement
        insurance in mitigating operational risk
        moral hazard
        adverse selection
      - stress testing
- Part 2
  - investment management and risk management
    - factor investing
      - factor theory
        CAPM
        assumptions
        make decisions solely in terms of expected values and standard deviations
        plan for the same single holding period
        have homogeneous expectations or beliefs
        An individual cannot affect the asset price by buying or selling action
        All assets are tradable, infinitely divisible
        Markets are frictionless, including no transaction cost and no taxes
        Short sale is allowed unlimitedly
        Unlimited lending and borrowing at the riskless rate
        only market factor
        all risky assets have risk premiums determined only by their exposure to the market portfolio
        Lessons
        hold the factors not the assets
        risk is factor exposure
        generalization of CAPM
        multi-factor
        factors
        Macro factors
        economic growth
        inflation
        volatility
        productivity, demographic, political
        dynamic factors
        size
        The SMB factor was designed to capture the
        outperformance of small firms relative to large firms
        value
        value investing: long value stocks and short growth stocks
        negative feedback strategy
        momentum
        trend investing
        positive feedback strategy
        Fama-French three factor model
        market index, firm size, book-to-market ratio
        A shock to a factor matters more than the level of the factor.
        can use data mining to find new factors
      - efficient market hypothesis
        weak
        market data
        semi-strong
        all publicly known and available information
        passive investing
        strong
        all information
      - Alpha
        excess return w.r.t a benchmark
        tracking error
        standard error of excess return
        information ratio
        $IR = \frac{active \ return }{tracking \ error} = \frac{\alpha}{\sigma}$
        Ideal benchmark
        Well defined, Tradable , Replicable, Adjusted for risk
        Grinold fundamental law
        IR = IC $\times \sqrt{BR}$
        IC: the correlation of the manager's forecast with the actual returnd
        BR(the breadth of the strategy): how many bets are taken
        assumption
        each active return is independent from the other active forecasts for that period and independent from the forecast for the security in subsequent periods
      - low-risk anomaly
        stocks with low betas and low volatility have high returns
        explanation
        data mining
        leverage constraints
        agency problems
        eg. long only
        preference
      - illiquid assets
        characteristics
        infrequent trading, small amounts being traded, low turnover
        source of illiquidity
        due to market imperfections
        participation costs
        transaction costs
        search frictions
        asymmetric information
        price impact
        large trade will move markets
        funding constraints
        biases on reported returns
        survivorship bias
        infrequent sampling
        smoothed price curve, underestimated volatility
        selection bias
        illiquidity risk premiums
        allocation across asset classes
        illiquidity premiums may not be true
        illiquidity biases
        reported data cannot be trusted
        ignored risk
        no market index for inliquid assets
        no comparison
        factor risk cannot be separated from manager skill
        investing in illiquid markets is always a bet on management talent
        choose securities within an asset class that are more illiquid
        assets from different classes are rarely treated consistently as a whole
        eg: treasuries on the run / off the run
        act as market maker at the individual security level
        supply liquidity by acting as an intermediary
        rebalancing
        force asset owners to buy at low prices when others want to sell
        counter-cyclical
    - portfolio risk management
      - inputs for construction
        Tasks
        assess the impact of practical issues in portfolio construction, such as determination of risk aversion, incorporation of specific risk aversion, proper alpha coverage
        describe portfolio revisions and reblancing, evaluate the tradeoff between alpha, risk, transaction costs and time horizon
        determine the optimal no-trade region for rebalancing with transaction costs
        alphas
        refining alphas
        motivation
        most active managers construct portfolio subject to certain constraints
        eg. no short, cash limit, liquidity
        scales the alphas
        $\alpha = volatility * IC * Score$
        score has mean 0 and standrad deviation 1
        volatility = residual risk
        $std(\alpha) = volatility * IC$
        trim alpha outliers
        very large positive or negative alphas can have undue influence
        closely examine all stocks with alphas greater in magnitude than three times the scale of the alphas
        neutralization
        remove alpha of benchmark
        active risk aversion
        general risk aversion
        $= \frac{IR}{2\sigma_{\alpha}}$
        utility function
        $= \alpha_p - \lambda_A \sigma_{\alpha}^2$
        $\alpha_p = IR * \sigma_{\alpha}$
        specific factor risk
        transaction costs
        marginal contribution to value added(MCVA)
      - construction techniques
        objective
        maximize active returns minus an active risk penalty
        screen
        steps
        rank the assets by alpha
        choose the top performing assets
        equal-weight or capitalization-weight
        advantage
        simple, easy to understand
        robust
        enhance alphas by concentrating the portfolio in the high-alpha stocks
        risk control
        include a sufficient number of stocks to avoid concentation in any single stock
        transaction costs are limited by controlling turnover
        disadvantage
        ignore all informations in the alphas except for the rankings
        exclude assets with lower alpha
        straification
        steps
        split the list of assets into mutually exclusive categories
        screen
        linear programming
        advantage
        take all information into consideration
        disadvantage
        hard to produce portfolios with a prescribed number of stocks
        quadratic programming
      - VaR measures
        assumptions
        Delta-normal model
        all individual security returns are assumed normally distributed
        traditional portfolio analysis
        based on variances and covariances
        individual VaR
        $= z \times \sigma \times V$
        z : z-score associated with the level of confidence
        $\sigma$ : the standard deviation of stock returns
        V : the market value of the stock
        VaR for several stocks
        correlation matters in large portfolio
        marginal VaR
        $= z \frac{cov(R_i,R_p)}{\sigma_p} = z \times \beta_i \times \sigma_p = z \times \rho_{ip} \times \sigma_i$
        incremental VaR
        the change of VaR owing to a new position
        $= VaR_{p+a} -VaR_{p}$
        or $\approx MVaR_i \times amount$
        component VaR
        indicate how much the portfolio VaR would change approximately if the given component is deleted
        = MVaR * component value
      - portfolio risk
        analytic methods
        only consider risk
        when the portfolio risk has reached a global minimum, all MVaRs or all betas must be equal
        consider risk and return
        maximize the Shape ratio with VaR
        $\frac{R_p-F_f}{VaR_p}$
        all ratios of excess return to marginal VaR(beta) $\frac{R_i-F_f}{MVaR_i}$ must be equal
      - VaR and risk budgeting
        DB vs DC
        surplus at risk(SaR)
      - risk monitering
        liquidity duration
      - protfolio performance measurement
        risk-adjusted performance measures
        Sharpe ratio
        $= \frac{R_p-R_f}{\sigma_p}$
        Treynor ratio
        $= \frac{R_p-R_f}{\beta_p}$
        for well diversified portfolio, Shaper ratio and Treynor ratio give the same ranking
        Jensen's alpha
        information ratio
        statistical significance of alpha
        Modiglinai-square measure
        $M^2 = \frac{\sigma_m}{\sigma_p}(R_p-R_f) - (R_m-R_f)$
        market timing
        include high(low) beta stocks if expect an up(down) market
        style analysis
        Regress fund returns on indexes reprensenting a range of asset classes. The regression coefficient on each index measures the fund's implicit allocation to that 'style'
        performance attribution
        asset allocation
        equity / bond
        selection
        sector selection
        security selection
    - hedge fund
      - strategies
      - agency problem
      - due diligence
  - market risk management and measurement
    - VaR and other risk measures
      - methods of VaR estimation
        parametric approach
        requires to explicitly specify the statistical distribution from which our data observations are drawn
        normal VaR
        $VaR_{\alpha} = - (\mu - z_{\alpha}\times \sigma)$
        lognormal VaR
        $VaR_{\alpha} = (1-e^{\mu - z_{\alpha}\times \sigma}) \times P$
        nonparametric approach
        see coherent risk measures
        standard errors of quantile estimators
        $SE(q) = \left[ \frac{p(1-p)}{nf(q)^2} \right]^{1/2}$
        p: the proportion for the quantile
        f(q): the corresponding probability density function
      - coherent risk measures
        Def
        weighted average of the quantiles of the loss distribution, where the weighting function is specified by the user
        expected shortfall(ES)
        probability-weighted average of tail losses
        Quantile-Quantile(QQ) plot
        a plot of the quantiles of the empirical distribution against those of some specified distribution and can be used to identify the distribution of our data by its shape
        non-parametric historical simulation
        basic historical simulation
        bootstrap historical simulation
        steps
        create a large number of new samples at random from our original sample with replacement
        each new 'resampled' sample gives a new VaR or ES estimate
        take the best estimate to be mean of these resample-based estimates
        non-parametric density estimation
        use histograms to simulate the probability density function
        weighted historical simulation
        semi-prametric
        age-weighted historical simulation(BRW)
        discount the older observations in favor of newer ones
        $w_i = \frac{\lambda^{i-1}(1-\lambda)}{1-\lambda^n}$
        volatility-weighted historical simulation(HW)
        correlation-weighted historical simulation
        filtered historical simulation
      - backtesting VaR
        difficulties
        the trading portfolio evolves dynamically in practice
        testing framework
        test statistic
        $z = \frac{x-pT}{\sqrt{p(1-p)T}}$
        x: number of exceptions
        the exceptions are assumed to be independent
        model verification
        unconditional coverage model
        ignore time variation in the data, so the exceptions should be evenly spread over time
        reject if $LR_{uc} > 3.84$ (95% confidence level)
        conditional coverage model
        $LR_{cc}=LR_{uc} + LR_{ind}$
        reject if $LR_{cc} > 5.991$
        type 1 and type 2 errors
        Basel rules
        daily exceptions of 99% VaR over the last year
        Basel penalty zones
        number of exceptions
        multiplicative factor k
        penalty
        green
        <= 4
        3
        no
        yellow
        5
        3.4
        model integrity or accuracy: should apply
        6
        3.5
        intraday trading: should be considered
        7
        3.65
        bad luck: no guidance
        8
        3.75
        9
        3.85
        red
        >=10
        4
        automatic penalty
      - VaR mapping
        similar to PCA
        choice of the set of general risk factors should reflect the tradeoff between better quality of the approximation and faster processing
        mapping process
        making all positions to market in current dollars
        the market value for each instrument is then allocated to the risk factors
        positions are summed for each risk factor
        applications
        mapping fixed-income portfolio
    - modelling dependence: correlation and copulas
      - financial correlation risk
        the risk of financial loss due to adverse movements in correlation between two or more variables
        eg
        during 2008 financial crisis, the correlation between default of bonds increases substantially, thus decrease the value of senior tranche
      - statistical correlation models
        Pearson correlation
        $\rho_{XY} = \frac{cov(X,Y)}{\sigma_X\sigma_Y}$
        limitations
        financial relationship are typically nonlinear
        zero correlation does not imply independence
        not invariant to tranformations
        Spearman rank correlation
        $\rho_{S} = 1 - \frac{6\sum d_i^2}{n(n^2-1)}$
        n: number of observations
        $d_i$ :the difference between the ranking for period i
        Kendall’s   $\tau$
        $\tau = \frac{n_c-n_d}{n(n-1)/2}$
        $n_c$ : number of concordant pairs
        concordanct pairs:    $sgn(X_i-X_j) = sgn(Y_i-Y_j), i\neq j$
        $n_d$ : number of discordant pairs
        discondant pairs:   $sgn(X_i-X_j) = -sgn(Y_i-Y_j), i\neq j$
        $X_i=X_j \text{ or } Y_i=Y_j$ : neither concordant nor discordant
        limitation of ordinal risk measures
        less sensitive to outliers
      - financial correlation modeling
        Def
        create a joint probability distribution between two or more variables while maintaining their individual marginal distributions
        Gaussian copula
    - term structure models of interest rate
      - binomial interest rate tree model
        backward induction valuation
        steps
        find the risk-neutral probabilities that equate the price of the underlying securities with their expected discounted values
        price the contingent claim by expected discounted
        value under these risk-neutral probabilities
        issues
        recombining vs non-recombining
        option-adjusted spread(OAS)
        size of time steps
      - models
        drift
        no drift
        constant drift
        Ho-Lee model: time-dependent drift
        $dr =\lambda_tdt + \sigma dW$
        Vasicek model: mean-reverting drift
        $dr =\kappa(\theta - r)dt + \sigma dW$
        volatility
        CIR model
        $dr = \kappa(\theta - \alpha r)dt + \sigma \sqrt{r} dW$
        Courtadon model
        $dr = \kappa(\theta - \alpha r)dt + \sigma r dW$
    - other related topics
      - empirical approaches to risk metrics and hedges
        DV01
        drawback
        the changes in yields on hedged portfolio and
        hedging instrument may not be one-for-one
        partial solution
        regression analysis based on historical data
      - volatility smiles
  - credit risk management and measurement
    - identification of credit risk
      - tasks
        define and explain credit risk
        explain the components of credit risk evaluation
        describe, compare and contrast various credit risk mitigants and their role in credit risk
      - four componets of credit risk
        obligor's capacity and willingness to repay
        external conditions
        attributes of bligation from which credit risk arises
        credit risk mitigants
        collateral
        guarantees promised by a third party to accept liability
      - quantitative measures of credit risk
        probability of default (PD)
        loss given default (LGD)
        exposure at default (EAD)
        expected loss(EL)
        = EAD * PD * LGD
        use cash to cover
        unexpected losses(UL)
        = credit VaR
        use capital to cover
        concentration risk
        sum of individual risks does not equal the portfolio risk
        measurement
        marginal contribution to portfolio UL
      - classifications of credit risk
        default-mode(loss-based) valuation
        default risk
        recovery risk
        exposure risk
        value-based valuation
        migration risk
        spread risk
        liquidity risk
    - measurement of credit risk
      - estimate probability of default (PD)
        experts-based approach
        agencies' rating
        migration matrix
        change on PD
        investment-grade credits, increase more than proportional with the horizion
        speculative-grade credits, increase less than proportional with the horizion
        internal credit rating
        infer from corporate bond prices
        risk-neutral measure
        $P= \frac{100}{1+YTM} = (1-PD)\frac{100}{1+R_f} + PD\frac{100(1-LGD)}{1+R_f}$
        credit spread: $YTM-R_f \approx PD\times LGD$
        real world
        $YTM-R_f - \text{risk premium} \approx PD\times LGD$
        different kinds of credit spread
        nominal spread
        $YTM_{\text{risky bond}}-YTM_{\text{benckmark government bond}}$
        i-spread (interpolated)
        linearly interpolated YTM on benchmark government bond or swap rate
        Z-spread (Zero volatility spread)
        Option adjusted spread (OAS)
        Z-spread adjusted for optionality of embedded options
        Z-spread = OAS if no option
        spread01(DVCS)
        Increase and decrease the z-spread by 0.5 basis points, reprice the bond for each of these shocks, and compute the difference
        infer from equity prices
        credit risk as option
        Equity: Long call option on firm value V, strike price= F
        Merton model
        $S_t = V_tN(d_1) -Fe^{-r(T-t)}N(d_2)$
        $d_1 = \frac{\ln(V/Fe^{-r(T-t)})}{\sigma\sqrt{T-t}}+\frac{\sigma\sqrt{T-t}}{2}, d_2 = d_1-\sigma\sqrt{T-t}$
        $N(d_2)$
        probability of exercising the call option = 1-PD = $1- N(-d_2)$
        credit spread
        $= -\frac{1}{T-t} \ln\left(\frac{D}{F}\right) -r_f$
        distance to default (DtD)
        $= -\frac{\ln(V/F)+(\mu-0.5\sigma^2)(T-t)-\text{ other payouts} } {\sigma\sqrt{T-t}}$
        limitations
        applicable only to liquid, publicly traded names
        subordinated debt
        firm value low
        like equity
        long call
        firm value high
        like senior debt
        short call
        KMV model
        overcome two shortcoming of Merton model
        all the debt matures at the same time
        the value of the firm follows a lognormal diffusion process
        default intensity models
        distributions
        Poisson distribution
        exponential distribution
        Hazard rate (or default intensity)
        survival probability
        $P(t^*>t) = 1-F(t) = e^{-\lambda t}$
        memoryless
        recovery rate (RR)
        $\lambda \approx \frac{z}{1-RR}$
        the spread is approximately equal to the default probability times the LGD
        credit scoring model
        retail credit risk
        three types
        credit bureau scores
        pooled models
        custom model
        cost: credit bureau scores < pooled Models < custom models
        other methods
        structural Approaches
        reduced Form Approaches
        statistical-based models
        supervised model
        LDA
        logistic regression
        unsupervised model
        clustering
        PCA
        heuristic approaches
        mimic human decision
        numerical approaches
        neural network
      - estimate LGD and EAD
        LGD
        the seniority of the OTC derivative claim
        timing of recovery
        credit exposure
        metrics
        Expected MtM
        the expected value of a transaction at a given point in the future
        Expected exposure (EE)
        the amount that is expected to be lost (positive MtM only) if the counterparty defaults
        Expected exposure is larger than expected MtM.
        Potential future exposure (PFE)
        the worst exposure that could occur at a given time in the future at a given confidence level
        Maximum PFE
        highest PFE value over a given time interval
        Expected positive exposure (EPE)
        the average exposure across all time horizons——the weighted average of the EE across time
        Negative Exposure
        represented by negative future values, from a counterparty’s point of view
        effective EE (EEE)
        a non-decreasing version of the EE profile
        Effective Expected Positive Exposure (EEPE)
        introduced for regulatory capital. It is the average of the effective EE (EEE)
        deal with two problems
        EPE neglect very large exposures
        EPE may underestimate exposure for short-dated transactions and not properly capture “rollover risk”
        factors
        future uncertainty
        bond and loan
        interest rate swap
        optionality
        credit derivative
        risk migrants
        netting
        netting benefit improves for a large number of exposures and low correlation
        netting factor
        $= \frac{\sqrt{n+n(n-1)\rho}}{n}$
        collateral
      - counterparty risk
        migrants
        netting
        create legal risk in cases where a netting agreement cannot be legally enforced
        expose other creditors to more significant losses
        collateral
        hedging
        central counterparties
        potentially create operational and liquidity risks, and also systemic risk
        credit limits and CVA
        credit limits
        counterparty risk can be diversified by limiting exposure to any given counterparty.
        credit value adjustment (CVA)
        price counterparty risk
        assume no wrong-way risk
        sum of discounted expected exposure
        as a spread
        = EPE * spread
        wrong-way risk and right-way risk
        stress test
      - portfolio credit risk
        default correlation
        single-factor model
        unconditional default distribution
        conditional default distribution
        credit risk portfolio models
        CreditRisk+(credit suisse)
        CreditRiskTM (JPMorgan)
        KMV model (Moody)
        Credit Portfolio View (McKinsey)
    - management of credit risk
      - credit derivative swap
        First-to-Default CDS
        Total Return Swap (TRS)
        Asset-Backed Credit-Linked Notes
        Vulnerable Option
        Swap with Credit Risk
        Collateralized Debt Obligations (CDOs)
      - structured products
        PD and correlation effect
        convexity effect
      - securitization
        seven frictions
  - operation risk management and measurement
    - OpRisk management framework
      - def
        risk of loss resulting from inadequate or failed internal processes, people and systems or from eternal events
        include legal risk, but exclude strategic and reputational risk
      - three common lines of defense
        business line management
        independent corporate operational risk management function
        independent review
      - principles for sound management of operational risk
        summary
        人和流程
        general principle
        1. The board of directors should take the lead in establishing a strong risk management culture
        2. Banks should develop, implement and maintain a Framework that is fully integrated into the bank’s overall risk management processes
        governance
        board of directors
        3. The board of directors should establish, approve and periodically review the Framework
        4. The board of directors should approve and review a risk appetite and tolerance statement for operational risk that articulates the nature, types, and levels of operational risk that the bank is willing to assume
        senior management
        5. Senior management should develop for approval by the board of directors a clear, effective and robust governance structure with well defined, transparent and consistent lines of responsibility
        risk management enviroment
        indentification and assessment
        6. Senior management should ensure the identification and assessment of the operational risk inherent in all material products, activities, processes and systems to make sure the inherent risks and incentives are well understood
        tools for assessing operating risks
        audit findings
        focus on control weaknesses and vulnerabilities
        internal loss data collection and analysis
        provide meaningful information for assessing exposure to operational risk and effectiveness of internal controls
        external data collection and analysis
        risk self assessment(RSA)
        assess the processes underlying its operations against potential threats, vulnerabilities, and consider their potential impact
        risk control self assessment(RCSA)
        evaluate inherent risk before controls are considered
        business process mapping
        risk and performance indicators
        key risk indicators(KRIs)
        key performance indicators(KPIs)
        scenario analysis
        measurement
        use the output of the risk assessment tools as inputs into a model that estimates operational risk exposure
        comparative analysis
        7. Senior management should ensure that there is an approval process for all new products, activities, processes and systems that fully assesses operational risk
        monitoring and reporting
        8. Senior management should ensure that there is an approval process for all new products, activities, processes and systems that fully assesses operational risk
        control and mitigation
        9. Banks should have a strong control environment that utilizes policies, processes and systems; appropriate internal controls; and appropriate risk mitigation and transfer strategies
        five components of strong control enviroment
        control enviroment
        risk assessment
        control activities
        information and communication
        monitoring activities
        business resiliency and continuity
        10. Banks should have business resiliency and continuity plans in place to ensure an ability to operate on an ongoing basis and limit losses in the event of severe business disruption
        role of disclosure
        11. A bank’s public disclosures should allow stakeholders to assess its approach to operational risk management
      - enterprise risk management
        def
        all risks viewed together within a coordinated and strategic framework
        macro level
        more strategic risks and opportunities should be put into core business
        micro level
        focus on decentralizing the risk-return trade-off in a company
        four-step conceptual framework
        management begins by determining the firm's appetite
        given the firm's target rating, management estimates the amount of capital it requires to support the risk of its operations
        management determines the optimal combination of capital and risk that is expected to yield its target rating
        top management decentralizes the risk-capital trade-off and makes investment and operating decisions optimize their trade-off
        technology risk
        risk appetite framework
        help strategic decisions and risk management
        articulate a clearly defined risk appetite for the firm: not only about risk management but also about their firms’ forward-looking business strategies
        The board of directors sets overarching expectations for the risk profile, while CEO, CRO, and CFO translate those expectations into incentives and constraints for business lines
        data aggregation
        data quality
        accuracy, completeness, consistency, reasonableness, currency, uniqueness
    - OpRisk data
      - scenario analysis
        common biases
        prentation bias
        context bias
        anchoring bias
        hundle bias or anxiety bias
        gaming
        参与者利益与整体利益有冲突，不肯暴露真实意图，却努力影响结果
        availablity bias
        over / under confidence bias
        inexpert opinion
    - modelling approaches
      - data features
        low frequency
        discrete
      - development
        people are not reliable
        little research of operational risk
        Basel 3
        return to basic approach
      - basic approach
      - standardized approach
      - advanced measurement approach
      - extreme value
        generalized extreme-value theory (GEV)
        peaks-over-threshold approach (POT)
      - model risk
        sources
        distribution of the underlying asset is stationary
        rates of return are normally distributed
        oversimplify a model
        assume perfect capital market exists
        liquidity is assumed to be ample
        misapplied to a given situation
    - risk capital attribution and risk-adjusted performance
    - capital planning and framework
      - effective capital adequacy process
        框架，方法论，流程，人
        sound foundational risk management
        effective loss-estimation methodologies
        solid source-estimation methodologies
        sufficient capital adequacy impact assessment
        comprehensive capital policy and capital planning
        robust internal controls
        effective governance
    - financing: liquidity and leverage
      - Repo rate
        GC rate
        special rate
      - liquidity risk
        constant spread approach
        exogenous spread approach
        endogenous price approach
        liquidity discount approach
        liquidity-at-risk (LaR) or cash-flow-at-risk (CFaR)
        5 factors affecting cash flow and LaR
      - source of liquidity risk
        transaction liquidity risk
        balance sheet risk
        systemic risk
    - banks failure
    - Basel Accords
  - current issues

FRM

​Part 1

foundation of ​risk management

​overview

what is risk

​expected loss

​unexpected loss

​risk manager's job

​uncover the source of risk and measure its impact: help make risk management decision

​balance risk and reward

​make risk transparent to key decision makers and stakeholders

​find the right relationship between business leaders and the specialist risk managment functions

​topology of risk

​market risk

​equity price risk

​interest rate risk

​tranding risk

​gap risk

​foreign exchange risk

​commodity price risk

​credit risk

​default risk

​bankruptcy risk

​downgrade risk

​settlement risk

​liquidity risk

​funding liquidity risk

​trading liquidity risk

​operational risk

​potential losses resulting from operational weaknesses including inadquate sysetms, management failure, faulty controls, fraud and human errors

​human factor risk

​technology risk

​legal and regulatory risk

​business risk

​strategic risk

​reputation risk

​systematic risk

​putting risk management into practice

​determining the objective

​mapping the risks

​instruments for risk management

​constructing and implementing a strategy

​performance evaluation

​ERM

comprehensive and integrated framework for managing key risks in order to achieve business objectives, minimise unexpected earnings volatility and maximise firm value

​Markowitz portfolio theory

​effective frontier

​CML

​CAPM

​SML

​Sharpe ratio E[Rp]−Rfσ(Rp)\frac{E[R_p] - R_f}{\sigma(R_p)}σ(Rp​)E[Rp​]−Rf​​

​Treynor ratio E[Rp]−Rfβp\frac{E[R_p] - R_f}{\beta_p}βp​E[Rp​]−Rf​​

​Sortino ratio E[Rp]−MAR1N−1∑t=1N(Rpt−MAR)2\frac{E[R_p] - MAR}{\sqrt{\frac{1}{N-1}\sum_{t=1}^N \left( R_{p_t} - MAR\right)^2 } }N−11​∑t=1N​(Rpt​​−MAR)2​E[Rp​]−MAR​

​MAR: minimum acceptable return

​Jensen's alpha

​information ratio E[Rp]−E[RB]σ(Rp−RB)\frac{E[R_p] - E[R_B]}{\sigma(R_p-R_B) }σ(Rp​−RB​)E[Rp​]−E[RB​]​

​ σ(Rp−RB)\sigma(R_p-R_B) σ(Rp​−RB​) is called tracking error

​APT

​factor model

​Fama-French Three-Factor model

​SMB: small minus big

​HML: high minus low

​market

examples of ​financial disasters

​quantitative analysis

​probability

​statistics

​moment, central moment, expected value, variance, s​​kewness, ​kurtosis, covariance, correlation

​sample mean, variance s2=∑i=1n(Xi−X‾)2n−1s^2 = \sum_{i=1}^n \frac{(X_i-\overline{X})^2}{n-1}s2=∑i=1n​n−1(Xi​−X)2​

​best linear unbiased estimator(BLUE)

​linearity

​unbiased

​has minimum variance

​Chebyshev's inequalityP(∣X−μ∣≤kσ)≥1−1k2P(|X-\mu|\leq k\sigma) \geq 1 - \frac{1}{k^2}

P(∣X−μ∣≤kσ)≥1−1k2P(|X-\mu|\leq k\sigma) \geq 1 - \frac{1}{k^2}P(∣X−μ∣≤kσ)≥1−k21​

distribution

​binomial

​poisson

​uniform

​normal

Part 1

foundation of risk management

overview

expected loss

unexpected loss

risk manager's job

uncover the source of risk and measure its impact: help make risk management decision

balance risk and reward

make risk transparent to key decision makers and stakeholders

find the right relationship between business leaders and the specialist risk managment functions

topology of risk

market risk

equity price risk

interest rate risk

tranding risk

gap risk

foreign exchange risk

commodity price risk

credit risk

default risk

bankruptcy risk

downgrade risk

settlement risk

liquidity risk

funding liquidity risk

trading liquidity risk

operational risk

potential losses resulting from operational weaknesses including inadquate sysetms, management failure, faulty controls, fraud and human errors

human factor risk

technology risk

legal and regulatory risk

business risk

strategic risk

reputation risk

systematic risk

putting risk management into practice

determining the objective

mapping the risks

instruments for risk management

constructing and implementing a strategy

performance evaluation

ERM

Markowitz portfolio theory

effective frontier

CML

CAPM

SML

Sharpe ratio $\frac{E[R_p] - R_f}{\sigma(R_p)}$

Treynor ratio $\frac{E[R_p] - R_f}{\beta_p}$

Sortino ratio $\frac{E[R_p] - MAR}{\sqrt{\frac{1}{N-1}\sum_{t=1}^N \left( R_{p_t} - MAR\right)^2 } }$

MAR: minimum acceptable return

Jensen's alpha

information ratio $\frac{E[R_p] - E[R_B]}{\sigma(R_p-R_B) }$

$\sigma(R_p-R_B)$ is called tracking error

APT

factor model

Fama-French Three-Factor model

SMB: small minus big

HML: high minus low

market

examples of financial disasters

quantitative analysis

probability

statistics

moment, central moment, expected value, variance, skewness, kurtosis, covariance, correlation

sample mean, variance $s^2 = \sum_{i=1}^n \frac{(X_i-\overline{X})^2}{n-1}$

best linear unbiased estimator(BLUE)

linearity

unbiased

has minimum variance

Chebyshev's inequality
$P(|X-\mu|\leq k\sigma) \geq 1 - \frac{1}{k^2}$

$P(|X-\mu|\leq k\sigma) \geq 1 - \frac{1}{k^2}$

binomial

poisson

uniform

normal

$1 \sigma: 68\% 1.65 \sigma: 90\%, 1.96\sigma:95\%, 2.58\sigma,99\%$ $1 \sigma: 68\%, 1.65 \sigma: 90\%, 1.96\sigma:95\%, 2.58\sigma: 99\%$

logmormal

Chi-Square distribution

t-distribution