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Objective 2 Flashcards

1
Q

What are 2 approaches to model extreme events?

(FERM12)

A
  1. Generalized Extreme Value Distribution
  2. Generalized Pareto Distribution
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2
Q

Describe the general principle behind the Generalized Extreme Value Distribution

(FERM12)

A
  • It considers the maximum observation (Xm) from each sample (of iid RVs), and pools them together to form an extreme loss data population
  • As the size of a sample increases, the distribution of the maximum observation converges to the generaized extreme value (GEV) distribution
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3
Q

What is the cummulative distribution function of the GEV?

(FERM12)

A
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4
Q

What are 2 methods to take extreme values?

(FERM12)

A
  1. Return level approach
    • Take the highest obervation in each block of data
  2. Return period approach
    • Set a level above which an observation could be regarded as extreme
    • Take the observations higher than the level in each block of data
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5
Q

What is a major drawback of the GEV distribution?

(FERM12)

A

By using only the largest value(s) in each block of data, it ignores a lot of potentially useful information

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6
Q

Describe the idea behind the Generalized Pareto Distribution

(FERM12)

A
  • G(x) is the distribution of a RV X in excess of a fixed hurdle u given that X is greater than u
  • Assume the observations are iid
  • As the threhold increases, the distribution of the conditional loss distribution converges to a GPD
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7
Q

What is the cummulative distribution of GPD?

(FERM12)

A
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8
Q

What is a key consideration when using GPD?

(FERM12)

A

Choosing the right omega threshold

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9
Q

What are 3 characteristics of financial time series?

(FERM14)

A
  1. Serial correlation does not exist to the extent that it is possible to make money from it
  2. There is strong serial correlation in a series of absolute or squared returns
  3. The distribution of market returns appears to be leptokurtic (i.e., extreme values tend to occur closely together)
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10
Q

What are 3 characteristics of multivariate return series?

(FERM14)

A
  1. Correlations do exist between stocks, and between asset classes and economic variables
  2. There is little evidence of cross-relation (i.e., change in stock price t has little effect on stock price t+1)
  3. The time series of extreme returns are individually leptokurtic and they have jointly fat tails
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11
Q

What are the 3 most common ways to measure spread?

(FERM14)

A
  1. Nominal Spread
  2. Static Spread
  3. Option-Adjusted Spread
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12
Q

What is nominal spread and how is it calculated?

(FERM14)

A
  • The difference between the gross redemption yields of the credit security and the reference bond (e.g., a treasury bond)
  • Quick and easy to measure/calculate

NS = rGY - rREF

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13
Q

What is static spread and how is it measured?

(FERM14)

A
  • The addition to the risk-free rate required to value cash flows at the market price of a bond
  • It considers the full risk-free term structure and the constant (spread) SC added to the yield at each duration

Bond Price = Sum over t [CashFlowt / (1 + rf,t + SC)]

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14
Q

What is the option adjusted spread and how is it applied to calculate Bond Price?

(FERM14)

A
  • Allows for a large number of stochastically generated interest rates (rf,t,sim) such that the expected yield curve is consistent with that seen in the market
  • It considers any options that are present in the credit security (OAS)
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15
Q

How are government bonds’ expected returns estimated?

A

Both domestic and overseas government bonds are risk-free

Returns are estimated from the gross redemption yield, an annual compound interest rate

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16
Q

What is the difference between corporate bonds and government bonds?

What is credit spread?

(FERM14)

A
  • Corporate bonds are not risk-free and have a chance of default, so their expected returns should consider a risk premium
  • Credit spread represents the additional return offered to investors with repect to the credit risk being taken
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17
Q

What are 5 reasons why the credit spreads are higher than historical studies’ findings?

(FERM14)

A
  1. Credit risk premium - reward for volatility relative to risk free securities
  2. Liquidity risk premium - reward for lower liquidity compared to government bonds
  3. Risk aversion premium - reward for possibility of extreme events and skeyness of bond payoff structure
  4. Tax premium - less favorable treatment compared to government
  5. Correlation premium - correlation between credit spreads and interest rates is typically negative
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18
Q

What CAPM and what is the formula behind it?

(FERM14)

A

Capital Asset Pricing Model

rx = r* + ßx (ru - r*) where

rx = rate of return on individual investment X

r* = risk-free rate of return

ru = market return

ßx = σxpx,u / σu

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19
Q

What are the 6 properties of a good benchmark?

(A good benchmark is important when considering market risk!)

(FERM14)

A
  1. Unambiguous (components and constituents should be well-defined)
  2. Investable (can buy components of a benchmark and track it)
  3. Measurable (can quantify the vaue of a benchmark with reasonable frequency)
  4. Appropriate (consistent with investor’s style and objectives)
  5. Reflective of current investment opinion (contains components about which investor has opinions)
  6. Specified in advance (known by all participants before the period of assesment)
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20
Q

What is are 8 specific criteria against which a benchmark can be measured in its appropriateness?

(A good benchmark is important when considering market risk!)

(FERM14)

A
  1. Proportion (should contain a high proportion of the securities held in the portfolio)
  2. Turnover (benchmark’s constituents’ turnover should be low)
  3. Allocations (shoud be investable position sized)
  4. Position (investor’s active position should be given relative to the benchmark)
  5. Variability (benchmark variability to the portfolio should be lower than market variability to portfolio)
  6. Positive correlation between rX - rU and rB -rU
  7. Zero correlation between rX - r<span>B</span> and rB -rU
  8. Style exposure (must be similar between portfolio an benchmark)
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21
Q

What is the Black-Scholes Model used for?

What are the formulas?

(FERM14)

A

Used to model European call and put options

Call = C0 = X0 N(d1) - Ke-r*T N(d2)

Put = P0 = Ke-r*t N(-d2) - X0 N(-d1)

where

d1 = [ln(X0/K) + (r* + σ2X/2)T] / [σX sqrt(T)]

d2 = d1 - σX sqrt(T)

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22
Q

What are two types of interest rates?

(FERM14)

A
  1. Spot rates
    • (1 + rt)-1 = e-st
  2. Forward rates
    • e-st = e -(f1 + f2 + … + fT)
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23
Q

What is the bootstrapping approach?

(FERM14)

A

Constructing a spot rate curve from the gross redemption yields on a series of bonds with a range of terms

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24
Q

What are 6 single-factor interest rate models?

(FERM14)

A
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25
Q

What is the difference between lognormal and normal movements?

(FERM14)

A
  • In lognormal,
    • Volatility term is proportional to the interest rate level r
    • Interest rates can never be negative
  • In normal,
    • Volatility term is independent of interest rate level r
    • Interest rates can become negative
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26
Q

What are the pros and cons of one-factor interest rate models?

(FERM14)

A
  • Pros
    • Works well for simulating a single interest rate, particularly short-term (3M or 1Y) spot rates
    • Can be used to simulate the movement of a different sport rate with a longer term
  • Cons
    • Not good for modeling different points on a yield curve
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27
Q

What are the relevant formulas to calculate modified duration (BDX), bond convexity (BCX), and change in bond price (BPX)?

How is interest rate calculated?

(FERM14)

A

Interest rate can be solved for in the change in BP formula.

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28
Q

What is the formula for interest rate parity?

(FERM14)

A
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29
Q

What are the two components of credit risk?

How does it relate to the purposes of modeling credit risk?

(FERM14)

A
  1. Probability of default (PD)
    • Model how likely the credit event is to occur
  2. Magnitude of loss given default (LGD)
    • Determine the extent of loss that will be incurred
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30
Q

What are 3 factors that affect the credit spread and default risk?

(FERM14)

A
  1. Debt seniority (more senior issues have an earlier call on assets remaining)
  2. Presence of collateral (a collateral lowers level of risk)
  3. Types of collateral (the more liquid, the better loan ters)
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31
Q

What are 3 broad types of quantitative credit models?

(FERM14)

A
  1. Credit scoring
    • Uses features of an entity (e.g., financial ratios) to arrive at a score that represents the likelihood of its insolvency
    • Examples:
      • Probit and logit
      • Altman’s Z-score
      • k-nearest neighbor approach
      • Support vector machines
  2. Structural form
    • Models the value of an entity (e.g., debt value or equity value)
    • Examples:
      • Merton
      • KMV models
  3. Reduced form
    • Uses credit rating to derive a probability of default
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32
Q

What is the general idea behind Probit and Logic models?

(FERM14)

A
  • Regression coefficients can be applied to give a soce representing the probability of default
  • They consider numberal characteristics of firms (e.g., financial leverage and income) that have defaulted or remained solvent
  • They offer highly effective approaches to determine risk
  • But, they do not allow the inclusion of qualitative factors
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33
Q

What is the most familiar discriminant analysis credit modeling approach?

(FERM14)

A
  • Altman’s Z-score
  • It gives each firm a score that represent the probability of default:
    • Zn > 2.99 : is safe
    • 1.80 <= Zn <= 2.99 : uncertain
    • Zn < 1.8 : risk of distress
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34
Q

What are the 2 reasons why Z-score uses financial ratios?

(FERM14)

A
  1. Ratios allow firms of different sizes to be compared on a consistent basis
  2. Ratios allow sensible comparisons to be made over time
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35
Q

What is the k-nearest neighbor approach?

(FERM14)

A
  • A non-parametric model that considers the characteristics of a number of firms that fall into one of two groups (solvent or insolvent)
  • When a new firm is considered, its distance from a number (k) of neighbors is assessed- the proportion of these neighbors that have become insolvent gives an indication of the likelihood that this firm will also fail
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36
Q

What is the support vector machines (SVMs) approach?

(FERM14)

A

An approach that uses a line to separate two groups (solvent vs insolvent) of data based on some measures (e.g., leverage, earnings cover)

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37
Q

Visually describe the k-nearest neighbor approach and SMVs

(FERM14)

A
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38
Q

What is the Merton model?

What is its core assumption?

(FERM14)

A
  • An equity-based approach to model credit risk
  • The firm value as a whole follows a lognormal random walk, and insolvency occurs when the firm value falls below the level of debt oustanding
  • Pr(XT<=B) = N(-d2)
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39
Q

What is the KMV model?

How is it calculated?

(FERM14)

A
  • It uses the capital structure of the firm to estimate the probability of default, where
    • The level of a firm’s debt B is replaced by a B~ = STD + 0.5 LTD
    • Values X0 and σX are derived rather than directly observable
  • Distance to default is calcualted as:

DD = ( X0 - B~) / (X0 σX)

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40
Q

How do credit migration models work?

(FERM14)

A
  • Uses transition matrices to infer default probabilities, produced by most crediting rating agencies produce these
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41
Q

What is a Martigale vs Markov Chain process?

(FERM14)

A
  • Martingale -
    • Today value is the same as the past value
    • t x probability
  • Markov Chain -
    • Today value is independent of hte past value
    • (More complicated)
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42
Q

What are some practical issues with credit migration models?

(FERM14)

A
  • Credit ratings do not give a high level of granularity
  • Different agencies can produce different ratings for the same firm
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43
Q

What are common shock models?

(FERM14)

A

Assumes bond defaults are linked by Poisson processes

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44
Q

What are Survival/Time-unti-default models?

(FERM14)

A

Survival function F(x) = e-ht

where the probability of default is 1 - F(x)

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45
Q

What are 2 reasons why liquidity is difficult to quantify?

(FERM14)

A
  1. Data on liquidity crises is limited
  2. Liquidity occurs in different ways (i.e., is firm-specific) so industry data is of little use for liquidity modeling
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46
Q

What is the most common approach used to asses liquidity risk?

(FERM14)

A

Stress testing

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47
Q

What are 7 scenarios tested for liquidity risk stress tests?

(FERM14)

A
  1. Rising interest rates
  2. Ratings downgrade
  3. Large operational loss
  4. Loss of control over a key distribution channel
  5. Impair capital markets
  6. Large insurance claim for a single or related events
  7. Sudden termination of a large reinsurance contract
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48
Q

What are 2 types of systemic risks?

(FERM14)

A
  1. Feedback risk
    • Returns exhibit some degree of serial correlation
    • e.g., a change in price will result in further changes in the same direction
  2. Contagion risk
    • Interaction between different financial series, better modeled using copulas
    • e.g., the risk that one firm fails results in further failures
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49
Q

What are 7 types of demographic risks?

(FERM14)

A
  1. Mortality
    • Level
    • Volatility - risk that mort experience will differ because there is a finite number of lives int he population considered
    • Catastrophe
    • Trend
  2. Longevity
    • Level
    • Volatility
    • Trend
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50
Q

What are 2 ways to determine the current level of mortality?

How are they combined?

(FERM14)

A
  1. Experience rating
    • From past mortality rates for a group of lives
    • Can be used to calculate (a) central rate of mortality or (b) initial rate of mortality
  2. Risk rating
    • From the underlying characteristics of a group of lives

Can be combined using credibility

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51
Q

What are two aspects in non-life insurance claims?

(FERM14)

A
  1. Incidence (frequency)
  2. Intensity (loss severity)
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52
Q

What are 2 statistical issues when analyzing claim frequencies?

(FERM14)

A
  1. Data set will often span a number of years
  2. Some policyholders will be included in the data set for each year, while others will not (i.e., different risk exposure)
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53
Q

What are 3 statistical issues when analyzing claim intensity?

(FERM14)

A
  1. Seasonaility
  2. Clustering
  3. Censoring
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54
Q

What are 3 approaches to determine claim reserves for high claim frequency classes?

(FERM14)

A
  1. Total Loss Ratio
    • Total estimated claims = (Total loss ratio) x (Earned premium)
  2. Chain Ladder
    • Use link ratio
  3. Bornhuetter Ferguson Method
    • Ultimate loss = (Claims reported) + (Premium)(Loss ratio) x (Claims Outstandingchain ladder approach)
    • Interim loss = (Claims reported) x (Link ratio)
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55
Q

What are 3 ways to assess operational risk?

(FERM14)

A
  1. Simple approach (income times a fixed percentage)
  2. Top-down method (residual income volatility)
  3. Risk capital (residual risk capital)
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56
Q

Why do unquantifiable risks arise?

(FERM14)

A

Not all risks can be quantified because the potential losses are difficult to assess (size and likelihood)

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57
Q

How can unquantifiable risks be assessed?

(FERM14)

A

Using a risk map:

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58
Q

What is VAR?

(VAR5)

A

The worst loss over a target horizon such that there is a low, prespecified probability that the actual loss will be larger

Pr(L>VAR) <= 1 - c

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59
Q

What are the steps in computing VAR?

(VAR5)

A
  1. Mark to market the current portfolio
  2. Measure the variability of the risk factor
  3. Set the time horizon, or the holding period
  4. Set the confidence level
  5. Report the worst potential loss by processing all the preciding information into a probability distribution of revenues, which is summarized by VAR
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60
Q

What are relative VAR and absolute VAR?

(Non-parametric)

(VAR5)

A
  • Relative VAR
    • Conceptually more appropriate because it views risk as a deviation from the mean
    • More consistent with definitions of unexpected loss
    • VAR(mean) = E(W) - W* = -W0(R*-µ)
  • Absolute VAR
    • VAR(zero) = W0 - W* = -W0 R***​
  • where
    • W0 = initial investment
    • R = rate of return
    • µ = expected rate of return
    • σ = volatility of return
    • W* = quantile of the distribution
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61
Q

What is the expected tail loss (ETL)?

(VAR5)

A

ETL = Average losses beyond VAR + mean

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62
Q

What are relative VAR and absolute VAR?

(Parametric)

(VAR5)

A
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63
Q

What are 4 desireble properties for risk measures?

(VAR5)

A
  1. Monotonicity
  2. Translation invariance
  3. Homogeneity
  4. Subadditivity
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64
Q

What are 5 uses of VAR?

(VAR5)

A
  1. as a Benchmark Measure
    • To comapre risks across different markets
  2. as a Potential Loss Measure
    • To give an idea of the worst loss an institution can suffer
  3. as Equity Capital
    • Used directly to set capital cushion for the institution
  4. as Criteria for Backtesting
    • Comapre backtested VAR against P&L
  5. Basel Parameters
    • Used to set minimum capital requirementf or regulatory purposes, equal to VAR(99, over 10 days)x3
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65
Q

What is maxVAR?

(VAR5)

A
  • The worst loss at the same confidence level but during the horizon period H
  • Must be greater than the usual VAR
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66
Q

What is the SE in the estimated mean and standard deviation?

(VAR5)

A
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67
Q

Describe the differences between parametric and non-parametric approaches on quantiles.

(VAR5)

A
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68
Q

Which method (parametric vs. non-parametric) is better?

(VAR5)

A
  • Parametric σ-based approach is more precise, given that it provides a narrower confidence interval
  • The sample standard deviation contains more information than sample quantiles
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69
Q

What are he formulas for EVT VAR and EVT ETL?

(VAR5)

A
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70
Q

What are the advantages and disadvantages of the parametric and non-parametric approaches to calculate VAR?

(VAR5)

A
  • Advantages:
    • easier to use
    • create more precise estimates of VAR
  • Disadvantages:
    • may not approximate well the actual distribution of P/L
    • (note: if using empirical quantiles, the effect of sampling variation or imprecision might be included in the VAR number)
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71
Q

When is it important to select confidence level and horizon for VAR?

(VAR5)

A
  • Arbitrary when:
    • VAR is used as a benchmark risk measure, only needs to be consistent across markets adn time
  • Importatn when:
    • VAR is used to decide amount of equity capital to hold
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72
Q

What is an alternative measure of risk to VAR?

(VAR5)

A

Expected tail loss (ETL), which has several advantages relative to VAR:

  • ETL is stable under addition (time aggregation)
  • Compensates for:
    • Underestimation of potential losses of normal distribution
    • Lack of data in the tails of empirical distributions
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73
Q

How do you calcualte portfolio VAR?

(VAR7)

A

where

  • w = [w1 w2 … wN] = unitless weights or linear exposures to a risk factor
  • Σ = N by N covariance matrix of N components
  • W = initial portfolio value

note that:

σ2p W2 = w’ Σ w W2 = x’ Σ x

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74
Q

How is individual VAR calculated?

(VAR7)

A
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75
Q

What is the relationship between portfolio risk, correlation and number of components?

(VAR7)

A
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76
Q

How is the diversification benefit calculated?

(VAR7)

A

DB = Undiversified VAR - Diversified VAR

where

Undiversified VAR = sum of all individual VARs

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77
Q

What is marginal VAR and how is it calculated?

(VAR7)

A

The change in portfolio VAR resulting from taking an additional dollar of exposure to a given component.

Note that

ßi = cov(Ri,Rp) / σ2p = pip σi / σp

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78
Q

How is beta calculated?

(VAR7)

A
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79
Q

What is incremental VAR and how is it calculated?

(VAR7)

A

It is the change in VAR owing to a new position.

A drawback is that it requires a full revaluation of portfolio VAR with a new trade.

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80
Q

What is the best hedge?

(VAR7)

A

The additional amount to invest in an asset to minimize the risk fo the total portfolio.

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81
Q

What is the component VAR and how is it calculated?

(VAR7)

A

The partition of the portfolio VAR that indicates how much the portfolio VAR would change (approximately) if the given component was deleted.

Component VAR = Marginal VAR x dollar position

Sum of Component VARs = Portfolio VAR

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82
Q

Describe a graphical illustration of the VAR decomposition

(VAR7)

A
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83
Q

How is the percent contribution to VAR calculated for a single component?

(VAR7)

A

Component VAR / Portfolio VAR

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84
Q

What is the sharpe ratio and how is it calculated?

(VAR7)

A

The metric that measures how much expected excess return a portfolio can give for each unit of risk.

SRp = Ep / σp

where

Ep = expected return of the portfolio

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85
Q

What are 3 reasons why volatility prediction is important for risk management?

(VAR9)

A
  1. If volatility increases, VAR increases, so investors may adjust their portfolio exposure for assets which volatility will increase
  2. Predictable volatility means that assets depending directly on volatility (e.g., options) will change value in a predictable fashion
  3. In a rational market, equilibtium asset prices will be affected by changes in volatility, and those who can predict volatility can control financial markets better
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86
Q

What is variance under the moving average method?

(VAR9)

87
Q

What are the drawbacks of using the moving average method?

(VAR9)

A
  1. It ignores dynamic ordering of observations (recent information has same weight as older observations)
  2. Volatility estimate may drop without reason (also known as the ghosting feature)
88
Q

What is variance under the GARCH method?

(VAR9)

A

Conditional variance ht uses information up to time t-1 and rt-1 as the previous day’s return

89
Q

What are the advantages and disadvantages of using GARCH?

(VAR9)

A
  • Advantages:
    • Offers a model with few parameters that fits the data well
    • Canb e used in the markets that display volatility clustering
    • Many variants of the GARCH model are available
  • Disadvantages
    • Nonlinearity
    • Parameters must be estimated using MLE
90
Q

What is variance under the EWMA approach?

(VAR9)

A

EWMA = exponentially weighted moving average

Variance ht is a weighted average of the previous variance forecast ht-1 and the latest squared innovation

91
Q

What are the advantages and disadvantages of using EWMA to estimate variance?

(VAR9)

A
  • Advantages
    • Easy to implement because it is an exponential model with one parameter
    • Recursive
  • Disadvantages
    • Difficult to estimate the parameter daily
    • The decay factor may vary (1) across series and (2) over time, losing consistency over different periods
    • Different values of lambda create incompabilities across the covariance terms and may lead to unreasonable values for correlations
    • The model does not allow for mean reversion
92
Q

How are the EWMA and GARCH models related?

(VAR9)

A

The EWMA model is a special case of the GARCH process where a0 is set to 0 and a1 and ß add up to unity

93
Q

How is implied volatility or implied standard deviation (IDS) calculated?

(VAR9)

A
  • Derived by setting the market price of an option equal to its model value (e.g., Black-Scholes)
    • The approach can be used to infer a term structure of ISDs, plotting the ISD against the maturity of the associated option (also known as volatility surface)
    • ISD refers to the average volatility over the life of the option, not instantaneous volatility
  • Implied correlations can be recovered from the options data (quanto options)
94
Q

What are the advantages and disadvantages of using Monte Carlo Simulation?

(VAR12)

A
  • Advantages:
    • Its flexibility makes it the most powerful approach to VAR
    • Accounts for wide range of risks and complex interactions
    • Accounts for nonlinear exposures and complex pricing patterns
    • Simulations can be extended to longer horizons
    • Can be used for operational risk measurement, and integrated risk management
  • Disadvantages:
    • Involves costly investments in intellectual and systems development
    • Requires substantially more computing power than simpler methods
95
Q

What are the steps of using the Monte Carlo simulation to calculate VAR?

(VAR12)

A
  1. Simualte repeatedly a random process for the financial variable(s) of interest covering a wide range of possible situations
  2. Draw these variables from pre-specified probability distributions that are assumed to be known, including the analytical function and its parameters
  3. Simulations recreate the entire distribution of portfolio values, from which VAR can be derived
96
Q

What is the stock price formula?

(VAR12)

97
Q

What are the advantages and disadvantages of using the Bootstrap Method to generate random numbers?

(VAR12)

A
  • Advantages
    • Can include fat tails jumps, or any departure from the normal distribution
    • Accounts for correlations across series because one draw consists of the simultaneous returns of N series
  • Disadvantages:
    • For small sample sizes M, the bootstrapped distribution may be a poor approximation of the actual one so sufficient data points are needed
    • Relies heavily on the assumption that returns are independent so by resampling at random, any pattern of time variation is broken
98
Q

What are the main drawbacks of Monte Carlo methods?

(VAR12)

A
  1. Computational time requirements (increasing one risk factor can increase the number of simulation runs significantly)
  2. Nested loops- a simulation within a simulation is needed for risk measurement on complex instruments (e.g., mortages or exotic options) since their valuation requires a simulation
99
Q

How is the relative error calculated?

How does it vary?

(VAR12)

A

RE = Standard Error / Expected Quantile

It depends on (1) the number of replications and (2) the shape of the distribution:

  • Higher RE for left-skewed distributions
  • Lower RE for right-skewed distribution
100
Q

What are acceleration methods?

(VAR12)

A

They aim to reduce the computational time per stochastic simulation.

  1. Antithetic variable technique
    • Easiest method, consists of changing the sign of all the random samples
    • Appropriate when the distribution is symmetric and to eliminate historical trends
  2. Control variates technique
    • We use another function to find the “error” between the sample and the closed form, and add the “error” term to our simulation
    • The estimator has much lower variance
  3. Important sampling technique
    • Most effective acceleration method, where we focus on simulations that generate observations in the target area
  4. Stratified sampling technique
    • Divides the distribution into zones and samples representative values from each
101
Q

What are deterministic simulations?

(VAR12)

A
  • a.k.a., Quasi-Monte Carlo
  • Constructed to provide a more consistent fill of the N-space
  • The choice must account for:
    1. Sample size
    2. Dimensionality of the problem
    3. Shape of the function being generated
  • Numbers are not independent but rather constructed as an ordered sequence of points
102
Q

What are the advantages and disadvantages of deterministic simulation?

(VAR12)

A
  • Advantages:
    • Faster computational time
    • SE shrinks at a rate of 1/K, faster than 1/sqrt(K)
  • Disadvantages:
    • Not independent draws, so accuracy cannot be assessed easily
    • QMC tends to cycle, which decreases performance
103
Q

What is the difference between equilibrium models and arbitrage models?

(VAR12)

A
  • Equilibrium models
    • Use stochastic process for some risk factors to generate a term structure
    • The output term structure will not fit exactly the current term structure
    • Not appropriate for fixed-income option traders
    • Useful for risk management, where it’s important to capture the richness in movements in the term structure, not necessarily to price today’s instruments precisely
  • Arbitrage models
    • Takes today’s term structure as an input and fit the stochastic process accordingly
104
Q

What are the advantages and disadvantages of using simulation methods?

(VAR12)

A
  • Advantages:
    • Flexible
    • Can postulate a stochastic process or resample from historical data
    • Allow for full valuation on the target date
  • Disadvantages:
    • Prone to model risk owing to the need to prespecify the distribution
    • Much slower and less transparent than analytical methods
    • Create sampling variation in the measurement of VAR
    • Greater precision requires a larger number of replications, increasing computing time
105
Q

What are the 3 main components of an EC definition?

(ERM101)

A
  1. Risk measure
  2. Probability threshold
  3. Time horizon
106
Q

What are the deficiencies of using correlation solely as a measure of dependency?

(ERM101)

A
  1. Correlation is a scalar measure of dependency
  2. Possible values of correlation depend on the marginal distribution of risks
  3. Perfectly positively (negatively) dependent risks do not necessarily have a correlation of 1 (-1)
  4. A correlation of zero does not imply independence between risks
  5. Correlation is not invariant under monotonic transformations
  6. Correlation is only defined when the variances of the risks are finite
107
Q

What is the most popular risk measure for EC?

(ERM101)

A

1-year 99.5% VaR

Used in UK’s individual capital assessment( ICA) and Solvency II

108
Q

What are the 3 pillars of Basel II?

What are the 3 major components of risk covered?

(ERM101)

A
  • The 3 pillars are
    • Specify a minimum capital amount
    • Supervisory review
    • Market discipline
  • The three major components of risk covered are
    • Credit risk
    • Operational risk
    • Market risk
109
Q

What are the 3 pillars of Solvency II?

What are the capital requirements for the first pillar?

(ERM101)

A
  • The three pillars are:
    • Quantitative requirements
    • Qualitative requirements
    • Supervisory reporting and disclosure
  • Capital requirements for the first pillar are:
    • Solvency Capital Requirement (SCR) = Var 99.5% over 1 year, which can be calculated using:
      • European Standard Formula
      • Firm’s own Internal Model to calculate EC
    • Minimum Capital Requirement (MCR), calculated using a simple formula below which urgent action would be required by the regulator
110
Q

What is the Internal Model Approval Process (IMAP) under SII?

(ERM101)

A
  1. Management must demonstrate understanding of the internal model
  2. Management must demonstrate understanding of the limitations of the internal model
  3. Timely calculation of results is essential
111
Q

What is the Pearson Correlation Coefficient?

(ERM101)

A
  • Statistical dependency between two risks described by a single number
112
Q

What are practical reasons for using the Pearson Correlation Coefficient?

(ERM101)

A
  1. Easy to calculate
  2. Easier to communicate with other professionals, since it’s covered in many statistical courses
  3. Given a vector of variables, it is simple to calculate the standard deviations and correlations
  4. The correlation matrix uniquely determines the dependence structure, int he context of elliptically contoured distributions
  5. Where many risks are correlated, the correlations form a correlation matrix
113
Q

What are the two most common types of rank correlation?

(ERM101)

A
  1. Spearman coefficient
    • Assess how well a monotonic function could describe the relationship between two variables, without making assumptions about the underlying distribution frequencies
    • Resolves limitations of Pearson correlation coefficient (2, 3, 5, 6)
  2. Kendall Tau correlation
    • Measures dependency as the tendency of two variables to move in the same direction (cordant vs discordant)
114
Q

List risk aggregation methodologies

(ERM101)

A
  1. Simple summation
  2. Fixed diversitifaction percentage
  3. Variance-covariance matrix
  4. Copulas
  5. Causal modeling
115
Q

What are the tradeoff to consider when choosing a risk aggregation methodology?

(ERM101)

A
  1. Model accuracy
  2. Methodology consistency
  3. Numerical accuracy
  4. Availability of data to perform a realistic calibration
  5. Intuitiveness and ease of communication
  6. Flexibility
  7. Resources
116
Q

What are the main advantages and disadvantages of the variance-covariance approach?

(ERM103)

A
  • Advantages:
    • Uses a limited number of inputs
    • It is simpler relative to other methods
    • Can be evaluated formulaically
    • Does not require fundamental information about lower-level risks
  • Disadvantages:
    • Imposes a simple dependency structure that is not accurate in the real world
    • Cannot deal with the cases in which standalone risks are not actually exclusive but integrated (e.g., market and credit risk)
117
Q

What are some methods to overcome the VarCovar limitations of linear correlations?

(ERM103)

A
  1. Stressed/tail correlation
    • Adjust historical correlations based on expert judgment
  2. Rank correlation
    • Does not preserve the assumptions required of Varcovar of uniform risk scaling
  3. Factor model
    • Estimate potential changes in the value of a risky asset based on its factor sensitivities, can be estimated using regression
118
Q

What are the main advantages and disadvantages of using distribution-based aggregation with copulas?

(ERM103)

A
  • Advantages:
    • Allow direct control over the distributional and dependency assumptions used
    • Use entire loss distributions as inputs to the aggregation process
    • Easy to implement from a computational stand point
  • Disadvantages:
    • Analytically complex and do not have closed-form solutions
    • Its specification is difficult to interpret for non-experts
    • Fitting the parameters of a copula is a difficult statistical problem
    • Aggregating group-wide across risk types would require different copulas
119
Q

What are the main advantages and disadvantages of scenario-based aggregation?

(ERM103)

A
  • Advantages:
    • Provides consistent approach to aggregation
    • Forces the firm to undertake a deeper understanding of its risks
    • The results can be interpreted easily
  • Disadvantages:
    • Judgment and expertise are key to identify risk drivers and deriving scenario sets
    • Simulation outcomes may be highly sensitive due to the uses of algorithms, processes, models and aggregation models
120
Q

What is the difference between correlation and dependency?

(ERM103)

A
  • Correlation
    • Quantifies a linear relationship (constant over time) between two random variables
  • Dependency
    • Quantifies a (linear or non-linear) relationship (varies overtime) between two variables
  • Correlation is a special case of dependency
121
Q

Compare the Gaussian, t and Archimedean copulas

(ERM103)

122
Q

Compare the following risk measures in terms of coherence:

Total exposure

Std Dev

VaR

TVar / CTE / Expected shortfall

(ERM103)

123
Q

What is parameter risk?

(ERM104

A
  • Uncertainty as to whether the parameters are appropriate for the phenomenon that we are attempting to model
  • Form of systematic risk that does not diversify with volume, although it may diversify across portfolios to some degree
124
Q

What are sources of uncertainty in modeling?

(ERM104)

A
  1. Sampling risk
    • Differences between the observed sample and the population
  2. Data bias
    • Data differences between the experience and exposure periods
  3. Actuarial model risk
    • Model may not be appropriate for the phenomenon being modeled
  4. Stochastic model risk
    • Risk that the wrong model is being estimated and applied
  5. Process risk
    • Uncertainty of the insurance claims process, which is diversified away as the volume increases
  6. Insufficient parameter identification
    • Results when we (a) fail to recognize relationships in our models, or (2) fail to recognize that certain elements of our model are subject to uncertainty
125
Q

What are 3 parameter estimation methods?

(ERM104)

A
  1. Regression analysis
  2. Maximum likelihood estimation
  3. Model-free methods (e.g., bootstrapping)
126
Q

What are potential pitfalls of using bootstrapping method to estimate a parameter?

(ERM104)

A
  1. Each parameter has its own number of degrees of freedom
  2. Unreliable in small samples (e.g., less than 40 per fitted parameter)
  3. Gives the distribution of the estimated parameters given the true parameters, but we need the opposite
  4. Model might not hold for the data in developing triangles
127
Q

What are two meanings of capital?

(ERM106)

A
  1. Required EC
    • Realistic amount a business believes it needs to meet future risks
  2. Available EC
    • Realistic or market-consistent amount the business actually has available
128
Q

Describe the different views on capital

(ERM106)

A
  • Statutory view of solvency
    • Supervisory systems use simple formulas to define the required minimum level of solvency
  • Economic (Fair Value) view
    • Market is moving towards this, where it equates to market-consistent valuation of liabilities
129
Q

What are the benefits of economic capital analysis?

(ERM106)

A
  1. Determine the appropriate level of capital
  2. Helps management better understand the capital requirements and form discussions with rating agencies
  3. Measures performance based on capital usage
  4. Helps shareholders assess the effective return on capital
  5. Helps regulators understand how well-capitalized companies are
130
Q

What are the weaknesses in formulaic approaches to calculate capital?

(ERM106)

A
  1. No link between the amount of required capital and the effectiveness of risk management
  2. Cannot deal with all types of risks
  3. Lack of transparency over the actual solvency level
  4. Does not cope with changes in the financial/insurance markets
  5. Does not allow for diversification benefit
131
Q

What are the advantages of using an internal model to calculate the SCR?

(ERM106)

A
  1. Encourages insurers to measure and manage their risks
  2. More flexible than industry-standard models
  3. Can be updated as financial markets and a company’s business evolve
  4. Represents the insurer’s business more closely than a rules-based standard approach
132
Q

What are some modeling considerations in EC?

(ERM106)

A
  1. VaR vs TVaR
  2. Stochastic Analysis vs Stress Test
  3. Real World vs Risk Neutral
  4. Diversification effect
  5. Time horizon to consider
  6. Whether to allow negative cumulative surplus int he middle of the time horizon
  7. Whether to account for future new business
133
Q

What type of risks should be considered?

(ERM106)

A
  1. UW / Insurance
  2. Credit
  3. Market
  4. Operational
  5. Liquidity
134
Q

What is model risk?

How can it be mitigated?

(ERM118)

A
  • Risk that:
    • The model is not providing appropriate output,
    • A model is being used inappropriately, or
    • The implementation of an appropriate model is flawed
  • Model validation is a key area of research that can help mitigate model risk, while model risk management includes elements of model development and governance.
    • Can help provide internal and external stakeholders a level of confidence that a model framework is sound and that results can be relied upon to inform decisions
135
Q

What are the elements of model risk management?

(ERM118)

A
  1. Model development
  2. Model validation
  3. Model governance
  4. Model use
136
Q

What are the key principles of model validation?

(ERM118)

A
  1. Model design and build need to be consistent with the models’ intended purpose
  2. Ensure that model validation is an independent process
    • Same company, different department
    • External party
    • External model validated by internal production team
  3. Establish an owner of the model validation
    • Escalates invalid results
    • Resolves issues arising through the validation process
  4. Ensure appropriateness of established model governance
  5. Make model validation efforts proportional to evidenced areas of materiality and complexity
  6. Validate the model components (inputs, calculation, output)
  7. Address limitations of model validations
  8. Document the model validation
137
Q

What types of validation tests are performed for the input component?

(ERM118)

A
  1. Static validation
  2. Back-test established distributions
  3. Reconcile risk driver distribution with other prevailing assumptions
  4. Benchmarking
  5. Expert judgment
138
Q

What types of validation tests are performed for the calculation component?

(ERM118)

A
  1. Sensitivity testing parameters
  2. Dynamic validation
  3. Validate appropriateness of modeled behavioral actions associated with stressed scenarios produced by risk models
  4. Dependencies between lines of business
139
Q

What types of validation tests are performed for the output component?

(ERM118)

A
  1. Historical back-testing
  2. Reconciliation with other reports
  3. Prudent deterministic scenario analysis
  4. Contribution analysis
  5. Parallel testing / version control
140
Q

What is the immediate stress approach?

(ERM119)

A
  • A single risk factor stress methodology that can be used to determined the EC required for each factor
  • Steps are:
    • A risk factor stress is applied at t=0
    • Assets and liabilities are re-valued and net change is calculated
    • Excludes receipt or payment of CFs, impact of future management actions, and dynamic hedging
  • Adopted under SST, ICA and Solvency II
141
Q

What are the advantages and limitations of the immediate stress approach?

(ERM119)

A
  • Advantages:
    • Simple approach to calculate and understand
  • Disadvantages:
    • Unable to capture risks and risk mitigation impacts arising from adverse scenarios that occur over time
142
Q

What is the projection scenarios approach?

(ERM119)

A
  • Used to determine EC required for each risk factor
  • Overcomes limitation of immediate stress
  • A scenario for each risk factor is postulated to occur over the shock application period (2 discrete time steps or smaller number of time steps)
  • Steps:
    • At each time step, value of assets and liabilities are recalculated
    • CFs are recalculated to project P&L
    • Amount of risk capital is set equal to the PV of net P&L
143
Q

What are the advantages and limitations of projection scenarios?

(ERM119)

A
  • Advantages:
    • Ability to capture dynamic interactions and risk management strategies
    • Provides significantly more flexible and realistic basis for specifying specific adverse scenarios
  • Disadvantages:
    • May require nested stochastic techniques, which are complex and computationally intensive
144
Q

What is the multivariate stress tests approach?

(ERM119)

A
  • Involves 2 or more risk factors at the same time, can be used for the aggregation of risk capital too
  • If an immediate stress methodology is used, can apply simultaneous risk factors
  • If a projection methodology is used, then an ESG should be used
145
Q

What are the advantages and limitations of multivariate stress tests?

(ERM119)

A
  • Advantages:
    • Allows for generation of the full distribution of P&L results under a range of assumptions and risk management bases
    • Has the potential to provide significant insight into the pros and cons of alternative risk management strategies, and the business management issues involved across the shock application period
  • Disadvantages:
    • Greater modeling complexity and computational requirements
146
Q

What is the difference between market-consistent and real-world approaches, in the context of ESG calibration?

(ERM119)

A
  • Market-consistent:
    • Models reproduce market prices of the instruments in the calibration set
  • Real-world:
    • Models reproduce a distribution of risk factor results aligned to expected realistic experience
    • Typically used for risk factor stresses
147
Q

What are the limitations of the correlation approach?

(ERM119)

A
  • Tail modeling-
    • Correlations tend to behave differently in extreme situations
  • Non-constant correlation-
    • Negative events might lead to a higher correlation between assets
148
Q

What is an important assumption under the correlation approach?

(ERM119)

A

Risks are normally distributed and dependence structure can be specified via the margins of a Gaussian distribution.

-> The combined risk distribution is multivariate normal

149
Q

What are 3 applications of capital allocation?

(ERM119)

A
  1. Pricing and Technical Provisions
    • Allow for cost of capital in pricing exercises
  2. Risk Budgeting and Capital Allocation
    • Risk budgeting - process where managers decide in which areas to accept risk
    • Determine where additional capital should be added or released
    • Develop risk mitigation strategies (e.g., reinsurance and hedging):
      • Fungibility of capital - ability of capital to absorb losses
      • Transferability of capital - ability of a BU to transfer assets/liabilities to the rest of the group
  3. Risk-Adjusted Performance Measurement
    • ROA, ROE, RAROC, RARORAC
150
Q

What are drawbacks of the traditional return measures, such as ROA and ROE?

(ERM119)

A
  • Focuses on assets and ignore leverage (ROA only, ROE does look at leverage)
  • No distinction between classes or riskiness of assets
  • Does not allow for risk being accepted to achieve the return generated
151
Q

What are some approaches to risk budgeting?

(ERM119)

A
  • Allocate capital to different BU, lines or products
    • Helps manage risk appetite
    • Useful as an important decision-making tool
  • Allocate capital to risk types
    • Useful to manage risks across subsidiaries
    • Less useful for management decision-making
  • Types of buffer between EC and risk-taking capacity:
    • A strategic buffer used for new business opportunities or regulatory requirements
    • A technical buffer used for business cycle impacts and volatility
152
Q

How is RARORAC calculated?

(ERM119)

A

RAROCRAC

= (revenues - costs - expected losses) / required EC

= (A x ra - L x rl - expected losses) / EC0

= EC1 / EC0

153
Q

What is the conceptual difference between RAROC and RARORAC?

(ERM119)

A
  • RAROC -
    • measures risk-adjusted return on the total capital supplied by the shareholders
  • RARORAC -
    • measures the return against the capital required to generate it
    • more for internal management purposes
154
Q

What are some capital allocation approaches, proposed by Milliman?

(ERM119)

A
  1. Marginal Distributions
    • Pro-rata or linear marginal contributions
      • Proportionally to each unit, does not penalize highly correlated portfolios
    • Discrete marginal contributions
      • Calculates VaR excluding X portfolio
      • A discrete marginal contribution is computed as the difference between VaR of portfolio minus total VaR excluding portfolio X
      • Scales the marginal contribution
    • Continuous marginal contributions
      • Negative contributions where negative correlations exist
    • Myers-Read allocation method
  2. Game Theory
    • Widely used in decision making in conflict situations, but calculation intensive
155
Q

Define a scenario

(ERM120)

A
  • Possible future environment, either at a point in time or over a period of time
  • Can be complex, involving changes to and interactions among many factors over time, perhaps generated by a set of cascading events
  • Useful for business planning or estimation of expected profits or losses, but not useful for assessing adverse scenarios (including rare and/or catastrophic future events)
    • A scenario with significant or unexpected adverse consequences is referred to as a stress scenario
    • Its purpose is to allow a firm to plan in advance about types of events or changes in conditions to be prepared if a catastrophe were to occur, rather than about predicting the future
156
Q

Define a sensitivity

(ERM120)

A
  • The effect of a set of alternative assumptions regarding a future environment (i.e., a scenario)
  • The scenario can be a result of a single or several alternative risk factors, over a short or long period of time
  • Uses as a tool to calculate volatilities (and other quantities) by the application of further assumptions on the underlying probability distributions, including non-linearity and inter-relationships between the parameters of the model
157
Q

Define a stress test

(ERM120)

A
  • Projection of the financial condition of a firm or economy under a specific set of severely adverse conditions, which may a result of:
    • Several risk factors and time periods with
    • One risk factor and short in duration
  • A common form of sensitivity testing, particularly useful for regulators and internal management
158
Q

What are 7 types of scenarios?

(ERM120)

A
  1. Reverse
    • Identify a scenario that is expected to give rise to a particular amount of financial loss
  2. Historical
    • Based on experience, i.e. an economic event
  3. Synthetic
    • Hypothetical conditions that have not been yet observed
    • Require more assumptions than history-based scenarios
  4. Company-specific
    • Tailored to the specific mix of risk exposures of a firm
  5. Single-event
    • Well-described triggering event
  6. Multi-event
    • Specific current or future events that lead to a cascade of future events
  7. Global
    • Useful to assess global interdependencies between financial institutions
    • Emphasis on systemic risk and linkages between different economic sectors
159
Q

What is credit exposure?

(ERM124)

A

The cost of replacing the transaction if the counterparty defaults (assuming a zero recovery value)

160
Q

What are two generic characteristics of counterparty risk?

(ERM124)

A
  1. They create credit exposure
  2. The credit exposure depends on one or more underlying market factors
161
Q

What are two broad classes of financial products?

(ERM124)

A
  1. OTC derivatives
    • Interest rate swaps
    • FX forwards
    • Credit default swaps
  2. Securities financing transactions
    • Repos and reverse repos
    • Securities borrowing and lending
162
Q

What is counterparty risk?

(ERM124)

A
  • Risk that a counterparty in a derivatives transaction will default prior to expiration of a trade and will not make the current and future payments required by the contract
  • OTC derivatives are subject to significant counterparty risk (but is smaller than an equivalent loan or bond because only net CFs are paid)
163
Q

What are two differences between counterparty risk from traditional credit risk?

(ERM124)

A
  1. The value of a derivatives contract in the future is uncertain (sign and magnitude)
  2. Counterparty risk is bilateral (in a derivatives transaction, each counterparty has risk to the other)
164
Q

What are three important aspects when considering future default probability?

(ERM124)

A
  1. Future default probability will have a tendency to decrease due to the change that the default may occur before the start of the period in question
  2. A counterparty with an expectation of deterioration in credit quality will have an increasing probability of default over time
  3. A counterparty with an expectation of improvement in credit quality will have a decreasing probability of default over time
165
Q

What is Mark-to-Market (MtM)?

(ERM124)

A
  • PV of all the payments an institution is expecting to receive, minus those it is obliged to make
  • Defines what could be potentially lost today with respect to a counterparty
  • Represent replacement cost
  • Counterparty exposure is based on the current MtM value of a transaction(s)
166
Q

What do a positive MtM and negative MtM mean?

(ERM124)

A
  • Positive MtM
    • When a counterparty defaults, they will be unable to make future commitments
    • The institution will have a claim on the positive MtM at the time of default
    • LGD = MtM - Recovery Value
  • Negative MtM
    • When a counterparty defaults, the institution is still obliged to settle this amount
    • The institution does not gain or lose from the counterparty’s default
167
Q

What is possible future exposure (PFE)?

(ERM124)

A
  • Defines possible exposure to a given confidence level, normally according to a worst-case scenario
  • Analogous to the traditional VaR measure
168
Q

What are ways to mitigate counterparty risk?

(ERM124)

A
  1. Diversification
    • Spread exposure across different counterparties
  2. Netting
    • Being able to legally offset positive and negative contract values (MtMs) with the same counterparty in the event of their default
  3. Collateralization
    • Holding cash or securities against an exposure
  4. Hedging
    • Trading instruments such as credit derivatives to reduce exposure and counterparty risk
169
Q

What are risk mitigation methods for counterparty risk?

(ERM124)

A
  1. Trading with high-quality counterparties
  2. Cross-product netting
  3. Close-out
    • Permits immediate termination of all contracts between an institution and a defaulted counterparty with netting of MtM values
    • If there is netting and no close-out feature, a default would have to be handled in court
  4. Collateralization (Margining)
  5. Walk-away (tear-up) features
    • Allows canceling transactions in the event that their counterparty defaults
  6. Monolines
    • Only trade with counterparties with very strong credit quality
  7. Diversification of counterparty risk
  8. Exchanges and centralized clearing houses
    • Which have no counterparty risk compared to OTC derivatives
170
Q

What is the risk derivative price?

(ERM124)

A

Risk-free price (assuming no counterparty risk) -

credit value adjustment (“CVA”)

  • No transaction will be refused directly but an institution needs to make a return more than the CVA
  • CVA can be defined:
    • Actuarial price = EPV of future cash flows with risk premium
    • Risk-neutral price = cost of an associated hedging strategy
  • The CVA should consider:
    1. Default of probability of the counterparty
    2. Default of probability of the institution
    3. Transaction in question
    4. Netting of existing transactions with the same counterparty
    5. Collateralization
    6. Hedging
171
Q

What are CDS and CCDS?

(ERM124)

A
  • Credit Default Swap (“CDS”)
  • Contingent Credit Default Swap (“CCDS”)
    • Tailored credit derivative products
    • CDS with the notional of protection indexed to the exposure on a contractually specified derivative
172
Q

What are

  • Expected MtM
  • Expected Exposure (EE)
  • Potential Future Exposure (PFE)
A
  • Expected MtM
    • This represents the forward or expected value of a transaction at some point in the future, so forward rates are a key factor when measuring it under the risk-neutral measure
    • Can be important due to the relatively long time horizons involved in measuring counterparty risk
    • May vary significantly from current MtM due to specifics of cash flows
  • Expected Exposure (EE)
    • This represents the amount expected to be lost if the counterparty defaults
    • Greater than the expected MtM since it concerns only the positive MtM values
  • Potential Future Exposure (PFE)
    • Defines the exposure worst-case gain that would be exceeded with a probability of no more than confidence level c
    • Exactly the same definition as traditional VaR with the exception of:
      • PFE may be defined at a point far in the future, whereas VaR refers to a short horizon
      • PFE refers to a number that will be associated with a gain, whereas VaR refers to a loss
173
Q

What is Skylar’s theorem?

(ERM125)

A
  • For any joint distribution function F, there is a unique copula C that satisfies F(x1,…,xd) = C[F1(x1),…,Fd(xd)]
  • Skylar’s theorem proves that in examining multivariate distributions, we can separate the dependence structure from the marginal distributions
    • Conversely, we can construct a multivariate joint distribution from (i) a set of marginal distributions and (ii) a selected copula
  • The dependence structure is captured in the copula function and is idenepdent of the form of the marginal distributions
174
Q

What is a copula?

(ERM125)

A
  • Contains the information about the structure of dependency:

F(x1,…,xd) = C[F1(x1),…,Fd(xd)]

175
Q

Describe the idea behind tail dependence

(ERM125)

A
  • We want to know that if one risk has a very large loss, is it more likely that another risk will also have a large loss?
  • Measures of tail dependence have been developed to evaluate how strong correlation is in the upper or lower tails
176
Q

What are the formulas behind upper tail dependence and lower tail dependence?

(ERM125)

177
Q

What are Archimedean copulas?

(ERM125)

A
  • Archimedean copulas of dimension d are those in the form:
  • C(u1,…,ud)=Ø-1[Ø(u1)+…+Ø(ud)]*
  • Where Ø(u) is called the generator, which is:
    • strictly decreasing
    • convex
    • continuous function
    • maps [0,1] into [0,infinity] with Ø(1)=0
  • The inverse of the generator Ø-1(t) must be completely monotonic on [0,infinity]
178
Q

What is the Gaussian copula?

(ERM125)

A

C(u1,…,ud)=ØP(Ø-1(u1)+…+Ø-1(ud))

  • Where
    • Ø(x) is the standard univariate normal cdf
    • ØP(x1,…,xd) is the joint cdf of the standard multivariate normal distribution with zero mean and variance of 1 for each component and a correlation matrix P
  • Contains d(d-1)/2 parameters, because this is the number of pairwise correlations
  • If all correlations in P are zero, the Gaussian copula reduces to the independence copula
179
Q

Why may the Gaussian Copula not be approriate for risk modeling?

(ERM125)

A
  • It has no tail dependence except in the special case with correlation = 1, resulting in indices of upper and lower tail dependence of 1
  • For example,
    • In a stable environment the degree of dependence between variables may be relatively small
    • However, a systemic event may affect all the variables and thus large values for one variable may be well associated with large values of the others
180
Q

What is the t Copula?

(ERM125)

A

C(u1,…,ud)=tv,<strong>P</strong>(tv-1(u1)+…+tv-1(ud))

  • tv(x) is the cdf of the standard univariate t distribution with v degrees of freedom
  • tv,<strong>P</strong>(x1,…,xd) is the joint cdf of the standard multivariate t distribution with v degrees of freedom for each component and P is a correlation matrix
181
Q

What are Extreme Value (EV) Copulas?

(ERM125)

A
  • This class of copulas is defined in terms of the scaling property of extreme value distributions
  • A copula is an EV copula if it satisfies:
    • C(u1n,…,udn) = Cn(u1,…,ud) for all (u1,…,ud) and for all n>0

Max-Stability

  • This scaling property results in the EV copula having the stability of the maximum (or max-stable) property. The copula of the maxima is given by:
  • Cmax(u1n,u2n) = Cn(u1,u2)*
  • where Cmax is the same as the original copula C
  • It also means that the copula associated with the random pair (MX, MY) is also C(x,y)
182
Q

What are ways to estimate parameters?

What are two issues with estimating parameters for copulas?

(ERM125)

A

Ways to estimate parameters:

  1. Parametric method (MLE)
    • Disadvantages:
      • # parameters = # parameters in marginals + # parameters in copula, resulting in high number
      • Maximization of a high-dimensional function can be challenging
  2. Semi-parametric method
  3. Non-parametric method
    • Advantages:
      • Does not depend on the values of parameters in marginal distributions
      • Removes uncertainty due to choice of marginal distributions
183
Q

What are ways to test the fit of a copula?

(ERM125)

A
  1. Chi-squared test of fit
  2. Univariate methods
184
Q

Describe the two-stage process for generating multivariate random variables

(ERM125)

A
  1. Use a method particular to the copula to generate a multivariate random variable from the copula distribution
  2. Use each element in this multivariate copula random variable to generate the value from the corresponding marginal distribution (can be done by using the inversion method)
185
Q

Describe arbitrage-free modeling

(ERM602)

A
  1. Adjust model time dependent parameters to fit market prices exactly
  2. Do not model the dynamics of the term structure
  3. It may look like the term structure today but it will not act like the term structure tomorrow
  4. The model is an interpolation system
186
Q

Describe equilibrium modeling

(ERM602)

A
  1. Model the behaviors of the term structure over time
  2. Employ a statistic approach
  3. Do not exactly match market prices today
  4. Do not contain time dependent parameters
187
Q

Describe risk-neutral scenarios

(ERM602)

A
  1. Assume all term premia (compensation for market risk) are zero
  2. Are not appropriate for all purposes
188
Q

Describe real-world scenarios

(ERM602)

A
  1. In the real world, term premia are not zero
  2. Reflects the changes of the real world
  3. Good for stress testing or reserve adequacy testing
189
Q

When should the 4 combinations of models be used?

(ERM602)

A
  1. Arbitrage-free + Risk-neutral
    • Current pricing
      • where current market price data is reliable
      • parameters are interpolated to match current market prices
  2. Equilibrium + Risk-neutral
    • Current pricing
      • where current market price data is sparse
      • captures global behavior of term over time
    • Horizon pricing
      • assumes future state of the market (~binomial trees)
  3. Arbitrage-free + Real-world
    • Not practically used
  4. Equilibrium + Real-world
    • Stress-testing
    • VaR
    • Reserves and AAT
190
Q

What are the focuses in the risk appetite framework?

(OR-RA)

A
  1. Protect and create value for the business
  2. Ensure the consistency between risk appetite and risk limits
  3. Integrate into business strategy and corporate culture
191
Q

What are traditional measures used by insurance companies to manage business?

(OR - RA)

A
  • New business volume/market share
  • Persitency rate
  • Embedded value of existing and new business
  • Appraisal value
  • Operating profit

These measures + conservative approach to reserving capital = may be too optimistic of a view

But risk appetite is more concerned about the tail events

192
Q

What are the 3 levels of detail in a risk appetite framework?

(OR - RA)

A
  1. Enterprise risk tolerance
    • Aggregate amount of risk the company is willing to take, in terms of:
      • Capital adequacy
      • Earnings volatility
      • Crediting rating target
    • Revised only if there are fundamental changes to the company’s financial profile
    • Prevents default by preserving capital position
  2. Risk appetite for each category
    • Allocated for risk categories and business activities
  3. Risk limit
    • Business operation
    • Translates above two levels into risk-monitoring measures
193
Q

What are quantitative measures of risk appetite?

(OR - RA)

A
  • Capital/equity at risk (e.g., VaR, TVaR)
  • Earnings at risk (under various accounting basis)
  • Embedded valuemarket consistent embedded value
194
Q

What are qualitative measures of risk appetite?

(OR - RA)

A
  • Credot ratomgs
  • Risk preferences
  • Franchise value
195
Q

What are the steps in setting up a risk appetite framework?

(OR - RA)

A
  1. Bottom-up analysis of the company’s current risk profile
  2. Interviews with the board of directors regarding the level of risk tolerance
  3. Alignment of risk appetite with the company’s goal and strategy
  4. Formalization of the risk appetite statement with approval from the baord of directors
  5. Establishment of risk policies, risk limit and risk-monitoring processes consistent with risk appetite
  6. Design and implementation of the risk-mitigation plan to be consistent with risk appetite
  7. Communication with local senior management for their buy-in
196
Q

What are five strategies considered to improve the risk profile and create value?

(OR - RA)

A
  1. Increase P&C market share
  2. Add MVA to pass through investment risk
    • Adjust product features to reduce liquidity risk
  3. Increase ERM investment
    • Enhance risk management policies and implementation
  4. Increase/decrease equity allocation
    • Balance between risky investments (higher returns) and less risky (less capital deficiency and earnings volatility)
  5. Hedge rho of VA business
    • Reduces earnings volatility and stabilitzes capital position
197
Q

How is risk appetite related to asset allocation?

What are two asset allocation strategies?

(OR - RA)

A

Risk appetite can help refine risk measures in strategic asset allocation and also help a company consider asset allocation from a holistic perspective.

Two aspects of it include:

  1. Strategic asset allocation (ASA)
    • Long-term policy portfolio
    • Reflects desired systematic risk exposure
  2. Tactical asset allocation (TAA)
    • Short-term market opportunities
    • Specifies allowable deviation from SAA
198
Q

What are two risk measures for new business budgeting?

(OR - RA)

A
  1. RAROC
    • Allows for consistent comparison of activities across different types of risks and businesses
  2. MCEV
    • Adjusts approach of traditional EV of new business
    • Takes into account non-hedgeable risks, cost of options and guarantees offered, and frictional cost of capital explicitly
199
Q

How does a risk appetite framework add value to an EC framework, in terms of capital allocation?

(OR - RA)

A

A risk appetite framework:

  • takes into account the specifics of the business, the investor’s risk tolerance, and all the contraints
  • provides the guideline for capital allocation with the baord’s risk tolerance (CaR, EC adequacy, EaR)
200
Q

What are some considerations on capital allocation in the risk appetite framework?

(OR - RA)

A
  1. Avaialble capital vs required capital
  2. Role of diversification in capital allocation, which happens at various levels:
    • for a given product, amount risk categories
    • within a BU, among products
    • across business types
  3. Capital fungibility
    • Capital is assumed to move freely among different legal entities
    • Might be subject to regulators’ approval
    • May be easier to do by paying the parent company in the form of dividend
201
Q

What is economic value added (EVA)?

(OR - RA)

A

EVA = MCEV earnings - opportunity cost x capital allocated

= MCEVt - MCEVt-1

It encourages senior management to take opportunity cost of capital into consideration and maximize shareholder’s value given their risk appetite

202
Q

What are two approaches to calculate MCEV?

(OR - RA)

A

~ How much would you be able to give your shareholders if you sold the business today

<> GAAP or Stat

203
Q

How is EVA decomposed?

(OR - RA)

A
  1. Investment
    • Achieve higher returns than expected
    • EVA<em>inv</em> = extra inv income over SAA - change in CoC
  2. Business management
    • Minimize the gap between asset and liability portfolio
    • EVA<em>bus</em> = MCEV of new business + expected return on RP + experience G/Ldue to nonfinancial factors - CoC
    • where CoC = CoC rate x required capital = CoC rate x max{Stat capital, Rating Agency-required capital, EC}
  3. ALM
    • Promote new business growth and gain/loss from nonfinancial arisk
    • EVA<em>ALM</em> = return on SAA - return on RPdue to ALM or interest rate impact
204
Q

What is extreme value theory (EVT)?

(OR - EVT)

A
  • Provides theoretical basis for a model that makes use of previous extreme values to offer information on the probability and magnitude of potential values more extreme than those seen previously
  • Quantifies the potential black swans hinted at by historical extremes
  • EVT’s main results characterize the distribution of the sample maximum or the distribution of values above a given threshold
205
Q

What are the steps to create a GPD function?

(OR - EVT)

A
  1. Rank the historical return data
  2. Select the threshold u
  3. Approximate the excess distribution Fu as a GPD with parameters s and k
  4. Repeat #3 for values of u farther out that left tail
  5. Choose the threshold that there is stability in the GPD parameter estimates from that point on
  6. Develop the GPD function
206
Q

What are the goals of measuring/assessing operational risk?

(OR - OpRisk)

A
  1. Strategic risk control optimization
  2. Strategic risk reward optimization
  3. Analyze principal-agent risk
  4. Estimate regulatory and economic capital for operational risk
207
Q

What is an actuarial approach to modeling operational risk?

(OR - OpRisk)

A
  • Decompose the aggregate loss distribution into two components:
    • Frequency N
    • Severity Xi
  • Aggregate loss will be equal to

Aggregate Loss = Sum over N of Xi

  • Aggregate Expected Loss = E(N) E(X)
208
Q

Describe and provide examples of hard data and soft data

(OR - OpRisk)

A
  • Hard data
    • Information that has been collected through a systematic process on a prospective basis
    • e.g., measure the height of ocean waves at a certain beach
  • Soft data
    • Information based on empirical observations, but where the data may not have been collected through a robust process and/or where the data may represent a proxy variable
    • e.g., make reasonable inferences about the height of waves based on some observations and scientific methods
209
Q

What are two good sources of external loss data?

(OR - OpRisk)

A
  1. External public data
    • Represents the data that has been collected from publicly available information, such as media reports, legal settlements, and judgments and corporate filings
    • Advantages:
      • Data is well documented
      • Some vendors provide relevant information for scaling losses
    • Disadvantages:
      • Suffers from a reporting bias (not all losses are reported)
      • Due to reporting bias, it is not advisable to use MLE methods
    • Solution:
      • Largest losses are less prone to reporting bias
  2. Consortium data
    • ​Represents the pooled data from member institutions who have agreed to share their internal loss data on an anonymus basis
    • Advantages:
      • Some of these initiatives have a large member base
    • Disadvantages:
      • Very little descriptive information is provided to preserve anonymity
      • Difficut to use the data where data may have been misclassified
    • Solution:
      • Use effective controls over data quality and consistency, consortium data can be a very useful source of information
210
Q

What are some challenges of modeling severity?

(OR - OpRisk)

A
  1. Lack of sufficient data
  2. Pool quality of data
  3. Truncated or censored data
  4. Sensitivity to low probability / high severity loss events
  5. Classification issues
  6. Need to incorporate both internal and external data
211
Q

What is the i.i.d. assumption?

(OR - OpRisk)

A
  • Assumption that the loss data are independent and identically distributed
  • Critical assumption underlying all models
  • When not valid, the models can produce spurious results
212
Q

What are some examples of frequency distributions?

(OR - OpRisk)

A
  1. Poisson
    • No variability
  2. Negative binomial
    • Excess variability
  3. Binomial
    • Very little variability
213
Q

Describe how to use a Monte Carlo simulation to estimate the aggregate Operational loss distribution

(OR - OpRisk)

214
Q

What are the formula standard error (FSE)?

(OR - VarCTE)

ERM Exam (6 decks)

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