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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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Answer 50 questions in 15 minutes.
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1. What does the OLS minimize?
SSR
Combine to form distribution with leptokurtosis (heavy tails)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generalized Auto Regressive Conditional Heteroscedasticity model - GARCH(1 -1) is the weighted sum of a long term variance (weight=gamma) - the most recent squared return (weight=alpha) and the most recent variance (weight=beta)

2. Multivariate Density Estimation (MDE)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
When one regressor is a perfect linear function of the other regressors
Low Frequency - High Severity events
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

3. LFHS
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Low Frequency - High Severity events
Least absolute deviations estimator - used when extreme outliers are not uncommon
Sum of n i.i.d. Bernouli variables - Probability of k successes: (combination n over k)(p^k)(1 - p)^(n - k) - (n over k) = (n!)/((n - k)!k!)

4. Econometrics
(a^2)(variance(x)) + (b^2)(variance(y))
When asset return(r) is normally distributed - the continuously compounded future asset price level is lognormal - Reverse is true - if a variable is lognormal - its natural log is normal
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Application of mathematical statistics to economic data to lend empirical support to models

5. Joint probability functions
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Probability that the random variables take on certain values simultaneously
Concerned with a single random variable (ex. Roll of a die)
Standard error of the regression - SER = sqrt(SSR/(n - 2)) = sqrt((ei^2)/(n - 2)) SSR - Sum of squared residuals - Summation[(Yi - predicted Yi)^2] - Summation of each squared deviation between the actual Y and the predicted Y - Directly related

6. Continuously compounded return equation
E(XY) - E(X)E(Y)
Based on a dataset
i = ln(Si/Si - 1)
Variance = (1/m) summation(u<n - i>^2)

7. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
OLS estimators are unbiased - consistent - and normal regardless of homo or heterskedasticity - OLS estimates are efficient - Can use homoscedasticity - only variance formula - OLS is BLUE
Variance(x)

8. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Sum of n i.i.d. Bernouli variables - Probability of k successes: (combination n over k)(p^k)(1 - p)^(n - k) - (n over k) = (n!)/((n - k)!k!)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
P(Z>t)

9. Biggest (and only real) drawback of GARCH mode
Expected value of the sample mean is the population mean
Nonlinearity
F(x) = (1/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared
Based on an equation - P(A) = # of A/total outcomes

10. Two ways to calculate historical volatility
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Summation((xi - mean)^k)/n
Only requires two parameters = mean and variance
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

11. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Among all unbiased estimators - estimator with the smallest variance is efficient
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Random walk (usually acceptable) - Constant volatility (unlikely)

12. Non - parametric vs parametric calculation of VaR
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

13. WLS
Independently and Identically Distributed
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Yi = B0 + B1Xi + ui
Returns over time for an individual asset

14. Limitations of R^2 (what an increase doesn't necessarily imply)

15. P - value
Expected value of the sample mean is the population mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
P(Z>t)

16. Consistent
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia
Variance(x) + Variance(Y) + 2*covariance(XY)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When the sample size is large - the uncertainty about the value of the sample is very small

17. Potential reasons for fat tails in return distributions
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Summation((xi - mean)^k)/n
Var(X) + Var(Y)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

18. Law of Large Numbers
F(x) = (1/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared
Sample mean will near the population mean as the sample size increases
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Random walk (usually acceptable) - Constant volatility (unlikely)

19. Four sampling distributions

20. Logistic distribution
Has heavy tails
More than one random variable
Random walk (usually acceptable) - Constant volatility (unlikely)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

21. Variance of X+b
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Population denominator = n - Sample denominator = n - 1
Variance(x)

22. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sampling distribution of sample means tend to be normal
Among all unbiased estimators - estimator with the smallest variance is efficient

23. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
P(Z>t)

24. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Nonlinearity
More than one random variable
Random walk (usually acceptable) - Constant volatility (unlikely)

25. Control variates technique
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Transformed to a unit variable - Mean = 0 Variance = 1
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

26. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
OLS estimators are unbiased - consistent - and normal regardless of homo or heterskedasticity - OLS estimates are efficient - Can use homoscedasticity - only variance formula - OLS is BLUE
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

27. Binomial distribution
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Easy to manipulate
Sum of n i.i.d. Bernouli variables - Probability of k successes: (combination n over k)(p^k)(1 - p)^(n - k) - (n over k) = (n!)/((n - k)!k!)
(a^2)(variance(x)

28. Simulation models
Flexible and postulate stochastic process or resample historical data - Full valuation on target date - More prone to model risk - Slow and loses precision due to sampling variation
Adjusted R^2 does not increase from addition of new independent variables -Adjusted R^2 = 1 - (n - 1)/(n - k - 1) * (SSR/TSS) = 1 - su^2/sy^2
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Sample mean will near the population mean as the sample size increases

29. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
(a^2)(variance(x)) + (b^2)(variance(y))
In EWMA - the lambda parameter - In GARCH(1 -1) - sum of alpha and beta - Higher persistence implies slow decay toward the long - run average variance
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

30. Inverse transform method
P(X=x - Y=y) = P(X=x) * P(Y=y)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Sum of n i.i.d. Bernouli variables - Probability of k successes: (combination n over k)(p^k)(1 - p)^(n - k) - (n over k) = (n!)/((n - k)!k!)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

31. Maximum likelihood method
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Rxy = Sxy/(Sx*Sy)
Transformed to a unit variable - Mean = 0 Variance = 1
Choose parameters that maximize the likelihood of what observations occurring

32. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Average return across assets on a given day
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

33. GARCH
Generalized Auto Regressive Conditional Heteroscedasticity model - GARCH(1 -1) is the weighted sum of a long term variance (weight=gamma) - the most recent squared return (weight=alpha) and the most recent variance (weight=beta)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Transformed to a unit variable - Mean = 0 Variance = 1

34. EWMA
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
P(X=x - Y=y) = P(X=x) * P(Y=y)
Average return across assets on a given day

35. Confidence ellipse
Statement of the error or precision of an estimate
Variance reverts to a long run level
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test

36. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Peaks over threshold - Collects dataset in excess of some threshold
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

37. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Transformed to a unit variable - Mean = 0 Variance = 1
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Based on a dataset

38. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Has heavy tails
Among all unbiased estimators - estimator with the smallest variance is efficient

39. Variance of aX + bY
Nonlinearity
(a^2)(variance(x)) + (b^2)(variance(y))
Yi = B0 + B1Xi + ui
Has heavy tails

40. Discrete random variable
More than one random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Nonlinearity
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

41. Sample variance
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

42. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
If variance of the conditional distribution of u(i) is not constant
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

43. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Yi = B0 + B1Xi + ui
Independently and Identically Distributed
F(x) = (1/stddev(x)sqrt(2pi))e^ - (x - mean)^2/(2variance) - skew = 0 - Parsimony = only requires mean and variance - Summation stability = combination of two normal distributions is a normal distribution - Kurtosis = 3

44. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Price/return tends to run towards a long - run level
Mean = np - Variance = npq - Std dev = sqrt(npq)

45. Two assumptions of square root rule
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Random walk (usually acceptable) - Constant volatility (unlikely)

46. Cross - sectional
Statement of the error or precision of an estimate
Average return across assets on a given day
i = ln(Si/Si - 1)
Confidence level

47. Covariance calculations using weight sums (lambda)
Generalized Auto Regressive Conditional Heteroscedasticity model - GARCH(1 -1) is the weighted sum of a long term variance (weight=gamma) - the most recent squared return (weight=alpha) and the most recent variance (weight=beta)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Observe sample variance and compare it to hypothetical population variance (sample variance/population variance)(n - 1) = chi - squared - Non - negative and skewed right - approaches zero as n increases - mean = k where k = degrees of freedom - Varia
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

48. POT
Peaks over threshold - Collects dataset in excess of some threshold
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance(y)/n = variance of sample Y
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

49. Continuous representation of the GBM
When asset return(r) is normally distributed - the continuously compounded future asset price level is lognormal - Reverse is true - if a variable is lognormal - its natural log is normal
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

50. Variance of sample mean
Random walk (usually acceptable) - Constant volatility (unlikely)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance(y)/n = variance of sample Y