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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. Variance - covariance approach for VaR of a portfolio
Choose parameters that maximize the likelihood of what observations occurring
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

2. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Distribution with only two possible outcomes
Choose parameters that maximize the likelihood of what observations occurring

3. Square root rule
E(XY) - E(X)E(Y)
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

4. Variance(discrete)
P(Z>t)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

5. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
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
Independently and Identically Distributed
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

6. Monte Carlo Simulations
Variance(x) + Variance(Y) + 2*covariance(XY)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

7. Time series data
(a^2)(variance(x)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Returns over time for an individual asset
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

8. Logistic distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Has heavy tails
When the sample size is large - the uncertainty about the value of the sample is very small
Based on an equation - P(A) = # of A/total outcomes

9. Cross - sectional
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Average return across assets on a given day
Variance(x)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

10. Normal distribution
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
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance ratio distribution F = (variance(x)/variance(y)) - Greater sample variance is numerator - Nonnegative and skewed right - Approaches normal as df increases - Square of t - distribution has a F distribution with 1 -k df - M*F(m -n) = Chi - s
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

11. Exact significance level
P - value
Confidence level
Rxy = Sxy/(Sx*Sy)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

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

13. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

14. Continuous representation of the GBM
Confidence level
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
Normal - Student's T - Chi - square - F distribution
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)

15. Adjusted R^2
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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

16. Sample mean
Contains variables not explicit in model - Accounts for randomness
Sample mean will near the population mean as the sample size increases
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Expected value of the sample mean is the population mean

17. LFHS
Returns over time for an individual asset
Low Frequency - High Severity events
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

18. Confidence interval (from t)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Based on an equation - P(A) = # of A/total outcomes
Sample mean +/ - t*(stddev(s)/sqrt(n))
Confidence level

19. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

20. Non - parametric vs parametric calculation of VaR
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

21. Discrete random variable
Only requires two parameters = mean and variance
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

22. Poisson Distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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

23. LAD
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!)
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)
Least absolute deviations estimator - used when extreme outliers are not uncommon

24. Simulating for VaR
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Confidence level
Transformed to a unit variable - Mean = 0 Variance = 1
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

25. Shortcomings of implied volatility
Application of mathematical statistics to economic data to lend empirical support to models
Model dependent - Options with the same underlying assets may trade at different volatilities
Easy to manipulate
Statement of the error or precision of an estimate

26. Four sampling distributions

27. Simulation models
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

28. Binomial distribution
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!)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

29. Empirical frequency
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Based on a dataset

30. Hazard rate of exponentially distributed random variable
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
Average return across assets on a given day

31. Implications of homoscedasticity
P(Z>t)
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
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

32. Law of Large Numbers
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Regression can be non - linear in variables but must be linear in parameters
Sample mean will near the population mean as the sample size increases

33. P - value
We reject a hypothesis that is actually true
Peaks over threshold - Collects dataset in excess of some threshold
Model dependent - Options with the same underlying assets may trade at different volatilities
P(Z>t)

34. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Probability that the random variables take on certain values simultaneously
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

35. Continuously compounded return equation
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Refers to whether distribution is symmetrical - Sigma^3 = E[(x - mean)^3]/sigma^3 - Positive skew = mean>median>mode - Negative skew = mean<median<mode - if zero - all are equal - Function of the third moment
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
i = ln(Si/Si - 1)

36. Biggest (and only real) drawback of GARCH mode
P(X=x - Y=y) = P(X=x) * P(Y=y)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Nonlinearity
Mean of sampling distribution is the population mean

37. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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

38. Sample correlation
Rxy = Sxy/(Sx*Sy)
Yi = B0 + B1Xi + ui
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
More than one random variable

39. Two assumptions of square root rule
If variance of the conditional distribution of u(i) is not constant
Random walk (usually acceptable) - Constant volatility (unlikely)
Choose parameters that maximize the likelihood of what observations occurring
95% = 1.65 99% = 2.33 For one - tailed tests

40. Historical std dev
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Price/return tends to run towards a long - run level
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

41. Expected future variance rate (t periods forward)
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Choose parameters that maximize the likelihood of what observations occurring
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

42. Potential reasons for fat tails in return distributions
Returns over time for an individual asset
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
When the sample size is large - the uncertainty about the value of the sample is very small

43. Standard normal distribution
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
Transformed to a unit variable - Mean = 0 Variance = 1
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

44. Test for statistical independence
Special type of pooled data in which the cross sectional unit is surveyed over time
P(X=x - Y=y) = P(X=x) * P(Y=y)
We accept a hypothesis that should have been rejected
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

45. Antithetic variable technique
Mean of sampling distribution is the population mean
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

46. F distribution
Variance ratio distribution F = (variance(x)/variance(y)) - Greater sample variance is numerator - Nonnegative and skewed right - Approaches normal as df increases - Square of t - distribution has a F distribution with 1 -k df - M*F(m -n) = Chi - s
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

47. WLS
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

48. Variance of X+Y assuming dependence
Sampling distribution of sample means tend to be normal
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance(x) + Variance(Y) + 2*covariance(XY)
We accept a hypothesis that should have been rejected

49. Direction of OVB
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Based on a dataset
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

50. Weibul distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
When the sample size is large - the uncertainty about the value of the sample is very small
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha