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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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Match each statement with the correct term.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. Tractable
Easy to manipulate
Confidence level
Z = (Y - meany)/(stddev(y)/sqrt(n))
Average return across assets on a given day

2. Time series data
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for an individual asset
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

3. Confidence ellipse
Statement of the error or precision of an estimate
Confidence set for two coefficients - two dimensional analog for the confidence interval
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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

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

5. Variance of X+Y
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
Based on an equation - P(A) = # of A/total outcomes
Var(X) + Var(Y)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

6. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Yi = B0 + B1Xi + ui
SSR

7. Weibul distribution
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Price/return tends to run towards a long - run level

8. Homoskedastic only F - stat
E(mean) = mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
We reject a hypothesis that is actually true

9. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
(a^2)(variance(x)) + (b^2)(variance(y))
Attempts to sample along more important paths
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

10. WLS
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

11. Standard error
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
For n>30 - sample mean is approximately normal
Population denominator = n - Sample denominator = n - 1
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

12. ESS
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
Distribution with only two possible outcomes

13. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(XY) - E(X)E(Y)
We accept a hypothesis that should have been rejected
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

14. Mean reversion in variance
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance reverts to a long run level

15. Implications of homoscedasticity
Variance(x) + Variance(Y) + 2*covariance(XY)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

16. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(y)/n = variance of sample Y
Among all unbiased estimators - estimator with the smallest variance is efficient
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

17. Sample covariance
Model dependent - Options with the same underlying assets may trade at different volatilities
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Normal - Student's T - Chi - square - F distribution

18. Cholesky factorization (decomposition)
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

19. R^2
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

20. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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)
Variance(y)/n = variance of sample 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

21. Standard error for Monte Carlo replications
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance reverts to a long run level
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

22. Statistical (or empirical) model
Application of mathematical statistics to economic data to lend empirical support to models
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
Yi = B0 + B1Xi + ui
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

23. GARCH
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
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!)
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)
Sample mean will near the population mean as the sample size increases

24. P - value
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
P(Z>t)
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
Price/return tends to run towards a long - run level

25. Unconditional vs conditional distributions
Special type of pooled data in which the cross sectional unit is surveyed over time
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

26. Least squares estimator(m)
Summation((xi - mean)^k)/n
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Nonlinearity

27. Joint probability functions
Variance(y)/n = variance of sample Y
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Probability that the random variables take on certain values simultaneously
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

28. Simulating for VaR
We accept a hypothesis that should have been rejected
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

29. Kurtosis
Var(X) + Var(Y)
When the sample size is large - the uncertainty about the value of the sample is very small
Yi = B0 + B1Xi + ui
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

30. Lognormal
SSR
Variance reverts to a long run level
When one regressor is a perfect linear function of the other regressors
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

31. Central Limit Theorem
Based on a dataset
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
If variance of the conditional distribution of u(i) is not constant
For n>30 - sample mean is approximately normal

32. Two ways to calculate historical volatility
Mean of sampling distribution is the population mean
Normal - Student's T - Chi - square - F distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

33. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance reverts to a long run level
Population denominator = n - Sample denominator = n - 1
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

34. Consistent
Statement of the error or precision of an estimate
When the sample size is large - the uncertainty about the value of the sample is very small
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
More than one random variable

35. Expected future variance rate (t periods forward)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Returns over time for an individual asset
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

36. Standard variable for non - normal distributions
Mean of sampling distribution is the population mean
Normal - Student's T - Chi - square - F distribution
Rxy = Sxy/(Sx*Sy)
Z = (Y - meany)/(stddev(y)/sqrt(n))

37. Beta distribution
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Peaks over threshold - Collects dataset in excess of some threshold
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

38. Heteroskedastic
Population denominator = n - Sample denominator = n - 1
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
When one regressor is a perfect linear function of the other regressors
If variance of the conditional distribution of u(i) is not constant

39. Chi - squared distribution
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Has heavy tails

40. Pooled data
Does not depend on a prior event or information
When one regressor is a perfect linear function of the other regressors
Easy to manipulate
Returns over time for a combination of assets (combination of time series and cross - sectional data)

41. Critical z values
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Among all unbiased estimators - estimator with the smallest variance is efficient
95% = 1.65 99% = 2.33 For one - tailed tests
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

42. Direction of OVB
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Based on a dataset
95% = 1.65 99% = 2.33 For one - tailed tests
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

43. Four sampling distributions

44. Biggest (and only real) drawback of GARCH mode
Nonlinearity
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Attempts to sample along more important paths
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

45. Test for unbiasedness
(a^2)(variance(x)) + (b^2)(variance(y))
Regression can be non - linear in variables but must be linear in parameters
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E(mean) = mean

46. Economical(elegant)
Sample mean will near the population mean as the sample size increases
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Only requires two parameters = mean and variance
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

47. Variance(discrete)
Does not depend on a prior event or information
Summation((xi - mean)^k)/n
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

48. Variance of aX
(a^2)(variance(x)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
For n>30 - sample mean is approximately normal
Combine to form distribution with leptokurtosis (heavy tails)

49. Significance =1
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Confidence level
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Random walk (usually acceptable) - Constant volatility (unlikely)

50. Logistic distribution
Has heavy tails
Application of mathematical statistics to economic data to lend empirical support to models
P - value
Model dependent - Options with the same underlying assets may trade at different volatilities