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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. Simplified standard (un - weighted) variance
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance = (1/m) summation(u<n - i>^2)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance(y)/n = variance of sample Y

2. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Rxy = Sxy/(Sx*Sy)

3. Tractable
Easy to manipulate
Among all unbiased estimators - estimator with the smallest variance is efficient
SSR
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

4. Central Limit Theorem
Probability that the random variables take on certain values simultaneously
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)
For n>30 - sample mean is approximately normal
Distribution with only two possible outcomes

5. Discrete random variable
Regression can be non - linear in variables but must be linear in parameters
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

6. Standard error
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)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

7. Variance of X+b
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Regression can be non - linear in variables but must be linear in parameters
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(x)

8. Logistic distribution
If variance of the conditional distribution of u(i) is not constant
Rxy = Sxy/(Sx*Sy)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Has heavy tails

9. GPD
Variance(X) + Variance(Y) - 2*covariance(XY)
When one regressor is a perfect linear function of the other regressors
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

10. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Random walk (usually acceptable) - Constant volatility (unlikely)
Confidence level
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

11. Discrete representation of the GBM
When the sample size is large - the uncertainty about the value of the sample is very small
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
For n>30 - sample mean is approximately normal

12. Kurtosis
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
SSR
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

13. Implications of homoscedasticity
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Returns over time for an individual asset
When one regressor is a perfect linear function of the other regressors
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

14. Priori (classical) probability
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
Distribution with only two possible outcomes
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Based on an equation - P(A) = # of A/total outcomes

15. i.i.d.
Independently and Identically Distributed
Model dependent - Options with the same underlying assets may trade at different volatilities
Distribution with only two possible outcomes
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

16. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Among all unbiased estimators - estimator with the smallest variance is efficient
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
P(Z>t)

17. Type II Error
For n>30 - sample mean is approximately normal
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Rxy = Sxy/(Sx*Sy)
We accept a hypothesis that should have been rejected

18. 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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

19. Chi - squared distribution
Variance = (1/m) summation(u<n - i>^2)
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Var(X) + Var(Y)

20. Beta distribution
Based on a dataset
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Statement of the error or precision of an estimate

21. Importance sampling technique
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
Attempts to sample along more important paths

22. Continuous representation of the GBM
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
P(X=x - Y=y) = P(X=x) * P(Y=y)

23. Variance of X - Y assuming dependence
i = ln(Si/Si - 1)
Yi = B0 + B1Xi + ui
Variance(X) + Variance(Y) - 2*covariance(XY)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

24. Confidence interval (from t)
Peaks over threshold - Collects dataset in excess of some threshold
Sample mean +/ - t*(stddev(s)/sqrt(n))
Has heavy tails
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

25. Standard normal distribution
Contains variables not explicit in model - Accounts for randomness
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Population denominator = n - Sample denominator = n - 1
Transformed to a unit variable - Mean = 0 Variance = 1

26. Sample correlation
Rxy = Sxy/(Sx*Sy)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

27. Regime - switching volatility model
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Distribution with only two possible outcomes
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

28. Single variable (univariate) probability
Variance = (1/m) summation(u<n - i>^2)
Concerned with a single random variable (ex. Roll of a die)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

29. Time series data
Sample mean will near the population mean as the sample size increases
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Returns over time for an individual asset
Least absolute deviations estimator - used when extreme outliers are not uncommon

30. SER
Variance(x) + Variance(Y) + 2*covariance(XY)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Population denominator = n - Sample denominator = n - 1
Variance(X) + Variance(Y) - 2*covariance(XY)

31. Test for statistical independence
Attempts to sample along more important paths
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance(x)
Variance reverts to a long run level

32. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Mean = np - Variance = npq - Std dev = sqrt(npq)
Low Frequency - High Severity events
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

33. Variance of sample mean
Confidence level
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance(y)/n = variance of sample Y
Transformed to a unit variable - Mean = 0 Variance = 1

34. Mean reversion in variance
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance reverts to a long run level
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

35. Inverse transform method
Price/return tends to run towards a long - run level
Statement of the error or precision of an estimate
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

36. Exact significance level
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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
P - value

37. BLUE
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Sample mean will near the population mean as the sample size increases

38. Persistence
Random walk (usually acceptable) - Constant volatility (unlikely)
Regression can be non - linear in variables but must be linear in parameters
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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

39. Continuous random variable
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Price/return tends to run towards a long - run level

40. Potential reasons for fat tails in return distributions
Regression can be non - linear in variables but must be linear in parameters
Population denominator = n - Sample denominator = n - 1
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

41. Shortcomings of implied volatility
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Model dependent - Options with the same underlying assets may trade at different volatilities
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

42. Monte Carlo Simulations
Variance = (1/m) summation(u<n - i>^2)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Confidence level
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

43. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
SSR
Sample mean +/ - t*(stddev(s)/sqrt(n))

44. Confidence ellipse
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Confidence set for two coefficients - two dimensional analog for the confidence interval
For n>30 - sample mean is approximately normal
Only requires two parameters = mean and variance

45. R^2
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sampling distribution of sample means tend to be normal
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

46. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Special type of pooled data in which the cross sectional unit is surveyed over time

47. Homoskedastic
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Statement of the error or precision of an estimate
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

48. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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

49. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
For n>30 - sample mean is approximately normal
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Easy to manipulate

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