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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

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. Significance =1
Confidence level
Statement of the error or precision of an estimate
Application of mathematical statistics to economic data to lend empirical support to models
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

2. Test for statistical independence
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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)
P(X=x - Y=y) = P(X=x) * P(Y=y)
E(XY) - E(X)E(Y)

3. Joint probability functions
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Probability that the random variables take on certain values simultaneously
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
Mean = np - Variance = npq - Std dev = sqrt(npq)

4. R^2
Based on an equation - P(A) = # of A/total outcomes
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
(a^2)(variance(x)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

5. Kurtosis
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
E(mean) = mean
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

6. Mean reversion
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Does not depend on a prior event or information
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

7. Confidence interval for sample mean
If variance of the conditional distribution of u(i) is not constant
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Average return across assets on a given day

8. Variance of X+Y
Var(X) + Var(Y)
Sampling distribution of sample means tend to be normal
Combine to form distribution with leptokurtosis (heavy tails)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

9. Maximum likelihood method
Contains variables not explicit in model - Accounts for randomness
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Choose parameters that maximize the likelihood of what observations occurring

10. Central Limit Theorem
Var(X) + Var(Y)
For n>30 - sample mean is approximately normal
Returns over time for an individual asset
Variance = (1/m) summation(u<n - i>^2)

11. Adjusted R^2
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

12. Mean reversion in variance
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance reverts to a long run level

13. Type I error
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
We reject a hypothesis that is actually true
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

14. Skewness
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

15. Economical(elegant)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Random walk (usually acceptable) - Constant volatility (unlikely)
Only requires two parameters = mean and variance
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

16. Discrete random variable
Summation((xi - mean)^k)/n
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
We accept a hypothesis that should have been rejected

17. Potential reasons for fat tails in return distributions
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
i = ln(Si/Si - 1)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Expected value of the sample mean is the population mean

18. Standard error for Monte Carlo replications
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Only requires two parameters = mean and variance

19. Variance of X+b
Sample mean +/ - t*(stddev(s)/sqrt(n))
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance(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

20. Continuously compounded return equation
Expected value of the sample mean is the population mean
Application of mathematical statistics to economic data to lend empirical support to models
i = ln(Si/Si - 1)
When one regressor is a perfect linear function of the other regressors

21. Homoskedastic
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Combine to form distribution with leptokurtosis (heavy tails)
Concerned with a single random variable (ex. Roll of a die)

22. Continuous representation of the GBM
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Regression can be non - linear in variables but must be linear in parameters
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)

23. Panel data (longitudinal or micropanel)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Special type of pooled data in which the cross sectional unit is surveyed over time
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)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

24. Hazard rate of exponentially distributed random variable
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Based on an equation - P(A) = # of A/total outcomes
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

25. Lognormal
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
95% = 1.65 99% = 2.33 For one - tailed tests
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

26. Unbiased
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Mean of sampling distribution is the population mean

27. Mean(expected value)
Summation((xi - mean)^k)/n
Random walk (usually acceptable) - Constant volatility (unlikely)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

28. Historical std dev
Price/return tends to run towards a long - run level
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

29. Multivariate probability
Sampling distribution of sample means tend to be normal
Attempts to sample along more important paths
More than one random variable
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

30. Empirical frequency
Based on an equation - P(A) = # of A/total outcomes
More than one random variable
Based on a dataset
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

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

32. Poisson distribution equations for mean variance and std deviation
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

33. Beta distribution
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

34. Marginal unconditional probability function
Does not depend on a prior event or information
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Transformed to a unit variable - Mean = 0 Variance = 1

35. Extreme Value Theory
Special type of pooled data in which the cross sectional unit is surveyed over time
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
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. Difference between population and sample variance
(a^2)(variance(x)) + (b^2)(variance(y))
Population denominator = n - Sample denominator = n - 1
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

37. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
We reject a hypothesis that is actually true
Variance reverts to a long run level
Confidence level

38. Normal distribution
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
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

39. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
SSR
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

40. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Model dependent - Options with the same underlying assets may trade at different volatilities
Expected value of the sample mean is the population mean
Variance(y)/n = variance of sample Y

41. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Distribution with only two possible outcomes
More than one random variable
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

42. Regime - switching volatility model
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
P - value
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Average return across assets on a given day

43. Central Limit Theorem(CLT)
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
Regression can be non - linear in variables but must be linear in parameters
Sampling distribution of sample means tend to be normal

44. 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
Population denominator = n - Sample denominator = n - 1
Variance(x)
Among all unbiased estimators - estimator with the smallest variance is efficient

45. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Var(X) + Var(Y)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Mean of sampling distribution is the population mean

46. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Has heavy tails
Sample mean +/ - t*(stddev(s)/sqrt(n))

47. Importance sampling technique
Attempts to sample along more important paths
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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)

48. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
More than one random variable
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

49. Unstable return distribution
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
If variance of the conditional distribution of u(i) is not constant

50. Persistence
Application of mathematical statistics to economic data to lend empirical support to models
Has heavy tails
E(XY) - E(X)E(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