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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. Economical(elegant)
Only requires two parameters = mean and variance
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Var(X) + Var(Y)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

2. Direction of OVB
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
E(XY) - E(X)E(Y)

3. Critical z values
Among all unbiased estimators - estimator with the smallest variance is efficient
E(XY) - E(X)E(Y)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
95% = 1.65 99% = 2.33 For one - tailed tests

4. Joint probability functions
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Probability that the random variables take on certain values simultaneously
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

5. Poisson Distribution
For n>30 - sample mean is approximately normal
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Summation((xi - mean)^k)/n
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

6. P - value
95% = 1.65 99% = 2.33 For one - tailed tests
Normal - Student's T - Chi - square - F distribution
P(Z>t)
When the sample size is large - the uncertainty about the value of the sample is very small

7. Two assumptions of square root rule
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Sampling distribution of sample means tend to be normal
Among all unbiased estimators - estimator with the smallest variance is efficient
Random walk (usually acceptable) - Constant volatility (unlikely)

8. Difference between population and sample variance
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Concerned with a single random variable (ex. Roll of a die)
Among all unbiased estimators - estimator with the smallest variance is efficient
Population denominator = n - Sample denominator = n - 1

9. Two requirements of OVB
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
When one regressor is a perfect linear function of the other regressors

10. Mean reversion
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
E(mean) = mean

11. Sample mean
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
Var(X) + Var(Y)
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
Expected value of the sample mean is the population mean

12. Unconditional vs conditional distributions
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Population denominator = n - Sample denominator = n - 1
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

13. Binomial distribution equations for mean variance and std dev
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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 = np - Variance = npq - Std dev = sqrt(npq)

14. Empirical frequency
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Based on a dataset
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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

15. Variance of X+Y assuming dependence
Application of mathematical statistics to economic data to lend empirical support to models
Use historical simulation approach but use the EWMA weighting system
Variance(x) + Variance(Y) + 2*covariance(XY)
Combine to form distribution with leptokurtosis (heavy tails)

16. Standard normal distribution
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
Transformed to a unit variable - Mean = 0 Variance = 1
Average return across assets on a given day
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

17. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

18. Type I error
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
We reject a hypothesis that is actually true
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
(a^2)(variance(x)

19. Maximum likelihood method
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Confidence set for two coefficients - two dimensional analog for the confidence interval
E(XY) - E(X)E(Y)
Choose parameters that maximize the likelihood of what observations occurring

20. Lognormal
Independently and Identically Distributed
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

21. Multivariate Density Estimation (MDE)
Summation((xi - mean)^k)/n
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

22. Standard variable for non - normal distributions
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Z = (Y - meany)/(stddev(y)/sqrt(n))
For n>30 - sample mean is approximately normal

23. Normal distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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

24. Perfect multicollinearity
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
When one regressor is a perfect linear function of the other regressors
P(Z>t)
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

25. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Normal - Student's T - Chi - square - F distribution
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

26. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Normal - Student's T - Chi - square - F distribution
Variance reverts to a long run level
(a^2)(variance(x)) + (b^2)(variance(y))

27. Reliability
Statement of the error or precision of an estimate
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Rxy = Sxy/(Sx*Sy)

28. Covariance
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
E(XY) - E(X)E(Y)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

29. Single variable (univariate) probability
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Concerned with a single random variable (ex. Roll of a die)
Variance(y)/n = variance of sample Y

30. i.i.d.
Average return across assets on a given day
Independently and Identically Distributed
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance(x)

31. Block maxima
95% = 1.65 99% = 2.33 For one - tailed tests
If variance of the conditional distribution of u(i) is not constant
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

32. Unstable return distribution
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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

33. Standard error for Monte Carlo replications
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)
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
Confidence level

34. Variance of sampling distribution of means when n<N
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

35. Conditional probability functions
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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
Independently and Identically Distributed
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

36. Stochastic error term
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Contains variables not explicit in model - Accounts for randomness
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

37. 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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

38. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Mean = np - Variance = npq - Std dev = sqrt(npq)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

39. Variance of X+b
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(x)

40. Gamma distribution
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Concerned with a single random variable (ex. Roll of a die)

41. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
When one regressor is a perfect linear function of the other regressors
Application of mathematical statistics to economic data to lend empirical support to models
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

42. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Distribution with only two possible outcomes
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

43. Variance - covariance approach for VaR of a portfolio
Returns over time for an individual asset
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Expected value of the sample mean is the population mean

44. Adjusted R^2
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Yi = B0 + B1Xi + ui
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

45. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Concerned with a single random variable (ex. Roll of a die)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

46. Chi - squared distribution
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance(x)
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
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

47. Central Limit Theorem
Price/return tends to run towards a long - run level
Use historical simulation approach but use the EWMA weighting system
For n>30 - sample mean is approximately normal
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

48. Historical std dev
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

49. Confidence interval (from t)
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
Average return across assets on a given day
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

50. Variance of weighted scheme
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
When the sample size is large - the uncertainty about the value of the sample is very small
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Yi = B0 + B1Xi + ui