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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. Test for statistical independence
Normal - Student's T - Chi - square - F distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

2. Variance - covariance approach for VaR of a portfolio
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Rxy = Sxy/(Sx*Sy)
Price/return tends to run towards a long - run level
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

3. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Var(X) + Var(Y)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Nonlinearity

4. Empirical frequency
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Based on a dataset
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
Sample mean will near the population mean as the sample size increases

5. Efficiency
If variance of the conditional distribution of u(i) is not constant
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
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
Among all unbiased estimators - estimator with the smallest variance is efficient

6. What does the OLS minimize?
Among all unbiased estimators - estimator with the smallest variance is efficient
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
SSR

7. Discrete representation of the GBM
Has heavy tails
Does not depend on a prior event or information
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

8. Standard normal distribution
Only requires two parameters = mean and variance
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Transformed to a unit variable - Mean = 0 Variance = 1
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

9. Unconditional vs conditional distributions
Price/return tends to run towards a long - run level
When one regressor is a perfect linear function of the other regressors
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

10. ESS
Variance(x) + Variance(Y) + 2*covariance(XY)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Nonlinearity

11. Poisson distribution equations for mean variance and std deviation
Does not depend on a prior event or information
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

12. Square root rule
SSR
Summation((xi - mean)^k)/n
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(y)/n = variance of sample Y

13. Implications of homoscedasticity
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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. Antithetic variable technique
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance(X) + Variance(Y) - 2*covariance(XY)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

15. Central Limit Theorem
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
For n>30 - sample mean is approximately normal
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

16. Hybrid method for conditional volatility
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Confidence set for two coefficients - two dimensional analog for the confidence interval
Use historical simulation approach but use the EWMA weighting system

17. Direction of OVB
Use historical simulation approach but use the EWMA weighting system
Has heavy tails
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

18. LFHS
Low Frequency - High Severity events
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[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

19. Lognormal
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Random walk (usually acceptable) - Constant volatility (unlikely)

20. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

21. Beta distribution
Yi = B0 + B1Xi + ui
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

22. Least squares estimator(m)
Only requires two parameters = mean and variance
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
More than one random variable
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

23. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
SSR

24. Exact significance level
Concerned with a single random variable (ex. Roll of a die)
i = ln(Si/Si - 1)
P - value
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. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Expected value of the sample mean is the population mean

26. Poisson Distribution
For n>30 - sample mean is approximately normal
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

27. Two drawbacks of moving average series
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Attempts to sample along more important paths
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

28. Sample variance
P(X=x - Y=y) = P(X=x) * P(Y=y)
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

29. Confidence ellipse
Contains variables not explicit in model - Accounts for randomness
Confidence set for two coefficients - two dimensional analog for the confidence interval
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

30. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

31. 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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Nonlinearity
(a^2)(variance(x)) + (b^2)(variance(y))

32. Sample mean
Mean of sampling distribution is the population mean
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Expected value of the sample mean is the population mean
Confidence set for two coefficients - two dimensional analog for the confidence interval

33. Potential reasons for fat tails in return distributions
Variance(y)/n = variance of sample Y
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Yi = B0 + B1Xi + ui

34. T distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Rxy = Sxy/(Sx*Sy)
Among all unbiased estimators - estimator with the smallest variance is efficient

35. Variance of sample mean
Variance(y)/n = variance of sample Y
SSR
If variance of the conditional distribution of u(i) is not constant
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

36. Mean reversion in asset dynamics
Rxy = Sxy/(Sx*Sy)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Returns over time for an individual asset
Price/return tends to run towards a long - run level

37. Economical(elegant)
Only requires two parameters = mean and variance
Does not depend on a prior event or information
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(X) + Variance(Y) - 2*covariance(XY)

38. Variance of aX
(a^2)(variance(x)
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
Summation((xi - mean)^k)/n

39. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

40. Hazard rate of exponentially distributed random variable
Among all unbiased estimators - estimator with the smallest variance is efficient
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Independently and Identically Distributed
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

41. Conditional probability functions
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^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)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

42. Binomial distribution equations for mean variance and std dev
Based on a dataset
Among all unbiased estimators - estimator with the smallest variance is efficient
Mean = np - Variance = npq - Std dev = sqrt(npq)
Probability that the random variables take on certain values simultaneously

43. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Peaks over threshold - Collects dataset in excess of some threshold
Population denominator = n - Sample denominator = n - 1

44. Kurtosis
Only requires two parameters = mean and variance
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Based on a dataset

45. Cross - sectional
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Average return across assets on a given day
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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)

46. Two requirements of OVB
When the sample size is large - the uncertainty about the value of the sample is very small
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Summation((xi - mean)^k)/n

47. Overall F - statistic
Normal - Student's T - Chi - square - F distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

48. Joint probability functions
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sampling distribution of sample means tend to be normal
Probability that the random variables take on certain values simultaneously
Confidence level

49. P - value
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
P(Z>t)
Transformed to a unit variable - Mean = 0 Variance = 1
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

50. Weibul distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Expected value of the sample mean is the population mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Statement of the error or precision of an estimate