QFC Quantitative Finance Statistical & Mathematical Analysis 3

Question 1

In the context of principal component analysis (PCA), what does the first principal component represent?

Accepted Answer

The direction of maximum variance in the data

Answer

The first principal component is the linear combination of original variables that captures the maximum variance in the dataset.

Question 2

What is the effect of increasing the number of Monte Carlo paths on the standard error of a price estimate?

Accepted Answer

Standard error decreases as 1/√N

Answer

Monte Carlo standard error equals σ/√N, so quadrupling paths halves the standard error — a slow convergence rate.

Question 3

Which of the following correctly defines the moment generating function (MGF) of a random variable X?

Accepted Answer

M_X(t) = E[e^tX]

Answer

The MGF is defined as M_X(t) = E[e^(tX)], and the k-th derivative evaluated at t=0 gives the k-th moment.

Question 4

In a linear regression with p predictors and n observations, how many degrees of freedom do the residuals have?

Accepted Answer

n - p - 1

Answer

Residual degrees of freedom equal n minus the number of estimated parameters (intercept + p slopes), giving n - p - 1.

Question 5

Which interpolation method is commonly used in building yield curves to ensure smooth forward rates?

Accepted Answer

Cubic spline interpolation

Answer

Cubic spline interpolation ensures continuity of the curve and its first two derivatives, producing smooth forward rate curves.

(QFC) Quantitative Finance Certification Practice Test