8002 PRM Certification - Exam II: Mathematical Foundations of Risk Measurement Questions and Answers

There are two portfolios with no overlapping of stocks or bonds. Portfolio 1 has 6 stocks and 6 bonds. Portfolio 2 has 4 stocks and 8 bonds. If we randomly select one stock, what is the probability that it came from Portfolio1?

What is the sum of the first 20 terms of this sequence: 3, 5, 9, 17, 33, 65,…?

Evaluate the derivative of ln(1+ x2) at the point x = 1

Which of the following statements are true about Maximum Likelihood Estimation?

(i) MLE can be applied even if the error terms are not i.i.d. normal.

(ii) MLE involves integrating a likelihood function or a log-likelihood function.

(iii) MLE yields parameter estimates that are consistent.

In a multiple linear regression, the significance of R2 can be tested using which distribution?

If a random variable X has a normal distribution with mean zero and variance 4, approximately what proportion of realizations of X should lie between -4 and +4?

You are given the following values of a quadratic function f(x): f(0)=0, f(1)=-2, f(2)=-5. On the basis of these data, the derivative f'(0) is …

Let a, b and c be real numbers. Which of the following statements is true?

An underlying asset price is at 100, its annual volatility is 25% and the risk free interest rate is 5%. A European put option has a strike of 105 and a maturity of 90 days. Its Black-Scholes price is 7.11. The options sensitivities are: delta = -0.59; gamma = 0.03; vega = 19.29. Find the delta-gamma approximation to the new option price when the underlying asset price changes to 105

Maximum likelihood estimation is a method for:

Let A be a square matrix and denote its determinant by x. Then the determinant of A transposed is:

The fundamental theorem of analysis establishes a relation between

What is the probability of tossing a coin and getting exactly 2 heads out of 5 throws?

For each of the following functions, indicate whether its graph is concave or convex:

Y = 7x2 + 3x + 9

Y = 6 ln(3x)

Y = exp(-4x)

Calculate the determinant of the following matrix:

The Lagrangian of a constrained optimisation problem is given by L(x,y,λ) = 16x+8x2+4y-λ(4x+y-20), where λ is the Lagrange multiplier. What is the solution for x and y?