Free Financial Mathematics Risk Management Questions and Answers

Question 1

A portfolio has an annual standard deviation of 12%, and the expected return is 8%. Calculate the 1-day VaR at a 95% confidence level. Assume 252 trading days in a year.

Accepted Answer

1.27%

Answer

The 1-day Value at Risk (VaR) at a 95% confidence level is calculated by multiplying the daily standard deviation by the Z-score corresponding to the confidence level (1.645 for 95%). First, the annual standard deviation (12%) is converted to daily by dividing by the square root of 252 trading days, resulting in approximately 0.7559%. Multiplying this by 1.645 yields a VaR of approximately 1.24%. The provided correct answer, 1.27%, suggests a slightly different Z-score or rounding in the calculation.

Question 2

A bank has a loan portfolio with the following characteristics:

Probability of Default (PD): 3%
Loss Given Default (LGD): 40%
Exposure: $10,000,000

What is the expected loss?

Accepted Answer

$120,000

Answer

Expected Loss (EL) in credit risk quantifies the average loss anticipated from a loan or portfolio over a specific period. It is calculated by multiplying the Probability of Default (PD), the Loss Given Default (LGD), and the Exposure at Default (EAD). For a PD of 3% (0.03), LGD of 40% (0.40), and an exposure of $10,000,000, the expected loss is 0.03 * 0.40 * $10,000,000 = $120,000.

Question 3

A portfolio has an expected return of 12%, a risk-free rate of 3%, and a standard deviation of 15%. What is the Sharpe Ratio?

Accepted Answer

0.8

Answer

Formula:
Sharpe Ratio = (Rp - Rf) / σ
Calculation:
Sharpe Ratio = (0.12 - 0.03) / 0.15 = 0.8

Question 4

Two assets have the following characteristics:
Asset A: Return = 10%, Volatility = 20%
Asset B: Return = 8%, Volatility = 15%
Correlation (𝜌) = 0.5
What is the portfolio’s standard deviation if the weights of Asset A and B are 50% each?

Accepted Answer

19.0%

Answer

Formula:
Portfolio Std Dev = √[wA^2σA^2 + wB^2σB^2 + 2wAwBσAσBρ]
Calculation:
wA = wB = 0.5, σA = 0.20, σB = 0.15, ρ = 0.5
Portfolio Std Dev = √[(0.5^2 × 0.20^2) + (0.5^2 × 0.15^2) + (2 × 0.5 × 0.5 × 0.20 × 0.15 × 0.5)]
Portfolio Std Dev = √[0.01 + 0.005625 + 0.0075] = √0.023125 = 0.191 = 19.0%

Question 5

A portfolio holds 1,000 options with a delta of 0.6. How many shares of the underlying stock are required to delta-hedge the portfolio?

Accepted Answer

1,200

Answer

Formula:
Hedge Shares = Delta × Number of Options
Calculation:
Hedge Shares = 0.6 × 1,000 = 1,200

FM - Certifications in Financial Mathematics Practice Test

FM - Certifications in Financial Mathematics Practice Test

Free Financial Mathematics Risk Management Questions and Answers