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Mathematical Finance

Description: This quiz covers the fundamental concepts and techniques used in Mathematical Finance.
Number of Questions: 14
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Tags: mathematical finance financial mathematics quantitative finance
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What is the formula for calculating the present value of a future cash flow?

  1. PV = FV / (1 + r)^n

  2. PV = FV * (1 + r)^n

  3. PV = FV / (1 - r)^n

  4. PV = FV * (1 - r)^n

Show answer
Correct Option: A
Explanation:

The present value (PV) of a future cash flow (FV) is calculated using the formula PV = FV / (1 + r)^n, where r is the interest rate and n is the number of periods.

What is the relationship between the price of a bond and its yield to maturity?

  1. As the price of a bond increases, the yield to maturity decreases.

  2. As the price of a bond decreases, the yield to maturity increases.

  3. The price of a bond and its yield to maturity are not related.

  4. The relationship between the price of a bond and its yield to maturity is unpredictable.

Show answer
Correct Option: A
Explanation:

There is an inverse relationship between the price of a bond and its yield to maturity. As the price of a bond increases, the yield to maturity decreases, and vice versa.

What is the formula for calculating the duration of a bond?

  1. Duration = (PV of cash flows / Price of bond) / (1 + r)

  2. Duration = (PV of cash flows / Price of bond) * (1 + r)

  3. Duration = (Price of bond / PV of cash flows) / (1 + r)

  4. Duration = (Price of bond / PV of cash flows) * (1 + r)

Show answer
Correct Option: A
Explanation:

The duration of a bond is calculated using the formula Duration = (PV of cash flows / Price of bond) / (1 + r), where r is the interest rate.

What is the Black-Scholes model used for?

  1. Pricing options

  2. Pricing stocks

  3. Pricing bonds

  4. Pricing commodities

Show answer
Correct Option: A
Explanation:

The Black-Scholes model is a mathematical model used to price options, which are financial instruments that give the holder the right to buy or sell an asset at a specified price on or before a specified date.

What is the formula for calculating the expected return of a portfolio?

  1. Expected return = (Weight of asset 1 * Expected return of asset 1) + (Weight of asset 2 * Expected return of asset 2) + ...

  2. Expected return = (Weight of asset 1 * Expected return of asset 1) - (Weight of asset 2 * Expected return of asset 2) - ...

  3. Expected return = (Weight of asset 1 / Expected return of asset 1) + (Weight of asset 2 / Expected return of asset 2) + ...

  4. Expected return = (Weight of asset 1 / Expected return of asset 1) - (Weight of asset 2 / Expected return of asset 2) - ...

Show answer
Correct Option: A
Explanation:

The expected return of a portfolio is calculated by multiplying the weight of each asset in the portfolio by the expected return of that asset and then summing the results.

What is the Sharpe ratio used for?

  1. Measuring the risk-adjusted return of an investment

  2. Measuring the volatility of an investment

  3. Measuring the correlation between two investments

  4. Measuring the beta of an investment

Show answer
Correct Option: A
Explanation:

The Sharpe ratio is a measure of the risk-adjusted return of an investment. It is calculated by dividing the excess return of an investment (the return above the risk-free rate) by the standard deviation of the investment's returns.

What is the formula for calculating the beta of an asset?

  1. Beta = Covariance(Asset returns, Market returns) / Variance(Market returns)

  2. Beta = Correlation(Asset returns, Market returns) / Variance(Market returns)

  3. Beta = Covariance(Asset returns, Market returns) / Standard deviation(Market returns)

  4. Beta = Correlation(Asset returns, Market returns) / Standard deviation(Market returns)

Show answer
Correct Option: A
Explanation:

The beta of an asset is a measure of its systematic risk, which is the risk that cannot be diversified away. It is calculated by dividing the covariance of the asset's returns with the market returns by the variance of the market returns.

What is the formula for calculating the value at risk (VaR) of a portfolio?

  1. VaR = (Expected return of portfolio - Minimum return of portfolio) * (1 + Confidence level)

  2. VaR = (Expected return of portfolio + Minimum return of portfolio) * (1 + Confidence level)

  3. VaR = (Expected return of portfolio - Minimum return of portfolio) / (1 + Confidence level)

  4. VaR = (Expected return of portfolio + Minimum return of portfolio) / (1 + Confidence level)

Show answer
Correct Option: A
Explanation:

The value at risk (VaR) of a portfolio is a measure of the maximum possible loss in the value of the portfolio over a given time period at a given confidence level. It is calculated by multiplying the difference between the expected return of the portfolio and the minimum return of the portfolio by one plus the confidence level.

What is the formula for calculating the expected shortfall (ES) of a portfolio?

  1. ES = (Expected return of portfolio - Minimum return of portfolio) / (1 - Confidence level)

  2. ES = (Expected return of portfolio + Minimum return of portfolio) / (1 - Confidence level)

  3. ES = (Expected return of portfolio - Minimum return of portfolio) * (1 - Confidence level)

  4. ES = (Expected return of portfolio + Minimum return of portfolio) * (1 - Confidence level)

Show answer
Correct Option: A
Explanation:

The expected shortfall (ES) of a portfolio is a measure of the average loss in the value of the portfolio that is expected to occur in the worst cases within a given confidence level. It is calculated by dividing the difference between the expected return of the portfolio and the minimum return of the portfolio by one minus the confidence level.

What is the formula for calculating the optimal portfolio weights using the mean-variance optimization approach?

  1. w = (Σ^-1 μ) / (μ^T Σ^-1 μ)

  2. w = (Σ μ) / (μ^T Σ μ)

  3. w = (Σ^-1 μ) / (μ^T Σ μ)

  4. w = (Σ μ) / (μ^T Σ^-1 μ)

Show answer
Correct Option: A
Explanation:

The optimal portfolio weights using the mean-variance optimization approach are calculated using the formula w = (Σ^-1 μ) / (μ^T Σ^-1 μ), where Σ is the covariance matrix of the asset returns, μ is the vector of expected asset returns, and w is the vector of optimal portfolio weights.

What is the formula for calculating the Sharpe ratio of a portfolio?

  1. Sharpe ratio = (Expected return of portfolio - Risk-free rate) / Standard deviation of portfolio returns

  2. Sharpe ratio = (Expected return of portfolio + Risk-free rate) / Standard deviation of portfolio returns

  3. Sharpe ratio = (Expected return of portfolio - Risk-free rate) / Variance of portfolio returns

  4. Sharpe ratio = (Expected return of portfolio + Risk-free rate) / Variance of portfolio returns

Show answer
Correct Option: A
Explanation:

The Sharpe ratio of a portfolio is a measure of its risk-adjusted return. It is calculated by dividing the difference between the expected return of the portfolio and the risk-free rate by the standard deviation of the portfolio returns.

What is the formula for calculating the Jensen's alpha of a portfolio?

  1. Jensen's alpha = Expected return of portfolio - (Risk-free rate + Beta of portfolio * Market risk premium)

  2. Jensen's alpha = Expected return of portfolio + (Risk-free rate + Beta of portfolio * Market risk premium)

  3. Jensen's alpha = Expected return of portfolio - (Risk-free rate - Beta of portfolio * Market risk premium)

  4. Jensen's alpha = Expected return of portfolio + (Risk-free rate - Beta of portfolio * Market risk premium)

Show answer
Correct Option: A
Explanation:

Jensen's alpha of a portfolio is a measure of its excess return over and above what would be expected given its risk. It is calculated by subtracting the risk-free rate plus the product of the portfolio's beta and the market risk premium from the expected return of the portfolio.

What is the formula for calculating the Treynor ratio of a portfolio?

  1. Treynor ratio = Excess return of portfolio / Beta of portfolio

  2. Treynor ratio = Expected return of portfolio / Beta of portfolio

  3. Treynor ratio = Excess return of portfolio / Standard deviation of portfolio returns

  4. Treynor ratio = Expected return of portfolio / Standard deviation of portfolio returns

Show answer
Correct Option: A
Explanation:

The Treynor ratio of a portfolio is a measure of its risk-adjusted return per unit of systematic risk. It is calculated by dividing the excess return of the portfolio (the return above the risk-free rate) by the beta of the portfolio.

What is the formula for calculating the Information ratio of a portfolio?

  1. Information ratio = (Expected return of portfolio - Benchmark return) / Standard deviation of portfolio returns

  2. Information ratio = (Expected return of portfolio + Benchmark return) / Standard deviation of portfolio returns

  3. Information ratio = (Expected return of portfolio - Benchmark return) / Variance of portfolio returns

  4. Information ratio = (Expected return of portfolio + Benchmark return) / Variance of portfolio returns

Show answer
Correct Option: A
Explanation:

The Information ratio of a portfolio is a measure of its excess return over and above a benchmark return per unit of risk. It is calculated by dividing the difference between the expected return of the portfolio and the benchmark return by the standard deviation of the portfolio returns.

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