VRM 1 Flashcards
What is the mean-variance framework?
A framework that describes the trade-off between risk and return, using mean and standard deviation of returns.
What does the efficient frontier represent?
The set of optimal portfolios that offer the highest expected return for a defined level of risk.
What are the limitations of the mean-variance framework?
It assumes normal distribution of returns, which does not account for fatter tails in financial variables.
Define Value-at-Risk (πππ ).
A measure of risk that estimates the maximum loss not expected to be exceeded over a specified time period at a given confidence level.
What are the two parameters that determine πππ ?
- Time horizon
- Confidence level
What is expected shortfall (πΈπ)?
A risk measure that quantifies the expected loss on days when there is a loss exceeding the Value-at-Risk.
How does expected shortfall (πΈπ) compare to πππ ?
πΈπ has more desirable theoretical properties and is a coherent risk measure, while πππ is not.
What is a coherent risk measure?
A risk measure that satisfies properties such as monotonicity, subadditivity, positive homogeneity, and translation invariance.
Why is πππ not considered a coherent risk measure?
Because it does not satisfy subadditivity, meaning the risk of a combined portfolio can exceed the sum of individual risks.
What is the standard deviation of returns used for?
To measure the risk associated with an investmentβs return variability.
What is the expected return for a risk-free asset?
The fixed return over a time period with a standard deviation of zero.
What is the formula for portfolio expected return (πp)?
πp = π1π€1 + π2π€2
What does the correlation coefficient (π1,2) indicate?
The degree to which the returns of two assets move in relation to each other.
What is the formula for portfolio standard deviation (πp)?
πP = β(π€1Β²π1Β² + π€2Β²π2Β² + 2π1,2π€1π€2π1π2)
What is the efficient frontier?
The curve representing the set of portfolios that offers the highest expected return for a given level of risk.
What happens when a risk-free investment is introduced?
The efficient frontier becomes a straight line, indicating a linear trade-off between expected return and risk.
What is the market portfolio?
The portfolio consisting of all investments in the market, weighted by their proportional amounts.
What assumptions are made in the efficient frontier analysis?
- All investors have the same expectations about returns
- Investors care only about mean and standard deviation
- Borrowing at the risk-free rate is possible
What characterizes the normal distribution in finance?
It is a convenient model for returns, requiring only mean and standard deviation for calculations.
How do actual financial variables differ from normal distributions?
They often exhibit fatter tails, meaning extreme events occur more frequently than predicted by normal distributions.
What is the significance of a 99% confidence level in πππ ?
It represents the maximum loss expected not to be exceeded with 99% certainty over a specified time frame.
What are some common types of probability distributions used in calculating πππ ?
- Normal distribution
- Uniform distribution
- Discrete distribution
How is πππ calculated for a normal distribution?
Using the NORM.INV function in Excel with the required percentile, mean loss, and standard deviation.
In a uniform distribution, how is πππ determined?
By identifying the range of equally likely outcomes and calculating the loss at the specified confidence level.
What is the approach for calculating πππ in a discrete distribution?
By analyzing the cumulative probability of different outcomes to determine loss thresholds.
What is a limitation of the πππ measure?
It does not account for how bad losses might be when they exceed the πππ level.
In the context of πππ , what does a 1% chance indicate?
The probability of losses exceeding the defined πππ level.
What is the Expected Shortfall (πΈπ)?
The expected loss after the πππ threshold has been breached.
How is the Expected Shortfall calculated?
As the probability weighted average of tail losses after the corresponding πππ level.
What is the formula for Expected Shortfall when losses are normally distributed?
Where π is the confidence level, and π is the point in the standard normal distribution that has a probability π% of being exceeded.
What are the four properties of a coherent risk measure?
- Monotonicity
- Translation Invariance
- Homogeneity
- Subadditivity
These properties ensure consistency in risk assessment.
Which properties does πππ satisfy?
- Monotonicity
- Translation Invariance
- Homogeneity
πππ may not satisfy the subadditivity property.
What type of risk measure is expected shortfall classified as?
A coherent risk measure.
What is a spectral risk measure?
A type of coherent risk measure where the weights assigned to percentiles increase reflecting risk aversion.
How are weights in a spectral risk measure calculated?
Proportional to:
Where π is the percentile (in decimal form) and Ξ³ is a constant reflecting risk aversion.
