Value at Risk (VaR) Fundamentals

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Questions and Answers

What is the primary purpose of Value at Risk (VaR)?

To estimate the maximum potential loss in a given time period. (correct)

To calculate the risk-free rate of return.

To measure the average return of a portfolio.

To predict the exact return of a portfolio in the future.

When portfolio returns are normally distributed, what two factors determine the VaR?

Portfolio size and volatility.

Portfolio size and average daily return.

Mean and standard deviation of the return distribution. (correct)

Risk-free rate and volatility.

In the historical method of calculating VaR, what information is crucial for determining the 1% VaR?

The standard deviation of daily returns over the past 500 days.

The average return over the past 500 days.

The lowest daily return over the past 500 days. (correct)

The highest daily return over the past 500 days.

What is the VaR using the historical method, given a portfolio value of €600 million and the lowest extreme daily return of -0.13%?

€780,000 (D) Signup and view all the answers

What is the Z-score corresponding to a 99% confidence limit?

2.33 (B) Signup and view all the answers

In the parametric method, what is the formula for calculating VaR in percentage terms?

VaR (%) = Z-score (%) x SDEV (A) Signup and view all the answers

If a stock's annual volatility is 30%, what is the daily volatility using the parametric method?

0.30% (C) Signup and view all the answers

Which of the following statements is TRUE regarding the three methods for calculating VaR discussed in the content?

The parametric method assumes that asset returns are normally distributed. (A) Signup and view all the answers

Which of the following factors contributes to the demand for funds in the economy?

Government net demand for funds (C) Signup and view all the answers

What does the real interest rate primarily measure?

The growth rate of your purchasing power (D) Signup and view all the answers

According to the Fisher hypothesis, how should nominal interest rates react to expected inflation?

They ought to increase one-for-one with expected inflation (A) Signup and view all the answers

Which of the following is a characteristic feature of nominal interest rates?

They include effects of inflation (A) Signup and view all the answers

Which mathematical expression closely represents the relationship between real and nominal interest rates?

r_real ≈ r_nom - i (C) Signup and view all the answers

Why can it be difficult to test the Fisher hypothesis definitively?

The equilibrium real rate changes unpredictably (C) Signup and view all the answers

What is typically expected when inflation rates are higher?

Higher nominal interest rates (D) Signup and view all the answers

Which of the following does NOT influence the supply of funds available in the economy?

Market interest rates (B) Signup and view all the answers

What does the Effective Annual Rate (EAR) represent?

The interest rate adjusted for compounding over a 1-year horizon (B) Signup and view all the answers

In the context of U.S. Treasury Zero Coupon Bonds, how is total risk-free return calculated?

Par value divided by price, minus 1 (B) Signup and view all the answers

Which statement about the Annual Percentage Rate (APR) is true?

It reflects simple interest over a year without compounding (C) Signup and view all the answers

What is indicated by a kurtosis greater than 3?

High probability of extreme values (D) Signup and view all the answers

What does a skewness value between 1 and 2 indicate?

High probability of extreme positive returns (C) Signup and view all the answers

Which of the following is NOT a characteristic of low kurtosis?

Higher average returns (B) Signup and view all the answers

How does the holding period affect total returns on investments?

Longer holding periods tend to provide greater total returns (D) Signup and view all the answers

What does a skewness value of 0 indicate in investment returns?

Equal probability of extreme positive and negative returns (B) Signup and view all the answers

What is the formula for calculating the 'N-day Volatility' using the variance-covariance method?

N-day Volatility = Daily Volatility x √N (B) Signup and view all the answers

What does VaR stand for in the context of financial risk management?

Value at Risk (B) Signup and view all the answers

What is the z-score used to calculate the 95% confidence level daily VaR in the second example of the text?

1.65 (C) Signup and view all the answers

If the annual volatility for a stock market portfolio is 25%, what is the daily volatility assuming 250 working days per year?

0.0158 (B) Signup and view all the answers

Calculate the 99% confidence level 10-day VaR for a portfolio with an annual volatility of 40% assuming 250 working days per year.

7.43% (D) Signup and view all the answers

Which of the following increases the confidence level in the VaR calculation?

Increasing the z-score (B) Signup and view all the answers

What is the main assumption used in the calculations provided in the text?

Stock returns are normally distributed (A) Signup and view all the answers

Which of the following is a potential limitation of using the variance-covariance method for calculating VaR?

It doesn't account for non-linear relationships between assets (A) Signup and view all the answers

Study Notes

Portfolio Management - Topic 2

Topic: Risk, Return, and the Historical Record
Pre-Readings: BKM Investments 11th Edition Book, Chapter 5, Sections 5.1, 5.2, 5.3, 5.4, 5.5, and 5.6
Interest Rate Determinants:
- Supply of funds from savers (primarily households)
- Demand for funds from businesses for investments (plant, equipment, inventories)
- Government's net demand for funds (modified by central bank actions)
- Expected rate of inflation

Real vs. Nominal Interest Rates

Nominal Interest Rate: The growth rate of your money
Real Interest Rate: The growth rate of your purchasing power
Formula: r_real = (r_nom - i) / (1 + i) where:
- r_nom = Nominal Interest Rate
- i = Inflation Rate

Approximating the Real Rate

An approximate real rate can be calculated as: Real rate ≈ Nominal rate - Expected inflation rate

Equilibrium Nominal Rate of Interest

Investors are concerned with real returns (increase in purchasing power)
Higher nominal rates are expected with higher inflation to maintain the expected real return.
Fisher Hypothesis: The nominal interest rate should increase one-for-one with expected inflation.
- Formula: r_nom = r_real + E(i)
The Fisher hypothesis implies that when real rates are stable, changes in nominal rates should predict changes in inflation rates.
Equilibrium real rate changes unpredictably over time.

Rates of Return for Different Holding Periods

U.S. Treasury Zero Coupon Bond:
- Par = $100
- Maturity = T
- Price = P
Total risk-free return r_f(T) = (100/P(T)) - 1

Annualized Rates of Return

Example Calculations using hypothetical zero-coupon Treasury prices (different time horizons/maturities).

Effective Annual Rate (EAR) and Annual Percentage Rate (APR)

Effective Annual Rate (EAR): The percentage increase in funds invested over a 1-year horizon.
- Formula: 1 + EAR = [1 + r_f(T)]^1/T
Annualized Percentage Rate (APR):
- Formula: APR = [(1 + EAR)^T - 1]/T

Example on EAR vs Total Return

Calculating EAR for 6-month and 25-year Treasury securities.

EAR (long-term) vs APR (short-term)

A table showing different compounding periods, implied/calculated values for the EAR, APR, and other relevant factors.

Risk and Risk Premiums

Rates of return (Single Period):
- HPR = (E(P₁) - P₀ + E(D₁))/P₀ where:
  - HPR= Holding period return
  - P₀= Beginning price
  - E(P₁) = Expected Ending price
  - E(D₁) = Expected Dividend during period one

Rates of Return: Single Period Example

Real-world example calculation with Expected Ending Price, Beginning Price, and Expected Dividend.

Expected Return and Standard Deviation (1 of 2)

Investors are uncertain about the future price and dividend of a share.
They can quantify their beliefs by considering various scenarios and their associated probabilities.
Formula: E(r) = ∑_s p(s) × r(s) where: - p(s) = Probability of a state - r(s) = Return if that state occurs - s = State

Scenario Returns: Example

Example showing calculations for expected return, E(r)

Expected Return and Standard Deviation (2 of 2)

Formula: σ² = Σ_s p(s) × [r(s) – E(r) ]²
Formula: STD = √σ²

Scenario VAR and STD Example

Calculations for variance and standard deviation using specific scenario data.

Excess Returns and Risk Premiums

Expected reward for risk involved in stocks (measured by the difference between the expected HPR and risk-free rate).
Risk premium = Expected(Index fund HPR) - Risk-free rate

Time Series Analysis of Past Rates of Return

Forward-looking scenario analysis: -Determine relevant scenarios and investment rates of return. -Assign probabilities and compute risk premium and standard deviation.
Time series of realized returns: -Do not explicitly use probabilities of different returns. -Return time series data only.

Returns Using Arithmetic and Geometric Averaging

Arithmetic Average: Historical data are treated as equally likely scenarios. -Formula: E(r) = (1/n) Σ r(s) -where n = number of observations
Geometric Average: Time-weighted average; considers compounded returns. -Formula: TV = (1 + r₁)(1 + r₂)..(1 + r_n) / Geometric Average = TV^1/n-1
Calculations of arithmetic and geometric averages using sample data

Concept Check

Practice question on calculating various measures of return & risk for a corporate bond investment.

Concept Check Solution

Provided solution to the practice question, showing computations and relevant calculations.

Normal Distribution

Possible outcomes cluster tightly around the mean for a lower standard deviation (SD).
More diffuse distributions for a higher SD.
Likelihood of outcomes is fully determined by mean and SD.

The Normal Distribution

Investment management with normal returns makes it easier, where a good measure of risk is standard deviation.
Mean and standard deviation will be enough to predict future scenarios.
Pairwise correlation coefficients will summarize the interdependence of return across securities.

Black Swans and Other Phenomena

The concept of unforeseen (rare) events that have a significant impact and disrupt the expected distribution of events.

Normality and Risk Measures

Normality greatly simplifies portfolio selection as standard deviation is a complete measure of risk and Sharpe ratio becomes a complete measure of portfolio performance.
Deviations from normality from asset returns may be potentially significant.

Normal and Skewed Distributions

Skewness: Asymmetry from the normal distribution, where extreme values (on one side) weigh heavier.
Distributions can be positively skewed (right tail extends further out) or negatively skewed (left tail extends further out)

Normal and Fat-Tailed Distributions

Kurtosis: The likelihood of extreme values on either side of the mean.
Distributions can be "fat-tailed" (higher kurtosis) or "thin-tailed" (lower kurtosis).
A "fat-tailed" distribution has heavier tails with more outliers.

Real formulas to measure skew & kurtosis

Real formulas to use for skew and excess kurtosis, for samples and for populations.

Value at Risk (VaR)

Risk measure (VaR) that shows a loss corresponding to a low percentile of return distribution (e.g., 5% or 1%).
VaR is fully determined by mean and SD of distribution, commonly estimated at 1%.

Historical Method

Sort returns in descending order
Match VaR to the number/percentile desired

Normal Distribution: Additional information

Tables showing typical numerical values to be used when dealing with the normal distribution.

VaR using the Parametric Method

Calculating VaR under the assumption of normality (95% confidence limit)
Calculating N-day volatility using daily volatility.

Expected Shortfall (ES) or Conditional Tail Expectation (CTE)

A more conservative measure of downside risk than VaR.
Takes into consideration the magnitudes of all potential losses even further out in the tail.

Evaluating and Interpreting Other Risk Measures

Lower Partial Standard Deviation (LPSD): Similar to standard deviation, but it only considers negative deviations from the risk-free return, addressing asymmetry in returns.
Sortino Ratio: The average excess return divided by the lower partial standard deviation (i.e., a better measurement of risk, specifically downside risk).

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your knowledge on the concepts and calculations related to Value at Risk (VaR). This quiz covers various methods of calculating VaR, including historical and parametric methods, and explores key factors influencing the outcomes. Ideal for finance students and professionals looking to assess their understanding of risk management.