Value at Risk (VaR) Fundamentals

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).