Measuring risk and return is crucial for investors to make informed decisions. We'll explore key metrics like holding period return, annualized returns, and the difference between nominal and real returns. These concepts help us understand how investments perform over time.

We'll also dive into risk assessment, covering systematic and unsystematic risks. We'll learn about tools like beta, variance, and standard deviation to quantify risk. Understanding these measures is essential for evaluating investments and building balanced portfolios.

Investment Return Metrics

Measuring Investment Returns

• Investment returns are measured as the total gain or loss on an investment over a specified time period, expressed as a percentage of the initial investment value
• The holding period return (HPR) is the total return earned from holding an investment over a specific period of time
  • Calculated as (Ending Price - Beginning Price + Dividends) / Beginning Price
• Annualized returns convert the holding period return into an annual rate, allowing for comparison of returns over different time periods
  • The formula for annualized return is (1 + HPR)^(1 / n) - 1, where n is the number of years in the holding period

Types of Investment Returns

• Nominal returns are the stated returns without adjusting for inflation
• Real returns account for the impact of inflation on purchasing power
• Arithmetic mean return is the simple average of a series of returns
  • Calculated by summing all returns and dividing by the number of observations
• Geometric mean return, also known as the compound annual growth rate (CAGR), accounts for the compounding effect of investment returns over time
  • Always lower than the arithmetic mean

Risk Assessment of Assets

Types of Investment Risk

• Risk in investments refers to the uncertainty or variability of returns and the potential for loss of principal
• Systematic risk, also known as market risk or undiversifiable risk, affects the entire market or economy and cannot be eliminated through diversification
  • Examples include interest rate risk, inflation risk, and political risk
• Unsystematic risk, also known as specific risk or diversifiable risk, is unique to a particular company, industry, or investment
  • Can be reduced through diversification

Quantifying Investment Risk

• The Capital Asset Pricing Model (CAPM) quantifies systematic risk using beta (β)
  • Beta measures the sensitivity of an asset's returns to market movements
  • A beta greater than 1 indicates higher volatility than the market, while a beta less than 1 indicates lower volatility
• Downside risk measures the potential for loss in an investment, focusing on returns that fall below a specified threshold or target return
• Value at Risk (VaR) estimates the maximum potential loss for an investment or portfolio over a given time horizon and confidence level
  • Expressed in monetary terms or as a percentage of the portfolio value

Variance and Standard Deviation

Measuring Dispersion of Returns

• Variance measures the dispersion of returns around the mean
  • Calculated as the average of the squared deviations from the mean return
  • A higher variance indicates greater risk
• Standard deviation is the square root of variance
  • Provides a measure of the typical deviation of returns from the mean in the original units of measurement (percentages for returns)

Risk-Adjusted Performance Metrics

• The coefficient of variation (CV) is a standardized measure of risk per unit of return
  • Calculated as the standard deviation divided by the mean return
  • Allows for comparison of risk across investments with different return levels
• Sharpe ratio measures risk-adjusted returns by comparing the excess return (return above the risk-free rate) to the standard deviation of returns
  • A higher Sharpe ratio indicates better risk-adjusted performance
• Tracking error measures the deviation of an investment's returns from its benchmark index
  • Calculated as the standard deviation of the differences between the investment and benchmark returns

Statistical Analysis of Risk and Return

Using Historical Data

• Historical data on asset prices, returns, and market indicators can be used to estimate risk and return parameters for individual assets and portfolios
• Calculating descriptive statistics such as mean, median, variance, standard deviation, and percentiles helps summarize the central tendency and dispersion of historical returns
• Probability distributions, such as the normal distribution, can be used to model the likelihood of different return outcomes based on historical data

Relationship Analysis

• Correlation analysis measures the strength and direction of the linear relationship between the returns of two assets or variables
  • Values range from -1 (perfect negative correlation) to +1 (perfect positive correlation)
• Regression analysis can be used to model the relationship between an asset's returns and one or more explanatory variables (market returns or economic factors)
• Monte Carlo simulation involves generating random return scenarios based on historical data
  • Estimates the probability distribution of future returns and assesses portfolio risk