Glossary

4.1 Measuring Risk and Return

4 min read•july 18, 2024

Measuring risk and return is crucial for making informed investment decisions. Investors use calculations to estimate potential gains, while quantifies risk. These metrics help evaluate individual assets and entire portfolios, guiding allocation strategies.

Understanding and tolerance is key to crafting suitable investment plans. Risk-averse individuals prefer safer options, while risk-tolerant investors may seek higher returns through riskier assets. Balancing risk and return is essential for achieving financial goals.

Risk and Return Measurements

Expected return and deviation calculation

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  • Expected return
    • Calculates weighted average of possible returns based on their probabilities
    • Uses formula E(R)=∑i=1npiRiE(R) = \sum_{i=1}^{n} p_i R_i where pip_i represents probability of each possible return and RiR_i represents each possible return
    • Example: Investment with 50% chance of 10% return and 50% chance of 5% return has expected return of 7.5% (0.5 * 10% + 0.5 * 5%)
  • Standard deviation
    • Quantifies dispersion of returns around the expected return
    • Calculated using formula σ=∑i=1npi(Ri−E(R))2\sigma = \sqrt{\sum_{i=1}^{n} p_i (R_i - E(R))^2}
    • Higher standard deviation indicates greater variability and risk
    • Determines weighted average of individual asset expected returns in a portfolio
    • Applies formula E(Rp)=∑i=1nwiE(Ri)E(R_p) = \sum_{i=1}^{n} w_i E(R_i) where wiw_i represents weight of each asset and E(Ri)E(R_i) represents expected return of each asset
    • Example: Portfolio with 60% stocks (expected return 10%) and 40% bonds (expected return 5%) has expected return of 8% (0.6 * 10% + 0.4 * 5%)
    • Accounts for individual asset standard deviations and correlations between assets
    • Uses formula σp=∑i=1nwi2σi2+2∑i=1n−1∑j=i+1nwiwjσiσjρij\sigma_p = \sqrt{\sum_{i=1}^{n} w_i^2 \sigma_i^2 + 2 \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} w_i w_j \sigma_i \sigma_j \rho_{ij}} where σi\sigma_i represents standard deviation of each asset and ρij\rho_{ij} represents correlation coefficient between assets ii and jj
    • can reduce portfolio standard deviation when assets are not perfectly positively correlated

Risk aversion and tolerance concepts

  • Risk aversion
    • Describes preference for lower risk investments when expected returns are equal
    • Risk-averse investors demand higher expected returns to compensate for taking on more risk
    • Example: Given choice between two investments with same expected return, risk-averse investor will choose the one with lower risk
    • Measures willingness to accept higher risk in pursuit of potentially higher returns
    • Factors such as age, income, and investment goals influence an individual's risk tolerance
    • Example: Young investor with long investment horizon may have higher risk tolerance than retiree relying on portfolio income
  • Impact on investment decisions
    • Risk-averse investors gravitate towards lower-risk investments like bonds and cash equivalents
    • Risk-tolerant investors are more likely to hold higher-risk assets such as stocks and derivatives
    • should align with individual risk preferences and financial objectives

Types of Risk and Risk-Return Tradeoff

Systematic vs unsystematic risk

  • (market risk)
    • Affects the entire market or economy and cannot be diversified away
    • Examples include interest rate changes, inflation, and political events
    • Investors must accept systematic risk as inherent to investing in the market
  • (firm-specific risk)
    • Unique to individual companies or industries and can be reduced through diversification
    • Examples include management changes, labor strikes, and product recalls
    • Holding a well-diversified portfolio minimizes exposure to unsystematic risk
  • Implications for investors
    • Diversification is effective in mitigating unsystematic risk but not systematic risk
    • Investors should prioritize managing systematic risk through asset allocation and risk management strategies
    • Understanding the distinction between systematic and unsystematic risk is crucial for constructing a resilient portfolio

Risk-return tradeoff analysis

  • Positive relationship between risk and expected return
    • Higher risk investments generally offer the potential for higher returns
    • Lower risk investments typically provide lower expected returns
    • Example: Stocks have historically offered higher returns than bonds but with greater volatility
    • Investors must strike a balance between the desire for higher returns and their tolerance for risk
    • Optimal portfolio depends on individual risk preferences and investment goals
    • Example: Conservative investor may favor a portfolio with 80% bonds and 20% stocks, while an aggressive investor may hold 80% stocks and 20% bonds
    • Describes the relationship between systematic risk and expected return for individual assets
    • Formula: E(Ri)=Rf+βi(E(Rm)−Rf)E(R_i) = R_f + \beta_i (E(R_m) - R_f) where RfR_f represents risk-free rate, βi\beta_i represents asset's sensitivity to systematic risk, and E(Rm)E(R_m) represents expected return of the market portfolio
    • Assets with higher values are expected to have higher returns to compensate for their greater systematic risk

Key Terms to Review (19)

Alpha: Alpha is a measure of an investment's performance on a risk-adjusted basis, representing the excess return generated above the expected return predicted by market movements. It helps investors assess how well an asset or portfolio is performing relative to a benchmark index, taking into account the risk taken to achieve those returns. A positive alpha indicates outperformance, while a negative alpha suggests underperformance.
Asset Allocation: Asset allocation is the process of distributing investments across various asset classes, such as stocks, bonds, and cash, to balance risk and return according to an investor's goals and risk tolerance. This strategy helps in managing overall portfolio risk and can significantly influence investment performance over time.
Beta: Beta is a measure of a stock's volatility in relation to the overall market, indicating how much the stock's price tends to move when the market moves. A beta greater than 1 signifies that the stock is more volatile than the market, while a beta less than 1 indicates lower volatility. Understanding beta helps investors assess risk and make informed decisions about portfolio management and investment strategies.
Capital Asset Pricing Model (CAPM): The Capital Asset Pricing Model (CAPM) is a financial model that establishes a linear relationship between the expected return of an asset and its systematic risk, measured by beta. It is used to determine the appropriate required rate of return for an investment, factoring in both the risk-free rate and the expected market return, providing insights into how risk influences investment decisions and valuation.
Diversification: Diversification is an investment strategy that involves spreading investments across various assets to reduce risk. By not putting all your eggs in one basket, it aims to lower the volatility of an investment portfolio, as different assets react differently to market changes. This concept is crucial for managing risk and achieving a more stable return on investments.
Efficient frontier: The efficient frontier is a concept in modern portfolio theory that represents a set of optimal investment portfolios that offer the highest expected return for a defined level of risk. It illustrates the trade-off between risk and return, showing investors the most efficient combinations of assets to achieve their financial goals while minimizing risk. This idea connects closely with measuring risk and return, portfolio diversification, and creating optimal portfolios.
Expected return: Expected return is the anticipated return on an investment based on its probable outcomes, weighted by their respective probabilities. It provides a way to measure the potential profitability of an investment, incorporating both the risk and reward associated with it. Understanding expected return is crucial when analyzing various investment opportunities and constructing a balanced portfolio.
Portfolio diversification: Portfolio diversification is the investment strategy of spreading investments across various financial assets to reduce risk. By holding a mix of asset types—such as stocks, bonds, and real estate—investors can minimize the impact of any single asset's poor performance on the overall portfolio. This strategy is crucial for managing both systematic and unsystematic risk, and it can influence cash management practices as well as derivative trading strategies.
Portfolio expected return: Portfolio expected return is the weighted average of the expected returns of the individual assets within a portfolio, reflecting the anticipated profit from holding those assets over a specified period. This concept is crucial for investors as it helps gauge potential earnings while considering the risk associated with different assets in the portfolio.
Portfolio standard deviation: Portfolio standard deviation is a statistical measure that quantifies the total risk of a portfolio by calculating the degree to which the returns on the investments within the portfolio deviate from their expected return. It captures both the individual volatility of the assets and how they interact with each other, allowing investors to understand the overall risk profile of their investment strategy.
Risk aversion: Risk aversion is a behavioral finance concept that describes an investor's tendency to prefer lower risk investments over higher risk ones, even if the potential returns are higher. This inclination reflects a preference for certainty and a desire to avoid potential losses, leading individuals to make investment choices that minimize risk. Understanding risk aversion is crucial as it influences how investors measure risk and return, build diversified portfolios, and apply modern portfolio theories.
Risk tolerance: Risk tolerance refers to the degree of variability in investment returns that an individual is willing to withstand in their investment portfolio. It reflects personal comfort levels with uncertainty and potential losses, influencing decisions on investments and financial planning. Understanding risk tolerance is crucial as it directly affects how individuals approach risk and return, guides their asset allocation, and informs corporate risk management strategies.
Risk-return tradeoff: The risk-return tradeoff is a financial principle that suggests the potential return on an investment increases with an increase in risk. Investors must balance the desire for the highest return against the potential for loss, leading them to seek investments that align with their risk tolerance and return expectations.
Sharpe Ratio: The Sharpe Ratio is a measure that helps investors understand the return of an investment compared to its risk. It is calculated by taking the difference between the investment's return and the risk-free rate, then dividing that by the investment's standard deviation. This ratio is useful in assessing risk-adjusted performance and helps in making informed investment decisions by allowing comparisons across different assets and portfolios.
Standard Deviation: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. It helps investors understand the degree of risk associated with an investment by indicating how much returns deviate from the expected average return. A higher standard deviation implies greater volatility, which can be linked to potential returns and risks in various financial contexts.
Systematic risk: Systematic risk refers to the inherent risk that affects an entire market or a large segment of the market, which cannot be mitigated through diversification. This type of risk is often linked to macroeconomic factors such as changes in interest rates, inflation, and political instability, impacting all investments across the board.
Total Return: Total return is the complete gain or loss made on an investment over a specific period, including both income generated and capital appreciation. It reflects not only the change in the investment's price but also any dividends, interest, or other income received during that time. Understanding total return is essential because it provides a more accurate picture of an investment's performance compared to just looking at price changes alone.
Unsystematic risk: Unsystematic risk refers to the risk that is unique to a specific company or industry, which can be mitigated through diversification in investment portfolios. This type of risk is not linked to the overall market movements and can arise from factors such as management decisions, product recalls, or regulatory changes impacting a particular organization.
Value at Risk (VaR): Value at Risk (VaR) is a statistical technique used to measure the risk of loss on an investment or portfolio over a defined period for a given confidence interval. It quantifies the potential loss in value that an asset or portfolio could experience under normal market conditions, allowing investors and risk managers to assess and manage financial risk effectively. VaR is especially important for understanding the potential downside risk in both individual investments and corporate financial strategies.
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