Return and Risk: Calculating and Understanding Financial Metrics

In the world of finance, understanding return and risk is crucial for making informed investment decisions. These two metrics help investors gauge the potential profitability of an investment and its associated risk. This article will delve into the methods of calculating return and risk, the factors influencing them, and practical examples to illustrate their importance.

Understanding Return

Return refers to the gain or loss generated by an investment relative to its initial cost. It is often expressed as a percentage and can be calculated using several methods. The most common forms include:

1. Absolute Return: This is the simplest form, calculated as:

   $\text{Absolute Return}=\text{Ending Value}-\text{Beginning Value}$Absolute Return=Ending Value−Beginning Value

   For example, if you bought a stock for $100 and it is now worth $120, the absolute return is $20.

2. Percentage Return: This metric provides the return relative to the initial investment:

   $\text{Percentage Return}=\left(\frac{\text{Ending Value}-\text{Beginning Value}}{\text{Beginning Value}}\right)\times 100$Percentage Return=(Beginning ValueEnding Value−Beginning Value​)×100

   Using the previous example, the percentage return would be:

   $\left(\frac{120-100}{100}\right)\times 100=20\%$(100120−100​)×100=20%

3. Annualized Return: This adjusts the return to reflect the time period of the investment:

   $\text{Annualized Return}={\left(\frac{\text{Ending Value}}{\text{Beginning Value}}\right)}^{\frac{1}{\text{Number of Years}}}-1$Annualized Return=(Beginning ValueEnding Value​)Number of Years1​−1

   If the investment took 3 years to grow from $100 to $150, the annualized return would be:

   ${\left(\frac{150}{100}\right)}^{\frac{1}{3}}-1\approx 14.47\%$(100150​)31​−1≈14.47%

Understanding Risk

Risk in finance refers to the potential for an investment to deviate from its expected return. It encompasses various forms, including:

1. Standard Deviation: This measures the amount of variability or dispersion around the average return. Higher standard deviation indicates greater risk.

   $\text{Standard Deviation}=\sqrt{\frac{\sum (X_i-\mu {)}^2}{N}}$Standard Deviation=N∑(Xi​−μ)2​​

   where $X_i$Xi​ is each individual return, $\mu$μ is the mean return, and $N$N is the number of returns.

2. Beta: This measures the investment’s sensitivity to market movements. A beta greater than 1 indicates higher volatility than the market.

   $\text{Beta}=\frac{\text{Covariance of the Investment and Market Returns}}{\text{Variance of Market Returns}}$Beta=Variance of Market ReturnsCovariance of the Investment and Market Returns​

   For example, a beta of 1.2 means the investment is expected to move 20% more than the market.

3. Value at Risk (VaR): This estimates the potential loss in value of an investment over a specified period for a given confidence interval. For instance, a 5% VaR of $1,000 means there is a 5% chance of losing $1,000 or more over a defined period.

Calculating Return and Risk with Practical Examples

Consider a portfolio with the following yearly returns over 5 years: 10%, 12%, -5%, 8%, and 15%. Here’s how to calculate the return and risk:

1. Average Return:

   $\text{Average Return}=\frac{10\%+12\%-5\%+8\%+15\%}{5}=8\%$Average Return=510%+12%−5%+8%+15%​=8%

2. Standard Deviation: First, calculate the variance:

   $\text{Variance}=\frac{(10\%-8\%{)}^2+(12\%-8\%{)}^2+(-5\%-8\%{)}^2+(8\%-8\%{)}^2+(15\%-8\%{)}^2}{5}$Variance=5(10%−8%)2+(12%−8%)2+(−5%−8%)2+(8%−8%)2+(15%−8%)2​

   Then, take the square root to get the standard deviation.

3. Beta Calculation: Compare the portfolio’s returns to a benchmark index like the S&P 500 to determine beta. For example, if the portfolio has a beta of 1.1, it’s 10% more volatile than the benchmark.

Risk-Return Tradeoff

Investors often face a tradeoff between risk and return. Generally, higher returns come with higher risk. This principle is illustrated through the Capital Asset Pricing Model (CAPM), which links expected return to risk:

$\text{Expected Return}=\text{Risk-Free Rate}+\text{Beta}\times (\text{Market Return}-\text{Risk-Free Rate})$Expected Return=Risk-Free Rate+Beta×(Market Return−Risk-Free Rate)

For example, if the risk-free rate is 3%, the market return is 8%, and the investment’s beta is 1.2, the expected return would be:

$3\%+1.2\times (8\%-3\%)=9\%$3%+1.2×(8%−3%)=9%

Conclusion

Calculating return and risk involves understanding various metrics and their implications for investment decisions. Investors need to balance their desired returns with their risk tolerance, utilizing tools like standard deviation, beta, and VaR to assess potential outcomes. By grasping these concepts, investors can make more informed decisions, aligning their portfolios with their financial goals and risk appetite.