Expected Return and Standard Deviation Calculator
Analyze the risk and return of your investments by calculating the expected return and standard deviation based on potential outcomes and their probabilities.
Investment Scenarios
Enter the potential return and its probability for up to five different economic scenarios.
E.g., 20 for a 20% gain.
The likelihood of this scenario occurring.
Use a negative number for a loss.
What is Expected Return and Standard Deviation?
When making investment decisions, two of the most critical metrics to understand are expected return and standard deviation. The expected return is the profit or loss an investor anticipates receiving on an investment, calculated as a weighted average of all possible outcomes. It provides a single value that represents the most likely return over time. However, this number doesn’t tell the whole story. To fully calculate expected return using standard deviation, you must also understand risk.
That’s where standard deviation comes in. In finance, standard deviation is the primary measure of an investment’s volatility or risk. It quantifies how much the actual return is likely to deviate from the expected return. A low standard deviation implies that returns are stable and cluster closely around the expected value, indicating a lower-risk investment. Conversely, a high standard deviation suggests that returns are spread out over a wider range, signaling higher volatility and risk. By analyzing both metrics together, investors can get a more complete picture of an investment’s risk-reward profile.
Formula and Explanation
To properly calculate expected return using standard deviation, you need two distinct formulas. The process involves first determining the expected return and then using that value to find the variance and standard deviation.
Expected Return Formula
The expected return, denoted as E(R), is the sum of the potential returns multiplied by their corresponding probabilities.
Where:
- E(R) is the Expected Return.
- Σ is the summation symbol, meaning you add up all the values.
- Ri is the return for the i-th scenario (e.g., Return in a ‘Boom’ economy).
- Pi is the probability of the i-th scenario occurring.
Standard Deviation Formula
The standard deviation (σ) measures the dispersion of a set of returns. It is calculated as the square root of the variance (σ²). The variance is the average of the squared differences from the Expected Return.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| Ri | Return of an Outcome | Percentage (%) | -100% to +∞% |
| Pi | Probability of an Outcome | Percentage (%) | 0% to 100% |
| E(R) | Expected Return | Percentage (%) | Varies based on inputs |
| σ | Standard Deviation (Volatility) | Percentage (%) | 0% to +∞% |
Practical Examples
Let’s walk through two examples to see how to calculate expected return using standard deviation in practice. For a deeper dive, consider using a portfolio volatility calculator.
Example 1: Conservative Stock
Imagine a stable, blue-chip stock with the following potential one-year returns:
- Inputs:
- Scenario 1: 15% return with a 30% probability (Strong Economy)
- Scenario 2: 8% return with a 50% probability (Normal Economy)
- Scenario 3: -2% return with a 20% probability (Recession)
Expected Return Calculation:
E(R) = (0.30 × 15%) + (0.50 × 8%) + (0.20 × -2%) = 4.5% + 4.0% – 0.4% = 8.1%
Standard Deviation Calculation (simplified steps):
1. Calculate Variance: 0.30(15-8.1)² + 0.50(8-8.1)² + 0.20(-2-8.1)² = 14.283 + 0.005 + 20.402 = 34.69
2. Calculate Standard Deviation: √34.69 = 5.89%
Results: The stock has an expected return of 8.1% with a standard deviation (volatility) of 5.89%. This represents a relatively low-risk investment.
Example 2: Tech Startup
Now consider a high-growth, volatile tech stock:
- Inputs:
- Scenario 1: 100% return with a 20% probability (Product Launch Success)
- Scenario 2: 10% return with a 60% probability (Meets Expectations)
- Scenario 3: -50% return with a 20% probability (Product Launch Failure)
Expected Return Calculation:
E(R) = (0.20 × 100%) + (0.60 × 10%) + (0.20 × -50%) = 20% + 6% – 10% = 16%
Standard Deviation Calculation (simplified steps):
1. Calculate Variance: 0.20(100-16)² + 0.60(10-16)² + 0.20(-50-16)² = 1411.2 + 21.6 + 871.2 = 2304
2. Calculate Standard Deviation: √2304 = 48%
Results: The tech stock has a much higher expected return of 16%, but its standard deviation is a massive 48%, indicating extreme risk and volatility. This highlights the importance of using an investment return calculator to weigh both factors.
How to Use This Expected Return Calculator
Our calculator is designed to simplify the process of financial analysis. Follow these steps:
- Enter Scenarios: In the input fields, provide at least two potential scenarios. For each, enter the potential “Return (%)” and the “Probability (%)” of that return occurring.
- Check Probabilities: Ensure the sum of all your ‘Probability’ fields equals 100%. The calculator will warn you if it doesn’t.
- Click Calculate: Press the “Calculate” button to perform the analysis.
- Interpret the Results:
- Expected Return: This is the main result, showing the probability-weighted average return you can anticipate.
- Standard Deviation: This crucial number represents the investment’s risk or volatility. A higher number means a wider range of possible outcomes. For a deeper understanding of risk, you might explore the modern portfolio theory.
- Breakdown Table & Chart: Use the table and chart to visualize how each scenario contributes to the final result and to see the distribution of potential outcomes.
Key Factors That Affect Expected Return and Standard Deviation
Several factors can influence these calculations. Understanding them is crucial for accurate risk and return analysis.
- Economic Conditions: Macroeconomic factors like GDP growth, inflation, and unemployment are primary drivers. A strong economy generally leads to higher return expectations, while a recession increases the probability of negative returns.
- Interest Rates: Changes in interest rates set by central banks affect the cost of borrowing and the attractiveness of other investments, shifting the expected returns for stocks.
- Market Sentiment: Investor psychology can create short-term volatility. Bullish sentiment can inflate returns, while bearish sentiment can depress them, increasing standard deviation.
- Company-Specific News: For individual stocks, events like earnings reports, new product launches, or management changes can dramatically alter the probabilities and potential returns of different scenarios.
- Industry Trends: A stock’s performance is often tied to its industry. Technological disruption, regulatory changes, or shifts in consumer behavior can affect all companies in a sector.
- Geopolitical Events: Global events, such as trade wars or political instability, can introduce systemic risk that impacts the entire market, widening the distribution of potential outcomes and thus increasing standard deviation.
Frequently Asked Questions (FAQ)
1. What is a good expected return?
A “good” expected return is relative and depends on the associated risk (standard deviation). A high expected return isn’t good if the risk is unacceptable. Investors often compare an investment’s expected return to a benchmark, like the S&P 500’s historical average (~10%), and consider its risk level.
2. Is a lower standard deviation always better?
Generally, for a given level of expected return, a lower standard deviation is preferred because it indicates less risk. However, investments with very low risk (like government bonds) also tend to have very low expected returns. The goal is to find a balance you’re comfortable with. Some investors use a Sharpe ratio calculator to measure risk-adjusted return.
3. What’s the difference between expected return and historical return?
Historical return is what an investment *actually* earned in the past. Expected return is a *forward-looking* estimate based on probabilities of future events. While historical data is often used to inform those probabilities, it’s not a guarantee of future results.
4. How do I estimate the probabilities for each scenario?
This is the most subjective part of the calculation. You can use historical data, analyst ratings, economic forecasts, or your own research. For example, if a company has a 75% success rate on product launches, you might use that as a probability for a “boom” scenario.
5. Can the expected return be negative?
Yes. If the probabilities of negative returns are high enough, the overall expected return for the investment can be negative, indicating you are statistically likely to lose money.
6. What is variance?
Variance is simply the standard deviation squared. It measures the same thing (dispersion of returns) but isn’t as intuitive because its unit is “percent squared.” Standard deviation is more commonly used because it is expressed in the same unit as the return (percent).
7. Why did my total probability show a warning?
The sum of the probabilities of all possible, mutually exclusive outcomes must equal 100%. Our calculator alerts you if your inputs don’t add up to 100% to ensure your analysis is mathematically sound.
8. What is the limitation of this model?
The primary limitation is that the output is only as good as the input. The accuracy of the calculated expected return and standard deviation depends entirely on the accuracy of your return and probability estimates for each scenario.