Introduction
Investment decisions often involve uncertainty. In such situations, investors rely on tools like expected value and variance to evaluate potential outcomes and associated risks. The expected value gives a measure of the average return one can anticipate, while the variance tells us how much the returns fluctuate around the average. In this answer, we will calculate both the expected value and the variance of a given investment scenario.
Given Data
The investor faces three possible outcomes with the following probabilities and returns:
| Probability | Return (₹) |
|---|---|
| 0.4 | 100 |
| 0.3 | 30 |
| 0.3 | -30 |
Step 1: Expected Value (Mean Return)
The expected value is calculated by multiplying each return by its probability and summing them up.
Formula: E(R) = Σ [P(i) × R(i)]
Where:
P(i) = Probability of outcome i
R(i) = Return of outcome i
Applying the formula:
E(R) = (0.4 × 100) + (0.3 × 30) + (0.3 × -30)
= 40 + 9 – 9 = ₹40
Expected Value = ₹40
Step 2: Variance of Returns
Variance measures the spread or risk of the investment. It is calculated by taking the weighted average of the squared deviations from the mean (expected value).
Formula: Var(R) = Σ [P(i) × (R(i) – E(R))²]
Let’s compute each squared deviation:
- For ₹100: (100 – 40)² = 60² = 3600
- For ₹30: (30 – 40)² = (-10)² = 100
- For ₹-30: (-30 – 40)² = (-70)² = 4900
Now multiply by the respective probabilities:
- 0.4 × 3600 = 1440
- 0.3 × 100 = 30
- 0.3 × 4900 = 1470
Add these up:
Variance = 1440 + 30 + 1470 = 2940
Variance = 2940
Step 3: Standard Deviation (Optional)
The standard deviation is the square root of the variance and provides a more interpretable measure of risk.
Standard Deviation = √2940 ≈ 54.23
Interpretation of Results
Expected Return: ₹40
This means that on average, the investor can expect to earn ₹40 from this investment. This is not a guaranteed return, but an average across many repetitions.
Variance: 2940
A high variance indicates that the returns are spread out and involve considerable risk. A variance of 2940 implies the returns fluctuate significantly around the mean.
Standard Deviation: Approximately ₹54.23
This tells us that the returns typically vary by around ₹54.23 from the average value of ₹40. It helps the investor understand the risk in more practical terms.
Why This Matters in Investment
Both expected value and variance are key tools for risk analysis. Investors use these concepts to:
- Compare different investment options
- Balance between risk and return
- Make informed decisions
- Construct diversified portfolios
For example, an investor may prefer a lower expected return with low variance over a high return with very high risk.
Conclusion
In this case, the expected value of the uncertain investment is ₹40, while the variance is 2940 and the standard deviation is approximately ₹54.23. These values give the investor a clear picture of the potential return and the level of risk involved. Understanding these concepts helps in making better financial decisions under uncertainty.