Given two investment opportunities with their returns represented by two random variables r1 and r2, other things being equal, the one with ________ variance is considered more risky.

Question

BlackTom AI · Accepted Answer

Variance measures how widely possible returns scatter around their expected value, so an investment with larger variance is considered more volatile and therefore riskier.

1. Relevant concepts and knowledge points
- Random variable returns (r1, r2) have an expected value (mean) and dispersion around that mean.  
- Variance (and its square root, standard deviation) quantifies dispersion: larger variance = larger typical squared deviation from the mean.  
- Under the usual finance assumption "other things being equal" (same expected return), investors view greater dispersion as greater risk because outcomes are less predictable and more likely to produce extreme losses.

2. Step‑by‑step reasoning
- If r1 and r2 have the same expected return, compare Var(r1) and Var(r2).  
- The investment with the larger variance has greater spread of possible outcomes and hence more uncertainty about what return will actually occur.  
- Greater uncertainty (volatility) means higher risk in the usual sense used in portfolio choice and risk measurement.

3. Calculations (concrete example)
- Example: r1 outcomes {5%, 5%, 5%} → mean = 5%. Deviations 0,0,0 → Var(r1)=0.  
- r2 outcomes {0%, 5%, 10%} → mean = 5%. Deviations -5,0,5 → squared deviations 25,0,25 → Var(r2) = (25+0+25)/3 = 50/3 ≈ 16.67 (percentage‑points squared).  
- Var(r2) > Var(r1), so r2 is riskier despite identical means.

4. Why the correct answer is correct
- Variance directly captures dispersion/volatility; higher variance implies more uncertainty about returns and a higher chance of outcomes far from the mean (including larger losses), which is why higher variance corresponds to greater risk.

5. Why the other option is incorrect
- "Lower" variance would mean returns are more tightly clustered around the mean and thus more predictable; that corresponds to lower risk, not higher. Therefore option a (lower) is incorrect.

6. Conclusion
- The investment with higher variance is considered more risky; this confirms the correct choice is b. higher.

Given two investment opportunities with their returns represented by two random variables r1 and r2, other things being equal, the one with ________ variance is considered more risky.单项选择题

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