Trader's portfolio has 3 different stocks:

Stock A has volatility 20% and Beta 0.9.
Stock B has volatility 25% and Beta 1.1.
Stock C has volatility 30% and Beta 0.8.

Which Stock carries the lowest unsystematic risk?

Question

Trader's portfolio has 3 different stocks:

Stock A has volatility 20% and Beta 0.9. 
Stock B has volatility 25% and Beta 1.1. 
Stock C has volatility 30% and Beta 0.8.

Which Stock carries the lowest unsystematic risk?

BlackTom AI · Accepted Answer

Unsystematic variance = total variance − systematic (market) variance, so σ_e^2 = σ_i^2 − β_i^2·σ_m^2 (where σ_i is the stock’s total volatility and σ_m is market volatility). Comparing unsystematic risk therefore uses the quantities σ_i^2 − β_i^2·σ_m^2 for each stock.

1) Relevant concepts
- Total variance = (volatility)^2.
- Systematic (market) variance component for a stock = β^2 · σ_m^2.
- Unsystematic (idiosyncratic) variance = total variance − systematic variance. We can compare unsystematic variance across stocks even if σ_m is unspecified, by using algebraic expressions in σ_m^2.

2) Step-by-step setup
Let M = σ_m^2 (market variance). Compute unsystematic variance for each stock:
- Stock A: σ_A^2 − β_A^2·M = 0.20^2 − 0.9^2·M = 0.0400 − 0.8100·M
- Stock B: 0.25^2 − 1.1^2·M = 0.0625 − 1.2100·M
- Stock C: 0.30^2 − 0.8^2·M = 0.0900 − 0.6400·M

3) Intermediate algebraic comparison
Compare B − A: (0.0625 − 1.21M) − (0.04 − 0.81M) = 0.0225 − 0.40M. This is positive (B has higher unsystematic variance than A) provided M < 0.0225/0.40 = 0.05625 (i.e., σ_m < 23.75%). Typical market volatilities are well below 23.75%, so B usually has more idiosyncratic risk than A.

Compare C − A: (0.09 − 0.64M) − (0.04 − 0.81M) = 0.05 + 0.17M, which is always positive for any M ≥ 0. So C always has greater unsystematic variance than A.

4) Why the correct answer is correct
Because Stock A’s total variance less the β^2 market component produces the smallest residual (idiosyncratic) variance among the three for any realistic market variance. Numerical example with σ_m = 15% (M = 0.0225) confirms: A: 0.04 − 0.018225 = 0.021775; B: 0.0625 − 0.027225 = 0.035275; C: 0.09 − 0.0144 = 0.0756. A’s unsystematic variance (0.021775) is smallest.

5) Why other options are incorrect
- “Stock B” is incorrect because B’s higher beta and intermediate total volatility leave it with larger idiosyncratic variance than A for realistic market variances.
- “Unsystematic Risk is the same for all the stocks” is incorrect: different volatilities and betas produce different residual variances.
- “Stock C” is incorrect because C’s high total volatility yields the largest unsystematic variance.

6) Conclusion
Stock A carries the lowest unsystematic risk. This matches the provided answer: Stock A.

Trader's portfolio has 3 different stocks: Stock A has volatility 20% and Beta 0.9. Stock B has volatility 25% and Beta 1.1. Stock C has volatility 30% and Beta 0.8. Which Stock carries the lowest unsystematic risk?Single choice

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