Expected Return, Variance, and Standard Deviation of a Portfolio
What is the expected return of a portfolio that invests 22.69% in Asset 1 (expected return = 5.12%) and the remaining 77.31% in Asset 2 (expected return = 1.88% )?
✅ Expected return = w₁×r₁ + w₂×r₂ = 2.62%.
What is the variance of the portfolio above, given correlation 0.62, standard deviation of Asset 1 = 26.32% and of Asset 2 = 20.2%?
✅ Variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ = 3.95.
What is the standard deviation of the portfolio above?
✅ Standard deviation = √(variance) = 19.89%.
Portfolio Weight Calculation for Multiple Stocks
Suppose you purchase 155 shares of Company A at $62.32 per share, 135 shares of Company B at $39.38 per share, and 91 shares of Company C at $55.37 per share.
✅ Weight on Company A = value_A / total_value × 100 = 48.26%.
Portfolio Variance and Standard Deviation Calculation
You invest 84.45% of your funds in Asset A (volatility = 9.95%) and 15.55% in Asset B (volatility = 23.61%). The correlation between the two assets is 0.35.
✅ Portfolio variance = w²×sd₁² + (1−w)²×sd₂² + 2×w×(1−w)×sd₁×sd₂×ρ = 105.51.
✅ Portfolio standard deviation = √(variance) = 10.27%.
Calculating Beta Using Covariance and Market Variance
Given that the covariance between a stock and the market is 4.69 and the variance of the market is 4.84,
✅ Beta = Cov(asset, market) / Var(market) = 0.97.
Risk Premium and Sharpe Ratio Calculation
An investment has an expected return of 6.26%, a risk‑free rate of 2.02%, and a volatility (standard deviation) of 14.87%.
✅ Risk premium = expected return − risk‑free rate = 4.25%.
✅ Sharpe ratio = risk premium / volatility = 0.29.
Combining Risky Asset and Risk‑Free Rate
You invest 93.62% of your funds in a risky portfolio (expected return = 10.41%, volatility = 21.77%) and the remainder 6.38% in the risk‑free asset (rate = 3.55%).
✅ Expected return = w×ret_risky + (1−w)×Rf = 9.97%.
✅ Since the risk‑free asset has zero volatility, the portfolio’s volatility equals w×σ_risky = 20.38%.
Beta of a Three‑Asset Portfolio
You hold a portfolio composed of three assets with betas 1.14, 1.11, and 0.23, invested in proportions 66.79%, 0.12%, and 33.09%, respectively.
✅ Portfolio beta = w₁×β₁ + w₂×β₂ + w₃×β₃ = 0.84.
Expected Return via CAPM
The risk‑free rate is 3.32%, the market risk premium is 7.61%, and the beta of the investment is 1.27.
✅ Expected return = Rf + β × (market risk premium) = 13%.
Variance and Volatility of a Three‑Asset Portfolio
You invest 1.21%, 26.06%, and 72.72% of your funds in three assets with volatilities 16.01%, 15.88%, and 19.11%, respectively. The pairwise correlations are ρ₁₂ = 0.07, ρ₁₃ = 0.09, and ρ₂₃ = 0.08.
✅ Variance = Σwᵢ²σᵢ² + 2Σ wᵢwⱼσᵢσⱼρᵢⱼ = 220.4.
✅ Standard deviation = √(variance) = 14.85%.
Risk‑Free Combination and Sharpe Ratio
You build a portfolio by combining the risk‑free asset (rate = 4.25%) with a tangent portfolio (expected return = 11.43%, volatility = 20.4%) using a weight of 88.82% in the tangent portfolio (weights greater than 100% imply leverage).
✅ Expected return = w×ret_tan + (1−w)×Rf = 10.63%.
✅ Standard deviation = |w|×σ_tan = 18.12%.
✅ Sharpe ratio = risk premium / volatility = 0.35.
Sharpe Calculation
An investment has an expected return of 8.31%, a risk‑free rate of 2.09%, and a volatility (standard deviation) of 15.9%.
✅ Risk premium = expected return − risk‑free rate = 6.22%.
✅ Sharpe ratio = risk premium / volatility = 0.391.