Module 3: Numeric Questions

For students

Last updated: 09/01/2026 12:44

The questions are based on or inspired by the following references:

• Berk & DeMarzo, Corporate Finance, 5th ed. (2020)
  
• Brealey & Myers, Principles of Corporate Finance, 13th ed. (2020)

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💡 You can also press Ctrl + P (or Cmd + P on Mac) to print or save your responses as a .pdf file.

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⚠️ These exercises are powered by AI-assisted technologies and may contain occasional formatting or logic errors. Please report any issues you encounter so I can improve the experience.

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📘 Part 1 (until Midterm)

Module	Chapter
3	ch11

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Answer the following questions based on the discussions in class.

Q1.

Expected Return, Variance, and Standard Deviation of a Portfolio

What is the expected return of a portfolio that invests 22.84% in Asset 1 (expected return = 13.62%) and the remaining 77.16% in Asset 2 (expected return = 6.74% )?

1. What is the expected return of the portfolio (in %)?

✅ Expected return = w₁×r₁ + w₂×r₂ = 8.31%.

What is the variance of the portfolio above, given correlation 0.68, standard deviation of Asset 1 = 30.17% and of Asset 2 = 19.02%?

2. What is the variance of the portfolio (in squared %)?

✅ Variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ = 4.

What is the standard deviation of the portfolio above?

3. What is the standard deviation of the portfolio (in %)?

✅ Standard deviation = √(variance) = 19.99%.

Q2.

Portfolio Weight Calculation for Multiple Stocks

Suppose you purchase 232 shares of Company A at $55.55 per share, 149 shares of Company B at $38.62 per share, and 26 shares of Company C at $51.42 per share.

1. What is the weight on Company A in your portfolio (in %)?

✅ Weight on Company A = value_A / total_value × 100 = 64.51%.

Q3.

Portfolio Variance and Standard Deviation Calculation

You invest 23.16% of your funds in Asset A (volatility = 14.97%) and 76.84% in Asset B (volatility = 18.62%). The correlation between the two assets is 0.61.

1. What is the variance of your portfolio (in squared %)?

✅ Portfolio variance = w²×sd₁² + (1−w)²×sd₂² + 2×w×(1−w)×sd₁×sd₂×ρ = 277.78.

2. What is the standard deviation of your portfolio (in %)?

✅ Portfolio standard deviation = √(variance) = 16.67%.

Q4.

Calculating Beta Using Covariance and Market Variance

Given that the covariance between a stock and the market is 4.47 and the variance of the market is 5.2,

1. What is the beta of the stock?

✅ Beta = Cov(asset, market) / Var(market) = 0.86.

Q5.

Risk Premium and Sharpe Ratio Calculation

An investment has an expected return of 7.32%, a risk‑free rate of 2.18%, and a volatility (standard deviation) of 16.25%.

1. What is the risk premium (in %)?

✅ Risk premium = expected return − risk‑free rate = 5.14%.

2. What is the Sharpe ratio (return per unit of risk)?

✅ Sharpe ratio = risk premium / volatility = 0.32.

Q6.

Combining Risky Asset and Risk‑Free Rate

You invest 40.29% of your funds in a risky portfolio (expected return = 5.69%, volatility = 12.4%) and the remainder 59.71% in the risk‑free asset (rate = 3.88%).

1. What is the expected return of your portfolio (in %)?

✅ Expected return = w×ret_risky + (1−w)×Rf = 4.61%.

2. What is the standard deviation of your portfolio (in %)?

✅ Since the risk‑free asset has zero volatility, the portfolio’s volatility equals w×σ_risky = 5%.

Q7.

Beta of a Three‑Asset Portfolio

You hold a portfolio composed of three assets with betas 0.53, 1.09, and 0.08, invested in proportions 9.37%, 76.54%, and 14.09%, respectively.

1. What is the beta of the portfolio?

✅ Portfolio beta = w₁×β₁ + w₂×β₂ + w₃×β₃ = 0.89.

Q8.

Expected Return via CAPM

The risk‑free rate is 2.98%, the market risk premium is 6.13%, and the beta of the investment is 1.08.

1. What is the expected return of the investment according to the CAPM (in %)?

✅ Expected return = Rf + β × (market risk premium) = 9.59%.

Q9.

Variance and Volatility of a Three‑Asset Portfolio

You invest 54.96%, 32.37%, and 12.67% of your funds in three assets with volatilities 14.07%, 23.99%, and 14.44%, respectively. The pairwise correlations are ρ₁₂ = 0, ρ₁₃ = -0.07, and ρ₂₃ = -0.09.

1. What is the variance of the portfolio?

✅ Variance = Σwᵢ²σᵢ² + 2Σ wᵢwⱼσᵢσⱼρᵢⱼ = 118.99.

2. What is the standard deviation of the portfolio (in %)?

✅ Standard deviation = √(variance) = 10.91%.

Q10.

Risk‑Free Combination and Sharpe Ratio

You build a portfolio by combining the risk‑free asset (rate = 3.98%) with a tangent portfolio (expected return = 11.8%, volatility = 17.64%) using a weight of 135.54% in the tangent portfolio (weights greater than 100% imply leverage).

1. What is the expected return of your combined portfolio (in %)?

✅ Expected return = w×ret_tan + (1−w)×Rf = 14.58%.

2. What is the standard deviation of your combined portfolio (in %)?

✅ Standard deviation = |w|×σ_tan = 23.92%.

3. What is the Sharpe ratio of your combined portfolio?

✅ Sharpe ratio = risk premium / volatility = 0.44.

Q11.

Sharpe Calculation

An investment has an expected return of 7.02%, a risk‑free rate of 2.15%, and a volatility (standard deviation) of 11.44%.

1. What is the risk premium (in %)?

✅ Risk premium = expected return − risk‑free rate = 4.87%.

2. What is the Sharpe ratio?

✅ Sharpe ratio = risk premium / volatility = 0.426.

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