Module 3: Numeric Questions

For students

Last updated: 23/09/2025 22:44

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	Slides	T/F	MCQ	Numeric	Long	Self-quiz
3	ch11	🎞️	✅	❓	🔢	📝	🧪

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Answer the following questions based on the discussions in class.

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Q1.

Expected Return, Variance, and Standard Deviation of a Portfolio

What is the expected return of a portfolio that invests 30.48% in Asset 1 (expected return = 11.18%) and the remaining 69.52% in Asset 2 (expected return = 3.27% )?

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

✅ Expected return = w₁×r₁ + w₂×r₂ = 5.68%.

What is the variance of the portfolio above, given correlation 0.21, standard deviation of Asset 1 = 32.38% and of Asset 2 = 15.2%?

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

✅ Variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ = 2.53.

What is the standard deviation of the portfolio above?

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

✅ Standard deviation = √(variance) = 15.9%.

Q2.

Portfolio Weight Calculation for Multiple Stocks

Suppose you purchase 226 shares of Company A at $34.83 per share, 118 shares of Company B at $48.58 per share, and 141 shares of Company C at $45.4 per share.

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

✅ Weight on Company A = value_A / total_value × 100 = 39.35%.

Q3.

Portfolio Variance and Standard Deviation Calculation

You invest 84.18% of your funds in Asset A (volatility = 18.26%) and 15.82% in Asset B (volatility = 22.76%). The correlation between the two assets is 0.65.

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 = 321.77.

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

✅ Portfolio standard deviation = √(variance) = 17.94%.

Q4.

Calculating Beta Using Covariance and Market Variance

Given that the covariance between a stock and the market is 5.02 and the variance of the market is 5.68,

1. What is the beta of the stock?

✅ Beta = Cov(asset, market) / Var(market) = 0.88.

Q5.

Risk Premium and Sharpe Ratio Calculation

An investment has an expected return of 8.63%, a risk‑free rate of 1.77%, and a volatility (standard deviation) of 13.36%.

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

✅ Risk premium = expected return − risk‑free rate = 6.86%.

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

✅ Sharpe ratio = risk premium / volatility = 0.51.

Q6.

Combining Risky Asset and Risk‑Free Rate

You invest 99.31% of your funds in a risky portfolio (expected return = 9.48%, volatility = 29.85%) and the remainder 0.69% in the risk‑free asset (rate = 2.92%).

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

✅ Expected return = w×ret_risky + (1−w)×Rf = 9.43%.

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 = 29.64%.

Q7.

Beta of a Three‑Asset Portfolio

You hold a portfolio composed of three assets with betas 0.99, 1.37, and 0.23, invested in proportions 90.45%, 7.26%, and 2.29%, respectively.

1. What is the beta of the portfolio?

✅ Portfolio beta = w₁×β₁ + w₂×β₂ + w₃×β₃ = 1.

Q8.

Expected Return via CAPM

The risk‑free rate is 3.09%, the market risk premium is 6.41%, and the beta of the investment is 1.32.

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

✅ Expected return = Rf + β × (market risk premium) = 11.54%.

Q9.

Variance and Volatility of a Three‑Asset Portfolio

You invest 37.95%, 39.78%, and 22.28% of your funds in three assets with volatilities 14.2%, 25%, and 35.92%, respectively. The pairwise correlations are ρ₁₂ = 0.79, ρ₁₃ = -0.13, and ρ₂₃ = 0.66.

1. What is the variance of the portfolio?

✅ Variance = Σwᵢ²σᵢ² + 2Σ wᵢwⱼσᵢσⱼρᵢⱼ = 369.81.

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

✅ Standard deviation = √(variance) = 19.23%.

Q10.

Risk‑Free Combination and Sharpe Ratio

You build a portfolio by combining the risk‑free asset (rate = 3.47%) with a tangent portfolio (expected return = 15.33%, volatility = 18.34%) using a weight of 60.48% 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 = 10.64%.

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

✅ Standard deviation = |w|×σ_tan = 11.09%.

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

✅ Sharpe ratio = risk premium / volatility = 0.65.

Q11.

Sharpe Calculation

An investment has an expected return of 8.13%, a risk‑free rate of 2.3%, and a volatility (standard deviation) of 15.97%.

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

✅ Risk premium = expected return − risk‑free rate = 5.83%.

2. What is the Sharpe ratio?

✅ Sharpe ratio = risk premium / volatility = 0.365.