Module 3: Numeric Questions

For students

Last updated: 17/08/2025 19:52

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
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 56.16% in Asset 1 (expected return = 19.2%) and the remaining 43.84% in Asset 2 (expected return = 21.3% )?

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

✅ Expected return = w₁×r₁ + w₂×r₂ = 20.12%.

What is the variance of the portfolio above, given correlation 0.84, standard deviation of Asset 1 = 24.23% and of Asset 2 = 15.77%?

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

✅ Variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ = 3.91.

What is the standard deviation of the portfolio above?

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

✅ Standard deviation = √(variance) = 19.77%.

Q2.

Portfolio Weight Calculation for Multiple Stocks

Suppose you purchase 155 shares of Company A at $63.81 per share, 164 shares of Company B at $36.44 per share, and 73 shares of Company C at $56.25 per share.

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

✅ Weight on Company A = value_A / total_value × 100 = 49.52%.

Q3.

Portfolio Variance and Standard Deviation Calculation

You invest 5.17% of your funds in Asset A (volatility = 11.31%) and 94.83% in Asset B (volatility = 22.93%). The correlation between the two assets is 0.77.

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 = 492.71.

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

✅ Portfolio standard deviation = √(variance) = 22.2%.

Q4.

Calculating Beta Using Covariance and Market Variance

Given that the covariance between a stock and the market is 4.74 and the variance of the market is 4.28,

1. What is the beta of the stock?

✅ Beta = Cov(asset, market) / Var(market) = 1.11.

Q5.

Risk Premium and Sharpe Ratio Calculation

An investment has an expected return of 9.81%, a risk‑free rate of 2.02%, and a volatility (standard deviation) of 15.52%.

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

✅ Risk premium = expected return − risk‑free rate = 7.79%.

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

✅ Sharpe ratio = risk premium / volatility = 0.5.

Q6.

Combining Risky Asset and Risk‑Free Rate

You invest 7.51% of your funds in a risky portfolio (expected return = 11.76%, volatility = 17.77%) and the remainder 92.49% in the risk‑free asset (rate = 3.12%).

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

✅ Expected return = w×ret_risky + (1−w)×Rf = 3.77%.

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 = 1.33%.

Q7.

Beta of a Three‑Asset Portfolio

You hold a portfolio composed of three assets with betas 0.52, 1.7, and 0.47, invested in proportions 36.99%, 26.48%, and 36.52%, respectively.

1. What is the beta of the portfolio?

✅ Portfolio beta = w₁×β₁ + w₂×β₂ + w₃×β₃ = 0.81.

Q8.

Expected Return via CAPM

The risk‑free rate is 4.03%, the market risk premium is 6.04%, and the beta of the investment is 1.06.

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

✅ Expected return = Rf + β × (market risk premium) = 10.43%.

Q9.

Variance and Volatility of a Three‑Asset Portfolio

You invest 20.44%, 29.79%, and 49.77% of your funds in three assets with volatilities 12.37%, 19.47%, and 18.22%, respectively. The pairwise correlations are ρ₁₂ = 0.3, ρ₁₃ = 0.48, and ρ₂₃ = 0.75.

1. What is the variance of the portfolio?

✅ Variance = Σwᵢ²σᵢ² + 2Σ wᵢwⱼσᵢσⱼρᵢⱼ = 231.84.

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

✅ Standard deviation = √(variance) = 15.23%.

Q10.

Risk‑Free Combination and Sharpe Ratio

You build a portfolio by combining the risk‑free asset (rate = 3.4%) with a tangent portfolio (expected return = 9.04%, volatility = 19.81%) using a weight of 137.86% 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 = 11.18%.

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

✅ Standard deviation = |w|×σ_tan = 27.31%.

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

✅ Sharpe ratio = risk premium / volatility = 0.28.

Q11.

Sharpe Calculation

An investment has an expected return of 8.37%, a risk‑free rate of 2.03%, and a volatility (standard deviation) of 11.47%.

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

✅ Risk premium = expected return − risk‑free rate = 6.34%.

2. What is the Sharpe ratio?

✅ Sharpe ratio = risk premium / volatility = 0.553.