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CMA Final SFM – Portfolio Management Master Tool

CMA Final SFM - Portfolio Management Master Tool | CMA Knowledge

CMA Final SFM - Portfolio Management Master Tool

Complete interactive resource with 100% accurate calculations and conceptual clarity

5 Complete Questions
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100% Accurate Solutions
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Key Formulas
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Latest CMA Final SFM Portfolio Management Questions 100% Accurate Solutions

Complete questions with detailed interpretations, step-by-step solutions, and conceptual explanations.

1 Portfolio Standard Deviation & Diversification Benefits

Problem: You are considering investing in a two-asset portfolio comprising Stock A and Stock B. Stock A has an expected return of 12% and standard deviation of 15%. Stock B has an expected return of 18% and standard deviation of 22%. The correlation coefficient between the two stocks is 0.25. If you decide to invest 60% of your funds in Stock A and 40% in Stock B, calculate:

  1. The expected return of the portfolio
  2. The portfolio standard deviation
  3. The diversification benefit achieved

Question Interpretation:

This question tests your understanding of portfolio theory, specifically how to calculate expected returns and risk for a two-asset portfolio. The key insight is that portfolio risk depends not only on individual asset risks but also on how they move together (correlation). A correlation less than +1 provides diversification benefits, reducing overall portfolio risk below the weighted average of individual risks.

Key Concepts:

  • Portfolio expected return is a weighted average of individual returns
  • Portfolio variance depends on weights, variances, and covariance
  • Diversification reduces risk when correlation < +1
  • Covariance = Correlation × σA × σB

Step-by-Step Solution:

1 Calculate portfolio expected return

E(Rp) = wA × E(RA) + wB × E(RB)
E(Rp) = 0.60 × 12% + 0.40 × 18%
E(Rp) = 7.2% + 7.2% = 14.4%

2 Calculate covariance between Stock A and Stock B

CovAB = ρAB × σA × σB
CovAB = 0.25 × 15% × 22%
CovAB = 0.25 × 0.15 × 0.22 = 0.00825

3 Calculate portfolio variance

σp2 = wA2σA2 + wB2σB2 + 2wAwBCovAB
σp2 = (0.60)2(0.15)2 + (0.40)2(0.22)2 + 2(0.60)(0.40)(0.00825)
σp2 = 0.0081 + 0.007744 + 0.00396 = 0.019804

4 Calculate portfolio standard deviation

σp = √0.019804 = 0.1407 or 14.07%

5 Calculate diversification benefit

Weighted average σ = wAσA + wBσB
Weighted average σ = 0.60 × 15% + 0.40 × 22% = 9% + 8.8% = 17.8%
Diversification benefit = 17.8% - 14.07% = 3.73%
Final Answers:
  • Expected Portfolio Return = 14.4%
  • Portfolio Standard Deviation = 14.07%
  • Diversification Benefit = 3.73% risk reduction
2 Capital Asset Pricing Model (CAPM) & Security Valuation

Problem: The risk-free rate of return is 6%. The expected market return is 13%. Stock X has a beta of 1.2 and is currently trading at a price that implies an expected return of 16%.

  1. Calculate the required rate of return for Stock X using CAPM
  2. Determine if Stock X is overvalued or undervalued
  3. Calculate the alpha of Stock X
  4. If the standard deviation of Stock X is 24% and the standard deviation of the market is 18%, calculate the systematic and unsystematic risk of Stock X

Question Interpretation:

This question tests your understanding of CAPM, security valuation, and risk decomposition. CAPM provides a theoretical required return based on systematic risk. Comparing this to the actual expected return helps identify mispriced securities. Alpha measures the excess return over the CAPM required return. Total risk can be decomposed into systematic and unsystematic components.

Step-by-Step Solution:

1 Calculate required return using CAPM

E(Ri) = Rf + βi(Rm - Rf)
E(RX) = 6% + 1.2(13% - 6%)
E(RX) = 6% + 1.2(7%) = 6% + 8.4% = 14.4%

2 Determine if stock is overvalued or undervalued

CAPM Required Return = 14.4%
Actual Expected Return = 16%
Since actual return > required return, the stock is UNDERVALUED

3 Calculate alpha

α = Actual Return - CAPM Required Return
α = 16% - 14.4% = 1.6%

A positive alpha indicates the stock is expected to outperform the market on a risk-adjusted basis.

4 Calculate systematic and unsystematic risk

Total Risk (Variance) = (24%)² = 0.0576
Systematic Risk = β² × Market Variance = (1.2)² × (18%)²
Systematic Risk = 1.44 × 0.0324 = 0.046656
Unsystematic Risk = Total Risk - Systematic Risk
Unsystematic Risk = 0.0576 - 0.046656 = 0.010944

Systematic Risk = 46.66%, Unsystematic Risk = 10.94% (of total variance)

Final Answers:
  • CAPM Required Return = 14.4%
  • Stock X is UNDERVALUED
  • Alpha = 1.6%
  • Systematic Risk = 46.66%, Unsystematic Risk = 10.94%
3 Sharpe Ratio & Portfolio Performance Evaluation

Problem: You have the following information about three mutual funds:

  • Fund A: Return = 15%, Standard Deviation = 12%, Beta = 1.1
  • Fund B: Return = 18%, Standard Deviation = 20%, Beta = 1.4
  • Fund C: Return = 12%, Standard Deviation = 8%, Beta = 0.9

The risk-free rate is 5%. Calculate and compare:

  1. Sharpe Ratio for each fund
  2. Treynor Ratio for each fund
  3. Jensen's Alpha for each fund (assuming market return is 11%)
  4. Rank the funds based on each performance measure

Question Interpretation:

This question evaluates your understanding of different portfolio performance measures. Sharpe Ratio measures risk-adjusted return using total risk, Treynor Ratio uses systematic risk, and Jensen's Alpha measures excess return over CAPM expectations. Each measure has different applications depending on the investor's perspective and the portfolio's characteristics.

Step-by-Step Solution:

1 Calculate Sharpe Ratio for each fund

Sharpe Ratio = (Rp - Rf) / σp

Fund A: (15% - 5%) / 12% = 10% / 12% = 0.833
Fund B: (18% - 5%) / 20% = 13% / 20% = 0.650
Fund C: (12% - 5%) / 8% = 7% / 8% = 0.875

2 Calculate Treynor Ratio for each fund

Treynor Ratio = (Rp - Rf) / βp

Fund A: (15% - 5%) / 1.1 = 10% / 1.1 = 0.0909
Fund B: (18% - 5%) / 1.4 = 13% / 1.4 = 0.0929
Fund C: (12% - 5%) / 0.9 = 7% / 0.9 = 0.0778

3 Calculate Jensen's Alpha for each fund

α = Rp - [Rf + β(Rm - Rf)]

Fund A: 15% - [5% + 1.1(11% - 5%)] = 15% - [5% + 6.6%] = 15% - 11.6% = 3.4%
Fund B: 18% - [5% + 1.4(11% - 5%)] = 18% - [5% + 8.4%] = 18% - 13.4% = 4.6%
Fund C: 12% - [5% + 0.9(11% - 5%)] = 12% - [5% + 5.4%] = 12% - 10.4% = 1.6%

4 Rank the funds based on each measure

Sharpe Ratio Ranking: Fund C (0.875) > Fund A (0.833) > Fund B (0.650)
Treynor Ratio Ranking: Fund B (0.0929) > Fund A (0.0909) > Fund C (0.0778)
Jensen's Alpha Ranking: Fund B (4.6%) > Fund A (3.4%) > Fund C (1.6%)
Final Answers:
  • Sharpe Ratios: A=0.833, B=0.650, C=0.875
  • Treynor Ratios: A=0.0909, B=0.0929, C=0.0778
  • Jensen's Alpha: A=3.4%, B=4.6%, C=1.6%
  • Rankings vary by measure due to different risk perspectives
4 Portfolio Beta & Market Risk

Problem: Mr. Verma has constructed a portfolio with the following securities:

  • Stock P: Investment ₹5,00,000, Beta 1.2
  • Stock Q: Investment ₹3,00,000, Beta 0.8
  • Stock R: Investment ₹2,00,000, Beta 1.5
  • Stock S: Investment ₹4,00,000, Beta 1.0

The risk-free rate is 6% and the market risk premium is 7%. Calculate:

  1. The portfolio beta
  2. The required return on the portfolio using CAPM
  3. If Mr. Verma wants to reduce the portfolio beta to 1.0 by adding a risk-free security, how much should he invest in the risk-free security?
  4. The new portfolio return after adding the risk-free security

Question Interpretation:

This question tests portfolio beta calculation and its application in CAPM. Portfolio beta is the weighted average of individual security betas. Adding risk-free assets to a portfolio reduces its beta, as risk-free securities have beta = 0. This concept is fundamental to portfolio construction and risk management.

Step-by-Step Solution:

1 Calculate total portfolio value and weights

Total Value = 5,00,000 + 3,00,000 + 2,00,000 + 4,00,000 = ₹14,00,000

Weights:
wP = 5,00,000 / 14,00,000 = 0.3571
wQ = 3,00,000 / 14,00,000 = 0.2143
wR = 2,00,000 / 14,00,000 = 0.1429
wS = 4,00,000 / 14,00,000 = 0.2857

2 Calculate portfolio beta

βp = wPβP + wQβQ + wRβR + wSβS
βp = 0.3571×1.2 + 0.2143×0.8 + 0.1429×1.5 + 0.2857×1.0
βp = 0.4285 + 0.1714 + 0.2143 + 0.2857 = 1.0999 ≈ 1.10

3 Calculate portfolio required return using CAPM

E(Rp) = Rf + βp(Rm - Rf)
E(Rp) = 6% + 1.10 × 7% = 6% + 7.7% = 13.7%

4 Calculate investment in risk-free security to achieve beta = 1.0

Let x be the weight in risk-free security (β=0)
New β = (1-x)×1.10 + x×0 = 1.0
1.10(1-x) = 1.0
1.10 - 1.10x = 1.0
1.10x = 0.10
x = 0.0909 or 9.09%

Investment in risk-free = 0.0909 × 14,00,000 = ₹1,27,260

5 Calculate new portfolio return

New Portfolio Return = (1-x)×13.7% + x×6%
New Portfolio Return = 0.9091×13.7% + 0.0909×6%
New Portfolio Return = 12.45% + 0.55% = 13.0%
Final Answers:
  • Portfolio Beta = 1.10
  • Required Portfolio Return = 13.7%
  • Investment in Risk-Free Security = ₹1,27,260
  • New Portfolio Return = 13.0%
5 Minimum Variance Portfolio & Efficient Frontier

Problem: Consider two stocks, Stock M and Stock N, with the following characteristics:

  • Stock M: Expected Return = 14%, Standard Deviation = 18%
  • Stock N: Expected Return = 20%, Standard Deviation = 25%

The correlation coefficient between the two stocks is 0.30. Calculate:

  1. The weights of each stock in the minimum variance portfolio
  2. The expected return and standard deviation of the minimum variance portfolio
  3. The weights for an optimal portfolio with target return of 17%
  4. Compare the risk of the minimum variance portfolio with an equally weighted portfolio

Question Interpretation:

This question explores portfolio optimization concepts. The minimum variance portfolio has the lowest possible risk for the given assets. The optimal portfolio lies on the efficient frontier, which represents the set of portfolios offering the highest return for each level of risk. Understanding these concepts is crucial for constructing efficient portfolios.

Step-by-Step Solution:

1 Calculate weights for minimum variance portfolio

wM = [σN2 - ρMNσMσN] / [σM2 + σN2 - 2ρMNσMσN]
wM = [0.252 - 0.30×0.18×0.25] / [0.182 + 0.252 - 2×0.30×0.18×0.25]
wM = [0.0625 - 0.0135] / [0.0324 + 0.0625 - 0.027] = 0.049 / 0.0679 = 0.7216
wN = 1 - 0.7216 = 0.2784

2 Calculate minimum variance portfolio return and risk

E(RMVP) = 0.7216×14% + 0.2784×20% = 10.10% + 5.57% = 15.67%

Portfolio Variance = wM2σM2 + wN2σN2 + 2wMwNρMNσMσN
= (0.7216)2(0.18)2 + (0.2784)2(0.25)2 + 2×0.7216×0.2784×0.30×0.18×0.25
= 0.0169 + 0.0048 + 0.0054 = 0.0271

σMVP = √0.0271 = 0.1646 or 16.46%

3 Calculate weights for target return of 17%

wM × 14% + (1-wM) × 20% = 17%
14wM + 20 - 20wM = 17
-6wM = -3
wM = 0.50
wN = 0.50

4 Compare with equally weighted portfolio

Equally Weighted Portfolio (wM=0.5, wN=0.5):
E(R) = 0.5×14% + 0.5×20% = 17%
Variance = (0.5)2(0.18)2 + (0.5)2(0.25)2 + 2×0.5×0.5×0.30×0.18×0.25
= 0.0081 + 0.0156 + 0.00675 = 0.03045
σ = √0.03045 = 0.1745 or 17.45%

Risk Reduction = 17.45% - 16.46% = 0.99%
Final Answers:
  • Minimum Variance Weights: M=72.16%, N=27.84%
  • MVP Return=15.67%, MVP Risk=16.46%
  • Target Return Weights: M=50%, N=50%
  • MVP reduces risk by 0.99% compared to equal weights

Key Portfolio Management Formulas & Applications

Portfolio Expected Return

Weighted average of individual asset returns

E(Rp) = w1E(R1) + w2E(R2) + ... + wnE(Rn)

Interpretation: The expected return of a portfolio is simply the weighted average of the expected returns of its individual assets. This linear relationship makes return calculation straightforward, but doesn't account for risk reduction through diversification.

Application: Used to estimate the overall return of a portfolio based on the expected performance of its components and their allocation weights.

Portfolio Variance (Two Assets)

Measures total portfolio risk considering correlation

σp2 = wA2σA2 + wB2σB2 + 2wAwBσAσBρAB

Interpretation: Portfolio variance depends not only on individual asset variances but also on their covariance (correlation). When correlation is less than +1, diversification benefits occur, reducing overall portfolio risk below the weighted average of individual risks.

Application: Critical for understanding how different asset combinations affect overall portfolio risk and for constructing efficient portfolios.

Portfolio Management Interactive Calculators

Calculate key portfolio metrics by entering values below. Experiment with different inputs to see how they affect your results.

CAPM & Performance Metrics Calculator

Calculation Results 100% Accurate

Model Question with Interactive Calculator

Model Question: Two-Asset Portfolio Optimization

Problem: Mr. Sharma is considering investing in two stocks: Stock X and Stock Y. Stock X has an expected return of 14% and standard deviation of 20%. Stock Y has an expected return of 18% and standard deviation of 25%. The correlation coefficient between the two stocks is 0.3. Mr. Sharma wants to invest 60% in Stock X and 40% in Stock Y.

Calculate the expected return, standard deviation, and diversification benefit of this portfolio.

Portfolio Inputs Live Calculator

Portfolio Calculation Results 100% Accurate

Step-by-Step Solution Methodology:

1 Calculate Portfolio Expected Return

E(Rp) = wXE(RX) + wYE(RY)

2 Calculate Covariance between Stocks

CovXY = ρXY × σX × σY

3 Calculate Portfolio Variance

σp2 = wX2σX2 + wY2σY2 + 2wXwYCovXY

4 Calculate Portfolio Standard Deviation

σp = √σp2

5 Calculate Diversification Benefit

Diversification Benefit = Weighted Average σ - Portfolio σ

Disclaimer

This educational tool is designed for CMA Final SFM students to enhance their understanding of portfolio management concepts. All calculations are based on standard financial formulas and provide 100% accurate results when correct inputs are provided. However, this resource should be used as a supplement to official ICMAI study materials and not as a replacement. Always refer to the latest ICMAI curriculum and consult with qualified instructors for definitive examination guidance. The calculations and examples provided are for educational purposes only and should not be used for actual investment decisions without professional financial advice.

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