What Is The Sharpe, Treynor Ratio, And Jensen's Alpha

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What is the Sharpe ratio, Treynor ratio, and Jensen’s alpha for each portfolio? (Negative values should be indicated by a minus sign. Leave no cells blank - be certain to enter "0" wherever required. Do not round intermediate calculations. Round your Sharpe ratio answers and Treynor ratio answers to 5 decimal places and Jensen's alpha answers to 2 decimal places. Omit the "%" sign in your response.) Portfolio RP σP βP Portfolio Sharpe Ratio Treynor Ratio Jensen's Alpha Cannot figure out the Treynor Jensen's or Market X 16.5 % 37 % 1.40 X % Y 15..15 Y % Z 7.4 22 .70 Z % Market 11..00 Market % Risk-free 5.

Paper For Above instruction

The goal of this analysis is to compute key performance ratios—Sharpe ratio, Treynor ratio, and Jensen's alpha—for multiple investment portfolios based on provided data. Accurate calculation of these ratios offers insight into the risk-adjusted returns of each portfolio, facilitating informed investment decision-making.

Firstly, understanding each ratio's definition and calculation methodology is essential:

  • Sharpe Ratio: Measures excess return per unit of total risk, calculated as (Portfolio Return – Risk-Free Rate) divided by the standard deviation of portfolio returns.
  • Treynor Ratio: Assesses risk-adjusted return based on systematic risk, calculated as (Portfolio Return – Risk-Free Rate) divided by beta (systematic risk).
  • Jensen's Alpha: Represents the abnormal return of a portfolio over the expected return predicted by the Capital Asset Pricing Model (CAPM), calculated as Actual Portfolio Return minus the CAPM predicted return.

Given the provided data, the calculations proceed as follows. For each portfolio, we will first determine the excess returns by subtracting the risk-free rate from the portfolio returns. Then, we'll compute the Sharpe ratio by dividing this excess return by the portfolio's standard deviation. The Treynor ratio involves dividing the excess return by beta. Jensen's alpha requires calculating the expected return using CAPM, based on beta, market return, and risk-free rate, then subtracting this from the actual portfolio return to find the alpha.

Applying the formulas and substituting the provided values:

Portfolio X:

  • Return: 16.5%; Beta: 1.40; Standard deviation: 37%; Market return: 11.00%; Risk-free rate: 5%.
  • Excess Return = 16.5% – 5% = 11.5%.
  • Sharpe Ratio = 11.5% / 37% ≈ 0.31081.
  • Treynor Ratio = 11.5% / 1.40 ≈ 0.08214.
  • Expected return via CAPM = Risk-Free + Beta (Market Return – Risk-Free) = 5% + 1.40 (11% – 5%) = 5% + 1.40 * 6% = 5% + 8.4% = 13.4%.
  • Jensen’s Alpha = Actual Return – Expected Return = 16.5% – 13.4% = 3.10%.

Portfolio Y:

  • Return: 15.15%; Beta: Y%; Standard deviation: Y%; Market return: 11%; Risk-free rate: 5%.
  • Since specific values of Y are missing, we would substitute actual figures when available, following the same computation method used above.

Portfolio Z:

  • Return: 7.4%; Beta: 0.70; Standard deviation: 22%; Market return: 11%; Risk-free rate: 5%.
  • Excess Return = 7.4% – 5% = 2.4%.
  • Sharpe Ratio = 2.4% / 22% ≈ 0.10909.
  • Treynor Ratio = 2.4% / 0.70 ≈ 3.42857.
  • Expected return via CAPM = 5% + 0.70 (11% – 5%) = 5% + 0.70 6% = 5% + 4.2% = 9.2%.
  • Jensen’s Alpha = 7.4% – 9.2% = -1.80%.

Note: Due to incomplete data, especially for Portfolio Y, some calculations are illustrative. When the actual data are available, the same formulas should be applied precisely.

In conclusion, these ratios provide a comprehensive assessment of each portfolio’s risk-adjusted performance. The Sharpe ratio evaluates total risk, the Treynor ratio focuses on systematic risk, and Jensen’s alpha indicates performance compared to CAPM expectations. Investors can use these insights to optimize portfolio allocations and enhance returns.

References

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