Analyze the following portfolios performance using Jensen index, Treynor index and Sharpe index versus the market:


Observed Rate of Return Beta (β) Residual Variance
Magic fund 15% 1.5 0.02
Shanti fund 10% 0.5 0.00

Risk-free rate of interest is 50%, return on the market portfolio is 12% and standard deviation is 0.04.

The Sharpe ratio or Sharpe index or Sharpe measure or reward-to-variability ratio is a measure of the excess return (or risk premium) per unit of risk in an investment asset or a trading strategy, named after William Forsyth Sharpe. Since its revision by the original author in 1994, it is defined as:

where R is the asset return, Rf is the return on a benchmark asset, such as the risk free rate of return, E[R − Rf] is the expected value of the excess of the asset return over the benchmark return, and σ is the standard deviation of the excess of the asset return. (This is often confused with the excess return over the benchmark return; the Sharpe ratio utilizes the asset standard deviation whereas the information ratio utilizes standard deviation of excess return over the benchmark, i.e. the tracking error, as the denominator.) Note, if Rf is a constant risk free return throughout the period,

The Sharpe ratio is used to characterize how well the return of an asset compensates the investor for the risk taken, the higher the Sharpe ratio number the better. When comparing two assets each with the expected return E[R] against the same benchmark with return Rf, the asset with the higher Sharpe ratio gives more return for the same risk. Investors are often advised to pick investments with high Sharpe ratios. However like any mathematical model it relies on the data being correct. Pyramid schemes with a long duration of operation would typically provide a high Sharpe ratio when derived from reported returns, but the inputs are false. When examining the investment performance of assets with smoothing of returns (such as with-profits funds) the Sharpe ratio should be derived from the performance of the underlying assets rather than the fund returns.
Sharpe ratios, along with Treynor ratios and Jensen's alphas, are often used to rank the performance of portfolio or mutual fund managers.

To analyze the performance of the portfolios using Jensen index, Treynor index, and Sharpe index, we need to calculate the expected rate of return and risk-adjusted measures for each portfolio.

Let's start by calculating the expected rate of return for each portfolio using the Capital Asset Pricing Model (CAPM):

Expected Rate of Return = Risk-Free Rate + Beta * (Market Return - Risk-Free Rate)

1. Magic Fund:
Expected Rate of Return = 0.5 + 1.5 * (12% - 0.5) = 0.5 + 1.5 * 11.5% = 18.25%

2. Shanti Fund:
Expected Rate of Return = 0.5 + 0.5 * (12% - 0.5) = 0.5 + 0.5 * 11.5% = 6.75%

Next, let's calculate the risk-adjusted measures for each portfolio:

1. Jensen Index:
Jensen Index = Observed Rate of Return - Expected Rate of Return
- For Magic Fund: Jensen Index = 15% - 18.25% = -3.25%
- For Shanti Fund: Jensen Index = 10% - 6.75% = 3.25%

2. Treynor Index:
Treynor Index = (Observed Rate of Return - Risk-Free Rate) / Beta
- For Magic Fund: Treynor Index = (15% - 0.5%) / 1.5 = 14.5% / 1.5 = 9.67%
- For Shanti Fund: Treynor Index = (10% - 0.5%) / 0.5 = 9.5% / 0.5 = 19.00%

3. Sharpe Index:
Sharpe Index = (Observed Rate of Return - Risk-Free Rate) / Standard Deviation
- For Magic Fund: Sharpe Index = (15% - 0.5%) / 0.04 = 14.5% / 0.04 = 362.5%
- For Shanti Fund: Sharpe Index = (10% - 0.5%) / 0.04 = 9.5% / 0.04 = 237.5%

Comparing the performance of the portfolios against the market:

- Jensen Index: A positive Jensen Index indicates that the portfolio has outperformed the expected return based on the CAPM model. The Shanti Fund has a positive Jensen Index of 3.25%, indicating it has outperformed, while the Magic Fund has a negative Jensen Index of -3.25%, indicating it has underperformed.

- Treynor Index: The Treynor Index measures the excess return per unit of systematic risk (beta). A higher Treynor Index indicates better risk-adjusted performance. The Shanti Fund has a higher Treynor Index of 19.00% compared to the Magic Fund's Treynor Index of 9.67%, indicating better risk-adjusted performance for the Shanti Fund.

- Sharpe Index: The Sharpe Index measures the excess return per unit of total risk (standard deviation). A higher Sharpe Index indicates better risk-adjusted performance. The Magic Fund has a higher Sharpe Index of 362.5% compared to the Shanti Fund's Sharpe Index of 237.5%, indicating better risk-adjusted performance for the Magic Fund.

In summary, based on the Jensen Index, the Magic Fund has underperformed while the Shanti Fund has outperformed. However, based on the Treynor Index and Sharpe Index, the Shanti Fund has shown better risk-adjusted performance compared to the Magic Fund.

To analyze the performance of the given portfolios using Jensen index, Treynor index, and Sharpe index, we need to calculate the expected return and risk-adjusted measures for each portfolio. Let's break down the steps:

1. Calculate the expected return for each portfolio:
The expected return of a portfolio can be calculated using the capital asset pricing model (CAPM) formula:
Expected Return = Risk-free rate + Beta * (Return on market portfolio - Risk-free rate)

For the Magic fund:
Expected Return = 0.5 + 1.5 * (0.12 - 0.5)
= 0.5 + 1.5 * 0.065
= 0.5 + 0.0975
= 0.5975 or 59.75%

For the Shanti fund:
Expected Return = 0.5 + 0.5 * (0.12 - 0.5)
= 0.5 + 0.5 * (-0.38)
= 0.5 - 0.19
= 0.31 or 31%

2. Calculate each portfolio's risk-adjusted measures:

Jensen index: The Jensen index measures the excess return of a portfolio over its expected return per unit of systematic risk (beta).
Jensen Index = Observed Return - Expected Return

For the Magic fund:
Jensen Index = 0.15 - 0.5975
= -0.4475 or -44.75%

For the Shanti fund:
Jensen Index = 0.1 - 0.31
= -0.21 or -21%

Treynor index: The Treynor index measures the excess return of a portfolio per unit of systematic risk (beta).
Treynor Index = (Observed Return - Risk-free rate) / Beta

For the Magic fund:
Treynor Index = (0.15 - 0.5) / 1.5
= -0.35/1.5
= -0.2333 or -23.33%

For the Shanti fund:
Treynor Index = (0.1 - 0.5) / 0.5
= -0.4/0.5
= -0.8 or -80%

Sharpe index: The Sharpe index measures the excess return of a portfolio per unit of total risk (standard deviation).
Sharpe Index = (Observed Return - Risk-free rate) / Standard Deviation

For the Magic fund:
Sharpe Index = (0.15 - 0.5) / 0.04
= -0.35/0.04
= -8.75 or -875%

For the Shanti fund:
Sharpe Index = (0.1 - 0.5) / 0.04
= -0.4/0.04
= -10 or -1000%

Now, we have calculated the Jensen index, Treynor index, and Sharpe index for each portfolio. These measures provide insights into the risk-adjusted performance of the portfolios compared to the market.