The passive management is a style of investing associated with mutual and exchange-traded funds (ETF) where an investor aims to mirror a market index.

We can build the passive strategy through the following steps :

- Choice of a panel of funds ;
- Ranking of the funds ;
- Choice of the funds in the panel with the best ranking.

The panel is chosen in accordance with particular requirements (filter by class, macrocategory, assets, country, risk, currencies and so on ; it depends on the investor's preferences : see the following link ANIMA sgr products).

For example, the aim of the analysis is to rank the funds of the system "Anima Italia" ; the fund ISIN codes are respectively : IT0001040051, IT0005158784, IT0004896541.

The asset allocation is composed by equities, largely ; the currency is EUR ; the equity country is Italy, substantially : for further information, please consult the portfolio breakdown and the fund profile.

The site provides a rating and a benchmark for each fund ; however, we can build our benchmark and our rating. For the benchmark, we choose the FTSE-mib index because it can be a good comparative parameter, given the structure of the funds. For the ranking, we use the classic portfolio performance indicators. In this way, we import the NAVs on a excel sheet and then we calculate the daily returns.

The time frame is from February 22, 2016 to date. I converted the daily returns into annual returns, for greater significance.

Historical data (hidden cells for space requirements) : data source ANIMA sgr. |

The same for the FTSE-mib index, aka benchmark.

Historical data : data source Investing.com. |

In the following chart, we can see the performance of each fund compared to the benchmark performance.

Now, we can calculate the performance indicators :

- The
: the ratio uses standard deviation to measure a fund's risk-adjusted returns ; it quantifies a fund's return in excess of our proxy for a risk-free investment. It is equals to :__Sharp's Measure__

(R - Rf) / Std Dev

R = average return of the fund ; Std Dev = standard deviation of the fund

Rf = risk-free rate (I assume the average return of the BTP 10Y ITA)

- The
: the ratio is equal to the previous one ; however, the risk is adjusted for the beta. The index is equal to :__Treynor's Measure__

(R - Rf) / Beta

Beta = beta of the fund

- The
: the index is a risk-adjusted measure that compares the average return of a fund to the estimated return of the Capital Asset Pricing Model (CAPM). The formula is equal to :__Jensen's Alpha__

R - [Rf + Beta*(Rm - Rf)]

Rm = average return of the benchmark (or market index)

- The
: it is a risk-adjusted measure ; it explains the surplus return of the fund compared to the risk-free investment, considering that the variability of the fund is equal to the variability of the benchmark. The formula is :__M Squared Measure__

(Sharp's Measure)*(Std Devm) + Rf

Std Devm = standard deviation of the benchmark (or market index)

- The
: the structure is the same compared to the previous one ; the difference is the risk, systematic risk or beta ; substantially, it calcualtes the surplus return compared to the risk-free rate, under the assumption that the systematic risk of the fund is equal to the systematic risk of the market. The formula is :__T Squared Measure__

[(1 / Beta)*(R - Rf) - (Rm - Rf)]

- The
: rather than considering premiums regarding the risk-free asset, the index explains the surplus return with a minimum accettable return ; then, about the risk, it considers a minimum accettable risk, aka down side risk (the variability not appreciated by the investor ; we calculate a semi-standard deviation, only the negative deviations from the mean). The ratio is equal to :__Sortino Index__

(R - Minimum Return) / Down Side Risk

For semplicity, we consider the minimum return equal to the risk-free return

Finally, the higher the ratios, the better fund past performance (we must note that the future performance is not linked to the past performance ; however, it is a good beginning).

In this way, we can calculate the ratios and rank the three funds (see the following table).

The ranking is :

1) Fund ISIN code IT0004896541 (the best) ;

2) Fund ISIN code IT0001040051 ;

3) Fund ISIN code IT0005158784 (the worst).

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