Saturday, May 5, 2018

Markowitz efficient frontier : backtesting.

The aim of the analysis is to build the efficient frontier that compares the average return (y) to the risk or standard deviation (x) of the top 100 Italian stocks (the market cap is the proxy of the size). Secondly, I will build one hundred portfolios (each portfolio is composed by one stock and each stock has the same weight, for simplicity), by decreasing order of best risk-reward ratio.
Thirdly, I will test the following assumption : the order of best risk-reward ratio should be the same also for the future performances (at least, substantially). 
In this way, the third step will be the backtesting that will cover a medium/long time frame, for obvious reasons and it will be a continuous updating.

The time frame of the past returns is 5 years : it means sixty returns (monthly returns), from April 30, 2013 to April 30, 2018. The source of the data is the following (historical data) :
The source of the stock screener is the following : Stock Screener -

About the data, I converted the monthly returns into annual returns, for greater significance. 

Average Return (annual) = Average Return (monthly) * 12
Standard Deviation (annual) = SQRT [(Standard Deviation (monthly) )^2*12]

So, I adjusted the return for the risk (Standard Deviation), through the following formula : 

Risk Adjusted Return = Average Return / Standard Deviation

Finally, I ranked the stocks, according to the abovementioned ratio (best risk-reward ratio). 
For simplicity, I built ten groups, always respecting the previous order. 
According to the portofolio theory (please consult the links : Markowitz efficient frontier ; images), the best stocks are those with the best ratio : they offers a greater return, given a risk rate or they offers a lower risk, given a return rate. Then, we have the following assumption : the expected returns and the expected risks are based on the past data. Of course, this is a limit and the aim of the analysis is also to implement a backtesting and to test the assumption. 

The following chart shows the efficient frontier. 

The following tables show the ten groups, by decreasing order of best risk-reward ratio (--> ranking).

In the future, I will test the assumption, as mentioned. The stocks with good past performance (with given risk) should outperform the others. 

Sunday, April 29, 2018

The passive management.

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 : 
  1. Choice of a panel of funds ;
  2. Ranking of the funds ; 
  3. 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

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 Sharp's Measure : 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 : 
(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 Treynor's Measure : the ratio is equal to the previous one ; however, the risk is adjusted for the beta. The index is equal to : 
(R - Rf) / Beta
Beta = beta of the fund

  • The Jensen's Alpha : 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 : 
R - [Rf  +  Beta*(Rm - Rf)]
Rm = average return of the benchmark (or market index)

  • The M Squared Measure : 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 :
(Sharp's Measure)*(Std Devm) + Rf
Std Devm = standard deviation of the benchmark (or market index)

  • The T Squared Measure : 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 :
[(1 / Beta)*(R - Rf) - (Rm - Rf)]

  • The Sortino Index : 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 : 
(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).