Index analysis

Before starting with the portfolio analysis, it may be important to define the indices I will use, and perform a deep analysis about them.

In this work, I use 16 different indices divided in 4 categories: equities, sovereign bonds, corporate bonds and alternative assets and I report them in Table 8 along with a simplified name and some features.

Two of them, namely MSCI WORLD and CGBI-WGBI WORLD, are indices used to construct the benchmark in order to run a comparison in the last part of my work. Both of them have a weight of 50% in the benchmark.

Table 15: Classification of the indices

Name Category Region Simplified name

MSCI USA

Agg Corporate Corporate Bond United Stated of

America CB EMU

BOFA ML EUR

Corp Corporate Bond European

Monetary Union CB JAP

Alternative World Hedge Fund

38 UK-DS Inv.

Trust Private Equity

Alternative World Private Equity MSCI World

Real Estate Alternative World Real Estate

The sample for these indices is from January 1997 to December 2014, for a total of 18 years and 216 monthly observations.

One of the major problem in selecting the indices was the time range of the sample, since most of the indices used in the current finance world are born recently and did not exist back in 1997.

Consequently, the range of possible choices offered by Datastream was quite narrow, in particular for the private equity index.

Moreover, in the case of real estate, there is no common index that can represent all the real estate prices around the world, like for commodities, where there are specific exchange markets and specific derivative instruments.

In addition, real estate prices are affected by many elements such as geographical area, the dimensions and the material used. Thus, for this case I decided to employ an equity index representing the real estate companies as a proxy.

4.2.1 Normality

The probability distribution of return is an important aspect when analysis like mine are undergone.

Using a distribution that does not represent the reality of the fact may affect the findings and may have serious consequences, in particular when we employ economic and statistical measures such as the Value at Risk.

In addition, different return distributions can lead to different portfolio optimizations when it comes to risk budgeting allocation.

For these reasons, using the software Matlab, I run some tests to verify whether the returns are normally distributed. Table 16 show the results on normality distribution using Jarque-Bera test and Anderson-Darling test with different level of confidence.

Table 16: Normality test

Asset Jarque-Bera Test Anderson-Darling Test

10% 5% 1% 10% 5% 1%

Eq. USA x x x x x x

39

x = reject the null hypothesis of normal distribution

From Table 16 we can draw some conclusion about the distribution.

Both tests show for all the three levels of confidence that SB EMU and CB JAP have returns that follow a normal distribution while SB JAP reject only at 10%.

Eq. USA, Hedge Fund, Private Equity, Real Estate, CB USA, SB EM, CB EM and CB EMU do not have returns that follow a normal distribution for any level of confidence while Eq. JAP and Eq. EM accept the null hypothesis only at 1%.

Commodity accepts the hypothesis only with Anderson-Darling test at 1% and Eq. EMU only with Jarque-Bera at 1%.

For SB USA we have contrasting results.

Eventually, both methods provide with almost the same result: with Anderson-Darling³⁶ test 75% of the indices have returns that do not follow a Gaussian distribution whereas with Jarque-Bera it is the 81.25% of them.

36 At 5% confidence level.

40

From Figure 5 to 8, I provide with two probability density function estimates for each asset.

For the first (green bars) I used the Matlab function histogram³⁷, employing an automatic binning algorithm that returns bins with a uniform width, chosen to cover the range of monthly returns and reveal the underlying shape of the distribution.

For the second I used the Matlab function ksdensity, employing an Epanechnikov kernel function³⁸.

Table 17 shows the skewness and kurtosis for each asset.

Most of the estimates show concordant results with the tests. SB EMU, CB JAP and SB JAP have the lowest levels of kurtosis while the alternative assets such as Hedge Fund, Commodity, Private Equity, and all the corporate bonds (with the exclusion of Japan) show the highest levels of kurtosis. In some cases also the skewness is very high, for instance CB EM, CB USA and Hedge Fund.

Figure 5: Probability density function estimates for Eq. USA, JAP, EM, EMU

37 The height of each bar is, (number of observations in the bin) / (total number of observations * width of bin).

The area of each bar is the relative number of observations.

38 The kernel of a probability density function (pdf) is the form of the pdf in which any factors that are not functions of any of the variables in the domain are omitted.

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Figure 6: Probability density function estimates for Commodity, Hedge Fund, Private Equity, Real Estate

Figure 7: Probability density function estimates for SB USA, JAP, EM, EMU

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Figure 8: Probability density function estimates for CB USA, JAP, EM, EMU Table 17: Skewness and kurtosis of all the indices

Asset Skewness Kurtosis Asset Skewness Kurtosis

Eq.USA -0,354 5,560 SB USA -0,192 4,572

Eq.JAP 0,248 3,860 SB JAP 0,363 2,864

Eq.EM -0,064 4,057 SB EM -1,893 16,098

Eq.EMU -0,410 3,854 SB EMU 0,108 2,912

Commodity -0,512 4,123 CB USA 2,576 34,991

Hedge Fund -0,561 5,691 CB JAP 0,208 2,973 Private Equity 0,189 9,205 CB EM -2,131 16,907 Real Estate -0,272 5,755 CB EMU 0.179 3,39