**Figure 2.18:** The cumulative proportion of
variance explained by the ﬁrst two principal
components for the three indices.

The proportion of variance explained by the ﬁrst component varies from 0.35 at the lowest to above 0.6 at the highest. The spikes for component one in 1996–1997, in 2003, and again from the middle of 2007 are coincid-ing with the times where the corre-lation between the MSCI ACWI and the JPM GBI is signiﬁcantly strength-ened. During the GFC, the correla-tion between the MSCI ACWI and the S&P GSCI also increased signif-icantly, which is why the ﬁrst compo-nent accounted for a larger proportion of the variance at the expense of com-ponent two. Besides the GFC, the

sec-ond principal component explains an almost constant proportion of the variance of about 0.35.

The correlations are stronger at the times of high market volatility and stress.

Thus, diversiﬁcation may not materialize precisely when an investor needs it the most. The fact that the correlation between the MSCI ACWI and the JPM GBI gets increasingly negative is not a problem, but the increase in the correlation between the MSCI ACWI and the S&P GSCI, in particular during the GFC, is unfavorable. The impact would be more signiﬁcant if it was two diﬀerent stock indices, then the correlations would increase even more around year 2000 and 2008. Although the correlations increased during the GFC, there are deﬁnitely diversiﬁcation beneﬁts between asset classes.

**2.6** **In-Sample Adjustment**

The bond index has clearly outperformed both the stock and commodity index in-sample following the signiﬁcant decline in the term structure of interest rates with corresponding high returns for long-maturity bonds. For other time periods, stocks and commodities are likely to show a more attractive risk/return proﬁle compared to bonds. Had the in-sample period ended in 2007 rather than 2008, then the Sharpe ratios would have been 0.67 and 0.46 for the MSCI ACWI and the S&P GSCI, respectively.

The end of a major ﬁnancial crisis, where the level of stress in the markets is at its highest, is a critical time at which to be estimating Sharpe ratios. The SR of 0.29 for the MSCI ACWI is not an unrealistic long-term level. If the SR is set too low, then it will not be proﬁtable to change allocation, and if the SR is

set too high, then it will appear proﬁtable to change allocation very often, as the excess return will quickly cover the transaction costs.

The assumption that will be made when calibrating time series models to the in-sample data to use for scenario generation and asset allocation is that the annual SR of the JPM GBI equals that of the MSCI ACWI. This in attempt to make a neutral assumption although there is no such thing as a neutral assumption in this context. Leverage aversion could be one reason why stocks and bonds should not be expected to yield the same risk-adjusted return in the long run (Asness et al. 2012, Frazzini and Pedersen 2014).

Then there is the dispute about the commodity risk premium. The point of departure will be to use the returns as they are, meaning that commodities will have a lower SR than stocks and bonds in-sample. Hence, it is only the in-sample returns of the JPM GBI that will be adjusted. This is done by subtracting 0.022% from the daily log-returns. In this way, the MSCI ACWI and the JPM GBI will both have an in-sample annual SR of 0.29. It should be emphasized that the data used for out-of-sample testing will remain untransformed.

### CHAPTER 3

### Markov-Switching Mixtures

The normal distribution is a poor ﬁt to most ﬁnancial returns. Mixtures of normal distributions provide a much better ﬁt as they are able to reproduce both the skewness and leptokurtosis often observed. An extension to Markov switching mixture models, also referred to as hidden Markov models (HMMs), is frequently applied to capture both the distributional and temporal properties of ﬁnancial returns.

In an HMM, the distribution that generates an observation depends on the state of an unobserved Markov chain. The transition probabilities of the Markov chain are assumed to be constant implying that the sojourn times are geometrically distributed. The memoryless property of the geometric distribution is not always appropriate. An alternative is the hidden semi-Markov model (HSMM) in which the sojourn time distribution is modeled explicitly for each hidden state.

If the observations are not equidistantly sampled then a continuous-time hidden Markov model (CTHMM) that factors in the sampling times of the observations can be applied. The advantages of a continuous-time formulation include the ﬂexibility to increase the number of states or incorporate inhomogeneity without a dramatic increase in the number of parameters.

The theory of HMMs in discrete time and their estimation is outlined in sec-tion 3.1. The HSMM is introduced in secsec-tion 3.2 and the CTHMM in secsec-tion 3.3.

The models are ﬁtted to the in-sample returns of the MSCI ACWI in section 3.4 and gradient-based methods are considered in section 3.5.