6.2 Non-linear time series models
8.1.3 Analysis of scenarios
One of the factors in risk management is the risk-return ratio. In gure8.5the normalized median of each index is plotted. In this plot the correlation in the indices is very clear, especially within an asset class. The stock indices are highly correlated and the corporate bond and the high yield bond have some of the same patterns. The Danish bonds are highly correlated, and have almost identical behaviour in the rst regime. This is naturally caused by the homogeneity of the weighting in the principal components in falling regimes. This plot can be used to compare the indices and determine which one has the highest potential return at a given time among all the indices. Overall, the JPGCCOMP index performs the best, and in general the bond indices have the highest relative return after ve years. Depending on the investing strategy, it might be an idea to invest in NDEAMO until 2013 and then switch to JPGCCOMP in order to get the highest return throughout the whole period. This plot only gives information about the return, and does not give any idea about the risk attached to the index. For this reason it would be interesting to look at the distribution of the end values of the indices, in order to get an idea of the risk.
0.60.81.01.21.41.6
Normalized 50% quantile scenarios
Time [Year]
Normalized index value
2012 2013 2014 2015 2016
Index KAXGI NDDUE15 NDDUJN NDDUNA NDUEEGF TPXDDVD CSIYHYI JPGCCOMP NDEAGVT NDEAMO
Figure 8.5: Normalized 50% quantile scenario for each index.
Figure 8.6 shows histograms of the end values for each index, together with markings of 5%, 50% and 95% quantiles. A 50% quantile at zero corresponds to a zero return on the median. The end value distributions look similar with in an asset class. The stock indices are positively skewed with a higher de-gree of deviation, because they are having some extreme scenarios with very high return. Again it is stated that the bonds indices have the highest return, some of the stock indices even have a lot scenarios with negative 5-year returns.
This plot together with table 8.1 is a nice tool for risk-return valuation for 5-year investments. In the table the standard deviation on the 5-year return is given together with the median return. As an example of the risk-return considerations, the JPGCCOMP can be used. It has the highest median re-turn at 63.1% change over 5 years corresponding to an annual eective rate on
√5
1 + 0.631−1 = 0.103 = 10.3%. But the standard deviation of the relative changes is σˆ = 0.671, which is the highest among the bond indices. Whether the investor should choose to accept this higher risk in order to get a higher return depends on the overall strategy. It is unusual, is that some of the in-dices with high standard deviation on the 5-year relative return, have a negative median return. The extreme scenarios, the fact that the simulation starts in a bearish market and that the simulation horizon only is ve years lead to this unfavourable risk return relationship. If the horizon was longer, the overall pos-itive trend in historical data would probably have aected results, such that
8.1 Scenario generation using ARCH models 85
Histogram of rela−
tive changes in 5 year simulations Density
−0.5 0.5 1.5
0.00.6
KAXGI
Density
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0.00.40.8
NDDUE15
Density
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0.00.8
NDDUJN
Density
−0.5 0.0 0.5 1.0
0.00.6
NDDUNA
Density
−0.5 0.5 1.5 2.5
0.00.4
NDUEEGF
Density
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0.00.8
TPXDDVD
Density
0.0 0.5 1.0
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CSIYHYI
Density
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JPGCCOMP
Density
0.1 0.2 0.3 0.4
02
NDEAGVT
Density
0.1 0.3 0.5
0.01.5
NDEAMO
Density
5% & 95% quantile 50% quantile
Figure 8.6: Histogram of the relative end values for the series.
the risk on the stock indices still would have been higher, but the return would have exceeded the return on bond indices. But this behaviour is anyway still desirable and most likely because of the market situation at the starting point.
Index KAXGI NDDUE15 NDDUJN NDDUNA NDUEEGF
Median 0.085 -0.191 -0.359 -0.174 0.103
ˆ
σ 1.090 1.314 0.565 0.803 1.948
Index TPXDDVD CSIYHYI JPGCCOMP NDEAGVT NDEAMO
Median -0.357 0.239 0.631 0.275 0.344
ˆ
σ 0.680 0.475 0.671 0.119 0.152
Table 8.1: Median 5-year relative return and the standard deviation of relative returns for all the scenarios.
Figure8.7shows a plot of the normalized scenarios containing the largest max-imum drawdown. This can be used in risk management as well , an in this case it is clear that the Danish bonds do not experience such extreme maximum
drawdowns as the other indices, actually the ve year return on the scenarios for these indices are slightly negative, where the loses on the other scenarios are considerably large. It is also clear that it is the same market scenario that holds the largest maximum drawdown for all the indices except for NDEAGVT.
This means that the indices have a high correlation in bear market as they are supposed to.
0.00.51.01.52.0
Normalized MDD scenarios
Time [Year]
Normalized index value
2012 2013 2014 2015 2016
Index KAXGI NDDUE15 NDDUJN NDDUNA NDUEEGF TPXDDVD CSIYHYI JPGCCOMP NDEAGVT NDEAMO
Figure 8.7: Normalized scenario with MDD for all the indices.
A plot of the drawdown and maximum drawdown from simulation startt0until tτ might give a clue of how fast the maximum drawdown occur. In gure8.8 it is seen that it occurs as early as stated above. The plot can also be used to estimate when a crisis at least would occur. For instance, let a crisis be dened as a 40% drawdown, then the median states when there is 50% chance that a crisis already has occurred. On the other way around, the chance that a crisis will hit before a given date, can also be stated looking at the quantiles. For example there is 50% chance that a crisis (or actually a drawdown equals to 40% cf. the crises denition) on the Copenhagen Stock Exchange (KAXGI) has occurred in 2014. At the same time it is more than 95% certain that there will not be a drawdown larger than 8% in NDEAGVT and 15% in NDEAMO. It is very certain (∼95%) that the stock indices will experience a drawdown larger than 15% within the ve-year period, but it is very uncertain that the bond indices will experience drawdowns at that size. This is fully consistent with the