In this example, the solution of a global minimum-variance portfolio is again uti- lized as a benchmark allocation for long-only investments. The characteristics of this
Portfolio Value (Index)
1998−01−01 2002−01−01 2006−01−01 2010−01−01
100150200250
Minimum−Variance Minimum−CVaR
Figure 12.5 Trajectory of minimum-CVaR and minimum-variance portfolio values.
k k Relative Performance Minimum−CVaR vs. Minimum−Variance
Percentages
1998−01−01 2002−01−01 2006−01−01 2010−01−01
0510152025
Figure 12.6 Relative performance of minimum-CVaR and minimum-variance portfolios.
solution are compared with the allocations of portfolios that are restricted by their maximum, average, and conditional draw-downs and with a minimum conditional draw-down at risk portfolio. The spectrum of assets covers the major equity and bond markets as well as investment in gold. TheRcode is presented in Listing 12.3.
First, the relevantRpackages are brought into memory. Then theMultiAsset data set is loaded, which is part of theFRAPO package. The data set covers the month’s-end prices of a number of equity indexes (S&P 500, Russell 3000, DAX (XETRA), FTSE 100, Nikkei 225 and iShares MSCI Emerging Markets) and fixed income indexes (Dow Jones CBOT Treasury, German REX Performance, and United Kingdom gilts—all maturities) as well as the price of the SPDR Gold Shares ex- change traded fund from 30 November 2004 to 30 November 2011. In lines 5–8 the discrete returns are computed and converted to atimeSeriesobject. Commenc- ing in line 11 the solution of the global minimum-variance portfolio is determined and the historic draw-downs are computed with the functionDrawdowns()con- tained in the packagePerformanceAnalytics. Plots of these are displayed through the commands in lines 17–21. The time series plot is created from scratch in order to make its appearance the same as the draw-down plots for the ensuing portfolio plots.
The result is provided in Figure 12.7. Incidentally, one could have used the function chart.Drawdown()instead, as mentioned in Section 12.5.4.
In the following lines the various draw-down portfolio solutions are computed, with the maximum draw-down of the GMV portfolio serving as anchor value. The conditional draw-down at risk portfolios are computed for a confidence level of 95%.
Next, the trajectory of the four historic draw-down series is depicted in the form of a(2×2)plot where for a better cross-comparison the ordinates share the same scale except for the average draw-down constrained portfolio (see Figure 12.8).
k k Rcode 12.3Comparison of draw-down and GMV portfolios.
l i b r a r y ( f P o r t f o l i o ) 1
l i b r a r y ( FRAPO ) 2
l i b r a r y ( P e r f o r m a n c e A n a l y t i c s ) 3
d a t a ( M u l t i A s s e t ) 4
# # R e t u r n c a l c u l a t i o n 5
R e t s <− r e t u r n s e r i e s ( M u l t i A s s e t , method = " d i s c r e t e " , 6
p e r c e n t a g e = FALSE , t r i m = TRUE) 7
R e t s <− t i m e S e r i e s ( R e t s , c h a r v e c = rownames ( R e t s ) ) 8
# # Benchmark p o r t f o l i o : GMV 9
gmvspec <− p o r t f o l i o S p e c ( ) 10
GMV <− m i n v a r i a n c e P o r t f o l i o ( d a t a = R e t s , s p e c = gmvspec , 11
c o n s t r a i n t s = " LongOnly " ) 12
GMVret <− t i m e S e r i e s ( R e t s %∗% g e t W e i g h t s (GMV) , 13
c h a r v e c = t i m e ( R e t s ) ) 14
GMVDD <− P e r f o r m a n c e A n a l y t i c s : : : Drawdowns ( GMVret ) 15
# # P l o t o f draw downs f o r GMV 16
y l i m s <− c (−6 , 0 ) 17
p l o t (GMVDD ∗ 1 0 0 , x l a b = " " , y l a b = " Draw Downs ( p e r c e n t a g e ) " , 18
main = " Draw Downs o f G l o b a l Minimum V a r i a n c e " , 19
y l i m = y l i m s ) 20
a b l i n e ( h = 0 , c o l = " g r e y " ) 21
g r i d ( ) 22
# # Max DD o f GMV 23
GMVMaxDD <− max (−1 . 0 ∗ GMVDD) 24
# # Draw Down P o r t f o l i o s 25
MaxDD <− PMaxDD ( M u l t i A s s e t , MaxDD = GMVMaxDD) 26
AveDD <− PAveDD ( M u l t i A s s e t , AveDD = GMVMaxDD) 27
CDaR95 <− PCDaR ( M u l t i A s s e t , a l p h a = 0 . 9 5 , bound = GMVMaxDD) 28
CDaRMin95 <− PCDaR ( M u l t i A s s e t , a l p h a = 0 . 9 5 ) 29
# # P l o t o f draw downs 30
o l d p a r <− p a r ( no . r e a d o n l y = TRUE) 31
p a r ( mfrow = c ( 2 , 2 ) ) 32
p l o t ( AveDD , main = " ( a ) AveDD" ) 33
p l o t ( MaxDD , y l i m = y l i m s , main = " ( b ) MaxDD" ) 34
p l o t ( CDaR95 , y l i m = y l i m s , main = " ( c ) CDaR" ) 35
p l o t ( CDaRMin95 , y l i m = y l i m s , main = " ( d ) Minimum CDaR" ) 36
p a r ( o l d p a r ) 37
As already hinted in Section 12.4, the solution of the average draw-down con- strained portfolio can imply the occurrence of large draw-downs. In this case the maximum draw-down of the average draw-down constrained solution is greater by roughly a factor of four than the maximum historic draw-down of the minimum conditional draw-down at risk portfolio. By cross-comparing the draw-downs with those implied by the GMV solution, the latter seems to fare reasonable well, at first glance.
k k Draw−downs of global minimum variance
Draw−downs (percentage)
2005−01−01 2007−01−01 2009−01−01 2011−01−01
−6−5−4−3−2−10
Figure 12.7 Draw-downs of GMV portfolio.
(a) AveDD
Draw−downs (percentages)
2005−01−01 2009−01−01
−20−100
(b) MaxDD
Draw−downs (percentages)
2005−01−01 2009−01−01
−6−4−20
(c) CDaR
Draw−downs (percentages)
2005−01−01 2009−01−01
−6−4−20
(d) Minimum CDaR
Draw−downs (percentages)
2005−01−01 2009−01−01
−6−4−20
Figure 12.8 Comparison of draw-downs.
The characteristics of the five portfolio solutions are next analyzed with respect to the different allocations and their associated risk contributions and diversification ratios. The calculation of these statistics is provided in Listing 12.4.
In line 2 of this listing, the names for the five portfolio types are defined; these will be used as column names. Next, the portfolio weights are extracted from the objects
k k Rcode 12.4Analysis of portfolio solutions.
# # P o r t f o l i o Names 1
Pnames <− c ( "GMV" , "MaxDD" , "AveDD" , " CDaR95 " , " CDaRMin95 " ) 2
# # P o r t f o l i o a l l o c a t i o n s 3
W e i g h t M a t r i x <− c b i n d ( g e t W e i g h t s (GMV) , 4
W e i g h t s (MaxDD) , 5
W e i g h t s ( AveDD ) , 6
W e i g h t s ( CDaR95 ) , 7
W e i g h t s ( CDaRMin95 ) ) 8
c o l n a m e s ( W e i g h t M a t r i x ) <− Pnames 9
# # E x p e c t e d S h o r t f a l l and c o m p o n e n t s 10
tmp <− a p p l y ( W e i g h t M a t r i x , 2 , f u n c t i o n ( x ) ES ( R e t s , w e i g h t s = x , 11
method = " g a u s s i a n " , 12
p o r t f o l i o _ method = " c o m p o n e n t " ) ) 13
# # ES 95% 14
PES <− u n l i s t ( l a p p l y ( tmp , f u n c t i o n ( x ) x [ [ 1 ] ] ) ) ∗ 100 15
# # M a r g i n a l C o n t r i b u t i o n s t o ES 16
PMES <− m a t r i x ( u n l i s t ( l a p p l y ( tmp , f u n c t i o n ( x ) x [ [ 3 ] ] ) ) , 17
nrow = n c o l ( R e t s ) ) ∗ 100 18
rownames (PMES) <− c o l n a m e s ( R e t s ) 19
c o l n a m e s (PMES) <− Pnames 20
# # M a r g i n a l C o n t r i b u t i o n s t o S t d D e v 21
V <− cov ( R e t s ) 22
PMRC <− a p p l y ( W e i g h t M a t r i x , 2 , mrc , Sigma = V) 23
rownames (PMRC) <− c o l n a m e s ( R e t s ) 24
# # D i v e r s i f i c a t i o n r a t i o 25
PDR <− a p p l y ( W e i g h t M a t r i x , 2 , dr , Sigma = V) 26
created earlier and assigned to the matrix objectWeightMatrix. This object can then be utilized to compute the portfolios’ ES at the 95% level and the percentage contributions of the underlying assets with the functionES(). The intermediate re- sult in the form of alistobject is assigned totmp. The relevant items can then be extracted easily by employing lapply()and using theunlist() function to assemble the results as a matrix. The marginal contributions to the portfolio risk are computed in line 20 onwards with the functionmrc()which is included in a call toapply(). In a similar manner the diversification ratios for the five portfo- lios are computed using the functiondr()in line 25. The results are summarized in Table 12.1.
The GMV portfolio is characterized by a high share allotted to German govern- ment bonds, due to the low volatility of this asset. Exposure to equities is roughly 10% and investments in UK gilts and gold amount to less than 1% in each case. Ac- cording to the GMV solution, the capital is split between six of the ten possible assets.
Overall, the empirical stylized fact of highly concentrated solutions for GMV portfo- lios is vindicated. This becomes even more apparent when the marginal contributions to the 95% ES and the overall portfolio risk are taken into account. However, when
k k Table 12.1 Comparison of portfolio allocations and characteristics.
Analysis GMV MaxDD AveDD CDaR95 CDaRMin95
S&P 500
Weight 4.89 0.00 0.00 0.00 0.00
MES 5.58 0.00 0.00 0.00 0.00
MPR 4.89 0.00 0.00 0.00 0.00
Russell 3000
Weight 0.00 3.00 0.00 6.40 0.00
MES 0.00 −0.93 0.00 −3.68 0.00
MPR 0.00 −0.44 0.00 −2.10 0.00
DAX (XETRA)
Weight 4.34 3.62 0.00 0.00 0.00
MES 3.46 −3.51 0.00 0.00 0.00
MPR 4.34 −1.83 0.00 0.00 0.00
FTSE 100
Weight 0.00 0.00 0.00 0.00 0.00
MES 0.00 0.00 0.00 0.00 0.00
MPR 0.00 0.00 0.00 0.00 0.00
Nikkei 225
Weight 1.73 0.91 0.00 0.00 0.00
MES 2.37 −0.50 0.00 0.00 0.00
MPR 1.73 −0.45 0.00 0.00 0.00
MSCI EM
Weight 0.00 0.00 0.00 0.00 0.00
MES 0.00 0.00 0.00 0.00 0.00
MPR 0.00 0.00 0.00 0.00 0.00
CBOT Treasury
Weight 0.00 48.03 0.00 16.62 0.00
MES 0.00 75.01 0.00 25.73 0.00
MPR 0.00 69.75 0.00 23.39 0.00
German REX
Weight 87.87 42.83 55.07 72.49 85.73
MES 87.47 28.09 3.73 67.97 54.50
MPR 87.87 30.21 6.93 66.94 54.97
UK Gilts
Weight 0.96 0.00 0.00 0.00 0.00
MES 1.12 0.00 0.00 0.00 0.00
MPR 0.96 0.00 0.00 0.00 0.00
Gold
Weight 0.21 1.62 44.93 4.49 14.27
MES −0.01 1.83 96.27 9.97 45.50
MPR 0.21 2.76 93.07 11.76 45.03
Overall
ES 95% 1.31 1.82 4.36 1.53 1.98
DivRatio 1.86 1.63 1.17 1.67 1.34
k k one compares the diversification ratio of this portfolio with the values taken by the
draw-down portfolios, the conclusion would be that the GMV yields the most favor- able asset allocation in terms of the degree of diversification. This artifact can again be primarily attributed to the low volatility of the REX, which is mirrored by a low figure for the ES.
The number of assets invested in is the same for the constrained maximum draw-down portfolio. However, this solution differs in two major points with respect to the asset allocation: first, the allotment to government bonds is now roughly equally split between US Treasuries and German Bunds; and second, the equity exposure is characterized by negative marginal contributions to the ES and the overall portfolio risk. Hence, from a risk perspective this outcome is more akin to a general understanding of risk diversification, even though the diversification ratio is less than that of the GMV and the ES is half a percentage point greater.
The solutions for the average draw-down and minimum conditional draw-down at risk constrained portfolios imply an asset allocation in the REX and gold. How- ever, these two solutions are quite distinct from each other. Whereas for the average draw-down portfolio the capital is shared almost equally between the two assets, re- sulting in a high marginal contribution of gold to the portfolio’s risk, for the minimum conditional draw-down portfolio the pattern is reversed. Here, the lion’s share of cap- ital would be invested in German Bunds and the implied marginal risk contributions are almost equal.
The asset allocation for the constrained conditional draw-down at risk portfolio yields an intermediate solution between the maximum draw-down constrained and the other draw-down portfolios discussed above. Its equity exposure is limited to the Russell 3000 only, which contributes negatively to the ES and the portfolio’s risk, on the margin. It is further characterized by bond exposure to US Treasuries and German Bunds, where the latter constitute almost three quarters of the capital invested. Compared to the GMV allocation, the risk contributions are more evenly spread between only four investments. The ES and the diversification ratio is pretty close to the corresponding figures for the GMV portfolio.