Computing optimal mean/downside risk frontiers: the role of ellipticity 197
other three risk measures. The grey lines represent the frontiers identified by the
algorithm. In this case, we use the portfolios weights for the optimal portfolios
in each bucket found by minimizing the sample standard deviation. The identi-
fied frontiers for the mean/standard deviation and mean/semi – standard devia-
tion are very close to that identified by the quadratic programming algorithm.
The two expected loss frontiers are superimposed and cannot be distinguished
on this diagram. By contrast, the deviations between the two value at risk fron-
tiers are relatively larger. The quality of the approximations for the 128,000
random portfolios is summarized in Table 8.2 , where 100 buckets of portfolios
have been used. In Panel A, we chose the optimal portfolio in each bucket as
that portfolio with the smallest standard deviation. The metric here is returns
Table 8.2 Summary statistics: distance between frontiers, eight assets, 128,000
simulated portfoliosStandard
deviation
Semi – standard
deviationValue at risk Expected lossPanel ASt Dev Average 0.0105 0.0072 0.0175 0.0001
(^) St Dev 0.0049 0.0036 0.0284 0.0008
(^) Min 0.0004 0.0003 0.0508 0.0015
(^) Max 0.0242 0.0158 0.0955 0.0020
Panel B
Semi – St
Dev
Average 0.0111 0.0069 0.0185 0.0000
(^) St Dev 0.0055 0.0035 0.0295 0.0008
(^) Min 0.0004 0.0003 0.0508 0.0015
(^) Max 0.0270 0.0166 0.0981 0.0023
Panel C
Va R Average 0.0218 0.0148 0.0106 0.0008
(^) St Dev 0.0123 0.0093 0.0187 0.0012
(^) Min 0.0001 0.0001 0.0508 0.0013
(^) Max 0.0614 0.0481 0.0817 0.0048
Panel D
Expected
loss
Average 0.0125 0.0082 0.0122 0.0003
(^) St Dev 0.0064 0.0050 0.0258 0.0007
(^) Min 0.0004 0.0003 0.0480 0.0015
(^) Max 0.0391 0.0276 0.0955 0.0014
Note : The units of measurement are percent per day. For standard deviation and semi – standard
deviation, we measure the deviation as the simulated portfolio value minus the quadratic
programming value. For value at risk and expected loss, we measure the quadratic programming
value minus the simulated portfolio value. The statistics reported here are based on 100 points
equally spaced along the respective frontiers from the minimum to the maximum observed sample
portfolio returns.