WITH HEAVY-TAILED CONDITIONAL DISTRIBUTIONS
4. Prediction of densities and downside risk
Decisions on financial investments are typically based on the expected return and the ex- pected risk of the assets under consideration. Rather than adhering to the conventional mean-variance criterion, recent risk management concepts for financial institutions focus on the downside risk or the value-at-risk of a financial position due to market movements.
In this context, a typical question would be: what is the probability that the value of a fi- nancial position will drop by 50% or more over the next period, i.e., Pr(rt+1<−0.50)?
Alternatively, one may ask what is the threshold or value-at-risk,−z(γ ), under which a position will not fall with a probability of 100(1−γ )%; i.e., find −z(γ ) such that Pr(rt+1<−z(γ ))=γ.
Under unconditional normality, it would be sufficient to simply predict the conditional mean and variance to answer such questions. However, for GARCH processes driven by nonnormal, asymmetric and, possibly, infinite-variance innovations, the predictive condi- tional density
fˆt+1|t(rt+1)=f
rt+1−à(θˆt) ct+1(ˆθt)
rt, rt−1, . . .
, (21)
needs to be computed. In (21),θˆtrefers to the estimated parameter vector using the sample information up to and including periodt; andct+1(ã)is obtained from the conditional-scale
Ch. 9: Prediction of Financial Downside-Risk 399
recursion (2) usingθˆt.3Multistep density predictions, fˆt+n|t(rt+n)=f
rt+n−à(θˆt) ct+n(ˆθt)
rt, rt−1, . . .
, (22)
are obtained by recursive application of (2) with unobserved quantities being replaced by their conditional expectations.
For each of the five currencies under consideration, we evaluate fˆt+1|t(rt+1), t = 2000, . . . , T −1, for theSα,βδ GARCH(1,1)andtνδGARCH(1,1)models, as well as the conventional GARCH(1,1) model with normal innovations.4We re-estimate (via ML esti- mation) the model parameters at each step, as would typically be done in actual applica- tions.
The overall density forecasting performance of competing models can be compared by evaluating their conditional densities at the future observed valuert+1, i.e.,fˆt+1|t(rt+1).
A model will fare well in such a comparison if realizationrt+1is near the mode offˆt+1|t(ã) and if the mode of the conditional density is more peaked. The conditional densities are determined not only by the specification of the mean and GARCH equations, but also by the distributional choice for the innovations.
Table 4 presents the means, standard deviations and medians of the density values fˆt+1|t(rt+1),t =2000, . . . , T −1, for each currency. Based on the means, values cor-
Table 4
Comparison of overall predictive performancea
British Canadian German Japanese Swiss
Mean
Normal 0.4198 1.1248 0.4064 0.4796 0.3713
t 0.4429 1.1871 0.4258 0.5207 0.3851
Sα,β 0.4380 1.1798 0.4213 0.5173 0.3820
Standard deviation
Normal 0.1934 0.5697 0.1888 0.1988 0.1620
t 0.2325 0.6802 0.2151 0.2782 0.1840
Sα,β 0.2189 0.6482 0.2016 0.2662 0.1771
Median
Normal 0.4291 1.0824 0.4178 0.5172 0.3942
t 0.4483 1.1500 0.4452 0.5261 0.4069
Sα,β 0.4493 1.1730 0.4477 0.5242 0.4041
a The entries represent average predictive likelihood values,T−1
t=2000fˆt+1|t(rt+1).
3 A conditionally varying location parameter,àt, would be handled analogously.
4 Since the sample sizes,T, of the five currencies vary, the number of forecasts ranges from 1,621 to 1,682.
400 S. Mittnik and M.S. Paolella
responding to the Sα,β and Student’s t assumptions are extremely close, with the Stu- dent’st values nevertheless larger in each case. Based on the medians, however, the stable Paretian model is (slightly) favored by the British, Canadian and German currencies. No- tice that this is contrary to the model selection based on the goodness of fit measures; both AICC and AD statistics favored use of stable Paretian innovations for the Japanese yen and Student’st innovations for the British pound.
Next, we examine how well the models predict the downside risk. Consider the value- at-risk implied by a particular model,M, namely
Pr
rt+1−zMt+1(γ )
=γ , t=2000, . . . , T −1. (23)
For a correctly specified model we expect 100γ% of the observedrt+1-values to be less than or equal to the implied threshold-values−zt+1(γ ). If the observed frequency
ˆ
γM:= 1
T −2000
T−1 t=2000
I(−∞,−zM
t+1(γ )](rt+1)
is less (higher) thanγ, then modelMtends to overestimate (underestimate) the risk of the currency position; i.e., the implied absolutezMt+1(γ )-values tend to be too large (small).
The predictive performance for assessing the downside risk achieved by the normal, Student’st and stable Paretian models are compared in Table 5 for the shortfall proba- bilitiesγ =0.01, 0.025, 0.05, 0.10. A comparison of the stable Paretian and Student’st
Table 5
Comparison of predictive performance for downside riska
100γ Model British German Canadian Japanese Swiss
Normal 1.9036 1.5051 1.3674 1.9124 1.4899
1.0 t 1.3682 0.9031 0.7134 1.4189 1.3707
Sα,β 1.3682 0.9031 1.3080 1.3572 1.2515
Normal 3.0339 2.6490 2.3187 2.8994 3.2777
2.5 t 2.8554 2.9500 2.1403 3.2079 3.3969
Sα,β 2.9149 2.9500 2.4970 2.5910 3.1585
Normal 4.7591 4.5756 3.6266 4.9969 4.7676
5.0 t 5.1160 5.2378 3.9834 5.7372 5.0656
Sα,β 5.1160 5.2378 5.0535 5.2437 5.0656
Normal 8.3879 9.2113 8.5612 8.0814 8.9392
10.0 t 9.8751 10.6562 9.9287 10.3023 10.8462
Sα,β 9.6966 10.4154 10.2259 9.8088 10.2503
a The entries show the observed frequenciesγˆM=(T−2000)−1T−1
t=2000I(−∞,−zM
t+1(γ )](rt+1)multiplied by 100. For a correctly specified model, we expectγˆM≈γ.
Ch. 9: Prediction of Financial Downside-Risk 401
GARCH models over the five currencies and four cutoff values,γ, shows that, in 4 out the 20 cases, the Student’stGARCH model outperforms that of the stable Paretian, while the latter is more accurate in 11 cases, sometimes considerably so (as for the Canadian dollar withγ=0.025 and 0.05). The remaining 5 cases are tied.
Table 6 presents summary measures5for the predictive performance of the three models across all five currencies in the form of the mean error
MEM(γ )=1 5
5 i=1
100 ˆ γiM−γ
,
mean absolute error MAEM(γ )=1 5
5 i=1
100γˆiM−γ
Table 6
Summary measures for the predictive performancea
100γ Model ME(γ ) MAE(γ ) MSE(γ )
Normal 0.6357 0.6357 0.4558
1.0 t 0.1549 0.3083 0.1080
Sα,β 0.2376 0.2764 0.0861
Normal 0.3357 0.4083 0.2209
2.5 t 0.4101 0.5540 0.3527
Sα,β 0.3223 0.3235 0.1633
Normal −0.4548 0.4548 0.4357
5.0 t 0.0280 0.4346 0.3302
Sα,β 0.1433 0.1433 0.0273
Normal −1.3638 1.3638 2.0195
10.0 t 0.3217 0.4002 0.2517
Sα,β 0.0794 0.2772 0.0830
Normal −0.2118 0.7156 0.7830
Aggregate
t 0.2287 0.4243 0.2607
Sα,β 0.1956 0.2551 0.0899
a Shown are the mean error (ME), mean absolute error (MAE) and mean squared error (MSE) of the observed extreme-tail frequencies from Table 5 across the five currencies. The bottom panel is the aggregate over all γ-values considered.
5 The measures are evaluated for 100γrather thanγbecause the resulting scales of the reported values enhance readability.
402 S. Mittnik and M.S. Paolella
and the mean squared error
MSEM(γ )=1 5
5 i=1
1002 ˆ
γiM−γ2
.
TheME’s for the normal show that it underestimates the probability of extreme down- turns (MENormal(γ ) >0 forγ=0.01, 0.025) and overestimates the probability of moderate downturns (MENormal(γ ) <0 forγ=0.05, 0.10). With one exception, theME’s of the sta- ble Paretian and Student’stGARCH models are smaller (in absolute terms) than those for the normal. However, they are always positive, indicating, on average, slight underpredic- tion of the downturn probabilities. Forγ =0.01 andγ=0.05, the Student’st model has smallerMEthan the stable Paretian model. This is due to the Student’stmodel’s offsetting prediction error for the Canadian dollar for theseγ-values.
While theME’s indicate possible systematic prediction bias, theMAEs andMSEs re- flect the size of the prediction error. With respect to both measures, the stable Paretian model dominates those of both the normal and the Student’st for allγ-values considered.
This is also evident from the bottom panel of Table 6, which aggregates the summary mea- sures over allγ-values considered. In the aggregate, the model using the stable Paretian innovation assumption outperforms those using the normal and Student’st in terms of all three summary measures.