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Here we will show and discuss simulations with sample size of 1,000,000 whereρ=0,0.3,0.6,0.9 and examine the tails by looking at QQ plots for n = 2,3,4,5,10,15 for ρ = 0 and n =

2,10,25,50,100,150 forρ= 0.3,0.6,0.9 againstn=1. The parameters are chosen as detailed at the top of this appendix.

Table C.7 shows what we expect to see from our theoretical results for each correlation. What we will have to consider is that our expected results are from a single measure for both tails, whereas when we read offof a QQ plot we can read both tails separately. Now let’s see how our expectations hold up.

First we look at ρ = 0. The QQ plots in Figures C.6 confirm the pattern we saw in our theoretical moments by simulation. What we can easily see here is that this is true for the

Figure C.3: Theoretical Moments of Sums of Correlated Standard Lognormal Random Vari- ables,ρ=0.6

Table C.4: Theoretical Moments of Sums of Correlated Standard Lognormal Random Vari- ables,ρ=0.9

n mean variance skewness kurtosis

1 1.6487 4.6708 6.1849 113.9364 2 1.6487 4.6708 6.1862 114.0147 3 1.6487 4.6708 6.1857 113.9875 4 1.6487 4.6708 6.1854 113.9704 5 1.6487 4.6708 6.1853 113.9603 6 1.6487 4.6708 6.1852 113.9540 7 1.6487 4.6708 6.1851 113.9499 8 1.6487 4.6708 6.1851 113.9471 9 1.6487 4.6708 6.1850 113.9450 10 1.6487 4.6708 6.1850 113.9435 11 1.6487 4.6708 6.1850 113.9424 12 1.6487 4.6708 6.1850 113.9415 13 1.6487 4.6708 6.1850 113.9408 14 1.6487 4.6708 6.1849 113.9402 15 1.6487 4.6708 6.1849 113.9397

Figure C.4: Theoretical Moments of Sums of Correlated Standard Lognormal Random Vari- ables,ρ=0.9

Table C.5: Theoretical Moments of Sums of Correlated Standard Lognormal Random Vari- ables,ρ=−0.2

n mean variance skewness kurtosis

1 1.6487 4.6708 6.1849 113.9364 2 1.6487 4.6708 9.6551 432.5010 3 1.6487 4.6708 13.6852 1196.1126 4 1.6487 4.6708 18.0499 2650.9388 5 1.6487 4.6708 22.6453 5066.7076

Table C.6: Maximum Number of Negatively Correltated Lognormal Random Variables

ρ max(n) −0.1 10 −0.2 5 −0.3 4 −0.4 3 −0.5 2 −0.6 2 −0.7 2

Table C.7: Expected Results of Simulations

ρ expected results

0 For n = 2 we expect the sum to have fatter tails than a single lognormal, and the fattness should increase withnat an increasing rate.

0.3 For n = 2 we expect the sum to have fatter tails than a single lognormal, the fatness should increase with n at a decreasing rate until it starts to noticeably flatten out around

n = 8. Eventually, around n = 20, the fatness will start to decrease. For very large n,n > 80, the sum will begin to have slightly thinner tails than the single lognormal.

0.6 Forn=2 we expect the sum to have fatter tails than a single lognormal, the fatness should increase with n untiln = 4. Then fatness will start to decrease. Betweenn= 7 andn= 8 the tails will be exactly the same as a single lognormal. For larger n,n > 8, the sum will have increasingly thinner tails than the single lognormal.

0.9 Forn = 2,3 we expect the sum to have the same tails as a single lognormal. At n = 4 there will be a jump, the sum will be fatter than the single, then fatness will start to decrease. Atn= 7 the tails will be the same as a single log- normal. For larger n,n > 7, the sum will have increasingly thinner tails than the single lognormal.

Figure C.5: Theoretical Moments of Sums of Correlated Standard Lognormal Random Vari- ables,ρ=−0.2

right tails, a summation of two lognormals has a fatter right tail than a single lognormal and the fatness increases as nincreases. It is difficult to see the left tails though. This is because lognormal random variables have a lower bound of zero. In figure C.7 we have zoomed in on the left tails. Now we can see that what we expected is also true for the left tails, a summation of two lognormals has a fatter left tail than a single lognormal and the fatness increases asn

increases.

Next is ρ = 0.3; figure C.8 is again hard to read very accurately, especially the left tail, so we will zoom in on the left tail, figure C.9, and examine that first. Now we can see that for small values ofnthe sum of nlognormals has somewhat of a fatter tail left than a single lognormal. As n increases this moves all the way from having the same left tail to having a slightly thinner tail than a single lognormal for sufficiently large n. When the zoom of this diagram is considered these fatnesses and thinnesses are slight. Since that made it significantly easier to read we will also zoom in on the right tails in figure C.10. We can see easily now that forn =2 the sum of lognormals has a slightly fatter right tail than a single lognormal and asn

increases the degree of fatness increases. However, for sufficiently largenthe fattening seems to level off, and remain at a constant level.

Now if we combine our left and right to get a combined level of fatness we can compare that to what we expected to see. For n = 2 together the tails of the sum of lognormals are fatter than a single lognormal. Asnincreases the right tail’s fatness increases and the left tail’s fatness decreases slightly, so for mid-level values ofnthe fatness does increase at a decreasing rate. Finally for sufficiently largen the right tail levels off and the right tail becomes thinner than a single lognormal, so overall the tail is thinning, though for our largestnshown still fatter

than a single lognormal. So compared to our theoretical prediction the pattern we hoped to see was observed but it developed more slowly than expected.

Now we will look atρ=0.6 in the same fashion.

From figure C.11 it looks as though for all of these values of n a sum ofn lognormals is distributed the same as a single lognormal. But let’s take a closer look at the left tails in figure C.12. Now we can be sure that the left tails of a sum of lognormals are distributed the same as a single lognormal. Let’s look at the right tails in figure C.13

From this figure we could say that maybe for n = 4 a sum of lognormals has a slightly thinner tail than a single lognormal, but we will not because our data points run parallel to the quantile line rather than curving away from it. Also, we are looking at the sixth standard deviation away from the mean so we will read this as following the same distribution. We will also read all of the others as a sum of lognormals having the same distribution as a single lognormal. So overall we can easily see that this resolves to a sum of lognormals following the same distribution as a single lognormal forρ=0.6.

Finally we will look at our most extreme case,ρ=0.9 in figure C.14

As before these plots seem to say that a sum ofnlognormals follow the same distribution as a single lognormal, but to be sure we will take a closer look at each tail. First the left tail, figure C.15. Now we have verified what we suspected for the left tail. Now to look at the right in figure C.16. There is some slight wiggle seen here but since it does not follow a defined pattern and it is over five standard deviations from the mean we will read this too as before. So overall we can say that forρ= 0.9 a sum of lognormal random variables follows the same distribution as a single lognormal random variable.