the amount of principal in year t, i is the interest rate, and d is the percent deduction in the principal following a triggering event. With an interest rate of 10%, the expected return is calculated to be 7.9%:

# E ( return ) = $ P_{t }* p _{0 }* (1 + i) + $ P_{t }* p_{1 }* ( − d )

E ( return ) = $ P_{t }* 0 .99 * 1 .10 − $ P_{t }* 0 .01 * 1 E ( return ) = $ 1 .089 P_{t }− $ 0 .01 P_{t }= $ 1 .079 P_{t }E ( return ) = ($ 1 .079 P_{t }/ $ P_{t }) − 1 = 7 .9 %

# The variance was estimated based the outcomes of all 1000 draws (n = 1000):

σ ^{2 }=[ (x_{1 }− x)^{2 }+(x_{2 }− x)^{2 }+...(x_{1000 }

− x)^{2 }]

(1000−1)

With coverage of only a single region, the variance is 3.85% with an expected return of 7.9% based on a 10% premium. Figure 5 illustrates how variance on returns decreases as regions are added. Dramatic reductions in the variance can be achieved by pooling only a few regions. As more regions are added to the portfolio, the likelihood of total loss of principal declines. With a single event, the probability of total loss of capital equals the probability of a catastrophic event, 0.01 in our example. When countries are pooled in a bond portfolio, the maximum payment per country is only a portion of the principal, P/n, where n is equal to the number of regions included in the portfolio (assuming the probability of a triggering event in each region are given equal weight). Given that the disaster events across countries are assumed to be independent, the likelihood that all included regions will experience a catastrophic event in the same year is the product of their individual disaster probabilities:

p t o t a l l o s s = ( p 1 ) * ( p 2 ) * ( p 3 ) … * ( p n )

Assuming our portfolio covers 1-in-a-100-year independent events in 5 countries the probability of total investor loss equals one in a billion:

p _{total loss }= (0.01) ^{5 }= 0.0000000001

As Figure 3 shows, after the addition of six or seven region-events, little more reduction in variance is realized. This is an important distinction as the benefits of pooling regions are shown under the assumption that the catastrophes are independent across these countries/regions; there is zero correlation between the occurrence of natural disasters in one area and another. Therefore, reducing investor catastrophe exposure would require identifying only a small group of countries possessing independent catastrophe exposures. The transaction costs of pooling countries into “bundled” catastrophe bond could be quite high. Still, the transaction costs would rise with each additional country, but there may be some economies of scale and the marginal cost of adding each additional country should decline. Likewise, with a smaller pool, risk exposure increases. Therefore, it is encouraging that grouping even a small number of countries greatly reduces investor risk exposure. The optimal portfolio could be determined by the intersection of the transaction costs (indicated by the dashed line in Figure 3) and the variance.

It may even be possible to create a portfolio of countries that experience negative correlation between certain events. For example, in El Niño years the likelihood of abnormally