"There is no right price of insurance; there is simply the transacted market price which is high enough to bring forth sellers, and low enough to induce buyers".At university, I learned that tradable risks and non-tradable risks are priced differently. Traditional insurance risks have been considered as non-tradable, and financial instruments generally as tradable. In this view, the two text-book approaches areFinn and Lane (1997)
The actuarial price of two equal cash-flows, however, is not necessarily the same. As an example, consider two reinsurance alternatives for the same risk. Alternative 1, consisting of two XL treaties, 2xs1 and 3xs3. And an alternative 2, consisting of only one 5xs1 treaty. Obviously, these two alternatives produce the same cash-flows (assuming large enough annual aggregate limits). Nonetheless, the actuarial approach may result in two different prices for these alternatives. In our example below, alternative 1 costs 0.51 + 0.16 = 0.67 which is more than the cost of alternative 2, 0.64. Unthinkable for tradable risks.
ILS are only one component of the "alternative risk transfer" market. It is however out of the scope of this post to describe alternative risk transfer in more detail. The take-away of this section is that the market for alternative risk transfer is growing, weakening the non-tradability assumption for insurance risks.
For tradable insurance risks, a myopic traditional actuarial pricing is dangerous. It bears the risk of being arbitraged out of the market. There is ample literature on the financial approach to pricing insurance risks. See, for example, the papers of Delbaen and Haezendonck (1989), Sondermann (1991) and Embrechts (1996).
To conclude, let's do a pricing. Consider three (fictitious) securities with binary trigger based on industry loss, as depicted below. Security B, for example, pays the investor a coupon of 7.5% and suffers a complete loss if industry loss of a hurricane reaches USD 50bn. What coupon would you come up with for security C?
The following three graphs show simulated values of two random variables A and B. Instead of the values, their percentiles are plotted. The point (0.3, 0.5), for example, means the simulated values correspond to the 30th percentile for A and to the 50th percentile for B. For each graph, a correlation (Spearman's Rho) was used to produce the plots. Can you guess the values of these correlations?
The answer is...
The way to go is to use copulas instead of correlations. Contact us if you wish to know more about this subject.
Within a continuous-time model we derive insurance demand and welfare as a function of personal wealth, the insurance company's wealth and the claims process and compare them to their counterparts in a static model. Besides discussing welfare-maximizing capital we provide some new insights on insurance demand.
We find that the optimal amount of risk capital depends on the size of the insurance company, the default risk and risk aversion of the insurance buyer, the probability to find a new insurer following default of an existing insurer, the default fine, external costs and the price of capital. Hence, within our model, it is not welfare-maximizing to require the same level of solvency from all insurers.
Separate the number at the decade position into two numbers as in the following three examples.
1) 1'199 becomes 11 and 99
2) 917'037 becomes 9'170 and 37
3) 264 becomes 2 and 64
If the sum of these two numbers is divisible by 11 then the original number can also be divided by 11.
It turns out that all of our examples are divisible by 11:
1) 11+99 = 110 is divisible by 11 and so is 1'199.
2) 9'170+37 = 9'207 is divisible by 11 because 92+07 = 99 is and therefore also 917'037 is divisible by 11.
3) 2+64 = 66 is divisible by 11 and so is 264.
We remember that the real numbers consist of rational (can be expressed as a fraction of two integers) and irrational (cannot be expressed as a fraction of two integers) numbers. An example of an irrational number is the square root of 2. Observe that squaring the square root of 2 gives us 2 which is rational. Not every irrational number yields a rational number when squared.
A transcendental number is an irrational number for which we cannot find a non-negative integer such that its power to that integer is rational. More exactly, a transcendental number is not a root of a finite, non-zero polynomial with rational coefficients. Pi is transcendental (the proof is not trivial, see Wikipedia) meaning that the square of Pi or Pi3 or Pi5 + 0.3*Pi2 etc. are all irrational. A number which is not transcendental is called algebraic. 0.25 is algebraic. The square root of 2 is also algebraic.
Pi being transcendental is the reason why squaring the circle is impossible, a problem which kept ancient geometers busy. Indeed, if it were possible to square a circle meaning that we could construct a square with the same area as a circle using only a finite number of steps with a circle and a straightedge then this would imply the existence of a polynomial for which Pi is a root which is a contradiction (intersecting lines and circles boils down to equating polynomials).
Where can we find transcendental numbers? Well, take any two different numbers and there is an infinity of transcendental numbers between them. Proof: Pi is transcendental and so is q*Pi for any rational q not equal to zero (because if q*Pi were not transcendental then it would be the root of a polynomial with rational coefficients which would automatically make Pi algebraic too which is a contradiction) and there is an infinity of q so that q*Pi lies between our two numbers.