Posted by Elsa - April 2019

Trading insurance risks has been made possible by innovations in the capital markets. It began in the 1990s when Catastrophe futures started trading on the Chicago Board of Trading (CBOT). And might be lifted to a whole new level by FinTech companies leveraging blockchains.

"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)

Actuarial pricing:

Price = Expected Loss + Loading
Financial pricing:

The crucial difference is - No arbitrage
- Find a new probability measure Q by changing the real world probabilities in order to give more weight to unfavourable events

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.

Example. Actuarial prices for loadings equal to 20% standard deviation of losses in the XL treaty, assuming a compound Poisson (λ=1.0)-Pareto (α=2.2) for claims exceeding 0.8.

ILS are fixed-income securities typically issued by an insurance company. The issuer receives investor capital to cover losses when a pre-defined event occurs. In return, the ILS investor usually receives a variable coupon payment which is a fixed margin plus a money market reference rate. If no triggering events occur, the capital usually sits in a collateral account until maturity.

ILS are tradable. To-date, trading is a manual affair with brokers matching buyers and sellers over the phone and email. But, as posted on Artemis.bm, the London Stock Exchange can make any ILS listed on its International Securities Market tradable. This has the potential to improve the market's liquidity.
You might read more about ILS in an older
article, written in collaboration with Synpulse, or read a beginner's introduction to ILS and alternative risk transfer in general on Artemis.bm.

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?

Posted by Elsa - March 2014

Be it for portfolio analysis, an internal model or for pricing aggregate covers, modelling dependencies can be crucial. A popular concept is the one of correlations. However, correlations are indeed a very vague concept.

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...

...80% for all three examples! This shows that knowing the correlation between two variates is generally not enough to say something about their joint distribution and answer questions like "if A is hit by a 1 in 10 years event, what is the probability that B is also hit by a 1 in 10 years event in the same year?".

The way to go is to use copulas instead of correlations. Contact us if you wish to know more about this subject.

Posted by Elsa - March 2014 - Published in "Insurance: Mathematics and Economics"

Our paper proposes two rational agent models to analyze welfare-maximizing capital requirements for insurance companies considering that capital is costly and therefore affecting the premium.

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.

**Highlights**

- We propose a tractable model for insurance demand in continuous time
- For exponential claims, we obtain closed-form solutions
- We show that it can be rational for a risk-neutral agent to buy insurance even if the premium is above the expected loss
- We provide a new insight on the Bernoulli principle within a static model

Posted by Elsa - April 2019

You might know that if the cross sum of a number is divisible by three then the number can be divided by three. Example: 85'2201 can be divided by 3 because 8+5+2+2+0+1=18 is divisible by 3. Can we find a similar test for detecting divisibility by 11? The cross sum doesn't work because 11*9 = 99 is divisible by 11 but 9+9 = 18 is not. However, the following works (and I don't claim to be the first to discover this)!

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.

I find transcendental numbers fascinating because they are what make the irrational numbers uncountable and gave ancient geometers headaches.

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 Pi^{3} or Pi^{5} + 0.3*Pi^{2} 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.