The paper analyzes the possibility of reaching an equilibrium in a market of marine mutual insurance syndicates, called Protection and Indemnity Clubs, or P&I Clubs for short, displaying economies of scale. Our analysis rationalizes some empirically documented findings, and points out an interesting future scenario. We find an equilibrium in a market of mutual marine insurers, in which some smaller clubs, having operating costs above average, may grow larger relative to the other clubs in order to become more cost effective, and where medium to larger cost efficient clubs may stay unchanged or some even downsize relative to the others. Some of the very large clubs suffering from diseconomies of scale may have a motive to further increase relative to the other clubs. According to observations, most clubs have, during the last decade, expanded significantly in size measured by gross tonnage of entered ships, some clubs have merged, but very few seem to have decreased their underwriting activity, in particular none of the really large ones. The analysis points to the following future scenario: The small and the medium to large clubs converge in size, while there is a possibility for some very large clubs to be present as well.

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## Scholarly Works (4 results)

In this paper we make use of option pricing theory to infer about historical equity premiums. This we do by comparing the prices of an American perpetual put option computed using two different models: One is the standard model with continuous, zero expectation, Gaussian noise, the other is a very similar model, except that the zero expectation noise is of Poissonian type. Since a Poisson random variable is infinitely divisible, by the central limit theorem it is approximately normal. The interesting fact that makes this comparison worthwhile, is that the probability distribution under the risk adjusted measure turns out to depend on the equity premium in the Poisson model, while this is not so for the standard, Brownian motion version. This difference is utilized to find the intertemporal, equilibrium equity premium. We apply this technique to the US equity data of the last century, and find an indication that the risk premium on equity was about two and a half per cent if the risk free short rate was around one per cent. On the other hand, if the latter rate was about four per cent, we similarly find that this corresponds to an equity premium of around four and a half per cent. The advantage with our approach is that we only need equity data and option pricing theory, no consumption data was necessary to arrive at these conclusions. We round off the paper by investigating if the procedure also works for incomplete models.

In order to find the real market value of an asset in an exchange economy, one would typically apply the formula appearing in Lucas(1978), developed in a discrete time framework. This theory has also been extended to continuous time models, in which case the same pricing formula has been universally applied. While the discrete time theory is rather transparent, there has been some confusion regarding the continuous time analogue. In particular, the continuous time pricing formula must contain a certain type of a square covariance term that does not readily follow from the discrete time formulation. As a result, this term has sometimes been missing in situations where it should have been included. In this paper we reformulate the discrete time theory in such a way that this covariance term does not come as a mystery in the continuous time version. It is shown that this term is also of importance in the equivalent martingale measure approach to pricing. In most real life situations dividends are paid out in lump sums, not in rates. This leads to a discontinuous model, and adding a continuous time framework, it appears that our framework is a most natural one in finance.

In this paper, we solve an optimal stopping problem with an infinite time horizon, when the state variable follows a jump-diffusion. Under certain conditions our solution can be interpreted as the price of an American perpetual put option, when the underlying asset follows this type of process.

We present several examples demonstrating when the solution can be interpreted as a perpetual put price. This takes us into a study of how to risk adjust jump-diffusions. One key observation is that the probabililty distribution under the risk adjusted measure depends on the equity premium, which is not the case for the standard, continuous version. This difference may be utilized to find intertemporal, equilibrium equity premiums, for example

Our basic solution is exact only when jump sizes can not be negative. We investigate when our solution is an approximation also for negative jumps.

Various market models are studied at an increasing level of complexity, ending with the incomplete model in the last part of the paper.