生命保険とリスク配分

最近生命保険を金融商品として考える記事を読みました。
Life insurance companies are financial intermediaries which can serve to distribute risk in two fashions. The first method is, by underwriting a large number of people and assuming most of them will not die, insurance policies can be written which cost substantially less than a policy written by one person for another. The second way is that, by raising funds in the capital markets from a large number of investors the company can provide a relatively low risk investment as part of a diversified portfolio.
The key variables in modeling life insurance include: Y (number of people who die that year), P (premium amount), c (amount paid in event of death), N (number of policies sold), p (probability any one person will die), C (total liability for the year, or cY), Z (the return on capital for the company), K (the amount of capital paid in).
The fundamental question to be asked in structuring the capital for this company is: given a number of policies, a desired confidence level for avoiding default, and a desired rate of return, what is the amount of capital which must be held?
The return rate is: Z= (R(NP+K)-C)/K, where R is the risk free rate of interest plus 1. Based on the capital asset pricing model, the demand for the asset must equal the supply, where the supply is K. Z-R = r(s)Cov(Z,Z(m)) where r(s) is (I believe) the rate of return for the whole portfolio in excess of the risk free rate and Z(m) is the rate of return for the market. This equation is based on the security market line and basically says that the excess return of the insurance stock over the risk free rate is equal to the covariance of the insurance with the market times the excess return of the market. The derivation for this is not understood and taken as given. This allows us to compute P and E(Z) in terms of c, W(per capita wealth), V (std deviation of y, the binary variable for a death), R, N, r(s), p, and K.
Next to determine the probability of default, we use:
prob(c*sum(yi) > RK + Ncp + r(s)cqV^2), where q = c/W.
This is the probability that the liabilities exceed paid in premiums and capital. As any value to the right of the above is in default, CDF(RK+Ncp+r(s)cqV^2) = 1-Prob(default), where CDF is the cumulative normal distribution function. We could then solve for the value or capital given the desired probability of default by taking the inverse norm. However, we can simplify calculation somewhat by rewriting the inequality so that the left hand variable is standard normally distributed (mean 0, variance 1). Then we can compute the number of standard deviations to the right of the mean we will be for a given probability of default and compute the necessary K in terms of this number of standard deviations.
In summary, if N= 100000000 (100 million policies sold), then the maximal liability (if each policy is for 30000) is 3 trillion dollars. However, if only 60000000 in capital is paid in, the probability of default is only 1/25000. The ratio of capital to number of policies is .6, or 60 cents per policy. The expected rate of return on the insurance stock with this probability of default is 0, though, and there is a .16 probability of a 15 million dollar loss. However, if there are 100000000 shares outstanding (1 share per policy), then this amounts to a loss of only 15 cents per share.