Measuring liability uncertainty

Pricing block transactions is a high-stakes business.  An insurer writing a bulk annuity has one chance to assess the price to charge for taking on pension liabilities.  There is a lot to consider, but at least there is data to work with: for the economic assumptions like interest rates and inflation, the insurer has market prices.  For the mortality basis, the insurer usually gets several years of mortality-experience data from the pension plan.

Data sets are finite, however, so there is always a degree of uncertainty over the mortality basis.  This is particularly the case where liabilities tend to be concentrated in small sub-groups with different mortality characteristics.  Insurers are therefore very interested in how uncertainty over the different mortality characteristics feeds through into uncertainty over the liability value.

We can build a statistical model to account for these mortality characteristics, say with a parameter vector \(\boldsymbol{\theta}\).  We can then use the pension plan's mortality-experience data to obtain an estimate, \(\boldsymbol{\hat\theta}\), of the unknown true parameter vector, \(\boldsymbol{\theta^*}\).  If the estimate is obtained by maximising the log-likelihood, then the Maximum Likelihood Theorem applies.  Viewed as a random variable, \(\boldsymbol{\hat\theta}\) has a multivariate normal distribution (MVN) with mean vector \(\boldsymbol{\theta^*}\) and covariance matrix \(\boldsymbol{\Sigma^*}\) (Cox and Hinkley, 1996, Chapter 9(iii)).  Since we don't know \(\boldsymbol{\theta^*}\) and \(\boldsymbol{\Sigma^*}\), we can replace them with their corresponding estimates from the log-likelihood function, i.e.

\[\boldsymbol{\hat\theta}\sim {\rm MVN}(\boldsymbol{\hat\theta}, \boldsymbol{\hat\Sigma})\qquad(1)\]

So far, so good.  We have a model for the pensioner mortality differentials, including any important sub-groups.  We have estimates for these differentials in \(\boldsymbol{\hat\theta}\), and knowledge of the uncertainty over these estimates in equation (1).  So what does this parameter uncertainty mean for the liability estimate?

We assume that we have a function, \(V(\boldsymbol{\theta})\), that uses knowledge of the mortality differentials to calculate a (scalar) value for financial liabilities.  However, if \(\boldsymbol{\hat\theta}\) is a random variable, as in equation (1), then so is \(V(\boldsymbol{\hat\theta})\), which means that our liability value (or price) has a probability distribution.  Specifically, \(V(\boldsymbol{\hat\theta})\), is a function of a multivariate random variable (I previously wrote about the univariate case in another context).  We can get an idea of the uncertainty over financial liabilities by using the multivariate delta method to approximate the variance:

\[{\rm Var}\left[V(\boldsymbol{\hat\theta})\right]\approx \boldsymbol{a}^T\boldsymbol{\hat\Sigma}\boldsymbol{a}\qquad(2)\]

where \(\boldsymbol{a}\) is the vector of first derivatives of \(V\) evaluated at \(\boldsymbol{\hat\theta}\), i.e. \(\boldsymbol{a} =  \displaystyle\left.\frac{\partial V(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}\right\vert_{\boldsymbol{\theta}=\boldsymbol{\hat\theta}}\).

There is an implicit assumption behind the delta method that the distribution of \(V(\boldsymbol{\hat\theta})\) is normal (Gaussian).  This might be close enough to the truth for many purposes.  For work involving the tails of the distribution of \(V(\boldsymbol{\hat\theta})\), however, it might be necessary to use the sampling approach of Richards (2016).