Mortality: Good things come to those who weight (II)

Maths!

This article is almost all maths.

If that’s not your thing then I suggest that you skip ahead to part III (to come).

In the previous article, I noted that the the ultimate objective when assessing base pensioner mortality for DB plans is to determine the present value of the liabilities, as opposed to selecting and calibrating an abstract model.

In this article, I’ll use a simple framework to express this mathematically.

Articles in this series

Measures matter
A over E
Log-likelihood
Proportional hazards
Suddenly AIC
Overdispersion and quasi-log-likelihood
Incoherent rating factors
Pensioner mortality variation
Good things come to those who weight: Part I, Part II and Part III (to come)

Starting point

These are our givens:

The objective is to determine the best estimate and uncertainty of the present value of liabilities.
Our mortality model has a single scalar parameter – see e.g. this previous article in relation to DB pensioner mortality.
We are provided with a measure of relevance – see the definition here. This quantifies the real life fuzziness around which data to rely on when estimating mortality.

To ease the notational burden, I’ll make the following simplifications:

Instead of indexing everywhere by individual \(i\) and time \(t\), use a combined index, e.g. \(a\). (For individuals in the valuation dataset, \(t\) is always the valuation ‘as at date’ and so these are already effectively indexed by individual.)
Instead of writing out integrals over time and sums over individuals for experience data, denote this using the discrete summation symbol, e.g. \(\sum_a \ell_a\)⁵.
Instead of writing log-likelihood or liability value as a function of a mortality, e.g. \(\ell(\mu(\beta))\) or \(v(\mu(\beta))\), write them as direct functions of the mortality model parameters, e.g. \(\ell(\beta)\) and \(v(\beta)\).

Some definitions:

We’ll initially assume our mortality model takes \(n\) parameters, represented as a vector, e.g. \(\beta\). (We’ll specialise to the \(n=1\) case later.)
\(\ell_a(\beta)\) is the log-likelihood of the E2R for \(a\in\text{Exp}\) using our mortality model, where \(\text{Exp}\) is the experience data.⁵
\(v_b(\beta)\) is the present value of liabilities for valuation individual \(b\in\text{Val}\) using our mortality model, where \(\text{Val}\) is the valuation data.
\(r_{ab}\in[0,1]\) is the relevance of \(a\) to \(b\), with \(r_{aa}=1\) and \(r_{ab}=r_{ba}\).
\('\) and \(''\) are the vector first derivative and matrix second derivative with respect to \(\beta\).
\(^\text{T}\) is vector or matrix transpose.

We’ll also impose the simplest non-trivial conditions to make things tractable:

\(\ell_a(\beta)\) is approximately quadratic, with \(\ell''_a\) negative definite, and
\(v_b(\beta)\) is approximately linear in \(\beta\), with \(v'_b\) positive¹.

Roadmap

To give you an idea of where we’re headed, this is a roadmap:

First, we’ll define best estimate liability taking account of relevance.
Then we’ll look for an equivalent overall parameter that gives the same answer.
To complete the circle, we’ll re-express the derivation of that equivalent overall parameter in terms of weighted log-likelihood.
Finally, we’ll address parameter uncertainty.

Best estimate liability

The relevant log-likelihood for each individual to be valued \(b\in\text{Val}\) is

\[L_b(\beta)=\sum_{a\in\text{Exp}}r_{ba}\ell_a(\beta)\]

This defines an estimate of the model parameter for \(b\in\text{Val}\) as

\[\hat\beta_b = \underset{\beta}{\arg\max}\, L_b(\beta)\]

or, equivalently, given that the \(\ell_a\) are approximately quadratic, we can solve the differential equation

\[L'_b(\hat\beta_b)=0\tag{17}\]

The best estimate of total liability value is the sum of each individual’s liabilities evaluated using the parameter fitted to the data relevant to that individual, i.e.

\[\hat V=\sum_{b\in\text{Val}}v_b(\hat\beta_b)\tag{18}\]

Equivalent overall parameter

Equation \((18)\) is fine in theory but it’d be a nightmare to implement because each individual in the valuation data requires their own mortality model to be calibrated and selected.

What if, instead, we could derive a single overall parameter vector estimate, \(\hat\beta\), that gave the same total value, i.e.

\[\hat V=\sum_{b\in\text{Val}}v_b(\hat\beta)\]

The assumed approximate linearity of \(v_b\) means that this is equivalent to the condition

\[
0
=\sum_{b\in\text{Val}}\left\{v_b(\hat\beta)-v_b(\hat\beta_b)\right\}
\approx\sum_{b\in\text{Val}} v'_b{}^\text{T}  \left\{\hat\beta-\hat\beta_b\right\}
\tag{19}
\]

\(v'_b\) is written without an argument because, by assumption, it is approximately constant.

So a single overall estimate \(\hat\beta\) that complies with equation \((19)\) will provide (approximately) the same overall value as the complex approach embodied in equation \((18)\).

As it stands, equation \((19)\) still requires that we calculate parameters \(\hat\beta_b\) separately per individual. Is there a way to sidestep this?

One approach is to try to sum up the log-likelihoods in such a way that the maximum of that sum is automatically at \(\hat\beta\). Specifically, we’d like a scalar weight, \(\tilde w_b\) for \(b\in\text{Val}\) such that equation \(19\) holds if

\[\hat\beta=\underset{\beta}{\arg\max} \sum_{b\in\text{Val}} \tilde w_b \, L_b(\hat\beta)\]

or, equivalently,

\[\sum_{b\in\text{Val}} \tilde w_b \, L'_b(\hat\beta)=0\tag{20a}\]

The assumption that \(\ell_a\) is approximately quadratic means that so is \(L_b\) and hence

\[L'_b(\hat\beta) 
\approx L'_b(\hat\beta_b)+L''_b \left\{\hat\beta-\hat\beta_b\right\} 
= -I_b \left\{\hat\beta-\hat\beta_b\right\}\tag{21}\]

where I’ve used equation \((17)\) and, for convenience, defined the ‘relevant information matrix’ as

\[I_b=-L''_b\]

\(I_b\), \(L''_b\) and \(\ell''_a\) are written without an argument because they are approximately constant (similar to \(v'_b\), as noted above).

This enables us to rewrite equation \((20a)\) approximately as

\[\sum_{b\in\text{Val}} \tilde w_b I_b \left\{\hat\beta-\hat\beta_b\right\}=0\]

and comparison with equation \((19)\) suggests¹²³

\[\tilde w_b=I_b^{-1} v'_b\tag{22}\]

The single constraint implied by (the scalar) equation \((19)\) does not define a scalar weight if there are \(n>1\) degrees of freedom in (the vector) equation \((20a)\). This is manifest in equation \((22)\), which defines \(\tilde w_b\) as a vector, not a scalar, and in which case equation \((20a)\) would need to be rewritten as

\[\sum_{b\in\text{Val}} \tilde w_b^\text{T} \, L'_b(\hat\beta)=0\tag{20b}\]

Weighted log-likelihood

While this approach is not fruitful in the \(n>1\) case, one of our givens above is that our mortality model has a single scalar parameter, i.e. \(n=1\), which means that, in our scenario, equation \((22)\) does define a scalar weight.

I used \(\tilde w_b\) to denote the weighting of relevant log-likelihood indexed by valuation individual \(b\) because I was saving \(w_a\) to denote the equivalent weight indexed by experience datum, i.e.

\[w_a =\sum_{b\in\text{Val}} r_{ab}\tilde w_b\]

Changing order of summation, we have

\[\sum_{b\in\text{Val}} \tilde w_b L_b(\beta)=\sum_{b\in\text{Val}} \tilde w_b \sum_{a\in\text{Exp}} r_{ba}\ell_a(\beta)= \sum_{a\in\text{Exp}} w_a\ell_a(\beta)\]

and hence the general weight to apply to the experience data is³

\[w_a=\sum_{b\in\text{Val}} r_{ab} \, I_b^{-1} v'_b\tag{23}\]

Insight 15. Weighted log-likelihood automatically estimates liabilities correctly for single scalar parameter models when provided with relevance

If (a) a mortality model has a single scalar parameter and (b) relevance is provided then maximising log-likelihood weighted by

\[w_{it}=\sum_{k\in\text{Val}} r_{itku} \, I_k^{-1} v'_k\]

automatically results in the best estimate of the present value of liabilities.

In the above,

\(r_{itku}\) is the relevance of the log-likelihood of the E2R of individual \(i\) at time \(t\) to individual \(k\) in the valuation data as at the valuation date \(u\),
\(I_k\) is the relevant information matrix for valuation individual \(k\), and
\(v'_k\) is derivative of liability value for valuation individual \(k\) with respect to the model parameter \(\beta\).

For further definitions, see article body.

[All mortality insights]

I suggest that the above means is that, in a DB pensioner context, if we already have a reasonable default mortality (so that we can calculate \(I\) and \(v'\)) then we can define in advance a weight such that maximising weighted log-likelihood results in the correct overall parameter for assessing liability value for a single parameter model.

I’ll expand on this in part III (to come).

Parameter uncertainty

Best estimates are meaningless without some measure of uncertainty. I’m not going to plow through yet more maths here and so instead I’ll simply state results⁴.

We can estimate the variance in best estimate liabilities as

\[
\text{Var}(\hat V)
\approx \sum_{b,c\in\text{Val}}v'_b{}^\text{T}\,\text{Cov}(\hat\beta_b,\hat\beta_c)\,v'_c
\approx-\!\sum_{a\in\text{Exp}}u_a^\text{T}\,\ell''_a\,u_a
\]

where \(u_a\) is the vector

\[u_a=\sum_{b\in\text{Val}} \sqrt{r_{ab}}\,I_b^{-1}v'_b\tag{24}\]

Contrast the uncertainty weight in equation \((24)\) with the weight used to derive the best estimate in equation \((23)\) above.

Given our assumptions, we have

\[\text{Var}(\hat V)\approx V'{}^\text{T}\text{Var}(\hat \beta)V'\]

where

\[V' = \sum_{b\in\text{Val}}v'_b = -\!\sum_{a\in\text{Exp}}\ell''_a w_a \]

In the \(n=1\) case, this defines an estimate of the variance of \(\hat\beta\) as

\[\text{Var}(\hat\beta) \approx (V')^{-2}\;\text{Var}(\hat V)\tag{25}\]

Insight 16. A different weight is required to determine uncertainty in the presence of relevance

The log-likelihood weight to determine uncertainty that corresponds to the best estimate weight in Insight 15 is

\[u_{it}=\sum_{k\in\text{Val}} \sqrt{r_{itku}} \, I_k^{-1} v'_k\]

[All mortality insights]

Summary

Let’s take stock: if our underlying mortality model has a single scalar parameter, as is typically the case for mortality models for DB pensioners, then

maximising log-likelihood weighted by \(w\) leads to the best estimate of liability value, and
the second differential of log-likelihood weighted by \(u^2\) estimates parameter (and liability) uncertainty.

Next article: Good things come to those who weight (III)

There remain questions of practicality and generalisability.

I’ll conclude this series by addressing those questions in part III (to come).

Althought I’ve assumed \(v'_b\) is positive, the sign of \(v'_b\) doesn’t matter provided it’s consistent: if \(v'_b\lt0\) then replace \(v'_b\) everywhere with \(-v'_b\). Indeed, because mortality models are usually arranged so that higher \(\beta\) increases mortality and the liability in a pensions context is an annuity, I’d expect \(v'_b\) to be negative. ↩↩
\(I_b\) is known to be invertible because it is positive definite, which in turn is because, by assumption, the \(\ell''_a\) are negative definite. ↩
Pointed out to me by Andy Harding – thank you. ↩↩
These results are unchecked, so please review before you apply them. ↩
Sums over infinitesimals using measures are usually written with \(\int\), not \(\sum\). But I think using a single \(\sum\) symbol makes it easier to see what’s going on compared with having to write sums over individuals and integrals over time everywhere. If it troubles you, imagine there’s a time grid with spacing \(\delta t\), replace \(\ell_a\) with \(\delta\ell_{it}\) and take the limit \(\delta t\rightarrow 0\). ↩↩