Graduation

Jake

27/04/2022

Graduation

  • Graduation is the process of smoothing crude estimates of \(\hat{q}_x\) or \(\hat{\mu}_x\)

  • There are many reasons for graduation:

    • It is intuitive that the values for these estimates would follow a smooth function
      • Allows for interpolation and extrapolation
    • Removes sampling noise, allowing to see trends easier
    • Incorporates information from adjacent ages for estimate, similar to pooling information
    • It is desirable for financial quantities to progress smoothly with age

  • There are also many desirable features of a graduation:

    • Smoothed or graduated rates \(\mathring{q_x}, \mathring{\mu}\) need to satisfy:
      • Smoothness
      • Goodness of fit/adherence to data
      • Suitability for application
    • Smoothness and Adherence typically conflict
      • If rates are too smooth/overgraduated, then the rates show little adherence to data
      • If rates follow the observed data too closely/overgraduated, it will have inadequate smoothness

  • After graduation consider the reasonability and financial risks

    • Over/under estimation of rates can lead to losses
    • Do reasonability checks, such as male \(>\) female mortality

Parametric Graduation

  • Choose a mathematical formula with unknown parameters

  • Estimate parameters

    • MLE
    • Minimise sum of squared standardised deviations
    • Minimise weighted least squares

  • Calculate graduated rates

  • Test Graduation

  • Note too many parameters may result in undergraduation (low bias, high variance)

Gompertz and Makeham

  • Gompertz considers an exponential approach
    • Assumes log-linear

\[ \mu_x = Bc^x,\quad\text{or }\quad \mu_x=\alpha e^{\beta x}\]

  • Makeham improves on Gompertz by also adding an intercept for young values, as it tends to be curved in the loglinear space

\[ \mu_x = A + Bc^x\]

  • We can generalise these approaches with the generalised makeham class of models:

\[ \mu_x = \sum^{r-1}_{i=0}\alpha_i x^i+\exp\left(\sum^{s-1}_{j=0}\beta_jx^j\right)\]

  • Heligman and Pollard (first) Law

    • has different components to account for non-linear progression of mortality with age

    \[ q_x = A^{(x+b)^C}+D^{-E(\ln(x)-\ln(F))^2}+\frac{GH^x}{1+GH^x}\]

  • Maxmise Maximum Likelihood Function

    • Either binomial (\(q_x\)) or poisson (\(\mu\))

  • Minimise \(\chi^2\) statistic:

\[ \sum\frac{(A-E)^2}{E}\]

  • Minimise weighted least squares
    • Use weight based on variance to give less weight to more variable ages (typically with less data)
      • Use inverse of variance

\[ \sum w_x(\hat{q_x}-\mathring{q}_z)^2 \]

  • Binomial Weights:

\[ w_x\approx \frac{E_x}{\hat{q}_x(1-\hat{q}_x)}\approx\frac{E_x}{\hat{q}_x}\]

  • Poisson Weights:

\[ w_x \approx \frac{E^c_x}{\hat{\mu}_x}\]

Graduation with Reference to Standard Table

  • Useful when there is not a lot of data
    • Assume that the overall progression of rates from age to age should be similar
  • Select appropriate standard table
  • Decide relationship between standard table and graduated rates \(\mathring{q}_x = f(q_x^s)\) or \(\mathring{\mu}_x = f(\mu_x^s)\). For example:
    • Can determine these relationships by plotting \(\hat{q}_x\) against \(q^s_x\)

\[ \mathring{q}_x = a + bq_x^s\]

  • Determine parameter values through MLE, least squares, weighted least squares
  • Test the graduation/goodness of fit
    • Can assume it will be relatively smooth since the standard rates themselves are smooth

Graduation with Splines

  • Useful and flexible curve fitting method
  • Knots that are used are not necessarily integers
  • Concentrate knots around the accident hump as this is the most variable point
  • General Spline Formula:

\[ q_y = a_0+a_1y+a_2y^2+a_3y^3+\sum^n_{j=1}b_j(y-x^{(j)})_+^3 + \epsilon_y\]

  • We often use weights in the least squares spline regression. We times both sides by \(\sqrt{w_x}\):

\[ \sqrt{w_x}q_x = \sum^3_{i=0}a_i\sqrt{w_x}x^3 +\sum^n_{j=1}b_j\sqrt{w_x}(y-x^{(j)})_+^3+\sqrt{w_x}\epsilon_x\]

Smoothing Splines

  • Non-parametric natural cublic spline that minimises the following equation:
    • \(\lambda\) impacts the level of smoothness, with \(\lambda = 0\) resulting in undergraduation to the extent of extrapolation

\[ \sum^n_{i=1}[y_i-f(x_i)]^2 + \lambda\int^{x_n}_{x_1}[f''(t)]^2dt \]

Comparison of Different Grad Methods

Parametric

  • Used when large amounts of data is available
  • Produces extremely precise results
  • Creates smooth enough results for a small enough number of parameters
  • Optimised through statistical fitting, therefore not subjective
  • Can use MLE parameters which have good statistical properties
  • However, it is hard to find a curve that fits an experience well at all ages

Reference to Standard Table

  • Relatively simple
  • Used with little data
  • Typically produces smooth rates
  • Easier to fit the high and low ages
  • Does not fully represent the data as it relies on a reference
  • Heavily depends on the standard table chosen

Testing for Smoothness

  • Can test for smoothness using finite differences of the data
  • First Differences:

\[ \Delta\mathring{q}= \mathring{q}_{x+1}-\mathring{q}_x\]

  • Second Differences:

\[ \Delta^2\mathring{q}= \Delta\mathring{q}_{x+1}-\Delta\mathring{q}_x\]

  • Third Differences:

\[\Delta^3\mathring{q}= \Delta^2\mathring{q}_{x+1}-\Delta^2\mathring{q}_x \]

  • Absolute value of third differences should be relatively small when compared to the graduated rates themselves

Statistical Tests for Adherence to Data of a Graduation

  • No single test will provide adequate coverage of every aspect of the fit
  • Assumptions
    • Lives are independent
    • No heterogenity in each age
    • Approximation (expected deaths should be >5 for normal approximation)
  • Null Hypothesis is that the actual data are consistent with the ones that are predicted by the graduated rates
  • Standardised Deviation:
    • If there is sufficient number of independent lives at each age x, by central limit theorem the standardized deviations are standard normal and mutually independent.

\[ \frac{A-E}{\sqrt{E}} \]

  • Under Poisson:

    • Number of deaths:

\[ D_x\sim N(E^c_x\mathring{\mu}_{x+0.5}, E^c_x\mathring{\mu}_{x+0.5})\]

  • Standardised Deviation:

\[ z_x = \frac{d_x-E^c_x\mathring{\mu}_{x+0.5}}{\sqrt{E^c_x\mathring{\mu}_{x+0.5}}}\]

  • Under Binomial:

    • Number of deaths:

\[ D_x\sim N(E_x\mathring{q}_x,E_x\mathring{q}_x(1-\mathring{q}_x))\]

  • Standardised Deviation

\[ z_x = \frac{d_x-E_x\mathring{q}_x}{\sqrt{E_x\mathring{q}_x(1-\mathring{q}_x)}}\approx \frac{d_x-E_x\mathring{q}_x}{\sqrt{E_x\mathring{q}_x}}\]

Chi-Square Test of Fit

  • General test for goodness of fit
    • Doesn’t give information on direction of any bias
  • Test Statistic:
    • Degrees of freedom \(n\) = Number of groups - Number of estimated parameters

\[ X = \sum z_x^2\sim\chi^2_n \]

  • Degrees of freedom:
    • Lose one per parameter estimated
    • When graduating using a standard table
      • Lose one degree for each parameter fitted
      • Also lose some degrees of freedom (2-3) due to constraints imposed on chosen table

Standardised Deviations Test

  • Normality test for standardised deviations
    • Checks whether standard deviations are too bunched, too spread out or in line with a standard normal distribution
    • Can also use any other normality test, such as chi-squared, qq plots
  • Roughly half of the deviations should fall between \((-\frac{2}{3},\frac{2}{3})\)

  • Portion into table with intervals and use chi squared statistic based on number of deviations in each group:
    • Note chisquared has n-1 degrees of freedom, where n is number of groups

\[ X=\sum\frac{\text{actual-expected}}{\text{expected}}\sim\chi^2_{n-1}\]

Sign Test

  • Tests balance between positive and negative deviations

    • Roughly half of the deviations should be positive and negative
    • Provides no indication of the extent of the discrepencies

  • Calculate the test statistic \(P\): the number of \(z_x\) that are positive

  • Find p-value of the value in relation to binomial.

  • Alternatively use the normal approximation (if m>20):

\[ P\sim N\left(\frac{1}{2}m,\frac{1}{4}m\right)\]

Cumulative Deviations Test

  • General goodness of fit
    • High value of test statistic indicates either:
      • The graduated rates are biased
      • Actual variance is higher than predicted by the assumed model for the range of ages considered (could be due to duplicate policies)
    • Detects overall bias or long runs of deviations of the same sign
  • Test Statistic:

\[ \frac{\sum A - E}{\sqrt{\sum Var(rate)}}\sim N(0,1)\]

  • Perform hypothesis test assuming standard normal distribution

Grouping of Signs Test (Stevens)

  • Tests for overgraduation and runs of the same sign
    • This can however lead to different results based on whether positive or negative is chosen
  • Compare the number of groups/runs with the number of groups that would be expected if the positive and negative deviations were arranged in a random order
  • Test statistic is:

\[ G = \text{ Number of groups of positive }z_x's\]

  • Perform hypothesis test with the hypogeometric distribution
  • For large \(m\) we can use the normal approximation:

\[ G\sim N\left(\frac{n_1(n_2+1)}{n_1+n_2},\frac{(n_1n_2)^2}{(n_1+n_2)^3}\right) \]

Serial Correlations Test

  • Tests for overgraduation
    • Overgraduated curves tend to stay on the same side of the crude rates for relatively long periods of time
    • Undergraduated curves will cross the crude rates frequently
      • One part of correlation can be cancelled by other parts of correlation, therefore this test is typically weaker than the signs/grouping of signs tests
  • Formulas:

\[ r_j=\frac{\sum^{m-j}_{i=1}(z_i-\bar{z}_1)(z_{i+j}-\bar{z}_2)}{\sqrt{\sum^{m-j}_{i=1}(z_i-\bar{z}_1)^2\sum^{m-j}_{i=1}(z_{i+j}-\bar{z}_2)^2}}\]

\[ \bar{z_1} = \frac{1}{m-j}\sum^{m-j}_{i=1}z_{i} \]

\[ \bar{z_2} = \frac{1}{m-j}\sum^{m-j}_{i=1}z_{i+j}\]

  • For large \(m\), \(\bar{z}_1,\bar{z}_2\) can be approximated by the total average

\[ r_j\approx\frac{\frac{1}{m-j}\sum^{m-j}_{i=1}(z_i-\bar{z})(z_{i+j}-\bar{z})}{\frac{1}{m}\sum^{m}_{i=1}(z_i-\bar{z})^2}\]

  • Test statistic:
    • Noting large positive test statistic indicates over graduation

\[ t_j = r_j\sqrt{m}\sim N(0,1)\]

Effect of Duplicate Policies

  • There is the potential for duplicate policies, which violates the assumption of independence of policies

    • Has no effect on bias (expected value)
    • Affects variability, increasing the variance

  • Assume N lives from age \(x\) to \(x+1\)

    • Assume proportion $$ lives own \(i\) insurance policies (these properties are unknown)
    • Total number of policies is:

    \[ \sum_i i\pi_i N\]

    • Assume the mortality rate for each life is \(q_x\)
    • Let \(D_i\) be the number of deaths among the \(\pi_i N\) lives each with \(i\) policies and \(C_i\) be the number of claims among these lives

  • Assuming the binomial model:

\[ D_i\sim Bin(\pi_iN, q_x)\] * Then we have:

\[ \mathbb{E}[C] = \left(\sum_i i\pi_i N\right)q_x \]

\[ Var(C) = \left(\sum_i i\pi_i N\right)q_x(1-q_x)\]

  • Duplicate policies increase the variance of the number of claims by the ratio:

\[ \frac{\sum_i i^2\pi_i}{\sum_i i\pi_i}\]

  • Should make an allowance for the increased variances in statistical tests