Introduction:

This report presents mortality modelling and projection for lifetime annuities offerings plan to adults turning 18. The modellings are calibrated under: Parametric and non-parametric curve fitting; Lee-Carter Model for projection fitting, which will be used to price the annuities offerings. The mortality model input is from Human Mortality Database (HMD) for Australia.

As the HMD rate is heavily skewed to the right, we take log transformation of rate to see the features clearer. The curve of mortality rate describes the three distinct patterns of mortality: 1)A child adapting to its new environment. Mortality fall rapidly in early childhood as the child gain immunity from diseases. 2) Senescent mortality. Mortality rate going up as the age increase, becoming straight line when age advances. 3) the superimposed accident mortality hump from age 18 to 25.

Parametric curve fitting-Makeham model:

Explanation:

The Makeham mortality law is: \[ux=A+B.C^x\] Where 𝜇𝑥 is mortality rate A,B,c is parameters to be estimated, representing distinct components of mortality. First term: A parameter is a constant, represent the base of mortality (ie: due to accident, pandemic, etc.) Second term: 𝐵.𝑐𝑥 is from Gompertz law (1825), indicating the geometric rise in mortality as human age. Technically, B represents senescent mortality at age x =0.

Parameter calibration method:

The parameter is estimated by optimizing loss function: \[L = log[mu/ov]^2 \] Where: mu: Estimated value ov: Observed value

The estimated model is: \(ux=1.3627e -0.5 +9.946e-0.2*1.07e-0.4^{x}\)

The graduated rates are then calculated using the fitted parameter estimates. Finally, we conduct statistical test to assess the graduation.

Shortcomings of Makeham model for our modeling purposes:

Our purpose is mortality modelling for lifetime annuities pricing, offering to adult from 18 years old. However, Makeham model proved by statistical tests above to over-graduate the mortality rates, emphasize too much on smoothness and do not adhere closely to the crude rates.

One of potential consequences is Makeham model overestimate mortality rate at the early childhood downward curve, leading to insufficient reserves for annuities. Although the overestimate happens before people can buy annuities, but estimate any age x carries information about adjacent ages, so this still potentially causes severe consequences.

The Makeham model also has clumpings, such as underestimate mortality at accident hump ages, which can cause excessive premium rates for young people; and overestimate mortality rate at around 50-70 age range, which is our target ages for annuities payout-Again, this will potentially causes insufficient reserves and inadequate premium rates.

Non-parametric curve fitting – Smoothing spline:

Fitting smoothing cubic spline for Mortality in 2016:

Cross Validation procedure to estimate lambda with validation data in 2017:
log.mx.17 <- log(AusMort[["rate"]][["total"]][,"2017"])

spline <- function(spar){
  res <- rep(0, length(x))
  mod <- fitted(smooth.spline(x, log(mx.16), spar = spar))
   mse <- mod - log.mx.17
  return(sum(mse^2)/length(mse))
}
spars <- seq(0, 1, by = 0.0001)

ss <- rep(0, length(spars))
for (i in 1:length(spars))
{
ss[i] <- spline(spars[i])
}

plot(spars, ss, 'l', xlab = 'spar', ylab = 'Cross Validation MSE' , main = 'CV MSE vs Spar')

From the graph above, hyper parameter spar has increasing monotonic relationship with the cross validation Mean Squared Error (MSE), so the optimal spar happens to be 0, which gives lowest Cross Validation MSE. This happen because the validation data set in 2017 has linear relationship and correlation is nearly equal to 1 with training data set in 2016, as shown in figure above.

Mortality Projection fitting to the year 2030 using Lee Carter model

AusMort.LC.T <- lca(AusMort, series = "total", max.age = 100)
plot(AusMort.LC.T)

AusMort.LC.T.forecast <- forecast(AusMort.LC.T,h=12) 
plot(AusMort.LC.T.forecast$kt.f)

The two age-specific parameter remains the same over time, so we only plot the time specific parameter to the year 2030 so we project the mortality rate in 2030 will adhere the downward sloping trend.

Implications of not including mortality improvement in new products:

Improvement in mortality rates, especially for after age 65 (post-retirement age) can impose financial impacts on annuities cost. Longevity risk, together with decreasing interest rate will amplify the concern, especially for guaranteed annuities provider. Given the declining trend in Australia interest rate (Reserve Bank Australia,2020), ignorance on improvements of mortality can cause a great risk of making loss for insurer.