Lee-Carter Mortality Model
The following model allows for forecasting mortality risk based on the state-space Lee-Carter model, which is presented in the research paper A State-Space Estimation of the Lee-Carter Mortality Model and Implications for Annuity Pricing.
The calculations are based on the stochastic Lee-Carter mortality model, which forecasts age-specific central death rates \(m_{xt}\), where \(x=x_1,...,x_p\) and \(t=1,..,n\) represent age and year (time) respectively. We use the state-space representation of the model where the process \(\mathbf{y}_t = (y_{x_1 t},..., y_{x_p t})'\) of the log central death rate, \(y_{xt} = \ln m_{xt}\), is combined with the unobserved latent time trend denoted
by \(\kappa_t\). This can be written as one dynamic system
\[
\mathbf{y}_t = \boldsymbol{\alpha} + \boldsymbol{\beta} \kappa_t + \boldsymbol{\epsilon}_t, \quad \quad \kappa_t = \kappa_{t-1} + \theta + \omega_t,
\]
where \(\boldsymbol{\epsilon}_t \sim \mathcal{N}(0, \sigma^2_\epsilon \mathbf{1}_p)\), \(\boldsymbol{\omega}_t \sim \mathcal{N}(0, \sigma^2_\omega )\), \(\boldsymbol{\alpha} = (\alpha_{x_1} ,... , \alpha_{x_p})'\), \(\boldsymbol{\beta} = (\beta_{x_1} ,... , \beta_{x_p} )'\), \(\boldsymbol{\epsilon}_t = (\epsilon_{x_1 t}, ..., \epsilon_{x_p t})'\), \(\mathbf{1}_p\) is the \(p\) by \(p\) identity matrix and \(\mathcal{N}(.,.)\) denotes the Gaussian distribution.
By setting \(\alpha_{x_1}\) and \(\beta_{x_1}\) to constants it ensures that the model becomes identifiable. The model then uses an efficient approach involving a combined Gibbs sampling conjugate model sampler for the marginal target distributions \(\pi(\boldsymbol{\Psi}|\kappa_{0:n}, \mathbf{y}_{1:n})\) of the static model parameters \(\boldsymbol{\Psi}:= (\alpha_{x_2:x_p}, \beta_{x_2:x_p} , \theta, \sigma^2_\epsilon, \sigma^2_\omega)\) along with a forward backward Kalman filter sampler for the latent process \(\kappa_{1:t}\). The predictive distributions of \(\mathbf{y}_{n+k}\), given \(\mathbf{y}_n\), to forecast death rates at \(k\) years ahead are obtained using the Monte Carlo Markov Chain samples after a burn-in period has been removed.
The data sets are obtained from the Human Mortality Database. The available data ends at 2014, and we therefore utilize the 1-year death rates for age 60-100 from year 1975-2014.
For more information, please refer to A State-Space Estimation of the Lee-Carter Mortality Model and Implications for Annuity Pricing.
| Mortality table |
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| Years to predict |
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Please enter an integer in range [1, 40]
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| Iterations |
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Please enter an integer in range [100, 1 000 000]
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| Burn rate (%) |
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Please enter a value in range [0, 0.2]
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Important! This model is for online demonstration purposes only. Calculations can take a long time if the number of iterations are high. The time-out for calculations has been set to 1 hour, after which the webpage will return an error. We suggest you start with low values for the iterations and then increase them if better accuracy is needed.