Excerpt

## Table of contents

List of Figures

List of Tables

List of Abbreviations

1 Introduction

2 Literature Review

3 Model Implementation

3.1 Estimation of A_1 and A_2

3.2 Prediction of A_1 and A_2

3.3 Predicted Mortality Rates for 2030

3.4 Pricing of an Immediate Pension Annuity

4 Conclusion

References

## List of Figures

Figure 1: Estimated Values of A_1(t) and A_2(t) from 1933 to 2010 for U.S. Males

Figure 2: Predicted Mortality Rates for 2030 from Ages 20 to 109

Figure 3: Comparison Log Mortality Ages 20 to 109 for the Years 1933, 2010 and 2030

Figure 4: t-period Survival Probability for U.S. Males: Age of 20

## List of Tables

Table 1: IPAF: Price of U.S. $1 Annual Lifetime Income for U.S. Males

## List of Abbreviations

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## 1 Introduction

Assurance companies face two main risk factors in the process of pricing annuity products namely the interest risk and the longevity risk. There are numerous products and possibilities for the insurers to hedge their interest risk using interest derivatives and long bonds. Hedging products against the longevity risk is uncommon but insurers have to take it into account when they are pricing their annuity products.

There are two types of longevity risks. On the one hand the idiosyncratic longevity risk and on the other hand the systematic longevity risk. With regards to the idiosyncratic longevity risk, individuals are faced with the issue that they need to invest in assets for their retirement in spite of an uncertain span of lifetime and thus an uncertain invest- ment horizon. Pricing of life annuities could be done according to corresponding mor- tality tables. If the clients of an insurer die on average according to mortality rates pro- vided by such tables, the revenues of the insurer should be sufficient to ensure the pay- ments for the clients who are still alive. The issue out of a pension fund perspective is that longevity has been improving over time and clients could live longer than anticipat- ed. These improvements occurred in an unpredictable way, especially at higher ages according to Cairns et al. (2006). Insurers therefore made false calculations of the insur- ance premium and suffered losses due to pensioners living longer than anticipated.

The systematic longevity risk is based on the stochastic variation of mortality. The future development of life expectancy will be highly unpredictable due to medical improvements or discoveries in genetic research. For that reason insurers need stochastic models to quantify the systematic mortality changes over time and to make a prediction about future mortality in order to prevent losses caused by longevity risk.

This paper will firstly discuss the literature regarding the Lee and Carter one factor model and the relevance of longevity risk for annuity pricing. Second this paper aims to estimate the stochastic two-factor model by Cairns, Blake and Dowd (2006) (CBD) for U.S. males from 1933 to 2010 by running a simulation to predict average mortality for the year 2030. In the further course will this stated prediction be used to price an annuity product followed by a brief conclusion and summary of results.

## 2 Literature Review

Lee and Carter (1992) developed an extrapolative stochastic one-factor model to make forecasts about future mortality, which has become one of the most used mortality fore- casting models in the world. This fundamental model enables us to easily incorporate medical advances and both behavioural as well as social influences on mortality trends. Furthermore, it captures the downward sloping trend observed for mortality over time. The combination of demographic aspects and statistical times series methods is the strength of this model.

Lee and Carter applied the model on United States mortality rates to forecast agespecific mortality rates from 1990 to 2065. At first they modelled the logs of the agespecific death rates (*mx,t* central death rate at age x and in year t) as a linear function of an unobserved time-varying index and parameters depending on age:

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where *ax * is an age-specific parameter that reflects the general shape of the mortality process. The factor *kt * is a time-varying parameter which represents the general speed of mortality improvement, the factor * bx * is an age-specific parameter that characterizes the sensitivity of *kt * at age x and *ε x,t * is the error-term. In the second step they applied the model to the central death rates in the U.S. from 1933 to 1987 and estimated the corre- sponding parameters using the singular value decomposition (SVD) method. The parameter *kt* indicates the intensity of mortality and is forecasted using the statistical time series method, specifically a random walk with drift:

Abbildung in dieser Leseprobe nicht enthalten

where *d* is the drift and *ε t * are the error-terms. Following this, it is possible to make a forecast of age-specific mortality and future life expectancy combining the forecast of *k * with the parameters *a* and *b*. According to the results, the authors anticipated that life expectancy would rise to 86.05 years in 2065, with a 95% confidence interval of +3.9 or -5.6 years according to Lee and Carter (1992).

The systematic longevity risk has a significant influence on the prices and benefits of annuity products and therefore has to be modelled when pricing such products. Maurer Mitchell and Rogalla (2013) studied how optimal household consumption patterns and portfolio allocation is influenced by variable investment-linked deferred annuities (VILDA). Firstly they took solely idiosyncratic risk into account and then in a second step they also considered the systematic longevity risk to show how results change due to stochastic development of mortality over time. The impact of systematic longevity risk on annuity prices and therefore also on household welfare depends mostly on how insurers hedge their products against the systematic longevity risk. There are various capital market instruments available to hedge against longevity risk, but there are also managing approaches available which do not need such instruments. Maurer et al. focus in their paper on two risk management approaches. First the insurers self-insurance and second the risk pool participation. Under the first approach the insurer is bearing the systematic longevity risk and thus require a higher price for the annuity upfront. The second approach is based on risk pool participation by the annuitants which enable the VILDA provider to transfer a fraction of the systematic longevity risk to its clients. The provider can achieve this by selling products which adjust their benefit payments to unanticipated mortality shocks.

The results show that households would benefit from VILDAs because of an increased flexibility in investment choice of their annuity assets and higher consumption over lifetime. The annuity provider would also benefit from selling VILDAs by allowing the transfer of investment risk to their clients according to Maurer et al. (2013).

## 3 Model Implementation

The data used in the following model are based on the human mortality database and specifically the United States of America life tables of males are used. Furthermore, the life tables and mortality rates from 1933 to 2010 for U.S. males from age 20 to 109 are taken into account according to the Human Mortality Database (2015). Mortalities for younger people younger than 20 contain a lot of extreme values and are not relevant for modelling mortalities at advanced ages. Moreover mortalities above the age of 109 are not considered.

The following is based on Cairns et al. (2006) and all calculations are implemented in Matlab. Cairns, Blake and Dowd developed a two-factor discrete time model that is based on the one-factor model of Lee and Carter (1992). The model aims to capture the stochastic development of mortality rates over time and thereby requires a discrete-time series model for the approach. The CBD model uses two stochastic factors to capture the random element in the development of mortality rates. The first factor A_1 influ- ences mortality at all ages in an equal way and the second factor A_2 has an age- dependent influence on mortality. These two factors are needed to achieve a sufficiently dynamic model which describes the stochastic development of the mortality term struc- ture. The model enables us to make a prediction about future survival rates and thereby makes it possible to model longevity risk according to Cairns et al. (2006).

### 3.1 Estimation of A_1 and A_2

The two factors A_1 and A_2 are most important for the implementation of the CBD model. At first the two factors were estimated based on the historical mortality data for U.S. males from 1933 to 2010. These estimated values can be used to make a prediction about future mortality.

The CBD model assumes that the mortality rates *q(x)*, which represent the probability of death between age x and x+1, develop according to:

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At first the *logit* of *qx,t* is calculated and after that A_1(t) and A_2(t) can be estimated for each *t* using the ordinary least squares method. The *logit* of *qx,t * is needed to simulate the development of mortality over ages, which is for young people relatively constant at the beginning, but then increases rapidly and is assumed to be constant again at the age of 110+. In a next step the two factors A_1 and A_2 are estimated using the ordinary least squares method. The input parameters consists of an age column vector, a constant column vector consisting of all parameters valued with one and the mortality matrix which consists of *logit* (*qx,t )* values corresponding to each year *t* from 1933 to 2010. The constant factor A_1 influences mortality at all ages in an equal manner and represents the intercept of the regression. The dynamic A_2 factor has an age-dependent influence on mortality, because mortality rates are higher for advanced ages. This factor repre- sents the slope of the regression line.

**[...]**

- Quote paper
- Lasse Erdweg (Author), 2015, Longevity Risk from a Pension Fund Perspective, Munich, GRIN Verlag, https://www.grin.com/document/310639

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