This manual presents solutions to all exercises from

It should be read in conjunction with the spreadsheets posted at the website www.cambridge.org\97811ø76ø8443 which contain details of the calculations required. However, readers are encouraged to construct their own spreadsheets before looking at the authors' approach. In the manual, exercises for which spreadsheets are posted are indicated with anE.

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1.1 The insurer will calculate the premium for a term or whole life insurance policy assuming that the policyholder is in relatively good health; otherwise, if the insurer assumed that all purchasers were unhealthy, the cost of insurance would be prohibitive to those customers who are healthy. The assumption then is that claims will be relatively rare in the first few years of insurance, especially since most policies are sold to lives in their 30s and 40s.

This means that the price is too low for a life who is very unwell, for whom the risk of a claim shortly after purchase might be 10 or 100 times greater than for a healthy life. The insurer therefore needs evidence that the purchaser is in good health, to avoid the risk that insurance is bought too cheaply by lives who have a much higher probability of claim.

The objective of underwriting is to produce a relatively homogeneous insured population when policies are issued. The risk that the policyholder purchases the insurance because they are aware that their individual risk is greater than that of the insured population used to calculate the premium, is an example of adverse selection risk. Underwriting is a way of reducing the impact of adverse selection for life insurance.

Adverse selection for an annuity purchaser works in the other direction – a life might buy an annuity if they considered their mortality was lighter than the general population.

3.1 Figures S3.1, S3.2 and S3.3 are graphs of μx, lx and dx, respectively, as functions of age x up to x = 100. Each graph has been drawn using the values from ELT 15, Males and Females.

(a) The key feature of Figure S3.1 is that the value of μx is very low until around age 55, from where it increases steeply. Numerically, μx is very close to qx provided qx is reasonably small, so that the features in Figure S3.1 are very similar to those shown in Figure 3.1 in AMLCR. The features at younger ages show up much better in Figure 3.1 in AMLCR because the y-axis there is on a logarithmic scale. Note that the near-linearity in Figure 3.1 in AMLCR for ages above 35 is equivalent to the near-exponential growth we observe in Figure S3.1.

(b) The key feature of Figure S3.2 is that, apart from a barely perceptible drop in the first year due to mortality immediately following birth, the graph is more or less constant until around age 55 when it starts to fall at an increasing rate before converging towards zero at very high ages. This reflects the pattern seen in Figure S3.1.

(c) The function dx is the expected number of deaths between exact ages x and x + 1 out of l0 lives aged 0. The relatively high mortality in the first year of life shows clearly in Figure S3.3, as does the increase in the expected number of deaths for males in the late teenage years – the so-called ‘accident hump’.

In several chapters we discussed parametric regression modeling for a moderate number of explanatory variables based on maximum likelihood methods. In some areas of application, however, the number of explanatory variables may be very high. For example, in genetics, where binary regression is a frequently used tool, the number of predictors may be even larger than the number of predictors. In this “p > n problem” maximum likelihood and similar estimators are bound to fail. Typical data of this type are microarray data, where the expressions of thousands of predictors (genes) are observed and only some hundred samples are available. For example, the dataset considered by Golub et al. (1999a), which constitutes a milestone in the classification of cancer, consists of gene expression intensities for 7129 genes of 38 leukemia patients, from which 27 were diagnosed with acute lymphoblastic leukemia and the remaining patients acute myeloid leukemia.

In high-dimensional problems the reduction of the predictor space is the most important issue. A reduction technique with a long history is stepwise variable selection. However, stepwise variable selection as a discrete process is extremely variable. The results of a variable selection procedure may be determined by small changes in the data. The effect is often poor performance (see, e.g., Frank and Friedman, 1993). Moreover, it is challenging to investigate the sampling properties of stepwise variable selection procedures.

An alternative to stepwise subset selection is regularization methods. Ridge regression is a familiar regularization method that adds a simple penalty term to the log-likelihood and thereby shrinks estimates toward zero.