Lump sum or annuity?
People are often faced with a decision whether to live off their savings or buy an annuity. Normally such decisions are made around the retirement ages of 60–65. However, an interesting counter-example has just been provided by eighteen-year-old Charlie Lagarde, the winner of a lottery in Canada. She had to decide between taking a C\$1million lump sum or an annuity of C\$1,000 each week for life. She opted for the latter, forswearing an opportunity to become an instant millionaire. But was it the right decision?
We obtained the most recent mortality tables for the population of Quebec and performed some simple calculations of the discounted present value of C$1,000 per week. The results are shown in Table 1.
Table 1. Discounted present value of C$1,000 per week for a female aged 18. Source: Own calculations using the 2013–2015 Quebec mortality tables.
| Discount rate |
Present value of C$1,000 per week |
|---|---|
| 0% | C$3.46m |
| 1% | C$2.51m |
| 2% | C$1.90m |
| 3% | C$1.49m |
| 4% | C$1.21m |
| 5% | C$1.01m |
| 6% | C$0.87m |
Table 1 shows that our teenager was well-advised. The only way she could have made the lump-sum prize more valuable than the annuity would be if she could guarantee to achieved a long-run net-of-tax investment return of over 5% per annum over the rest of her life. Given the low rates of interest that prevail in Canada and much of the developed world, this seems unlikely.
How might the calculations have looked for an older lottery winner? Table 2 shows a different picture for a female aged 65.
Table 2. Discounted present value of C$1,000 per week for a female aged 65. Source: Own calculations using the 2013–2015 Quebec mortality tables.
| Discount rate |
Present value of C$1,000 per week |
|---|---|
| 0% | C$1.14m |
| 1% | C$1.01m |
| 2% | C$0.90m |
| 3% | C$0.81m |
| 4% | C$0.73m |
| 5% | C$0.66m |
| 6% | C$0.61m |
Table 2 shows that if a sixty-five-year-old female can guarantee to earn over 1% net of tax on her investments, she would be better off taking the lump-sum prize option. At ages 69 and older, the lump-sum prize would be the better option for the average Québécoise even if her investments earned no return at all.
We end with an important note of caution: all of the preceding calculations are based on the assumption that the winner is presented with two alternative options defined in terms of fixed benefits. However, they do not apply to individuals who already have a lump sum in the form of a pension pot and are considering annuitisation, where the annuity amount is determined by the size of the pension pot and various risk factors, rather than being set at a fixed amount irrespective of age (or any other risk factors). Charlie Lagarde consulted a financial adviser to work out the best prize option for her, as covered in a subsequent BBC article. Those approaching retirement should follow her example when it comes to annuities and pensions.
Previous posts
The Curse of Cause of Death Models
Stephen's earlier blog explained the origin of the very useful result relating the life-table survival probability \({}_tp_x\) and the hazard rate \(\mu_{x+t}\), namely:
\[ {}_tp_x = \exp \left( - \int_0^t \mu_{x+s} \, ds \right). \qquad (1) \]
To complete the picture, we add the assumption that the future lifetime of a person now aged \(x\) is a random variable, denoted by \(T_x\), and the connection with expression (1) which is:
Analysis of VaR-iance
In recent years we have published a number of papers on stochastic mortality models. A particular focus has been on the application of such models to longevity trend risk in a one-year, value-at-risk (VaR) framework for Solvency II. However, while a small group of models has been common to each paper, there have been changes in the calculation basis, most obviously where updated data have been used.
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