# Can Pension Funds Partially Manage Longevity Risk by Investing in a Longevity Megafund?

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## Abstract

**:**

## 1. Introduction

## 2. Potential Size of Longevity Risk

#### 2.1. Solutions Derived from Biology of Aging Are Reaching Clinics

#### 2.2. Model of Longevity Risk

#### 2.3. Implication for a Pension Fund in Terms of Needed Prudential Capital

#### 2.4. Conclusion of This Section

## 3. Potential Rate of Return of a Longevity Megafund

#### 3.1. Rate of Returns of Pharmaceutical Developments Today

#### 3.2. Evolution of Biomedical Megafund Returns with Longevity: “Linkage 1”

#### 3.3. Evolution of Longevity Megafund Returns with Longevity: “Linkage 2”

#### 3.4. Evolution of Megafund Equity Returns with Longevity

## 4. In Which Cases Will a Pension Fund Benefit from Investing in a Megafund?

#### 4.1. What Type of Retirement Systems Would Benefit from Investing in a Longevity Megafund?

- “Defined benefits” means that the pension benefits are defined and guaranteed by the pension fund. The fund is reponsible for the investments and bears the longevity risk. The capital may be handed over to an insurance company which will bear the longevity risk. The benefits are typically defined as a percentage of the pensionable salary, for example, 20% of the average salary over the last three years of work. Investing in a longevity megafund could be a way for defined benefit pension funds to partially hedge their longevity risk: in case of strong longevity improvements, investment returns should be greater so that the accumulated capital can pay benefits longer than the prospective life expectancy calculated at time of retirement.
- In defined contribution plans, employer and employees provide contributions during their working years. Often, employees make investment choices to build their capital for retirement. The amount of accumulated capital depends on how well investments perform. Often, the accumulated capital is transferred to an insurance company that pays annuities and bears the longevity risk. The risk can also be at the level of the pensioners in case of a lump sum, but they can also buy annuities. In recent decades, a shift has occurred from defined benefit plans to defined contribution plans in order to avoid the financial risk borne by the fund during the capital build up period.

- c.
- In pay-as-you-go pension plans, there is no, or very little, investment: the contributions from employers and employees directly pay the benefits to retired persons. In such a system, that is for example widely used in France, the lack of investment makes the longevity megafund of no use for the pay-as-you-go pension plan itself. However, retirees may ask an insurer to annuitize their wealth and the insurer would benefit from investing it in a longevity megafund.
- d.
- In conclusion, from a first qualitative perspective the equity part of a longevity megafund makes sense for defined benefit pension plans and for the investment of retirement capital by any stakeholder, but a priori not for the investment of contributions by defined contribution pension plans that bear the longevity risk, as it would be increased.

#### 4.2. Impact of Investing in a Megafund on Needed Prudential Capital

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Choice of the Mortality Model

**Figure A1.**Historical mortality rates, for three countries and as modeled. Annual mortality rates are shown in log scale as function of age, for the USA, France and Japan (graphs from top to bottom), in 1990 and in 2015 (graphs on the left and on the right). Each graph shows the mortality of males in a (blue) continuous line, the mortality of females in (red) dots, the mortality of both in (black) dashes and the mortality of the model used in this article in a gray line that is visually straight up to age 85. For the graphs on the right, the mortality of the model in 1990 is added to help visualize the change of mortality between 1990 and 2015.

**a**can be understood as the initial life expectancy level (or initial mortality level but mortality rates increase when a is decreased). Knowing that what matters most for the article is to simulate various longevity scenarios, we choose one reference country, Japan in our case, and we choose a = 11.3: this provides a reasonable estimate of life expectancy at birth and at age 65 of the Japanese general population in 2020, as shown in the two first graphs of Figure 1 and in greater details in Figure A2.

**Figure A2.**Static life expectancy at birth and at age 65, for three countries and as modeled. Life expectancy at birth (graphs on the left) and at age 65 (graphs on the right) are shown as function of calendar year, for Japan, France and the USA (graphs from top to bottom). Each graph shows the mortality of males in a (blue) continuous line, the mortality of females in (red) dots, the mortality of both in (black) dashes and the mortality of the model used in this article in portions of gray straight lines. More precisely, $\mathsf{\phi}$ = 20% is used for the thick straight line and $\mathsf{\phi}$ = 50% and $\mathsf{\phi}$ = 100% is used for the two thin straight lines that start in 2020.

## Appendix B. Investigation of the Current Rate of Return of Pharmaceutical Developments

**Figure A3.**Estimated gain per successful drug development based on Royalty Pharma sales. The x-axis of the two red vertical bars are the mean (2.48 bn$, on the right) and the median ($1.03 bn, on the left) of the sales amounts, in $M. The light blue shade is the density of the sales amount, using a smoothing gaussian kernel whose bandwidth is half of the sales standard deviation. The blue vertical bar, between the two red horizontal bars, is the gain we consider for investors in the megafund, under current longevity trends ($1.5 Bn).

^{0.1}-1 = 15.4%. Using a slightly lower success rate as seen above, of 12%, and a greater cost to include management fees and carried interest, as reminded by Phalippou (2010), of $60 M, leads to an annualized return of 11.6%. This is not far from the 11.9% annualized return suggested by Fernandez et al. (2012) in the case of a cancer megafund.

**Figure A4.**Pharmaceutical drug development indicators based on open data: clinicaltrials.gov, clinicaltrialsregister.eu, [email protected], and accessdata.fda.gov. The three first graphs on the left show numbers of USA clinical trials per calendar year, t, for respectively Phase I, Phase II, and Phase III reporting by the pharmaceutical industry according to clinicaltrials.gov. The Y-axis is a logarithmic scale. For each of them, the top (black) thick and continuous curve is the number of clinical trials started each year (first participant enrolled). The other thick continuous curve (in blue) is the number of clinical trials ended each year (primary completion data: last participant providing data for the primary outcome measure). A decay of 3 or 4 years of the first (black) curve is represented in (black) squares to see how well it compares with the second (blue) curve: the resulting estimated length of clinical trials is indicated in the (red) arrow pointing the right. The (black) thin dashed curve that plunges on the right side is the number of clinical trials started each year AND completed before 2018. The plateaus observed with the first two lines and the latter two lines, underlined (in red), differ due to the non-100% completion rate, as measured in the arrows pointing down (with a “<” sign with respect to success rate). The three graphs on the right side are equivalent slides but for Europe and the UK, based on clinicaltrialsregister.eu; clinicaltrialsregister.eu is more recent than clinicaltrials.gov and does not provide the same filters, so we only rely on clinical trials started each year, completed since or not. The last graph shows the accepted drugs of different types in the USA, by the FDA, every year. The top (blue) line represents any type of new drug accepted (United States Government Accountability Office, 2017). The middle yellow curve represents approvals for some specific types of treatments: New Molecular Entities (NME), New Drug Applications, Biologic License Applications (types 1–8 in ([email protected] 2018)). The bottom green curve represents NMEs only. The number of NME submissions (accessdata.fda.gov, 2018) is shown in green squares.

## Appendix C. List of Figures, Tables, and Variables in the Article

Theme | Figure | Name |
---|---|---|

Longevity model | 1 | Characteristics of the chosen longevity model |

A1 | Historical mortality rates, for three countries and as modeled | |

A2 | Static life expectancy at birth and at age 65, for three countries and as modeled | |

developments | A3 | Estimated gain per successful drug development based on Royalty Pharma sales |

pharmaceutical | A4 | Pharmaceutical drug development indicators based on open data: clinicaltrials.gov, clinicaltrialsregister.eu, [email protected] and accessdata.fda.gov |

Returns of | 3 | Megafund annualized return r as a function of longevity trend φ |

4 | Annualized equity return i as a function of longevity trend φ | |

Pension fund needed capital | 2 | Needed prudential capital depending on the future longevity trend (or lack of) expressed as a proportion of the initial wealth |

5 | Needed prudential capital, expressed as a proportion of the initial wealth, depending on the future longevity trend and on investments in a longevity megafund |

Theme | Table | Name |
---|---|---|

Pension fund needed capital | 1 | Contributions by age tranche |

2 | Needed prudential capital, expressed as a proportion of the initial wealth, depending on investments in a megafund |

Theme | Variable | Name | Definition |
---|---|---|---|

Longevity model | $\mathrm{a}$ | Level of longevity | Equation (1) |

$\mathrm{b}$ | Ageing rate | Equation (1) | |

$\mathrm{x}$ | Age in years | Equation (1) | |

$\mathsf{\phi}$ | Longevity trend | Equation (1) | |

$\mathrm{t}$ | Year (t = 0 corresponds to 2020) | Equation (1) | |

${\mathrm{q}}_{\mathrm{x},\mathrm{t}}$ | Annual mortality rate at age x and time t | Equation (1) | |

$\mathrm{s}$ | Standard deviation of potential longevity trends | Equation (2) | |

${\mathrm{L}}_{\mathrm{x},\mathrm{t}}$ | Prospective life expectancy at age x and time t | Equation (A1) | |

${\mathrm{e}}_{\mathrm{x},\mathrm{t}}$ | Static life expectancy at age x and time t | Equation (A2) | |

${C}_{0}$ | initial investment | Equation (12) | |

${Y}_{10}$ | gain ten years later | Equation (12) | |

$p$ | probability of success | Equation (12) | |

Returns of pharmaceutical developments | $\rho $ | 10-year return | Equation (12) |

$r$ | annualized return | Equations (12) and (16) | |

${\rho}_{L}$ | evolution of $\rho $ with longevity | Equations (14) and (15) | |

$A$ | parameter to ajust the expected level of $\rho $ | Equation (14) | |

$B$ | correlation between $\rho $ and longevity | Equation (15) | |

$\epsilon $ | residual performance of the megafund | Equation (14) | |

I | annualized return of the equity tranche | Equation (17) | |

$\alpha $ | equity percentage of investments in the megafund | Equation (17) | |

Pension fund needed capital | ${\mathrm{N}}_{\mathrm{x},\mathrm{t}}$ | number of persons aged x at time t | Equation (3) |

${\mathrm{C}}_{\mathrm{x},\mathrm{t}}$ | accumulated capital for the group of persons aged x at t | Equations (6) and (7) | |

i1, i2, i3 | expected annual return of contributions by age tranche | Table 1, Equation (4) | |

σ1, σ2, σ3 | standard deviation of the annual returns by age tranche | Table 1, Equation (4) | |

i_{k,t} | annual return of contributions by age tranche and year | Table 1, Equation (5) | |

${\mathrm{W}}_{0}$ | initial wealth of the pension fund | Equation (8) | |

${\mathrm{B}}_{65,\mathrm{t}}$ | annual benefit paid to the workers who retire at time t | Equation (9) | |

${\mathrm{K}}_{\mathsf{\phi},\left\{{\mathrm{i}}_{\mathrm{x},\mathrm{t}}\right\}}$ | needed prudential capital for a given scenario | Equation (10) | |

K | needed prudential capital | Equation (11) | |

p1, p2, p3 | percentage of investments in the equity tranche by age tranche | Section 4.2 |

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1 | The 50% correlation is not critical for the model as longevity risk is the main risk; setting the correlation to 0% did not materially affect results |

**Figure 1.**Characteristics of the chosen longevity model. In the first two graphs, remaining life expectancy at respectively birth and age 65 is computed from year 1990 to 2040 based on the model (see Equation (1)), in gray lines using different values of φ as written, and from 1990 to 2015 for specific countries based on historical data (more precisely based on the mortality rates estimates provided by the Human Mortality Database): Japan in (red) short dashes, France in (blue) continuous lines, and the USA in black long dashes. As expected, life expectancies are greater in Japan than in France and lower in the USA. The model fits life expectancy and mortality rates of Japan. The same style of lines is used for the next two graphs, that represent the mortality rate at different ages in log scale, for the years 1990 and 2015, respectively. The model matches the Japanese mortality well from age 40 to 85; mortality rates at these ages are important to model life expectancy at age 65 and below (see Debonneuil et al. 2017). In the fourth graph, the model at 1990 is additionally shown to provide a visual reference on how mortality rates have evolved from 1990 to 2015. The fifth graph shows the chosen lognormal probability density of φ.

**Figure 2.**Needed prudential capital depending on the future longevity trend (or lack of) expressed as a proportion of the initial wealth. In the two graphs, the y-axis shows the present value of remaining wealth after paying retirement benefits for current employees, divided by the initial wealth of the pension fund: “−1” means that 100% of additional wealth would be needed today, i.e., a needed prudential capital of the size of the initial wealth. In the first graph, the x-axis shows different scenarios for the longevity trend φ and each dot in the graph is the result of a scenario. In the second graph, the x-axis is the level at which to compute the VaR that represents the needed prudential capital: for example, a 90% VaR over all scenarios.

**Figure 3.**Megafund annualized return $r$ as a function of longevity trend φ. The square represents the central scenario with a future longevity trend of 20% and an annualized return of 11.9%. If the megafund target a very wide variety of pharmaceutical developments, returns are expected to depend on the longevity trend as show by the continuous (blue lines; the thick line is the best estimate, the thin lines represent a range of uncertainty): “linkage 1”. If the megafund targets mortality-linked diseases, the expected link with longevity is the (green) dashed line.

**Figure 4.**Annualized equity return, $i$, as a function of longevity trend φ. This graph is similar as Figure 3 except that (black) curves are superimposed to should annualized equity returns: the annualized returns of investing in the equity part of the fund, rather than the whole megafund annualized return.

**Figure 5.**Needed prudential capital, expressed as a proportion of the initial wealth, depending on the future longevity trend and on investments in a longevity megafund. The first graph is for a megafund that uses 50% equity investments, the second graph is for a megafund that uses 25% equity investments. Each dot in the graph is the result of a scenario of the future. In black, no investment is performed in the megafund (p1 = p2 = p3 = 0). In light gray, the pension fund invests in the longevity megafund. In gray, the pension fund invests in the generic biomedical megafund.

Age | Salary | Individual Annual Contribution | Annual Investment Return Rate | |
---|---|---|---|---|

20 to 34 | 30,000 | 3000 | i1 = 5% | σ1 = 4% |

35 to 49 | 45,000 | 4500 | i2 = 4% | σ2 = 3% |

50 to 64 | 60,000 | 6000 | i3 = 2% | σ3 = 1% |

_{1,t}= 5% ± 4%.

**Table 2.**Needed prudential capital, expressed as a proportion of the initial wealth, depending on investments in a longevity megafund.

85% VaR | 90% VaR | 95% VaR | ||||
---|---|---|---|---|---|---|

No megafund | 1.4 | 2.8 | 8.5 | |||

Longevity megafund that has 25% equity | 0.3 | 0.6 | 0.6 | 1.4 | 2.4 | 4.6 |

Biomedical megafund that has 25% equity | 0.1 | 0.6 | 0.5 | 1.4 | 3.3 | 5.1 |

Longevity megafund that has 50% equity | 0.3 | 0.8 | 0.9 | 1.6 | 3.1 | 4.9 |

Biomedical megafund that has 50% equity | 0.3 | 0.7 | 0.9 | 1.6 | 4.0 | 5.8 |

Longevity megafund that has 100% equity | 0.5 | 0.9 | 1.2 | 1.8 | 3.9 | 5.4 |

Biomedical megafund that has 100% equity | 0.5 | 0.9 | 1.2 | 1.9 | 4.6 | 6.4 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Debonneuil, E.; Eyraud-Loisel, A.; Planchet, F. Can Pension Funds Partially Manage Longevity Risk by Investing in a Longevity Megafund? *Risks* **2018**, *6*, 67.
https://doi.org/10.3390/risks6030067

**AMA Style**

Debonneuil E, Eyraud-Loisel A, Planchet F. Can Pension Funds Partially Manage Longevity Risk by Investing in a Longevity Megafund? *Risks*. 2018; 6(3):67.
https://doi.org/10.3390/risks6030067

**Chicago/Turabian Style**

Debonneuil, Edouard, Anne Eyraud-Loisel, and Frédéric Planchet. 2018. "Can Pension Funds Partially Manage Longevity Risk by Investing in a Longevity Megafund?" *Risks* 6, no. 3: 67.
https://doi.org/10.3390/risks6030067