Hedging Longevity Risk: Deal or No Deal On the Demand- and Supply-Side Pricing in the Incomplete Market of Longevity-Linked Securities Arthur Majtlis S2527855

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On the Demand- and Supply-Side Pricing in the Incomplete Market of

Longevity-Linked Securities

Arthur Majtlis

S2527855

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Master’s Thesis Actuarial Studies

Supervisor University of Groningen: Dr. E.L. Kramer Supervisor Ortec Finance: Rients Miedema

Second Assessor University of Groningen: Prof. dr. L. Spierdijk

Acknowledgments:

I would like to gratefully thank my supervisor, Dr. B. (Bert) Kramer, for helpful comments, dis-cussions, for answering my questions when necessary and for guiding me through the process of writing this thesis. It was a pleasure working with you.

I would also like to express my gratitude towards Rients Miedema, my supervisor at Ortec Fi-nance, with whom I have made many phone calls in search for the information we needed, and who occasionally pointed me in the right direction. Special thanks also goes out to Pieter Kloek at Ortec Finance, who guided me through some heavy material regarding risk-neutral valuations. To all my colleagues at Ortec Finance: I had a great time this past half year while writing this research paper and as a working student. Thanks for everything.

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On the Demand- and Supply-Side Pricing in the Incomplete Market of

Longevity-Linked Securities

University of Groningen

Faculty of Economics and Business

Master’s Thesis EORAS

Name: Arthur Majtlis

Student number: S2527855

Supervisor: B. Kramer

Second Assessor: L. Spierdijk

February 18, 2019

Abstract

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1 Introduction

“We have a critical problem.” With these words David Blake, renowned and frequently cited re-searcher in the field of longevity, addressed the audience at the annual Longevity Congress this past September in Amsterdam1. He was referring to the current state of retirement systems around the world. According to Blake and many others, these systems are not well prepared to handle the consequences of the demographic changes humanity has undergone over the last century.

Since the end of the First World War, the life expectancy of Western-European males has gone up by 30 years. For females this increase has been even higher; their life expectancy has gone up by almost 35 years (Mackenbach, Hu, & Looman,2013). The total number of years a Dutch male is expected to live after retirement has increased from 17,4 years in 2009 to 20,3 years in 20182_.

This is a substantial increase of more than 3 years of pension for each Dutch male retiree over the past decade.

While this is of course a great improvement for individual human beings, for many institutions this poses a great threat. Over the last decade, pension funds and other annuity providers have had to deal more and more with longevity risk. This risk refers to the uncertainty around realized mortality rates of the participants in a portfolio. More specifically it refers to the risk that annuity providers will face higher costs when individuals live longer than expected. A small increase in the life expectancy, and the additional liabilities associated with that, can have a drastic effect on a fund’s funding ratio. To give an idea, one extra year added to the average life expectancy of someone aged 65 can add 3-4% to the total present value of this person’s pension liabilities (Biffis & Blake, 2009).

Longevity risk is not a new phenomenon. At the end of the 20th _{century pension funds were}

already dealing with prediction errors when it came to life expectancies. Financially, these er-rors were less significant back then because long-dated discount rates were higher. An increase in the survival rates would be offset by the simultaneously increasing interest rates: higher liabili-ties were discounted at a higher rate, keeping the net present value of these liabililiabili-ties more or less at the same level. With today’s discount rates being as low as they are, this is not the case anymore.

One area that is currently particularly exposed to longevity risk is that of Defined Benefit (DB) pension plans. DB plans guarantee the payment of predetermined fixed benefits over the lifetime of the pension plan holder. To a large extent due to this risk, many DB plans have been closed over the past decade, especially in the UK. This means that no new employees can enter into these funds, and that the respective pension fund will completely shut down once the last person in it has passed away. Existing DB plans however, do have to take care of their enrolled participants. In an attempt to manage the longevity risk affecting these plans, over the last years multiple hedging techniques have been deployed that aim to transfer the risk from the annuity providers to the reinsurance market or to the capital market. One category of financial instruments that has been

1_{The Fourteenth International Longevity Risk and Capital Markets Solutions Conference, Thursday 20th and}

Friday 21st September 2018, Amsterdam

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1 INTRODUCTION

developed to deal with this risk are so called Longevity-Linked Securities (LLS). The pricing of these LLS is a challenging and complex process. As a consequence of the lack of transparency in the pricing of LLS, the uptake of these instruments is still rather limited.

Throughout this research we will mainly focus on the pricing of these LLS. Depending on the roll that a certain actor plays in the market around the transfer of longevity risk, also referred to as the Life Market (Blake, 2018), a different pricing methodology must be used. For example, a capital market investor will demand a certain excess return over the risk free rate on his or her investment, whereas a pension fund will be concerned with the amount of capital that is relieved once the transaction has been made. By taking the points of view from both the investor that wants to take on the longevity risk, and the pension fund or annuity provider seeking to hedge this risk, we will find out if a theoretical common ground can be reached which could form the basis of a more complete Life Market.

The supply-side pricing will be calculated using the Sharpe ratio approach. For the demand-side pricing we will use the Cost-of-Capital approach. In order to fully understand the proposed pricing methodologies it is important to first get an understanding of the Life Market as a whole, and the different factors that have to be taken into account when pricing longevity transactions. First the most popular hedging techniques that are currently available in the market will therefore be explained. Of these techniques, we will subsequently focus on the above-mentioned LLS, and in particular on the Index-Based Longevity Swap. One other product that we will discuss is the more innovative Out-Of-The-Money Longevity Swap, also known as Tail-Risk protection. Because these transactions are taking place more frequently every year, it is important that decision mak-ers get a better undmak-erstanding of the main dynamics underlying their pricing. We believe that understanding how longevity risk can be quantified and, following that, how to value the different LLS, will provide more insight into whether or not a certain price offer should be accepted.

Whereas many papers focus solely on either the pricing of a certain product, or on comparing different mortality forecasting models, this thesis adds to the literature a start to finish overview of the process foregoing a longevity risk transaction. In addition, the specific pricing methodologies that are proposed in this paper have not been compared before in any other research. Based on Dutch mortality data, this paper will give a practical application of some of the most frequently used longevity products. Finally, this is the first paper to do this using the most recently pub-lished mortality projections, based on a model presented by the Dutch Royal Actuarial Society in September 2018 (KAG,2018).

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risk that society is currently facing. Section 5 goes into the pricing of the Index-Based Longevity Swap. First the recent literature in this area will be discussed, after which we will apply the differ-ent pricing approaches. Section 6 compares the potdiffer-ential benefits of the swap with the associated costs, and discusses under what circumstances entering into the swap can be a beneficial solution. Section 7 will go into the pricing of a more innovative product, known as the Out-Of-The-Money Longevity Swap. In section 8 some final conclusions are drawn.

2 Introduction to the Life Market and Research Objectives

In this section, first a review is given of the past literature on the topic of longevity risk. After this, the current state of the Life Market shall be described. The aim of this section is to explain what has been done in this field, to give a clear idea of where the market is today, and finally, to further explain what the main focus of this thesis will be.

The first notable sign of the existence of an existential threat to annuity providers, in the form of longevity risk, came with the fall of the Equitable Life Assurance Society (ELAS) in December 2000. Throughout the second half of the 20th century, the ELAS had been selling annuities with so-called ‘Guaranteed Annuity Rates’, annuities with rates fixed at the interest and mortality as-sumptions made at the time. An extremely high increase in the value of these annuities in the late 90’s, due to unforeseen mortality improvements and fallen interest rates, posed such heavy financial difficulties to the ELAS that they eventually had to close down. Blake, Cairns and Dowd elaborate on this event in the beginning of their paper ‘Living with mortality’ (Blake, Cairns, & Dowd,2006). In this paper they discuss the different ways in which they believe annuity providers can manage longevity risk. After they give a clear overview of the stakeholders in this market (namely hedgers, general investors, speculators, regulators and the government), they explain the workings of different longevity solutions. As we will see in the remainder of this section, not all of the solutions they proposed in 2006 turned out to be adopted by the market as they expected.

2.1 Literature on Longevity Linked Securities

During the first decade of the 21st _{century, deals involving the transfer of longevity risk were}

very uncommon. Longevity bonds, such as the EIB/BNP3_{bond (2004) and the Swiss Re}

mortal-ity catastrophe bond (2003) have extensively been discussed (Cairns, Blake, Dowd, & MacMinn,

2006), (Blake et al., 2006), (Bauer, Börger, & Ruß,2010). In particular the EIB/BNP bond has received a lot of attention because it never hit the market. The overall conclusion in these papers is that the bond-based life market has more or less failed.

The need for a new longevity market was felt. This time the hopes were set on a derivative-based market, trusting that such a market will emerge for multiple reasons (Blake et al., 2006).

3_{The European Investment Bank (EIB) together with BNP Paribas constructed a 540 million, 25 year}

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2 INTRODUCTION TO THE LIFE MARKET AND RESEARCH OBJECTIVES

First of all, they were convinced that the reinsurance market alone does not have enough capacity to take on the entire demand for longevity risk transactions. Secondly, they argued that the overall awareness of the longevity problem is of such a magnitude that it will have to be addressed one way or the other. They added that the products sold in this derivative-based market will have to be transparent.

Others, bringing the discussion back to the bond-based market, plead for government interfer-ence in this market, claiming that the government is pre-eminently the party that should be issuing these bonds (Blake, Boardman, & Cairns,2014). They give two reasons for this claim. Their first argument is that they believe that the government should have a great interest in ensuring that there is an efficient annuity and an efficient capital market for longevity risk transfers. Secondly, they argue that the government is the best placed party to engage in inter-generational risk shar-ing, such as providing tail risk protection against systematic trend risk. Thirdly, in their eyes the government can provide a hedge with almost no counterparty risk, which would add value to the hedge for for example (re)insurance companies. To this day governments have not interfered in the longevity bond market in any way. One can imagine why this is the case. Many governments are already exposed to a significant amount of longevity risk through public pensions and social health care programs. Looking at the amount of longevity risk to which for example the Dutch Government is already exposed, by providing social security such as AOW, it would make perfect sense that they would not be very eager to take on much more longevity risk on top of this. Besides, if the pension fund’s refuse to solve their longevity problems, and more elderly fall into poverty because of this, it is most probably going to be the government that in the end will have to step in and cover for the longevity risk.

In the paper ‘Still living with mortality’ (Blake, Cairns, Dowd, & Kessler, 2018), the authors update their paper from 2006, using the knowledge they obtained by witnessing the longevity mar-ket coming to life over the intervening period. They conclude that over the years, insurance-based solutions have won in popularity over capital markets solutions, contrary to what they had pre-dicted in 2006. In table1the currently available longevity solutions have been divided over these two categories. The general idea behind them will be explained in the next section. First we will discuss the current status of the Life Market.

Insurance-Based Solutions Capital Market-Based Solutions

Bulk Annuity: Buy-in

-Bulk Annuity: Buy-out

-Indemnity-Based Longevity Swap Index-Based Longevity swap

- Longevity-Spread Bonds

- q-forwards

- S-forwards

- Tail-Risk Protection

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2.2 The Current State of the Life Market

Over the last decade, the market for longevity linked products has started to emerge and slowly develop into a global risk transferring business. In 2017 alone there have been $12.4 billion worth of buy-outs and buy-ins of UK pension funds. Aside from these deals there have been $4.6 billion worth of longevity swaps in the UK in the same year. The total size of the longevity risk transfer market is estimated at 60-80 trillion dollars’ worth of liabilities. The underlying longevity risk is estimated at 10%-12.5% of the total market value. This means that due to unforeseen improvements in the life expectancy, the total market value of these liabilities can increase by around 8 trillion dollars (Michaelson & Mulholland,2015).

Figure 1: The total size of the Life Market in the U.K., U.S. and Canada, over the period 2007-2017 (in billions (USD)). Since 2007, other countries besides these three have participated in the Life Market as well, though on a smaller scale. These transactions are not included in this graph.

Source: LIMRA, Hymans Robertson, LCP and PFI analysis as of June 2017

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2 INTRODUCTION TO THE LIFE MARKET AND RESEARCH OBJECTIVES

2.3 Research Objectives

So far we have discussed the recent literature on the topic of LLS and have given a broad overview of the current state of the Life Market. What needs to be stressed is that even though the total traded volume of insurance-based longevity solutions has been rapidly growing over the last few years, the total market is far from being mature and complete. Since the capacity of the coun-terparties taking on the risk in these insurance-based transactions is limited, these products alone do not form a sustainable solution for the longevity problem (Michaelson & Mulholland, 2015). Capital market investors therefore need to be attracted to the Life Market if the high demand for longevity risk transfers is to be met. Two solutions that could attract these investors are the Index-Based Longevity Swap, and the Out-of-the-Money Longevity Swap. Since both products are relatively new, and their traded volume still low, they cannot be priced by mark-to-market principles. Other pricing approaches must therefore be used.

In this thesis, the Index-Based Longevity Swap will be priced from the perspective of the cap-ital market investor as well as from the perspective of a (Dutch) pension fund and an insurance company4_{. The Out-of-the-Money Longevity Swap will only be priced from the perspective of an}

insurance company.

Our main research question is:

Under what assumptions is entering into an Index-Based Longevity Swap with a cap-ital markets investor a potentially beneficial solution for a pension fund or insurance company?

Our secondary research question is:

Under what assumptions is entering into an Out-of-the-Money Longevity Swap a po-tentially beneficial solution for an insurance company?

To answer these questions we will need to answer the following subquestions:

- How does one quantify the longevity risk that annuity providers are currently facing?

- How does one put a price on an Index-Based Longevity Swap from the perspective of a capital market investor?

- How does one put a price on an Index-Based Longevity Swap/Out-of-the-Money Longevity Swap from the perspective of the annuity provider seeking to hedge its longevity risk?

Before addressing these questions, the following section will discuss the different products that have been dominating the longevity market to this date.

4_{The mortality model used in this paper is focused on the Dutch population. Furthermore, Dutch regulations}

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3 Longevity Solutions

There are different tools available in the search for an efficient longevity hedge. The aim of this section is to describe the most important longevity solutions that are used in the Life Market today. We will start with briefly explaining the dominant insurance-based solutions, after which we will go into some of the capital market-based solutions. The first type of solution we will discuss is the bulk annuity.

3.1 Bulk Annuities

As mentioned earlier, a pension scheme can protect itself from longevity risk by entering into an insurance contract. A bulk annuity is an insurance policy covering the benefits of a group of members of a pension scheme. In a bulk annuity contract, the longevity risk as well as the investment risk are passed on to an insurer. This contrasts with a longevity swap (to be discussed later on) in which only the longevity risk is passed on to the insurer, whereas the scheme’s assets remain invested by the fund itself, thereby holding on to the investment risk. There are two types of bulk annuities: a Buy-in and a Buy-out.

3.1.1 Bulk annuity: Buy-in

In the case of a Buy-in, a pension fund buys annuity policies covering some or all scheme members. The annuities are assets on the fund’s balance sheet. In this form of de-risking the insurer does not take on any contractual responsibilities towards the fund members. The fund pays a premium to the insurer, often in the form of assets under management, and in turn receives payments to cover for the liabilities of some or all of their members. The pension fund is still responsible for paying out the pension premiums to its members. Insurance companies providing the annuities are often required to put down some sort of collateral, in case of a default, covering for the counterparty default risk that is present in these transactions. This collateral will be based on the present value of the fund’s liabilities.

3.1.2 Bulk annuity: Buy-out

In the case of a Buy-out, an insurance company issues individual annuity policies directly to the in-dividuals in the pension fund. The pension fund in return pays premiums to the insurer, which can also be in the form of an asset transfer from the fund to the insurer. After the Buy-out of each plan member the fund has completely removed the pension liabilities from its balance sheet, after which the scheme is ready to be wound up. This is considered as the ultimate form of de-risking. 5 _The

biggest difference with a Buy-in is that, after a Buy-out, the insurer is responsible for paying out

5 _{Blake et al. give a very illustrative, somewhat simplified, example of a full pension-buyout: Say company AB}

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3 LONGEVITY SOLUTIONS

the pension premiums to the scheme members. It is therefore also more expensive than a Buy-in. The structures of both types of bulk annuities are illustrated in figure2. Having explained the two types of bulk annuities we will now go into a different type of longevity solution: the longevity swap.

Figure 2: Structure of a Buy-out compared to a Buy-in

Source: "The Financial Impact of Longevity Risk", IMF Report, April 2012

3.2 Longevity Swaps

Unlike in the case of a Buy-in or a Buy-out, when both the investment risk and the longevity risk of a pension fund are insured, a longevity swap solely covers against longevity risk. Longevity swaps are therefore an appealing option for pension schemes who want to keep on the risk and reward of investing all the pension scheme’s assets, but want to be protected against an increasing life expectancy of their members. Historically, longevity swaps have been initiated only by large pension schemes (> 500 million euro’s in liabilities), but the market has recently been opening up to smaller schemes as well, mainly due to an increased affordability of the product. There are two types of longevity swaps to consider: the Indemnity-Based Longevity Swap and the Index-Based Longevity Swap. We will discuss both types below.

3.2.1 Indemnity-Based Longevity Swap

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3.2.2 Index-Based Longevity Swap

In Index-Based Longevity Swaps the floating rate which the insurer pays to the fund is tied to a mortality index. However, they are not customized to precisely match the pension fund’s realized mortality rate. As mentioned above, Indemnity-Based Swaps are very illiquid. Liquidity of the product however, is essential if the market for this product wishes to develop. The main purpose behind Index-Based Longevity Swaps is to satisfy the hedgers and investors who want to buy a liquid and standardized product. There cannot be too many different types of standardized hedges in the market however. This is because the more standardized contracts exist, the more effective the funds can use them as a hedge, the lower, however, their liquidity will be (Blake et al., 2018).

Choosing an appropriate index is of major significance for the effectiveness of the hedge, as it will determine the amount of basis risk embedded in the swap. For example, there is a high probability that the mortality rates in a UK pension fund for a group of lawyers substantially dif-fers from the mortality rates of the entire UK, on which the indexed rates could be based. When entering into an index-based swap, the hedger will have to take this basis risk, however big in size, for granted. Needless to say, the bigger the basis risk, the lower the effectiveness of the hedge. The advantages of these type of swaps are that they are more transparent, more liquid and that they are cheaper, since funds do not have to constantly keep track of the realized mortality rates. In addition these swaps often have a shorter maturity than the customized swaps. Pension funds can hedge their longevity risk for 10 years for example, after which they reconsider their hedging strategy. This means a shorter exposure to counterparty default risk, and thus a lower amount of this type of risk in total (Coughlan et al.,2007). In figure 3 the structure of a longevity swap is illustrated. The actual benefit payments depend on the type of the swap.

Figure 3: Structure of a Longevity Swap. In an Indemnity-Based Swap the actual payments are based on the realized mortality within a pension fund. In an index-based swap, these payments are based on a predetermined mortality index.

Source: "The Financial Impact of Longevity Risk", IMF Report, April 2012

In addition to the Index-Based Swap, more capital market-based solutions have been introduced over the last few years, as can be seen in table1. The final part of this section will focus on these products.

3.3

4 Quantifying Longevity Risk

In this section, the fundamental building blocks for quantifying longevity risk, known as mortality forecasting models, will be discussed. First, an introduction into the topic of mortality forecasting is given. After this, stochastic mortality models, and in particular the one used in this paper, the Li-Lee (2005) model, will be explained. Finally, the results of the future mortality simulations that have been made are shown, which will later be used when pricing the different LLS.

4.1 Mortality Forecasting Models

Over the last century tremendous improvements in mortality rates have taken place. The pre-dictions that were made on the human life expectancy, however, systematically underestimated the actual outcomes (Bongaarts & Bulatao, 2000). In the fall of 2018, for the first time the new projections for Dutch life expectancy were lower than those of the previous year, indicating that the growth in life expectancy was slowing down (KAG,2018). This illustrates the great amount of unpredictability when it comes to forecasting life expectancies. Since each additional year added to the life expectancy of a 65 year old increases the value of this person’s pension liabilities by 3-4%, incorrectly quantifying longevity risk can have dramatic consequences.

In the broadest sense, longevity risk can be split up into two parts: systematic longevity risk and non-systematic longevity risk. Non-systematic longevity risk is the risk that the mortality rates of a specific fund diverge from the best estimate assumption for this fund, given its members’ characteristics. This divergence is caused by the random nature of death in combination with the size of the fund. As its name implies, non-systematic longevity risk is diversifiable. With a large enough portfolio size, this risk is non-existent (Olivieri & Pitacco,2009), which is what is assumed throughout this paper. Systematic longevity risk is the undiversifiable risk resulting from incorrect assumptions on future mortality improvements, i.e. the risk that the realized mortality rates differ from the expected trend. Mortality forecasting models are used to quantify this systematic part of longevity risk, also known as the trend risk. Specifically, mortality models include one or more time-dependent trend parameters which capture this trend risk. The uncertainty embedded in these specific parameters comprises almost all the uncertainty in the mortality model itself.

One can divide mortality forecasting models into three groups (Blake et al.,2018): 1) ‘Process-based models’, which model the causes of death of the processes underlying mortality, 2) ‘Ex-planatory models’, which use exogenous ex‘Ex-planatory variables, such as GDP growth, inflation and unemployment, to model death, and 3) ‘Extrapolative models’, which are totally data-driven and heavily based on the notion that past trends will continue. The most famous and most com-monly cited forecasting model in this last category is the Lee-Carter (1992) model, followed by the Renshaw-Haberman (2006) and the Cairns-Blake-Dowd (2006) models. Lee and Carter use the central mortality rates of the USA to describe the mortality of the population, using only a single time trend parameter (Lee & Carter,1992). Cairns et al. give a detailed overview of a wide range of stochastic mortality forecasting models, including these three models (Cairns, Blake, & Dowd,

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4 QUANTIFYING LONGEVITY RISK

Since the main focus of this research is not on comparing different mortality forecasting models but on using the forecasts to price the different products, one mortality forecasting model has been chosen: the Li-Lee (2005) multi-population mortality forecasting model. Recently, this model has been growing in popularity, and for example has already become the default model for forecasting mortality rates in the Netherlands and in Belgium (Antonio, Devolder, & Devriendt,2015). It is also the model that the Dutch Actuarial Society, which we will throughout this paper refer to as the AG, uses to calculate their best estimate forecast for the mortality rates of the next 100 years. Once every two years the AG publishes a newly updated best estimate. Later in this section we will use the AG adaption of the Li-Lee (2005) model to quantify the longevity risk that annuity providers are facing today.

4.2 Stochastic Mortality Models

The mortality model that will be used in this paper is the Li-Lee multi-population stochastic mortality forecasting model (Li & Lee,2005). It is an extension of the original Lee-Carter (1992) model in a way that it allows for age-specific death rates of multiple populations to be coherently estimated, whereas the original model bases its estimates on a single population. To get a good understanding of the multi-population model proposed by Li-Lee (2005), first some basic specifics of the Lee-Carter (1992) model and of mortality forecasting in general shall be listed.

4.2.1 Lee-Carter model

Lee and Carter calculate the central death rate of a person of age x in year t in a log linear model. This approach is based on extrapolating past mortality rates into the future. The main equation is:

ln(mx,t) = αx+ βxκt+ x,t (1)

The variables in this equation can be defined as follows:

• mx,t: the raw central death rate, defined as

mx,t=

Dx,t

Ex,t

where Dx,t is the number of individuals aged x that died in year t, and Ex,tis the exposure

to risk, which denotes the number of individuals aged x exposed to the risk of death in year t.

• αx is a time independent parameter that gives the average of ln(mx,t) over the observed

period.

• βx is also constant over time and gives the magnitude of the change in mortality per age.

• κt is a time dependent parameter capturing the evolution of mortality over time, which is

the same for every age group.

• x,tis the white noise error-term of the model representing the non-systematic shocks, which

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Due to identification issues, in other words to ensure the uniqueness of the parameters, two constraints have been set on this model:

X

βx= 1 and

X κt= 0

The Lee-Carter (1992) model is defined for the central death rate mx,t. We can apply this rate

to model the annual mortality rate. qx,tdenotes the probability that an individual of age x who is

alive at the start of year t will die before the start of year t + 1. This is known as the mortality rate at age x in year t, or the one-year death probability. Assuming that mx+s(t + s) = mx(t) for

all 0 ≤ s < 1, we get:

qx(t) = 1 − e− R1

0mx+s(t+s)ds= 1 − e−mx(t)

A stochastic model that is described in terms of mx,t can thus be described in terms of one-year

mortality rates using this formula. If we take the complement of these one-year death probabilities we get the one-year survival rate px,t, i.e. the probability that an individual aged x at time t

reaches the age of x + 1:

px,t= 1 − qx,t

Using these one-year survival probabilities we can finally estimate the expected remaining lifetime for a person of age x in year t, denoted as ex,t. This estimate is obtained by summing the

probability that an individual survives for a given amount of years τ , given byτpx,t= τ −1

Y

i=0

px+i,t+i.

We then get for the expected remaining lifetime of a person aged x in year t:

ex,t= X τ ≥1 τpx,t or ex,t= ∞ X i=0 i Y j=0 (1 − qx+j,t+j) 4.2.2 Li-Lee model

The Li-Lee (2005) model adds to the Lee-Carter (1992) model that it is based on multiple popula-tions. The reason one would want to coherently estimate the mortality rates of multiple populations is that future mortality rates of closely related populations tend to converge over a longer period of time (Li & Lee,2005). When estimating the mortality rates for each population separately, the opposite appears to happen. On the grounds of biological reasonableness, a way of reasoning used to identify causality between two variables that is in line with the current medical knowledge, mor-tality rates of related populations should not diverge over the long term (Coughlan et al., 2011). To solve for this undesirable property, Li and Lee developed this mortality model, which identi-fies a common trend between multiple populations, but also accounts for individual population’s deviations in the short and medium term. In contrast to the Lee-Carter model, the Li-Lee model is defined for the force of mortality µx(t), instead of the raw central death rate mx,t. The force of

mortality, also known as the instantaneous death rate, describes the behaviour of the central death rate mx,tover an infinitely small duration n. In practice, the only difference with the Lee-Carter

model is in the definition, the variable µx(t) is constructed in the exact same way as mx,t. Hence,

throughout this paper we assume that:

µx(t) = mx,t

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ln(µx) = αx+ BxKt+ βxκt+ x,t (2)

Before we define the variables in this model, let us introduce a specific application, as proposed by the AG.

Actuarieel Genootschap - Projection Table 2018

In September of 2018, the AG published their latest best estimate forecast for the Dutch mortality rates of the next 100 years. The stochastic model used to obtain this forecast is an application of the Li-Lee model in which the Dutch population (NL) serves as the main population of interest. The population that serves as the closely related population in the Li-Lee model is a combination of the populations of the 13 European Union countries (EU ) with the highest GDP, with the ex-ception of the Netherlands6. In the general specification of the Li-Lee model, these two populations (NL and EU ) can be interchanged for any two arbitrary populations. Let us now take a closer look at the specifications of this specific model:

For ages x ∈ {0, 1, 2, .., 90}, and both sexes g ∈ {M, F }, the model is described as:

ln µgx(t) = ln µg,EUx (t) + ln µg,N Lx (t) (3) ln µg,EU_x (t) = Ag_x+ B_xgK_tg (4) ln µg,N Lx (t) = α g x+ β g xκ g t (5)

where the additivity principle is used in equation (3)7. (Agx, αgx) and (Bxg, βgx) can be interpreted

in the same way as αx and βxin the Lee-Carter model, respectively. Ktand κtare dynamic trend

factors for each sex g and year t ≥ 20188_{, with normally distributed error terms (}g t, δ

g

t), and can

be defined by the following time-series:

K_tg= K_t−1g + θg+ g_t (6)

κg_t = γgκg_t−1+ δg_t (7)

In equation (3), µg

x(t) is the force of mortality for the population of the Netherlands, µg,EUx for

the group of Western-European countries, and µg,N Lx is the deviation of the Netherlands relative

to the other Western-European countries (EU-group), i.e. the quotient of µgx(t) and µg,EUx . For

the EU-group time-series Kt, the model uses an ARIMA(0, 1, 0) random walk with drift. For

time-series κt, describing the deviation of the Dutch mortality from the EU-group over time, a

first-order auto-regressive AR(1) model without a constant term is used. The unknown parame-ters in equations (4) and (5) are found by means of maximum likelihood estimation. For further details on this methodology we refer to the 2018 AG projection table (KAG,2018).

6_{The countries taken into account are: Austria, Belgium, Denmark, Finland, France, Germany, Iceland, Ireland,}

Luxembourg, Norway, Sweden, Switzerland, United Kingdom

7_{log(A · B) = log A + log B}

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4.3 Forecasting Mortality Rates

Now that we have a better understanding of the fundamentals of the Li-Lee mortality forecasting model, we will proceed with explaining how we obtain the actual forecasts. When we take a close look at equations (4) and (5), we see that the only time-dependent variables are Kt and κt. The

uncertainty regarding the future mortality rates is therefore a consequence of the uncertainty em-bedded in the future values of these time-trend parameters. Especially the long term estimates for Ktand κtexhibit large amounts of uncertainty.

We want to know how big the uncertainty around the expected lifetime of current and future retirees is in order to get an idea of what the longevity risk is that an annuity provider is facing. What is needed for this are stochastic scenarios of future mortality rates. These are obtained by generating a unique set of stochastic time-series Ktand κtfor each scenario. Each simulated series

of Ktand κtgives a completely different mortality table. Since the only time-dependent elements

in these time-series are captured by the error terms (gt, δ g

t), a stochastic simulation of these error

terms will give us our future mortality scenarios. We do this by randomly drawing the error terms (g_t, δg_t), for each simulation, from a normal distribution with mean (0,0) and covariance matrixP. This matrix is based on the variance in the trend parameters over the preceding period 1970-2017. To obtain the so-called ‘Best Estimate’ of the future mortality rates, we leave out the error term completely. Doing so gives us a single fixed scenario for Ktand κt, which is referred to as the Best

Estimate mortality forecast and which is used as the mortality rate in the fixed leg of the longevity swap that we are going to price later on.

Using the country specific mortality data from the Human Mortality Database, in combination with the stochastic simulation technique as described above, 10.000 simulated mortality tables have been generated with mortality rates for the ages from 0 to 90, for the years 2018 to 2117. The reason we have initially only estimated the mortality rates up to the age of 90 is because the AG model provides parameters up till this age. Since the parameters for higher ages are not provided, we have used the Kannistö methodology to close the mortality table, in other words, to obtain the mortality rates for the ages from 91 up till 120.9

In the remains of this section we will present the results of the simulated mortality forecasts, and look at what the different outcomes would mean for the present value of a homogeneous fund’s liabilities.

4.3.1 Simulated Mortality Results

Using the results of the simulated mortality tables we will now take a look at the possible paths for future mortality rates for Dutch males and females. We will also examine what these different possible outcomes will mean for the liabilities of a pension fund. Before we look at the forecasts, a difference must be noted between the two forms in which life expectancy is most commonly ex-pressed: period life expectancy and cohort10_{life expectancy. Period life expectancy tables assume}

that probabilities of dying at a specific age do not change during a person’s lifetime. Cohort life

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expectancy gives the probability that an individual from a certain cohort dies at a specific age over the course of his or her lifetime. The most important difference is that cohort life expectancy takes the improvements in mortality rates that occur during a person’s lifetime into account.

In figure4we see the period life expectancies as estimated by our model, for males and females at birth from 1970-2070. This expectation has historically always been a few years higher for females than for males, as can also be seen here. The period life expectancies at birth in 2018 are 79,9 and 83,0 years for males and females respectively. If we then look at the spread in the future paths of the period life expectancy, we see that in 2040 the 95% confidence interval for females ranges from 84.6 years to 87.3 years whereas for males this ranges from 81.6 to 84.5 years.

Figure 4: The period life expectancies at birth for females and males from 1970-2070. From 2018 onward the results were obtained by simulating the trend parameters of the Li-Lee model.

The total amount of years which males and females are expected to be in retirement, and thus receiving benefits, is of crucial importance for pension funds and other annuity providers. The expected period life expectancies of 6511 _{year old males and females have therefore been displayed}

in figure5. In 2018, the expected period lifetime for 65 year old males and females is 17.3 and 19.7 years, respectively. In 2040, the 95% confidence interval for this same measure ranges from 18.2 to 21.3 years for males, and from 21.2 to 23.9 years for females. The spread of the confidence interval is larger for males than for females, indicating more uncertainty with respect to the mortality rate.

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Figure 5: The period life expectancies at the pensionable age of 65 for females and males from 1987-2067.

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Figure 6: The cohort mortality rates for the 25, 45, 65 and 85 year old male cohorts, from 1970-2065. The red lines indicate the realized mortality rates and the dotted black lines indicate the 95% confidence interval.

4.4 Impact on Liabilities

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Figure 7: The European Swap Curve of Dec ’17, used to obtain the present value of the liabilities throughout this research. On the x-axis the maturity in years is denoted, on the y-axis the swap rate.

We calculate the expected time-zero value of the liabilities (L(n)x ) in scenario n for the x aged

cohort, given the information up to and including time t, by multiplying the discounted probability that a person from this cohort is still alive t years after the start of his pension (vt * tp

(n) x ) with

the cash flow he or she is supposed to receive in that year (CF), and finally summing up all of these expected cash flows:

E[L(n)x ] = E X∞ t=1 vttp(n)x ∗ CF Ft (8)

Here, vt_{is the discount vector according to the European swap curve from December 2017,} tpxis

the probability that somebody aged x is still alive in t years, and Ftis the mortality information

up to and including time t. To this we add that the value of the liabilities at t=0 is a non-random amount, since it is a function of the fixed parameters K0 and κ0. Using this formula, the present

value of the liabilities for the x aged cohort is calculated for each simulated scenario (N = 10.000). The best estimate present value of the liabilities, LBE

x , is found by using the the best estimate

survival probabilities tp (BE)

x in equation (8) for the specific cohort x. For multiple cohorts, this

best estimate value is given in table2. In this table, also the standard deviation of the simulated liabilities (SL) are given, both in absolute terms and relative to the best estimate liabilities (LBE

x ). σSL_x = v u u u u t 10.000 X n=1 (L(n)− LBE x ) 2 10.000 (9) σSL_x (%) = σ SL x LBE x (10) L(n) _{with n ∈ {1, ..., 10.000}}

Table 2 also shows per cohort and gender the 99,5% Value-at-Risk (or 0.995 quantile) of the simulated liabilities. We define the Value-at-Risk for the α quantile of a distribution Fx(x) as:

V aRα(x) = inf {x ∈ R : Fx(x) > α}. In other words, VaRα is the value such that the probability

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Age cohort Best Estimate (LBE

x ) 99.5% VaR Liabilities σSLx σSLx (%)

45 (Males) 7.338 8.159 0.224 3,1%

45 (Females) 8.243 8.912 0.295 3,6%

65 (Males) 14.886 15.595 0.278 1,9%

65 (Females) 16.900 17.420 0.439 2,6%

Table 2: Different measures of the liabilities of the 45 and 65 year old cohorts.

As expected, the present value of the liabilities for a younger cohort are much lower than that of a cohort at retirement. This is because the benefits that will have to be paid to the 45 age cohort can be discounted for a much longer period, and there is a probability that some individuals will not make it to retirement, which lowers the present value of their liabilities even further. The rela-tive volatility of the liabilities is higher for younger cohorts, illustrating the longevity risk. To give a better understanding of how these these estimates should be interpreted, figure8has been added.

The height of the bars in figure8 represents the discounted annual payments that an annuity provider is obliged to pay to a male that turned 65 in 2018, assuming this person annually receives e1 at the end of each year that he is alive. We see three different scenarios. The line separating the red and green bars is the amount that the annuity provider is expected to pay according to the best estimate mortality rates. The red bars indicate the height of the payments according to the 99,5% VaR scenario of mortality rates. In the case that the fund decides to hedge its longevity risk, this would be a loss for the counterparty taking on the risk. The blue bars indicate the 0,5% VaR mortality scenario. The length of the green bars therefore would be a gain for the counterparty in the case of a hedge. As time goes by we see how the differences between these scenarios become more substantial, which illustrates the general concept of longevity risk in a portfolio of annuity providers.

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5 Pricing Longevity-Linked Securities

In this section we will go into the actual pricing of LLS. First, an overview of what has recently been done in this field is given. Then, the multiple pricing methodologies will be explained and applied.

5.1 Literature on Pricing Longevity-Linked Securities

Within the topic of longevity risk hedging, the pricing of the different products is seen as one of the most difficult aspects. To this day, there has not been a consensus on which pricing methodology to use when pricing LLS. The most important notion to be made here is that the Life Market can still be seen as an incomplete market. An incomplete market is a market where non-hedgeable and non-tradable claims exist. The non-hedgeable part is due to the systematic nature of the longevity risk, and the non-tradable part comes forth from the illiquidity of the products. Because of this incompleteness, a classical arbitrage-free pricing methodology can not be used, as it relies upon the idea of risk replication. The replication technique only works in markets with high liq-uidity and for deeply traded assets (Barrieu et al., 2012). Consequently, many different pricing methodologies have been proposed in papers on this topic. A key notice that is made through-out all of these papers is that the (potential) market for longevity contains more short than long investors. In other words, there is a higher supply of longevity risk than there is demand for it (Loeys, Panigirtzoglou, & Ribeiro,2007)12_{. This oversupply of the risk, in combination with the}

illiquid nature of the product, induces a premium which is added to the price of the longevity derivative. Throughout the rest of this paper this premium will be referred to as the risk premium. The modeling of this risk premium poses the greatest challenge when it comes to the pricing of LLS.

Where this risk premium is usually priced by the market, this is not the case in the longevity market due to the absence of liquid securities (Bauer et al.,2010). Different methods have therefore been proposed for pricing longevity risk. Bauer et al. (2010) compare and comment on different pricing approaches for LLS, mainly focused on the earlier mentioned EIB/BNP bond. They em-phasize the close relation between the pricing and the way in which the stochastic future mortality is modeled. Two methods stand out in their overview. The first is the Sharpe ratio-based pricing approach used by Loeys et al. (2007). In this approach, the Sharpe ratio can be seen as the excess payoff above the expected payment, divided by the standard deviation of the risky payment (Cui,

2007). The second method for pricing the risk premium that Bauer et al. (2010) discuss, is the risk neutral Wang-approach, used in multiple papers (Lin & Cox, 2005), (Cairns et al.,2006). In this method the forecasted mortality rates undergo a so-called Wang-transformation. This transfor-mation converts the mortality probabilities into their risk-neutral equivalents. This methodology is also used by Boyer & Stentoft (2013) in their paper on pricing longevity swaps, forwards and options (Boyer & Stentoft,2013).

Cui (2007) proposes a different risk premium pricing methodology than those mentioned above. In her efforts to overcome the obstacle that keeps the life market from becoming liquid she

pro-12_{With the supply of longevity risk, Loeys et al. refer to the risk that pension funds and annuity providers have}

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5 PRICING LONGEVITY-LINKED SECURITIES

vides a framework based on the equivalent utility pricing principle (Cui, 2007). Considering an incomplete market setting, she states investors should be compensated for the additional market risk they incur when taking on longevity risk. Due to this incompleteness, the risk premium then depends on the payoff structure of the security. She finally stresses the large impact that a natural hedge can have on the price.

Putting more focus on the fact that the market for longevity can be seen as an incomplete market, we will discuss two more pricing methodologies here, both proposed in the same paper (Pelsser,2011). The first method is the Good Deal Bound (GDB) method, which originates from earlier research (Cochrane & Saa-Requejo,2000). This methodology looks at the risk/return trade-off of non-hedgeable risk. The name comes from the notion that the absolute value of the Sharpe ratio on any non-hedgeable portfolio should be bounded, ruling out deals that are ’too good’. In a different paper it is proven that the upper good deal bound is equal to the value of a life annuity obtained through the instantaneous Sharpe ratio approach (Bayraktar, Milevsky, Promislow, & Young,2009). This will therefore the pricing approach used for the supply-side of the LLS market. The second methodology proposed by Pelsser (2011) is the Cost-of-Capital (CoC) method, aimed solely at pricing longevity risk by looking at the reduction in capital charges that can be obtained with it. This is the pricing approach that will be used for the demand-side of the market.

Before we look into these methodologies, some assumptions must be made. Firstly, following (Cairns et al.,2008), we will in the remainder of this paper assume that the term structure of mor-tality is independent of the term structure of interest rates, and that the market price of risk for non-systematic longevity risk is zero. Secondly, EIOPA states that an additional risk premium for counterparty default risk does not have to be added in the case of derivative contracts. Including this risk premium is commented on as being: "... extremely laborious and complex calculations, especially in view of the fact that the charge demanded for counterparty risk by the SCR standard formula is quite limited" - (EIOPA,2011)). To this we add the assumption that the counterparty taking on the longevity risk in this paper has a highly credible, AAA rating, in which case the counterparty default risk is negligible. For these reasons we will exclude the element of counter-party default risk in our pricing calculations.

In the following subsections, multiple methods are proposed to put a price on an Index-Based Longevity Swap. Before we look at these different methods, we will describe the basic elements of interest when it comes to the pricing of a longevity swap of a specific age cohort x. In general, we will refer to the party hedging its longevity risk as the hedger or annuity provider, and the party taking on the risk as the counterparty.

5.2 Basic Elements of a Longevity Swap

A longevity swap consists of the following payments: the hedger paying at time t a fixed amount to the counterparty (F ixed(x, t)) based on the best estimate mortality rates qBEx,t, and the

coun-terparty paying a floating payment to the hedger (F loating(x, t)), based on the realized mortality rates qr

x,t. In the case of an Index-Based Longevity Swap, these realized mortality rates are linked

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paper the floating or realized mortality rates are based on a set of simulations of the Dutch pop-ulation, and the fixed, best estimate, mortality rates are based on the AG model without error terms, both as described in section 4.

The Life Market is net short in longevity (Loeys et al.,2007). A pension fund seeking to transfer its longevity risk will therefore have to pay a premium, which is defined as π, on top of the best estimate expected mortality rate, in order to attract parties willing to take on this risk. We shall refer to the fixed payment plus this risk premium as the adjusted fixed leg. In practice, only the net differences between the adjusted fixed leg and the floating leg are paid. At initiation, t = 0, this risk premium (π) added to the fixed leg of the swap, is set such that the following equation holds:

NPV(F loating(x, t)) = NPV(Adjusted F ixed(x, t)) (11)

= (1 + π) · NPV(F ixed(x, t)) (12) π = NPV(F loating(x, t)) NPV(F ixed(x, t)) − 1 (13) π = NPV(q r x,t) NPV(qx,tBE) − 1 (14)

The following sections will describe different methodologies in which this premium can be set, in each case considering an Index-Based Longevity Swap with a maturity of T years for the age group of 65 year old males.

5.3 Pricing the Swap Through the Payoff Function

The first approach we propose is to price the risk premium on the LLS by looking at the payoff function V (x, t) of the product. Using the simulations from section 4, we calculate the payoff of the longevity swap for each of these N simulations. We then discount these payoffs to find their present value and find the premium πx that sets the value of the swap for the x aged cohort to

zero at t = 0. To use this approach we will, however, have to assume that the market for longevity is complete and arbitrage free. Following the lines of (Shreve, 2004), the value of a security in a complete market is given by:

V (t) = 1 D(t)

˜

E[D(T )V (T )|F (t)] (15)

where ˜E is the expectation under the risk-neutral measure, D is the discount factor, and F is the information set as defined in section 4. Since we assume throughout this paper that the term structure of mortality is independent of the term structure of interest rates, we can write the same equation as:

V (t) = 1 D(T )

˜

E[D(T )|F (t)] ·E[V (T )|F (t)]˜ (16)

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premium πx. This premium can be found by setting the value of the swap to zero at t = 0 and

solving the equation for the unknown factor πx. Let us first define the payoff function V (x, t), at

t years after the initiation of the swap, for the x aged cohort, assuming a notional amount ofe1: V (x, t)(N )= Floating(x, t) − Fixed(x, t) (17)

= E[qx,t] − qx,tBE (18)

= (q_x,tr )(N )− qx,tBE (19)

where q_x,tr is the realized, forecasted mortality value (we have as many forecasted values for age group x at time t, as simulations N ), and qBE

x,t is the best estimate mortality rate. These calculations

will give us N different payoffs. In figure9 we can see the payoff structure of the longevity swap in two different scenarios: the 99.5% and the 0.5% quantile scenarios of the mortality simulations. The absolute differences between the floating payments and the fixed payments are increasing for the first 25 years, after which the differences with the best estimate mortality scenario become smaller again.

Figure 9:The payoff structure of a longevity swap for the 65 year old male cohort. The green bars illustrate the swap payoffs in the 99.5% quantile scenario of all the simulated mortality paths. In this scenario a longevity swap would (net) save money as the realized mortality is much higher than the best estimate case, in which case the counterparty pays the difference to the hedger. The red bars show the payoffs relative to the best estimate scenario for the 0.5% quantile of mortality rates. In this scenario a longevity swap would (net) cost the hedger money since they will still pay the best estimate rate while the realized mortality rate would have been less expensive.

Combining the fact that the present value of the swap for cohort x should be zero at t = 0, with equation (16), where D(t) = (1 + r(t))t_{, we can then find the premium payment π}

xthat completes

the following equation:

N P V [Vswap(x)(N )] = T X t=1 (V (x, t)(N )_{− π}(N ) x ) (1 + r(t))t = 0 (20)

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t = 0 to discount the future payoffs because we have assumed that interest rates and mortality are uncorrelated. For each year t into the swap, this formula subtracts the risk premium πx from

the swap payoff V (x, t) and discounts this difference back to t = 0. The sum of these discounted payoffs should equal 0 at t = 0. For each simulated scenario N , we solve this equation to find a risk premium π(N )x that sets the present value of the swap to zero. This gives us N different values

of the risk premium. We finally take the average of all these estimated risk premiums to determine the overall risk premium on the swap.

Averaging the payoffs over all the simulated paths gives a value of πx≈ 0 for the risk premium.

This is independent of which cohort x we take into account. Since we simulated the mortality scenarios under a risk-neutral measure, and the mortality rate is uncorrelated with the interest rate, the market does not put a premium on the value of the swap. This is because a risk-neutral measure is constructed in such a way that when used to price a derivative in a complete market, the price is equal to the expected value of the discounted future payoffs.

In reality, however, investors want to be compensated for the fact that the product in which they invest is very illiquid. This can be in the form of, for example, a certain expected Sharpe Ratio on their investment (Cern`y, 2009). In the following sections we will dismiss the assumption of a complete market and look into two other methods to price the risk premium on a longevity swap: the Sharpe ratio approach for the supply-, and the Cost-of-Capital approach for the demand-side perspective.

5.4 LLS Supply-Side Pricing: The Sharpe Ratio Approach

In this section we will determine the risk premium on an Index-Based Longevity Swap necessary for an investor to obtain his or her desired return, given by their expected Sharpe ratio on the investment. On behalf of JPMorgan, Loeys et al. are the first to use the Sharpe ratio pricing procedure to price a longevity derivative (Loeys et al.,2007). They believe this approach has the potential to become the standard pricing methodology in the LLS market that JPMorgan aimed to launch in 2007. Loeys et al. use the Sharpe ratio approach to price a 10 year q -forward for a 65 year old male cohort in the US. In this section the same general approach will be used to price an Index-Based Longevity Swap, which can be constructed as a combination of q -forwards, one for each year of the swap (Boyer & Stentoft,2013). Before we construct this swap, we will look at how the price of the q -forward is set.

At the basis of this pricing approach is the Sharpe ratio (SR), developed by Nobel prize winner William F. Sharpe. It is a measure of risk-adjusted performance of a portfolio, generally defined as:

SR = Rp− Rf σp

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Here, Rp is the return on a certain investment portfolio, Rf the risk free rate, and σp the

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that was taken. An investment with a high SR delivers relatively large amounts of excess return per unit of risk that is taken. In this longevity pricing procedure, the investor should see the cash flow of the longevity derivative in year t as an excess-return on investment. An important notice here is that due to the fact that the market is short in longevity, and hence investors want to be compensated for the liquidity risk that they are taking, the forward mortality rate (qf_x,t) that is used in the pricing of the q -forward lies below the expected mortality rate (qBE

x,t ). This ‘discount’

is the above mentioned excess return, and is also referred to as the risk premium π. In figure10

this risk premium is illustrated. The forward mortality rate qx,tf is comparable to forward rates

used for currencies and bonds, locking in a certain expected future rate today.

Figure 10: The risk premium, visualized as a discount on the best estimate mortality rate, on a 10 year q-forward.

Source: "Longevity: A Market in the Making", Loeys et al., 2007

In a market that is liquid, the Sharpe ratio is used as a benchmark when determining the risk premium on an investment. Since the market for longevity is not liquid, the Sharpe ratio cannot be observed, and will therefore have to be set to a certain value. In several researches, (Milevsky, Promislow, & Young,2006), (Loeys et al.,2007), this Sharpe ratio is set to 0.25. It is argued that because longevity risk is uncorrelated to other assets, its SR should fall below the SR of riskier markets, such as equities, but above government bonds, since it is very illiquid. Following the lines of Loeys et al., we will now look at how to derive the risk premium on a q -forward with a maturity of t years for the cohort of x year old males, taking the SR of 0.25 as a starting point.

First, we will have to calculate the annualized expected return of the q -forward, E(Rt). This is

the difference between the expected mortality rate (qBE

x,t) and the forward mortality rate (q f x,t),

divided by the number of years it took to realize this return, i.e. the maturity of the forward (t):

E(Rt) = (qBE x,t − q f x,t) t (22)

Secondly, we will need to calculate the absolute risk, σx,t. This variable denotes the absolute

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σx,t= σq,x+t· qBEx,t (23)

Here, σq,x+tis the historic volatility of the year-on-year changes in the mortality rates of the (x + t)

aged cohort, in percentages of the mortality rate itself. Depending on the maturity (t) of the q -forward, the volatility of the (x + t) aged cohort is used in this formula. This historic volatility is calculated per age cohort. More precisely, we take the standard deviation of the relative year-on-year changes in the mortality rate of the specific age cohort, using mortality data from 1970 till 2017. The results of these calculations for the age cohorts 65-95 can be seen in figure11. It is very important to note that these values do not represent the volatility in the year-on-year mortality rates, which is strictly increasing in the age x, but the volatility in the changes in the year-on-year mortality rates. As we see, the volatility in these year-on-year changes is increasing up to the age of 81, after which it decreases again.

Figure 11: Historic volatility of the Year-on-Year changes in mortality rates per age group.

If we take the annualized expected return E(Rt) from equation (22) as the excess return, we can

combine this with the general formula for the Sharpe ratio as follows:

SR = E(Rt) σx,t =(q BE x,t − q f x,t)/t σq,x+t· qx,tBE (24) This gives: qfx,t= (1 − t · SR · σq,x+t) · qBEx,t (25)

where qBE_x,t is the expected rate of mortality and q_x,tf is the forward rate of mortality. The absolute (negative) premium, πabsolute

t , derived by Loeys et al. (2007), is the difference at time t between

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5 PRICING LONGEVITY-LINKED SECURITIES π_tabsolute= q_x,tf − qBE x,t = (1 − 1 − t · SR · σq,x+t) · qx,tBE (26) = (−t · SR · σq,x+t) · qx,tBE π_trelative= q f x,t− qx,tBE qBE x,t = (−t · SR · σq,x+t) (27)

The relative risk premium in equation (27) only depends on the maturity t, the Sharpe ratio and the historic volatility (σq,x+t) of the mortality rates for the x + t aged cohort. As we are mainly

interested in the additional costs that an annuity provider is paying above the fixed, best estimate mortality rate, we will refer to the positive equivalent of the risk premium, such that the premium proposed by Loeys et al. (2007) in equation (27) can be compared with the premium in equation (14).

Example q -forward

For a 10-year q -forward for the 65 year old male cohort, using the Sharpe ratio pricing methodology, an investor demanding a Sharpe ratio of 0.25 would then require the following relative risk premium:

πrelative₁₀ = (−t · SR · σq,65+10) = (−10 · 0.25 · 0.0265) = −0.0664

According to these calculations, on the 10-year q -forward the hedger will have to pay a risk premium of 6.64%. This risk premium can also be seen as a discount on the best estimate mortality rates, demanded by the party taking on the longevity risk.

5.4.1 From a q -forward to a Longevity Swap

Using a combination of q -forwards with different maturities, we will now construct a longevity swap with maturity T . Since the floating and fixed payments differ at each year into the swap (t), the value of the premium for each year is also different. We calculate the size of the absolute premium (πabsolute

t ) for each year into the swap. Then we can sum the total amount paid in premiums

and divide it by total sum of the best estimate mortality rates to obtain the size of the relative premium over the entire swap, π_(swap)relative:

πrelative_(swap) = PT i=1π absolute i PT i=1q BE x,i (28)

Sharpe Ratio approach: Example Swap

Let us now take the Sharpe ratio pricing methodology and use it to find the risk premium on an Index-Based Longevity Swap. The swap is for the 65 year old cohort of Dutch males, and has a maturity of 20 years. The 20 year swap can be seen as a combination of 20 one-period swaps with the same notional amount but with varying maturities t, with t ∈ {1, 2, ..., 20}. The relative risk premiums the hedger has to pay at each year t, given by πrelative

t , are presented in figure12.

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maturity. We do assume however, that the maturities are discrete. Finally, using equation (28), the relative risk premium over the maturity of the swap is given by: πrelative

(swap) = 9.15%, denoted by

the horizontal blue line in the right side of figure12.

Figure 12: In the left figure we see πabsolute

t , i.e. the annual absolute risk premiums (in percentage points)

on the swap of the 65 year old male cohort, using a Sharpe ratio of 0.25. These are interpreted as absolute reductions in the mortality rates, which is the discount demanded by the investor. In the right figure we see πrelativet , for the same Sharpe ratio, i.e. the annual discount relative to the size of the best estimate

liabilities. The horizontal blue line indicates the relative premium over the entire swap, πrelative(swap) .

As we see in figure12, both the absolute and the relative risk premiums are not a perfect linear function over time. They are both increasing with the maturity, where the absolute risk premium, also known as the mortality forward rate, has a much more convex curvature than the relative risk premium. This is because the absolute annual risk premium πabsolute

t is linearly dependent on

the best estimate mortality rate qx,tBE, which increases over time. As the maturity increases, these

mortality rates have a bigger and bigger impact on the absolute risk premium, hence the shape of the curve. By construction, the size of the relative risk premium is moderated by the increasing mortality rates.

5.4.2 Sensitivity Analysis

We will now test the sensitivity of our results to changes in the input parameters. The parame-ters for which we test are the Sharpe ratio demanded by the investor, the maturity of the swap, and the time frame used to determine the historical volatility in the year-on-year mortality changes.

Sharpe Ratio

The longevity premium calculated by this approach is subject to numerous assumptions, among which the predetermined Sharpe ratio of 0.25 can be seen as the most controversial. In previous papers in which the Sharpe ratio pricing approach is used,(Milevsky et al., 2006)(Loeys et al.,

Hedging Longevity Risk: Deal or No Deal On the Demand- and Supply-Side Pricing in the Incomplete Market of Longevity-Linked Securities Arthur Majtlis S2527855

On the Demand- and Supply-Side Pricing in the Incomplete Market of

Longevity-Linked Securities

Arthur Majtlis

S2527855

On the Demand- and Supply-Side Pricing in the Incomplete Market of

Longevity-Linked Securities

University of Groningen

Faculty of Economics and Business

Master’s Thesis EORAS

Name: Arthur Majtlis

Student number: S2527855

Supervisor: B. Kramer

February 18, 2019

Contents

1

Introduction

2

Introduction to the Life Market and Research Objectives

2.1

Literature on Longevity Linked Securities

2.2

The Current State of the Life Market

2.3

Research Objectives

3

Longevity Solutions

3.1

Bulk Annuities

3.2

Longevity Swaps

3.3

Other Capital Market Solutions

4

Quantifying Longevity Risk

4.1

Mortality Forecasting Models

4.2

Stochastic Mortality Models

4.3

Forecasting Mortality Rates

4.4

Impact on Liabilities

5

Pricing Longevity-Linked Securities

5.1

Literature on Pricing Longevity-Linked Securities

5.2

Basic Elements of a Longevity Swap

5.3

Pricing the Swap Through the Payoff Function

5.4

LLS Supply-Side Pricing: The Sharpe Ratio Approach