Equity-linked insurance or annuity (ELIA) is the generic term to designate a life insurance or annuity that includes benefits tied to the return of a reference portfolio of assets. For example, instead of adjusting the death benefit with inflation (or some predetermined rate of return), the benefit could increase with the returns of a financial index, e.g. the S&P 500. An ELIA is a hybrid between a life insurance (or annuity) policy and a pure investment product as it gives the policyholder an opportunity to benefit from the upside potential while being protected against the downside risk. It competes with mutual funds¹ with the important distinction that ELIAs include various guarantees at the maturity of the contract and/or on the death of the policyholder.

During their active years, policyholders invest their savings in preparation for retirement. Then, at retirement, they annuitize their savings, i.e. the accumulated savings are used to buy an annuity-type product that provides a regular stream of income until death. These two steps are known as the accumulation and the annuitization phases. ELIAs, depending on their features, can be used for one or both phases.

It is important at this point to mention that in actuarial and financial mathematics, there is a clear distinction between a life insurance contract and an annuity. The former provides a benefit at death whereas the latter provides a regular stream of income until death. But in the investment business, the term annuity refers to the entire investment package for both the accumulation and annuitization phases. Consequently, in this chapter, an annuity will be known as a savings product that has an insurance component in case of death.

Example 8.0.1 Equity-linked insurance tied to the S&P 500

A 60-year-old policyholder buys an equity-linked insurance, maturing 8 years from now, with a benefit of $100,000 indexed to the returns of the S&P 500. The contract will pay, at maturity, the greatest value between: (1) $100,000 credited with returns of the S&P 500, or (2) the guaranteed benefit of $100,000. Assume that the S&P 500 index is currently at 1500.

Consider the following scenario: in 8 years, the index will be at 2000. In this case, the accumulation factor is . The insured being entitled to the greatest value between $133,333 and $100,000 (the guaranteed benefit), the benefit paid in this scenario would be $133,333.

Because an ELIA’s value is contingent on the value of a reference index or portfolio, one can interpret an ELIA as a derivative sold by life insurance companies. Just like optionality for vanilla options, guarantees embedded in ELIAs come at a price. Therefore, we should expect contributions, premiums or other form of payments to be made by the investor, in order to benefit from these embedded guarantees.

However, there are four important differences between typical financial derivatives (such as call and put options) and ELIAs:

1. Life contingent cash flows: life insurance benefits are paid upon death or at maturity of the contract and annuities are paid periodically until the death of the policyholder. It is the single most important difference with common derivatives.
2. Very long maturities: whereas financial derivatives have fixed maturities below 1 year (generally a few months), ELIAs have effective maturities between 10 and 50 years. We use the term effective maturity because ELIAs are contingent upon death.
3. A premium is paid up-front to own a derivative but guaranteed benefits in ELIAs are generally paid for by giving up some of the upside potential on the reference portfolio.
4. ELIAs are held by small investors (savings for/during retirement) whereas financial derivatives are held mainly by institutional investors (such as banks, insurance companies, investment banks).

This chapter is not meant to be a thorough review of ELIA contracts and their practical aspects. The main objective of this chapter is to introduce the reader to a large class of insurance products known as ELIAs and to link their cash flow structure to financial derivatives and options. To classify ELIAs, we will use the most popular names in the United States, i.e. equity-indexed annuities and variable annuities. This simple classification has been chosen to ease the presentation and it is not necessarily the one prevailing on the market.

More specific objectives are to:

• understand the relationships and differences between ELIAs and other derivatives;
• understand the following three indexing methods: point-to-point, ratchet and high watermark;
• compute the benefit(s) of typical guarantees included in ELIAs;
• recognize how equity-indexed annuities and variable annuities are funded;
• analyze the loss tied to equity-indexed annuities and variable annuities;
• explain how mortality is accounted for when risk managing ELIAs.

Because maturity benefits are important building blocks for a full analysis of ELIA contracts, we first focus on this type of benefit in Sections 8.1-8.4. Mortality risk will be incorporated only in Section 8.5 with a formal treatment of death benefits.

Other applications of ELIAs will be spread out throughout the book once different models for the underlying asset’s price (the binomial model, Black-Scholes-Merton, etc.) are presented.

8.1 Definitions and notations

Before we proceed to the analysis of specific policies, let us introduce some notation and define important quantities to better understand how benefits are computed.

First, a policyholder investing in an ELIA has to select an index or a reference investment portfolio as the underlying asset of the contract. As before, let us use S_t to denote the time-t value of the chosen index or reference portfolio, where once again time 0 stands for the contract’s inception. Recall that for a fixed time t, the price S_t is a random variable. As for options, let T be the maturity time of a given ELIA contract. The underlying asset’s value is the main source of risk in such contracts.

Second, ELIAs include a guarantee at maturity and/or provide a guaranteed income. Let G_t be the time-t value of this guaranteed amount (at maturity). This guaranteed amount is usually tied to the initial investment I made by the policyholder and, at maturity time T, is given by

(8.1.1)

where 0 < φ < 1 is the fraction/percentage of the initial investment I that is guaranteed and γ is the guaranteed minimum return, also known as roll-up rate. Of course, if T is expressed in years, then γ is an annual rate. In other words, for each dollar invested, an amount of φ(1 + γ)^T is guaranteed at maturity. Usually, the guarantee at maturity is a deterministic (non-random quantity), i.e. its value is already known at inception. We will come back to guaranteed income later in this chapter.

Example 8.1.1 Equity-linked insurance tied to the S&P 500 (continued)

Consider again the contract of example 8.0.1 and let us identify the parameters of the maturity guarantee as defined in (8.1.1). For this contract, the maturity time is T = 8 and we have

This means that φ = 1 and γ = 0. The maturity benefit was implicitly given by

where S₈ is the (random) value of the S&P 500, 8 years from now.

As mentioned above, guarantees in ELIAs are funded or paid for by single or multiple contributions/payments made by the policyholder during the life of the contract (between time 0 and time T) and/or by a limited access to the upside gains of the underlying asset. These are expected to cover the value of the guarantee, in addition to management fees, operating expenses, taxes, etc. When we analyze more specific products, we will be more precise as to how these guarantees are paid for.

In the next two sections, we will study two types of ELIAs. We will use the names equity-indexed annuities and variable annuities to classify an ELIA according to the method used to pay for the embedded guarantee. As mentioned earlier, these names may differ from the ones used in the industry and in some countries.

8.2 Equity-indexed annuities

In what follows, we will use the name equity-indexed annuity (EIA) for an investment product sold by insurance companies providing:

1. a participation in the growth of an index or a reference portfolio;
2. a financial guarantee at maturity.

EIAs form a popular sub-class of ELIAs mostly used by investors in the accumulation phase, as they usually provide a single benefit at maturity time T (the maturity benefit). Of course, that amount can then be annuitized with another financial product (issued at time T or later).

EIAs are classified according to their indexing methods, i.e. how the maturity benefit is tied to the underlying asset. There are three popular methods of indexation:

• point-to-point;
• ratchet;
• high watermark.

8.2.1 Additional notation

Let R_t be the (periodic) cumulative returns of the underlying asset S between time 0 and time t. Mathematically,

For time points 0 = t₀ < t₁ < t₂ < … < t_n = T, the return earned by S during the k-th period, i.e. between time t_{k − 1} and time t_k, is simply

(8.2.1)

which can also be written as

For example, if we are interested in annual returns, we only need to choose t_k = k, for each k. Note that for a fixed time t, since the price S_t is a random variable, then the returns R_t and y_t are also random variables.

8.2.2 Indexing methods

As we will see, equity-indexed annuities have a similar payoff structure to investment guarantees studied in Chapter 6. Recall that the payoff of an investment guarantee is given by max (S_T, K). Consequently, it provides a full participation in the asset, if its final price S_T is larger than K, or a guaranteed cash amount of K, otherwise. In exchange, a premium must be paid.

Equity-indexed annuities are very similar to investment guarantees: the full participation in the asset will be replaced by another way of participating in the returns of the asset, while the guaranteed amount will be denoted by G_T, as defined in equation (8.1.1), instead of K. We will look at three ways of participating in the returns, generally known as indexing methods.

The point-to-point (PTP) indexing method yields a maturity benefit given by

(8.2.2)

where β is known as the participation rate and is typically such that 0 < β < 1, and where G_T is given in equation (8.1.1). In other words, for each dollar invested, an EIA with a PTP indexing method provides a maturity benefit of

The policyholder thus gives up part of the upside potential in exchange for a protection against the downside risk. Indeed, contrarily to financial derivatives such as options, no upfront premium is required to be entitled to the guarantee embedded in an EIA, but the participation rate being lower than 100% (β < 1) serves that purpose. In this case, finding the fair value of the participation rate β, i.e. its no-arbitrage value for a given maturity guarantee G_T, is similar to finding the fair price of an option. We will come back to this issue in Chapter 18.

Example 8.2.1 Point-to-point indexing method

Suppose that in example 8.0.1, the policyholder is entitled to 80% of the accumulated returns on the S&P 500 with a guarantee applying to 100% of the initial investment. Using the same scenario (values of the index), let us compute the maturity benefit.

For this PTP EIA, the maturity benefit of equation (8.2.2) is such that we have β = 0.8 and I = G₈ = 100000. Using the same scenario as in example 8.0.1, in 8 years we would have R₈ = 0.333 and therefore the maturity benefit would be

Another popular type of indexing is the (compound periodic) ratchet indexing method. In a T-year EIA contract with a ratchet indexing scheme applied at time points 0 = t₀ < t₁ < t₂ < … < t_n = T, the maturity benefit is given by

(8.2.3)

where the periodic returns y_k, k = 1, 2, …, n, were defined in (8.2.1). Again, β is known as the participation rate and is typically such that 0 < β < 1. The ratchet indexing method mostly differs from the PTP indexing method as a minimal return applies periodically, usually annually, rather than at maturity only.

Example 8.2.2 Ratchet indexing method

Let us consider a 2-year EIA whose ratchet indexing scheme applies annually. Moreover, the participation rate is 90% and the initial investment is fully guaranteed, i.e. G₂ = I. Therefore, the maturity benefit of (8.2.3) is here given by

Recall that y₁ and y₂ are random variables; therefore, at inception, their values are unknown.

Today, say we invest I = 250000 in this ratchet EIA. Consider the following scenario: in the first year, the return is 10% and during the second year, it is negative. Then, in this scenario

and

Therefore, in this scenario, the maturity benefit would be equal to

2 years from now, as G₂ = I = 250000.

Finally, the high watermark indexing method applies the maximum cumulative return, observed over the life of the EIA, to the whole life of the contract. Borrowing notation from lookback options, as seen in Chapter 7, we have that the maturity benefit of a high watermark EIA is given by

(8.2.4)

where is the maximum cumulative return, where is the discretely monitored maximum value of the underlying asset/portfolio (as already defined for lookback options in Chapter 7). The maximum is generally monitored at some pre-specified dates 0 < t₁ < … < t_n = T, usually yearly or bi-annually. Again, β is known as the participation rate and is usually such that 0 < β < 1. Note the similarity with the maturity benefit of a PTP EIA as given in (8.2.2).

Since G_T is usually smaller than the maximal cumulative return, in that case the maturity benefit will simplify to

Example 8.2.3 Comparison of indexing methods

Consider a 10-year EIA with a participation rate of 80% (for each of the indexing methods) and an initial investment guaranteed at 100% at maturity. For simplicity, assume this initial investment is 100. Therefore, T = 10, β = 0.8 and I = G₁₀ = 100.

Suppose that over the course of the next 10 years the reference portfolio evolves as follows: 1000, 1022.16, 1042.98, …, 1317.11, 1237.97 (details are provided in the next table). In this scenario, let us compute the maturity benefit for the three indexing methods.

k	Sk	Rk	yk	max (yk, 0)
0	1000
1	1022.16	0.02216	2.22%	2.22%
2	1042.98	0.04298	2.04%	2.04%
3	1008.88	0.00888	−3.27%	0.00%
4	1010.83	0.01083	0.19%	0.19%
5	1067.89	0.06789	5.64%	5.64%
6	1102.42	0.10242	3.23%	3.23%
7	1051.51	0.05151	−4.62%	0.00%
8	1125.86	0.12586	7.07%	7.07%
9	1317.11	0.31711	16.99%	16.99%
10	1237.97	0.23797	−6.00%	0.00%

For the PTP scheme, we have

For the compound annual ratchet, the computation is tedious, but we have

Thus, the maturity benefit is

Finally, the maximum value of the reference index is observed at time 9 (at the end of the 9-th year), i.e. M^S₁₀ = 1317.11 and hence the high watermark benefit is

8.3 Variable annuities

In what follows, we will use the name variable annuity (VA) for another popular sub-class of ELIA contracts. They are constructed with a separate account, also known as the sub-account of the contract, where the initial investment is deposited and credited with returns from a reference portfolio (underlying asset). The policyholder is then allowed to withdraw from this sub-account either for liquidity purposes during the accumulation phase or simply to purchase an annuity at retirement (annuitization phase). This sub-account is subject to a set of guarantees that apply at maturity or death, making VAs typical hybrids between insurance and investment.

One of the most important difference between EIAs and VAs is how guarantees are financed, i.e. how they are paid for by the policyholder. As mentioned before, in an EIA contract, the cost of the guarantee is paid implicitly through the participation rate β that lowers the upside potential. In a VA contract, premiums are withdrawn periodically from the sub-account and are usually set as a fixed percentage of the sub-account balance, acting again like a penalty on the credited returns. As before, these premiums also include management fees, operating expenses, taxes, etc. The sum of these expenses is known as the management and expense ratio (MER).²

The generic name for this type of ELIA contract is separate account policy. In the U.S., it is known as a VA, in Canada, as a segregated fund, and in the U.K., as a unit-linked contract. They borrow their name from typical regulations that prevent insurance companies from mixing assets backing variable annuities with other investments.

8.3.1 Sub-account dynamics

The insured’s initial investment A₀ = I is deposited in a sub-account. This sub-account is then credited with the returns of the reference portfolio. Also, it is from this sub-account that fees are deducted and that policyholder withdrawals will be made.

Let the sub-account value at time k (after k months or years) be denoted by A_k, where k = 1, 2, …, n. The sub-account value is adjusted periodically and it is not allowed to become negative so that deductions (withdrawals, fees) must be such that A_k ⩾ 0 for each k. With this notation, the index n corresponds to the maturity time T. More specifically, the sub-account dynamic is given by

(8.3.1)

where α is the fee/premium rate deducted from the sub-account at the end of the period and where ω_k is the amount withdrawn by the policyholder, at the end of the k-th period (as long as it is less than the sub-account value prior to the withdrawal).

We can describe equation (8.3.1) as follows: the sub-account balance A_k is obtained by crediting returns to the previous sub-account balance A_{k − 1}, then deducting the proportional periodic premium (done by multiplying by (1 − α)) and finally deducting the policyholder’s withdrawal ω_k. Again, unless stated otherwise, all the parameters and quantities are given and computed on an annual basis. Finding the fair value of α is a typical problem for insurance companies and is a concept related to finding the no-arbitrage price of an option: more details in Chapter 18.

If we set , i.e. if A_{k −} is the value of the sub-account just after deducing the fee but just before the k-th withdrawal, then we can rewrite (8.3.1) as follows:

Note that depending on the VA policy, there might be restrictions on withdrawals and guarantees might be adjusted based on the amounts withdrawn.

Example 8.3.1 Sub-account balance after 1 year

An investor puts $100 in the sub-account tied to a VA policy. The periodic premium is set to 1% per annum. The investor plans on withdrawing $5 at the end of the year, if the funds are available.

Assume the following scenario: the reference portfolio will grow by 4% over the next year. Let us compute the value of the sub-account balance at the end of the year, i.e. after the withdrawal, in this scenario.

Here, we have A₀ = 100, ω₁ = 5, α = 0.01. In this scenario, we have S₁/S₀ = 1.04 and therefore the sub-account balance at year-end would be

Note that we were allowed to withdraw ω₁ = 5 from the sub-account since ω₁ = 5 < A_{1 −} = 102.96.

In this scenario, in the second year, the return of the reference portfolio would be applied to a sub-account balance of 97.96.

Example 8.3.2 Sub-account balance during a year

A retiree puts $120 in the sub-account tied to a VA policy. If the premium paid at the end of each month is 0.1% of the (monthly) balance and if the contract allows for withdrawals of $10 at the end of each month (or the remaining balance), let us determine the sub-account balance at the end of the year in the scenario depicted in the following table:

Month k	Sk	Ak
0	1000	120.00
1	966	105.80
2	1015	101.06
3	999	89.37
4	950	74.90
5	992	68.13
6	953	55.39
7	865	40.22
8	837	28.88
9	808	17.85
10	898	9.82
11	981	0.72
12	1012	0

Here, A₀ = 120, α = 0.001 (monthly) and ω_k = min (10, A_{k −}) for all k = 1, 2, …, 10.

For example, here is how A₁ was computed:

Note that in the above scenario (given by the table), ω_k = 10, for each k = 1, 2, …, 11, while ω₁₂ = A_{12 −} = 0.74, as this is the sub-account balance just after crediting the returns of the 12th month (i.e. ). In other words, in this scenario, there are not enough funds left at time 12 to withdraw $10, so the remaining balance of $0.74 is withdrawn instead.

8.3.2 Typical guarantees

In this section, we will look at two popular guarantees included in VAs. The first one is a guarantee that applies on the sub-account balance at maturity and is known as a guaranteed minimum maturity benefit (GMMB) rider. The second guarantee assures the policyholder can withdraw predetermined amounts from the sub-account and is known as a guaranteed minimum withdrawal benefits (GMWB)³ rider. The first guarantee is widely used in the accumulation phase, while the second guarantee is designed for the annuitization phase.

8.3.2.1 GMMB

Let us begin by assuming the policyholder cannot withdraw from the sub-account, so that ω_k = 0, for each k. Therefore, the sub-account balance at time k is given by

In a GMMB, the maturity benefit is given by

(8.3.2)

where as in (8.1.1) the maturity guarantee is given by G_T = I × φ(1 + γ)^T and where 0 < φ ⩽ 1 and γ > 0. However, in practice, we have typically 0.75 < φ ⩽ 1 and 0 < γ < 0.02, i.e. only 75–100% of the initial investment is subject to a minimum guaranteed return of 0–2%.

As shown in equation (8.3.2), at maturity, the guarantee G_T is compared to the final sub-account balance A_T, and not to the underlying asset value S_T. In a scenario where the final balance A_T ends up below G_T, the guarantee will apply and the policyholder will receive G_T.

Example 8.3.3 Guaranteed minimum maturity benefit

Suppose that a 10-year GMMB is issued and the reference portfolio is tracking the Dow Jones 30 index. A premium of 1.5% per year is paid to fund the maturity guarantee, which is 105% of the initial investment. The value of this index is 10,000 at issuance of the contract.

In the scenario that the index reaches a value of 12,000 at maturity, let us determine the payoff of the GMMB for an initial investment of $500.

The guaranteed amount at maturity is given by:

In the given scenario, the final sub-account balance is easy to compute:

Consequently, the maturity benefit of this GMMB would be

8.3.2.2 GMWB

A GMWB rider guarantees a regular stream of income to the (retired) investor. If returns on the reference portfolio are favorable, then the investor may also receive the final balance of the sub-account.

The nature of a GMWB is to allow the policyholder to withdraw from the sub-account when needed. However, for modeling purposes, we assume that withdrawals are constant and predetermined. More specifically, the initial investment I = A₀ can be withdrawn uniformly, i.e. ω_k = ω = A₀/n at the end of each period, for each k = 1, 2, …, n, if there are n periodic withdrawals allowed. These withdrawals are guaranteed, no matter the value of the sub-account on a given year-end.

Let us consider a T-year GMWB that allows for annual withdrawals (it is easy to generalize to monthly or periodic withdrawals). Then, after the k-th withdrawal, the sub-account balance is

or, written differently,

for each k = 1, 2, …, n.

If returns are favorable, i.e. if A_T > 0, then the policyholder will recover this final sub-account balance A_T, in addition to the guaranteed withdrawals already collected.

Figure 8.1 shows the sample path of the sub-account balance after periodic withdrawals. In the case of this scenario, the sub-account balance being positive at maturity, this amount would be returned to the policyholder.

It is important to note that even if the sub-account balance were to reach zero before the maturity of the contract (due to unfavorable returns), the policyholder would still be entitled to the remaining guaranteed withdrawals. In that case, the insurer would suffer a loss. This will be discussed at length in the next section.

The next two examples illustrate the cash flows of a GMWB in two market scenarios.

Example 8.3.4 Sub-account of a GMWB – favorable returns

A policyholder invests $500 in a 5-year GMWB with annual fee payments and withdrawals. If the annual fee rate is 2% and if the underlying index evolves as follows during the next five years, i.e. 1000, 1174, 957, 1048, 1220, 1296, then let us determine the cash flows of this GMWB’s sub-account.

The following table illustrates the cash flows in this scenario:

year k	Sk	Withdrawal ω	Ak
0	1000		500
1	1174	100	475.26
2	957	100	279.67
3	1048	100	200.13
4	1220	100	128.32
5	1296	100	33.59

Recall that with a GMWB, no matter what happens, the policyholder is guaranteed to receive $100 at the end of each year.

After one year, the sub-account balance is

For the second year, the returns are credited on the previous account balance of 475.26, so that

The rest of the table is computed similarly.

Finally, because the final sub-account balance is positive (in this scenario), the policyholder is entitled to an extra A₅ = $33.59 at maturity.

Example 8.3.5 Sub-account of a GMWB – unfavorable returns

Consider the same GMWB rider as in the previous example, but suppose now that the index will evolve instead as follows: 1000, 977, 978, 1396, 954, 879.

year k	Sk	Withdrawal ω	Ak
0	1000		500
1	977	100	378.73
2	978	100	271.54
3	1396	100	279.84
4	954	100	87.41
5	879	100	0

Again, the policyholder is entitled to $100 at the end of each year, as this is guaranteed by the contract design and is not affected by the scenario.

However, the returns credited to the sub-account are not enough to cover the withdrawals. The policyholder will receive all five withdrawals of $100, but she will not receive any additional amount at maturity since

In this scenario, the insurance company would suffer a loss of 100 − 78.93 = 21.07.

Should I stay or should I go: the incentive to lapse

To lapse or to surrender a VA policy is a right given to the policyholder to cease premium payments and forego the underlying protection. We have ignored this feature in the above discussion.

We know that derivatives and variable annuities differ in various aspects, but one very important element is that an option is paid by an upfront premium whereas periodic premiums are paid throughout the life of a VA policy. This may raise an interesting question when the sub-account balance is well over the guarantee: should a policyholder continue to pay premiums for a guarantee that is unlikely to be triggered? In this case, there is an incentive to lapse.

Suppose 2 years before maturity, the sub-account balance is 200 and the minimum guaranteed maturity benefit is 100. If the investor believes it is nearly impossible for the sub-account balance to go below 100 during the next 2 years, then it might be profitable to stop paying premiums for such a guarantee.

In conclusion, depending on the market conditions and the structure of the contract, it might be optimal for a policyholder to lapse a VA. This is an American-like feature of a VA in the sense that an informed investor may make a decision that optimizes her wealth just as for American put options.

Insurance companies try to prevent lapses by having a periodic premium structure that discourages early surrenders (to recoup initial costs) or by including riders that reset the guaranteed amount whenever market returns are favorable (automatic and voluntary resets).

8.4 Insurer’s loss

Before we analyze the insurer’s loss tied to an ELIA contract, let us look again at an investment guarantee, a product that we presented first in Chapter 6. Recall that its payoff can be decomposed as

or

In the first case, we see that writing an investment guarantee is equivalent to selling a share of stock in addition to writing a put option. If the company already owns or buys a share of the stock, then it is liable to provide, at maturity, the following put payoff:

In the second case, we see that writing an investment guarantee is equivalent to selling risk-free bonds with a total principal value of K in addition to writing a call option. If the company already owns or buys the risk-free bonds, then it is liable to provide, at maturity, the following call payoff:

In both cases, the remaining risk arising from these long-term option payoffs needs to be actively managed using adequate investment strategies, i.e. with hedging portfolios. We will come back to this issue later in the book.

8.4.1 Equity-indexed annuities

EIAs are typically branded as a product offering participation in the upside potential of an index or investment portfolio. This is because, in practice, insurance companies typically invest the policyholder’s initial investment I in risk-free bonds and can then easily provide a payoff of G_T, i.e. the initial investment accumulated at some fixed rate, usually below the risk-free rate, if needed.

Therefore, the insurer’s loss on an EIA is the maturity benefit paid in excess of G_T. Mathematically speaking, the loss on a PTP EIA is defined as

Using the properties of the maximum operator, the loss can be written as follows:

This loss can be further rewritten (with some work) as

In conclusion, the insurer’s loss arising from a PTP EIA has the same value as a certain number of call options. More precisely, the loss is equivalent to units of a call option (written on the same underlying asset) with a strike price of .

Calculating the insurer’s loss from an EIA with a ratchet or high watermark indexing scheme is similar.

Example 8.4.1 Loss on several EIAs

Recall the policies considered in example 8.2.3 and the given scenario. For the PTP indexing method, the maturity benefit was 119.0376 (in that scenario) whereas the guaranteed amount was 100. Therefore, the loss for this policy would be 19.0376 (in that scenario). Similarly, the loss on the compound annual ratchet is 33.30 and 25.3688 on the high watermark benefit.

8.4.2 Variable annuities

A VA is a separate account policy and, as such, the sub-account belongs to the policyholder. It is held in her name by the insurance company. At maturity, the policyholder will cash the final sub-account balance A_T if it is positive. For the insurer, this is a riskless component of the contract. Therefore, it is natural to define the loss on a VA policy as the benefit net of the sub-account balance.

Mathematically, the insurer’s loss on a GMMB is defined as

Again, it can be rewritten as follows:

The loss has the same payoff structure as that of a put option written on the sub-account value and with a strike price of G_T, even though such a derivative is not traded.

Example 8.4.2 Loss on a GMMB

We continue example 8.3.3 by computing the loss of this specific contract with the observed trajectory of the reference portfolio. We know that the maturity benefit in that scenario would be equal to

Since the final sub-account value would be 516.425, the loss (for this specific trajectory) would then be

The loss of a GMWB is more complex than that of a GMMB because the guarantee allows the policyholder to withdraw the specified periodic amounts no matter what occurs on the financial markets. Let us contrast two scenarios, depending on whether or not the sub-account is sufficient to fund the withdrawals:

• Sub-account is sufficient to fund all withdrawals: the returns credited on the sub-account were sufficient to cover both the withdrawals and the premiums and hence the insurer suffers no loss. The sub-account remaining balance is paid back in full to the policyholder, providing the policyholder with upside potential.
• Sub-account is not sufficient to fund the withdrawals: the sub-account balance reaches zero before maturity and the policyholder is still entitled to the remaining withdrawals. The insurer suffers a loss equivalent to the sum of all withdrawals the sub-account could not provide. In this scenario, the guarantee has kicked in and provided the policyholder with downside protection on the withdrawals.

To summarize, the loss on a GMWB is the sum of withdrawals owed in excess of the sub-account balance. It is possibly best illustrated with a numerical example.

Example 8.4.3 Loss of a GMWB

We compute the loss of the GMWB in both the favorable and unfavorable market scenarios of example 8.3.4 and example 8.3.5.

In the favorable market scenario, the sub-account balance at the end of the contract is positive and therefore the sub-account balance A₅ = 33.59 is returned in full to the policyholder. Since all withdrawals were funded by the sub-account, the insurance company has suffered no loss.

In the unfavorable market scenario, the returns were insufficient to fund the final withdrawal. In that scenario, the loss would be equal to 21.07, i.e. the portion of the final withdrawal that would not be covered by the sub-account.

8.5 Mortality risk

Although we have only treated maturity benefits so far, most ELIAs offer a protection until maturity or death, whichever comes first. The typical death benefit is very similar to the maturity benefit in the sense that a similar guarantee may apply at the time of death, if death occurs during the life of the contract. We will begin by analyzing general investment guarantees with death benefits and apply the same principles to EIAs and VAs.

Let τ be the (random) time of death of some given individual. The lifetime distribution of τ depends on the age of the policyholder and other characteristics such as gender, smoking habits, etc. The effective maturity time of an ELIA contract is given by the following random variable:

In other words, the contract lasts until one of the following two possible events occurs: death of the policyholder at the random time τ, which is less than the maturity time T, or survival until time T.

Example 8.5.1 Investment guarantee applying at death or at maturity

A policyholder enters into an investment guarantee that protects 100% of the initial capital in case of death and 105% of the initial investment at maturity (in 10 years). Today, the reference index is worth 1000. Suppose the insured invests $500 in this product.

Let us consider the following two scenarios:

1. the reference index is worth 980 after 3 years and 1350 after 10 years, and the policyholder dies at the end of the third year;
2. the reference index is worth 980 after 3 years and 1350 after 10 years, and the policyholder is alive after 10 years.

Let us compute the value of the payoff of this contract in each scenario.

In scenario (a), the effective maturity is at time min (T, τ) = min (10, 3) = 3, i.e. the guarantee applies at time of death (at time 3), and the corresponding death benefit is . Note that in this scenario, the fact that the reference index will reach level 1350 at time 10 does not have any impact on the benefit.

In scenario (b), the effective maturity is at time 10 and, since the accumulated investment is larger than the guaranteed 5% return, the maturity benefit is .

For the rest of this section, we will discuss how death benefits are priced and managed at the level of the insurance company. We will assume the insurer manages a large portfolio of ELIA contracts sold to (independent) individuals having the same age, living habits, etc., and thus having the same mortality distribution. The insurance company is exposed to two risks: mortality and fluctuations of the (common) reference portfolio. As seen in Chapter 1, the risk associated with the reference portfolio is a systematic risk for the insurance company because it is common to all contracts, whereas the mortality risk is assumed to be fully diversifiable. As usual, we assume that those two risks are independent.

Even if an insurer cannot exactly identify the time of death of each individual, diversification of the mortality risk in a very large portfolio allows the insurer to approximate the number of policyholders who will die every year. As seen in Chapter 1, this is a direct consequence of the law of large numbers. Therefore, even if at the individual level the effective maturity of the policy is random (it is the random variable denoted above by T^⋆), at the portfolio level the insurer can estimate the number of policies maturing after 1 month, 2 months, …, after 1 year, 2 years, etc., based upon the expected number of people who will die after 1 month, 2 months, etc.

Therefore, the insurer can approximate its portfolio by considering it as a pool of ELIA contracts all having various deterministic maturities

as if each of these contracts did not include the mortality component. This way, we can use all the previous results of this chapter. The following example will illustrate this very important idea.

Example 8.5.2 Risk management of investment guarantees with a death benefit

An insurance company sells an investment guarantee contract to 100,000 insureds assumed to have the same age and living habits. The actuary has determined that approximately 1% of the 100,000 individuals will die in each year over the next 10 years.

Let us explain how to manage this portfolio of investment guarantees protecting 100% of the capital in case of death and 105% of the capital until maturity if the policy matures in 10 years.

Out of the 100,000 policies issued, the actuary expects⁴ that 1,000 will terminate in each year due to death (over 10 years), and therefore cashing in the death benefit, leaving 90,000 policies to terminate with the maturity benefit. In other words, 1,000 policyholders will die during the first year, another 1,000 during the second year, …, and finally a last 1,000 during the last year; all those policies will receive the death benefit equal to 100% of the initial investment. The other policies, i.e. 90% of the 100,000 in this portfolio, will receive the maturity benefit whose guarantee is 105% of the initial capital.

Therefore, the insurance company can approximate its actual portfolio by considering that it holds the following portfolio of 11 different investment guarantees with deterministic maturities:

• 1,000 contracts maturing in 1 year with a maturity benefit corresponding to 100% of the investment;
• 1,000 contracts maturing in 2 years with a maturity benefit corresponding to 100% of the investment;
• …
• 1,000 contracts maturing in 10 years with a maturity benefit corresponding to 100% of the investment;
• 90,000 contracts maturing in 10 years with a maturity benefit corresponding to 105% of the investment.

As we did before, we can decompose each investment guarantee’s maturity benefit max (S_T, K) as the sum of a unit of the underlying S_T and a put option max (K − S_T)₊, where T = 1, 2, …, 10. Then, the approximate portfolio is equivalent to holding

• 1,000 put options maturing in k year(s) with a strike of 100% × S₀, for each k = 1, 2, …, 10;
• 90,000 put options maturing in 10 years with a strike of 105% × S₀;
• 100,000 shares of the underlying stock.

EIA and VA contracts also include death benefits that are very similar to the maturity benefits described previously. Therefore, the point-to-point, the compound annual ratchet and the high watermark indexing schemes may also apply at the time of death, if it occurs before maturity.

Similar to a GMMB, a guarantee that applies at death is known as a guaranteed minimum death benefit (GMDB). Therefore, to receive a benefit at death or maturity, whichever comes first, we need to combine a GMMB and a GMDB. Much like a GMWB, a guaranteed lifetime withdrawal benefit (GLWB) allows the retiree to withdraw guaranteed amounts until death whereas the remaining sub-account balance (if any) is paid upon death to the beneficiaries.

Example 8.5.3 Illustration of a GLWB

A 65-year-old retiree has accumulated savings of $20,000 which she invests in a policy allowing withdrawals of $1,000 per year until death and subject to an annual premium of 2% of the sub-account balance.

Consider the following scenario: the policyholder will die after 8 years, while the evolution of the index will be 1, 1.1047, 1.0559, 1.3563, 1.0952, 0.9798, 1.0774, 1.1667, 0.9904 at the end of those 8 years. Let us compute the death benefit of this contract.

We have computed the sub-account balance for the next 8 years, according to equation (8.3.1), in this scenario. The results are presented in the next table.

In this scenario, the policyholder will die in the 8th year and the sub-account balance at the end of that year will be 10082.4157. Therefore the policyholder's heirs will receive $10082.42 at that time, while the retiree will have received $1000 per year until death.

Clearly, a GLWB provides annuity payments in addition to participation in the upside potential of the reference portolio. The insurance company will thus provide protection if the insured dies very old and/or if returns are unfavorable. This contrasts with basic annuities that cease at death and do not provide any right to favorable market returns (i.e. the sub-account balance at time of death).

Similarly, as in example 8.5.2, the loss tied to an ELIA is a portfolio of call/put options having different maturities whose quantities are determined by the expected number of people who will die each year.

Example 8.5.4 Loss of two ELIAs with death benefits

Suppose we have a pool of 100,000 insureds having the same age and living habits. The actuary has determined that, on average, 1.5% of those insureds will die in each of the following 6 years (1500 per year), whereas the rest will survive the next 6 years. Describe the loss of a 6-year PTP EIA and the loss of a 6-year policy combining a GMDB and GMMB.

We found earlier that the liability tied to a PTP EIA with maturity T (and no death benefit) is equivalent to units of a call option on the stock with a strike price of . However, we can approximate the annual number of deaths, so that the total loss is

• units of call options maturing in T years, with strike , for each T = 1, 2, …, 6;
• units of call options maturing in 6 years, with strike .

Recall that the loss tied to a GMMB with maturity T (and no death benefit) is a put option on the sub-account balance with strike price equivalent to the guaranteed amount G_T. Similarly, the total loss of the GMMB combined with a GMDB both issued to 100,000 policyholders is

• 100000 × 0.015 put options maturing in T years with strike G_T, for each T = 1, 2, …, 6;
• 100000 × 0.91 put options maturing in 6 years with strike G₆.

Depending on the clauses of the policies, note that G₆ could be different whether the insured died in the 6th year or survived the 6 years.

In conclusion, it is important to remember that risk management of death benefits in ELIAs is based upon diversification, i.e. by approximating the expected number of people dying/surviving every year. Then, the insurance portfolio can be treated as a pool of contracts each having fixed maturities (i.e. no death benefit).

8.6 Summary

Equity-linked insurance or annuity (ELIA): generic name to designate a life insurance/annuity including living and/or death benefits tied to the returns of a reference portfolio of assets.

Differences between an ELIA and other derivatives

• Life contingent cash flows.
• Very long maturities.
• No premium paid up-front.
• Held by small investors (savings for/during retirement).

General notation and terminology

• Time-t price of the underlying asset/portfolio: S_t.
• Maturity date: T.
• Guaranteed amount at maturity: G_T.
• Time-t value of the guaranteed amount: G_t.
• Initial investment: I
• Percentage of I that is guaranteed: φ ∈ (0, 1).
• Roll-up rate (guaranteed minimum return): γ
• Guaranteed amount at maturity:
  
• Cumulative returns of S between time 0 and time t:
  
• Periodic monitoring: 0 = t₀ < t₁ < t₂ < … < t_n = T.
• Return earned by S during the k-th period:
  
• Maximum price of S: .
• Maximum cumulative return: .

Equity-indexed annuities (EIAs)

• Specific features:
  • – Participation rate: β.
  • – Guarantee is paid for (by the policyholder) because β is less than 1.

• Maturity benefits for different indexing methods/schemes:

  • – point-to-point:
    
  • – ratchet:
    
  • – high watermark:
    

Variable annuities (VAs)

• Sub-account increases: initial investment, credited returns.
• Sub-account decreases: fees/premium payments, withdrawals.
• Sub-account value at time k: A_k.
• Initial investment: I = A₀.
• Fee/premium rate: α.
• Withdrawal (if allowed) at the end of the k-th period: ω_k.
• Value of the sub-account after the fee but before k-th withdrawal:
  
• Sub-account dynamics:
  
• Guaranteed minimum maturity benefit (GMMB):
  • – No withdrawal, i.e. ω_k = 0, for all k.
  • – Sub-account balance at time k:
    
  • – Maturity benefit: max (A_T, G_T).

• Guaranteed minimum withdrawal benefit (GMWB):

  • – Guaranteed withdrawals.
  • – Assume ω_k is constant and equal to ω = A₀/n.
  • – After the k-th withdrawal:
    
    or, written differently,
    
  • – If A_T > 0, then policyholder receives A_T.

Insurer’s loss

• For an EIA, the loss is the maturity benefit paid in excess of G_T:
  
• Loss on a PTP EIA: same value as a number of call options.
• For a VA, the loss is the benefit net of the sub-account balance:
  • – Loss for a GMMB: max (A_T, G_T) − A_T = (G_T − A_T)₊ (payoff of a put option).
  • – Loss for a GMWB: sum of the withdrawals in excess of the sub-account balance.

Mortality risk and death benefits

• Time of death (random) of an individual: τ.
• Effective maturity: min (T, τ).
• Guaranteed minimum death benefit (GMDB): like a GMMB, but guarantee applies at time τ.
• GMMB + GMDB: benefit at death or at maturity.
• Guaranteed lifetime withdrawal benefits (GLWB): like a GMWB, but ends at time τ.
• Mortality risk can be diversified: approximate the expected number of people dying/surviving every year.
• Risk management: treat the portfolio as a pool of contracts with fixed maturities and no death benefit.

Notes