2.1. Data

We consider mortality data from eight female populations in Europe (see Table 1), provided by the Human Mortality Database. For consistency reasons, the same sample period (1900–2018) and age range (40–89) are used for all populations under consideration. The age range includes typical ages of pension plan members (active and retired), but excludes extreme ages for which data are often extrapolated and smoothed. We set the beginning point of the sample period to 1900, as a longer time-series facilitates the study of long memory.

Table 1. A summary of the mortality data used in this paper

2.2. Mortality model

Our modeling work is based on the well-known Li–Lee (LL) model [Li and Lee (Reference Li and Lee2005)]. Let $m^{( i) }_{x, t}$ be the central rate of death at age x in year t for population i, where $i \in \mathbb{P}$ with $\mathbb{P}$ being the set of populations under consideration. The LL model is specified as

(1)$$\ln m^{( i) }_{x, t} = a_x^{( i) } + B_x K_t + b_x^{( i)} k_t^{( i)} + e^{(i)}_{x, t},$$

where

• • $a_x^{( i) }$ is an age-specific parameter indicating the i-th population's average level of mortality at age x,

• • K _t is a time-varying index that is shared by all populations in $\mathbb{P}$,

• • B _x is an age-specific parameter indicating the sensitivity of $\ln ( m^{( i) }_{x, t})$ to K _t,

• • $k_t^{( i) }$ is a time-varying index that is specific to the i-th population,

• • $b_x^{( i) }$ is an age-specific parameter indicating the sensitivity of $\ln ( m^{( i) }_{x, t})$ to $k_t^{( i) }$, and

• • $e^{( i) }_{x, t}$ is the error term that captures all remaining variations.

It is well-known that the LL model is subject to an identifiability problem. To stipulate parameter uniqueness, the following constraints are used:

$$\eqalign{\sum_x B_x & = 1,\; \quad \sum_t K_t = 0,\; \cr \sum_x b_x^{( i) }& = 1,\; \quad i\in\mathbb{P} \ \hbox{and} \sum_t k_t^{( i) } = 0,\; \ i\in\mathbb{P},\; }$$

where the summations are taken over the whole sample period or age range. Following Li and Lee (Reference Li and Lee2005), we estimate the parameters in Equation (1) using a singular value decomposition.

In applications of the LL model, {K _t} is typically assumed to follow a random walk with drift:

(2)$$K_t = \mu + K_{t-1} + \epsilon_t,\; $$

where μ is the drift term, and $\epsilon _t$ is the time-t random innovation that is normally distributed with a zero mean and a constant standard deviation σ.

On the other hand, $\{ k_t^{( i) }\}$ is often assumed to follow an ARMA(P,Q) process:

(3)$$\Phi^{( i) }_P( B) k^{( i) }_t = \Theta^{( i) }_Q( B) \epsilon^{( i) }_t,\; $$

where $\epsilon ^{( i) }_t$ is the time-t random innovation that is normally distributed with a zero mean and a constant standard deviation σ ⁽ⁱ⁾, $\Phi ^{( i) }_P( B) = 1 - \phi _1^{( i) }B - \phi _2^{( i) }B^2 - \ldots - \phi _P^{( i) }B^P$, and $\Theta ^{( i) }_Q( B) = 1 - \theta _1^{( i) }B - \theta _2^{( i) }B^2 - \ldots - \theta _Q^{( i) }B^Q$, with B being the backshift operator (i.e., $Bk^{( i) }_t = k^{( i) }_{t-1}$) and the roots of both $\Phi ^{( i) }_P( B)$ and $\Theta ^{( i) }_Q( B)$ lying outside the unit circle. Given how the ARMA model is specified, the long-term mean of $k^{( i) }_t$ is finite and equals zero.Footnote ¹ As a result, the long-term expected mortality differential (in log scale) between populations i and j is

(4)$$\ln m^{( i) }_{x, t} - \ln m^{( j) }_{x, t} = a_x^{( i) } - a_x^{( j) },\; $$

a constant that is free of t, so that the coherence property is achieved.

In the rest of this subsection, we investigate whether long memory exists in {K _t} and $\{ k_t^{( i) }\}$, and suggests suitable processes for modeling {K _t} and $\{ k_t^{( i) }\}$ accordingly.

2.2.1. Dynamics of K _t

Figure 1 shows the estimates of K _t, the time-varying index shared by the eight populations under consideration. Given the steady trend in the estimates, we apply first differencing and examine whether long memory exists in {K _t − K _t−1} using the modified R/S test [Lo (Reference Lo1991)], of which the null hypothesis is that the underlying process is a short memory one.

Figure 1. Estimates of K _t within the sample period of 1900–2018.

Reported in Table 2, the values of the R/S test statistic indicate that the null hypothesis cannot be rejected for lags 0–5, suggesting that long memory is not significant in {K _t − K _t−1}.Footnote ²

Table 2. Values of the modified R/S test statistic for {K _t − K _t−1} at lags 0, 1, …, 5

At 2.5%, 5%, and 10% significance levels, the critical values of the modified R/S test are 1.862, 1.747, and 1.620, respectively, according to Lo (Reference Lo1991).

Given the results of the modified R/S test, we use a random walk with drift to model {K _t} throughout the rest of this paper. While more sophisticated processes, such as those that feature conditional heteroskedasticity [Zhou and Li (Reference Zhou and Li2020)], may provide a better fit to {K _t}, we choose to use a simple random walk with drift so that our discussions can focus on the issue of short and long memory. Furthermore, as K _t is shared by all populations under consideration, the process for K _t has no direct relevance to population basis risk.

2.2.2. Dynamics of ${\rm k}^{\rm ( i) }_{\rm t}$

Figure 2 shows the estimates of $k^{( i) }_t$ for the eight populations under consideration. Compared to {K _t}, the trends in $\{ k^{( i) }_t\}$ are not as apparent. To better understand the dynamics of $\{ k^{( i) }_t\}$, we consider the sample autocorrelation functions (ACF) of $\{ k^{( i) }_t\}$. The lag-k sample ACF of $\{ k^{( i) }_t\}$ is defined as the sample correlation between $k^{( i) }_t$ and $k^{( i) }_{t-k}$ over the sample period.

Figure 2. Estimates $k^{( i) }_t$ for all $i\in \mathbb{P}$ within the sample period of 1900–2018.

The nature of a time-series can be told from its sample ACF plot (the plot of sample ACF against lag). In particular, the ACF of a long memory time series converges more slowly than that of a short memory one (which decays exponentially), but is not as extreme as that of an integrated (difference stationary) process which does not converge. To illustrate, in Figure 3 we display the sample ACF plots of time-series generated from three processes: (a) an AR(1) process with an autoregressive parameter of 0.7 and a volatility parameter of 1, (b) an ARFIMA(1,d,0) with d = 0.4 and the same autoregressive and volatility parameters as those of (a), and (c) an ARIMA(0,1,0) process with the same volatility parameter as the other two processes. It can be observed that the ACF of the ARFIMA-generated series clearly has a lower convergence rate than that of the AR-generated series, while the ACF of the ARIMA(0,1,0) process does not converge.

Figure 3. Sample ACF plots of time-series generated from an AR(1) process (left panel), an ARFIMA(1, 0.4, 0) (middle panel), and an ARIMA(0, 1, 0) process (right panel); the autoregressive parameter for the first two processes is 0.7, and the volatility parameter for all three processes is 1.

Figure 4 shows the sample ACF plots of the estimated series of $k^{( i) }_t$ for the eight populations under consideration. All of the eight sample ACF plots behave like that of a typical ARFIMA process that features long memory.

Figure 4. Sample ACF plots of the estimated series of $k^{( i) }_t$ for the eight populations under consideration.

We further use the modified R/S test to confirm the existence of long memory in the eight estimated series of $k^{( i) }_t$. As shown in Table 3, for all lags up to 5, the values of modified R/S test statistic are strictly greater than the critical value at 5% significance level, indicating that long memory in the eight estimated series of $k^{( i) }_t$ is statistically significant.

Table 3. Values of the modified R/S test statistic for $\{ k^{( i) }_t\}$ at lags 0, 1, …, 5

At 2.5%, 5%, and 10% significance levels, the critical values of the modified R/S test are 1.862, 1.747, and 1.620, respectively, according to Lo (Reference Lo1991).

2.3. The ARFIMA process

To capture the long memory in $k^{( i) }_t$, we consider an ARFIMA process for $k^{( i) }_t$:

(5)$$\Phi_P^{( i) }( B) ( 1-B) ^{2d} k^{( i) }_t = \Theta_Q^{( i) }( B) \epsilon^{( i) }_t,\; $$

where d is the fractional difference parameter, $\epsilon ^{( i) }_t$ is the time-t innovation which follows a normal distribution with a zero mean and a standard deviation of σ ⁽ⁱ⁾, and $\Phi _P^{( i) }( B)$ and $\Theta _Q^{( i) }( B)$ are the autoregressive and moving average operators as defined in the previous subsection, respectively. As the identifiability constraint $\sum _t k^{( i) }_t = 0$ for all $i\in \mathbb{P}$ is used, we consider an ARFIMA specification that has a zero unconditional mean.

The ARFIMA process captures the long-term serial dependence in $\{ k^{( i) }_t\}$ through the fractional difference parameter d, and the short-term serial dependence in $\{ k^{( i) }_t\}$ via the autoregressive and moving-average operators. The fractional difference parameter d can take any real value between −0.5 and 0.5 inclusive. If d = 0, then the ARFIMA process reduces to an ARMA process with short-term memory only. If d = 0.5, then Equation (5) becomes an ARIMA process with first-order differencing. For d ∈ (0,0.5), the ARFIMA process is said to have long memory or persistence, which implies that a high value of $k^{( i) }_t$ is likely to be followed by high values $k^{( i) }_s$ for s > t over a prolonged period of time. Finally, if d ∈ (−0.5,0), then the process is said to have anti-persistence, which means that the series will likely to switch between high and low values in adjacent time points for a prolonged period of time.

2.4. Estimation

We estimate AR(1), ARMA(P,Q), and ARFIMA(P,d,Q) processes to the eight estimated series of $k_t^{( i) }$, using R package “forecast.” We consider AR(1), as it is frequently used to model $\{ k_t^{( i) }\}$ in the LL model including the original work of Li and Lee (Reference Li and Lee2005). When fitting an ARMA(P,Q) process, the optimal values of P and Q are identified by the “auto.arima” function; and when fitting an ARFIMA(p,d,q) process, the optimal values of P, d, and Q are found using the “arfima” function.

In what follows, we report the estimation results for English and Welsh female population (EW). The estimation results for the other seven populations under consideration have similar properties, and are therefore not shown for the sake of space. Table 4 reports the estimates of the parameters in the processes for $\{ k^{( {\rm EW}) }_t\}$. For both ARMA and ARFIMA processes, the optimal values of P and Q are 1. The estimate of d in the ARFIMA(1,d,1) process is 0.3787, indicating that there is strong long-term serial dependence in $\{ k^{( {\rm EW}) }_t\}$. All of the parameters are significant, as indicated by their p-values.

Table 4. Estimates of the parameters in the AR, ARMA, and ARIMA processes for $\{ k^{( {\rm EW}) }_t\}$

Figure 5 displays the sample ACF plots of the residuals from the three fitted processes for $\{ k^{( {\rm EW}) }_t\}$. For the AR process, the sample ACF plot of the residuals has significant spikes at lag 1 and 10, indicating that some short-term serial dependence is not adequately captured by the process. Both the ARMA and ARFIMA processes seem to have provided adequate provision for short-term serial dependence, but the former takes no account of long memory which is found to be statistically significant in section 2.2.2.

Figure 5. Sample ACF plots of the residuals from the fitted processes for $\{ k^{( {\rm EW}) }_t\}$: AR(1) (left panel), ARMA(1,1) (middle panel), and ARFIMA(1,d,1) process (right panel).

2.5. Forecasting

We now turn to forecasting. Figure 6 shows the mean forecasts and predictive intervals of $k^{( {\rm EW}) }_t$ produced by the three estimated processes. The trajectories of the mean forecasts underscore an important property of ARFIMA processes: the rate of convergence implied by an ARFIMA process is smaller than the corresponding AR and ARMA processes. Another important property of ARFIMA processes can be inferred from the width of the predictive intervals: an ARFIMA process implies a higher level of forecast uncertainty compared to the corresponding AR and ARMA processes.

Figure 6. Mean forecasts and predictive intervals of $k^{( {\rm EW}) }_t$ generated from the fitted AR process (left panel), ARMA process (middle panel), and ARFIMA process (right panel). Each fan chart shows the 10% predictive interval with the heaviest shading, surrounded by the 20%, 30%, …, 90% predictive intervals with progressively lighter shadings.

To obtain deeper insights about the rate of convergence, we consider additionally the female population of Italy (IT). For the reader's information, the estimated AR, ARMA, and ARFIMA processes for $\{ k^{( {\rm IT}) }_t\}$ are summarized in Table 5.

Table 5. Estimates of the parameters in the AR, ARMA, and ARIMA processes for $\{ k^{( {\rm IT}) }_t\}$

Figure 7 shows the mean forecasts of $\ln m_{x, t}^{( {\rm EW}) }$ and $\ln m_{x, t}^{( {\rm IT}) }$ for x = 40, 50, 60 when AR, ARMA, and ARFIMA processes are used to model $\{ k^{( {\rm EW}) }_t\}$ and $\{ k^{( {\rm IT}) }_t\}$. It can be observed that all three types of processes (including ARFIMA) produce coherent mortality forecasts; that is, for a given age x, the mean forecasts of $\ln m_{x, t}^{( EW) }$ and $\ln m_{x, t}^{( IT) }$ do not diverge over the long run.

Figure 7. Mean forecasts of $\ln m_{x, t}^{( EW) }$ and $\ln m_{x, t}^{( IT) }$ at x = 40 (top row), x = 50 (middle row), and x = 60 (bottom row) when $\{ k^{( {\rm EW}) }_t\}$ and $\{ k^{( {\rm IT}) }_t\}$ are modeled by AR processes (left column), ARMA processes (middle column), and ARFIMA processes (right column).

Noting that all of the three processes specified for $k^{( i) }_t$ have a zero unconditional mean, it can be easily deduced that in the long-run equilibrium the difference between the log central death rates for EW and IT populations at any given age x is always $a_x^{( {\rm EW}) } - a_x^{( {\rm IT}) }$ regardless of which of the three processes are used. Figure 7 therefore reveals that three types of processes lead to very different rates of convergence to the long-term equilibrium. In particular, the rate of convergence to the long-term equilibrium is the fastest (slowest) when AR (ARFIMA) processes are used to model $\{ k^{( {\rm EW}) }_t\}$ and $\{ k^{( {\rm IT}) }_t\}$.

3.1. Set-up

Let $S^{( i) }_{x, t}( T)$ be the ex post probability that an individual from population i who has survived to age x at time t would have survived to time t + T for T = 1, 2, …. Under the LL model, $S^{( i) }_{x, t}( T)$ is given by

(6)$$S^{( i) }_{x, t}( T) = \exp\left(-\sum_{s = 1}^{T}\exp \left(a^{( i) }_{x + s-1} + B_{x + s-1}K_{t + s} + b^{( i) }_{x + s-1}k^{( i) }_{t + s} \right)\right).$$

As a shorthand, we write

(7)$$S^{( i) }_{x, t}( T) = e^{- W^{( i) }_{x, t}( T) },\; $$

where

(8)$$W^{( i) }_{x, t}( T) = \sum_{s = 1}^{T}\exp( Y^{( i) }_{x, t}( s) ) $$

and

(9)$$Y^{( i) }_{x, t}( s) = a^{( i) }_{x + s-1} + B_{x + s-1}K_{t + s} + b^{( i) }_{x + s-1}k^{( i) }_{t + s}.$$

Let ${\cal F}_t$ be the information about the evolution of mortality of all populations under consideration up to and including time t. Then, the survival probability for an individual from population i aged x at time t to survive to time t + T given ${\cal F}_t$ can be expressed as

$$\hbox{E} \left[e^{- W^{( i) }_{x, t}( T) } \big\vert {\cal F}_t \right].$$

Under the LL model, dependence across populations is driven exclusively by {K _t}. To stress the role of {K _t}, we define the following:

$$p^{( i) }_{x, t}( T,\; K_{t}) \, \colon = \hbox{E} \left[e^{- W^{( i) }_{x, t}( T) } \big\vert {\cal F}_t \right].$$

Note that the expression above depends on K _t but not K _t−1, K _t−1, … due to the Markovian property of the assumed process for {K _t}.

To construct a delta-neutral longevity hedge, we need the longevity delta for $p^{( i) }_{x, t}( T,\; K_{t})$, which is defined as the first-order partial derivatives of $p^{( i) }_{x, t}( T,\; K_{t})$ with respect to K _t:

$$\eqalign{\Delta^{( i) }_{x, t}( T) & \, \colon = {\partial \over \partial K_t} p^{( i) }_{x, t}( T,\; K_{t}) \cr & = -\sum_{s = 1}^T B_{x + s-1} \hbox{E} \left[\exp\left(Y^{( i) }_{x, t}( s) -W^{( i) }_{x, t}( T) \right)\big\vert {\cal F}_t \right].}$$

We assume that the hedger is an annuity provider and the hedging instrument is an S-forward. In the next two subsections, we derive the time-t values and longevity deltas of a life annuity and S-forward.

3.1.1. Life annuities

Let us consider a τ-year deferred T-year temporary life annuity issued to an individual from population i who is aged x at time t. Assuming a constant interest rate of r, the sum of all discounted cash flows from this life annuity is given by

$${\cal L}^{( i) } = \sum_{s = 1}^{T}( 1 + r) ^{-( \tau + s) } S^{( i) }_{x, t}( \tau + s) .$$

As a linear combination of $S^{( i) }_{x, t}( \tau + s)$ for s = 1, …, T, the time-t value of ${\cal L}^{( i) }$ given ${\cal F}_t$ can be written as

$$L^{( i) }( K_t) \, \colon = \hbox{E}\left[{\cal L}^{( i) } \big\vert {\cal F}_t\right] = \sum_{s = 1}^{T}( 1 + r) ^{-( \tau + s) }p^{( i) }_{x, t}( \tau + s,\; K_t) .$$

The longevity delta of this life annuity is defined as the first-order partial derivative of L ⁽ⁱ⁾(K _t) with respect to K _t; that is,

$$\Delta^{( i) }_L\, \colon = {\partial L^{( i) }( K_t) \over \partial K_t} = \sum_{s = 1}^{T} ( 1 + r) ^{-( \tau + s) } \Delta^{( i) }_{x, t}( \tau + s) .$$

3.1.2. S-forwards

A S-forward is a zero-coupon swap with a fixed leg proportional to a fixed forward rate S ^f that is determined when the contract is written at time t, and a floating leg proportional to a random survival rate. For a S-forward with a reference population i, a time-to-maturity T and a reference age x, the floating leg is proportional to $S^{( i) }_{x, t}( T)$, as defined in Equation (6).

The value of $S^{( i) }_{x, t}( T)$ is completely realized at time t + T when the S-forward matures. If the realized value of $S^{( i) }_{x, t}( T)$ turns out to be higher than expected (i.e., realized mortality is lighter than expected), then the fixed-rate payer of the S-forward receives a positive net payment from the floating-rate payer. To hedge their longevity exposures, pension and annuity providers may participate in a S-forward as a fixed-rate payer, so that in case mortality turns out to lighter than expected, the net payment received from the floating-rate payer can be used to offset their correspondingly larger liabilities.

Assuming a constant interest rate of r, the discounted payoff of the above-described S-forward from the fixed-rate payer's perspective is given by

$${\cal H}^{( i) } = ( 1 + r) ^{-T}( S^{( i) }_{x, t}( T) -S^f)$$

per $1 notional. Given ${\cal F}_t$, the time-t value of this S-forward can be expressed as

$$H^{( i) }( K_t) \, \colon = \hbox{E}\left[{\cal H}^{( i) } \big\vert {\cal F}_t \right] = ( 1 + r) ^{-T} \left(\,p^{( i) }_{x, t}( T,\; K_t) - S^f \right).$$

The longevity delta is defined as the first-order partial derivative of H ⁽ⁱ⁾(K _t) with respect to K _t; that is,

$$\Delta^{( i) }_H\, \colon = {\partial H^{( i) }( K_t) \over \partial K_t} = ( 1 + r) ^{-T} \Delta^{( i) }_{x, t}( T) .$$

3.2. Calibration and evaluation

Suppose that the liability being hedged is the life annuity described in section 3.1.1 and the hedging instrument is the S-forward described in section 3.1.2. Then, the unhedged position is simply ${\cal L}^{( i) }$, while the hedged position can be expressed as ${\cal L}^{( i) } - u {\cal H}^{( j) }$, where u is the hedge ratio (the notional amount of the S-forward purchased). When i ≠ j, the life annuity and S-forward are associated with different populations, and thus population basis risk exists. We consider two methods to choose u.

For a delta-neutral hedge established at time t, the value of u is determined such that the longevity delta of the hedged position is zero; that is,

$$\Delta^{( i) }_L - u \Delta^{( j) }_H = 0,\;$$

the solution to which is given by

(10)$$u^{( i, j) }_\Delta = {\Delta^{( i) }_L\over \Delta^{( j) }_H}.$$

For a variance-minimizing hedge established at time t, the value of u is chosen such that the following is minimized:

$$\hbox{Var} \left({\cal L}^{( i) } - u {\cal H}^{( j) }\vert {\cal F}_t \right).$$

It can be shown that the solution to the above minimization problem is

(11)$$u^{( i, j) }_V = { \hbox{Cov} \left({\cal L}^{( i) } ,\; {\cal H}^{( j) } \vert {\cal F}_t \right)\over \hbox{Var} \left({\cal H}^{( j) } \vert {\cal F}_t \right)}.$$

Finally, to quantify hedge effectiveness, we consider the reduction in variance between the hedged and unhedged positions:

(12)$$\hbox{HE} = 1 - {\hbox{Var} \left({\cal L}^{( i) } - u {\cal H}^{( j) } \big\vert {\cal F}_{t_0} \right)\over \hbox{Var} \left({\cal L}^{( i) } \big\vert {\cal F}_{t_0} \right)},\; $$

where u represents the hedge ratio, which is calculated using Equation (10) when a delta-neutral hedge is used and Equation (11) when a variance-minimizing hedge is used.

4.1. Case study I

The following assumptions are used for case study I:

• • The current time is t ₀ = 2018 (i.e., the end of the last year of the sample period).

• • The liability being hedged is a 25-year deferred 25-year temporary life annuity issued to an individual from population i who is aged 40 at time t ₀ = 2018. The annuity pays $1 at the end of each year, starting from age 65. Payment ceases when the annuitant dies or reaches age 90, whichever is the earliest. The annuitant's mortality experience is identical to that of IT.

• • The hedger's annuity portfolio is large enough so that diversifiable risk can be ignored.

• • The hedger establishes a delta-neutral longevity hedge with a freshly launched S-forward at time t ₀ = 2018. No adjustment is made to the hedge after time t ₀.

• • When calibrating the hedge, the hedger assumes that {K _t} follows a random walk with drift, and assumes $\{ k_t^{( {\rm EW}) }\}$ and $\{ k_t^{( {\rm IT}) }\}$ follow either ARFIMA or ARMA processes. Using ARMA processes means that the hedger ignores or is not aware of the empirical fact that there exists long memory in mortality differentials.

• • At time t ₀ = 2018, freshly launched S-forwards with a reference age 40 and times-to-maturity up to 50 years are available. The reference population of these S-forwards is EW. Therefore, the hedge is subject to population basis risk.

• • The hedge effectiveness is gauged using HE as defined in Equation (12), with i = IT, and $u = u^{( {\rm IT}, {\rm EW}) }_\Delta$. The evaluation model (from which realizations of ${\cal L}^{( {\rm IT}) } - u^{( {\rm IT}, {\rm EW}) }_\Delta {\cal H}^{( EW) }$ and ${\cal L}^{( {\rm IT}) }$ given ${\cal F}_{t_0}$ are simulated) is assumed to be the same as the one used for calibrating the hedge. In other words, if the hedger overlooks long memory when calibrating the hedge, he/she also ignores long memory when evaluating the hedge.

• • When discounting future cash flows, a constant interest rate of $r = 2\%$ per annum is used for all durations.

Figure 8 reports the resulting values of $\hbox {HE}$ for S-forward times-to-maturity ranging from 25 to 45 years. Regardless of whether long memory is taken into account, the value of $\hbox {HE}$ is the highest when the time-to-maturity of the S-forward is around 40 years.

Figure 8. The values of $\hbox {HE}$ for a delta-neutral hedge constructed using a S-forward with a time-to-maturity ranging from 25 to 45 years, when $\{ k_t^{( {\rm EW}) }\}$ and $\{ k_t^{( {\rm IT}) }\}$ are modeled by ARMA (solid line) and ARFIMA (dot-dashed line) processes.

For any S-forward time-to-maturity, the value of $\hbox {HE}$ when long memory is ignored (ARMA processes are used) is higher than that when long memory is taken into account (ARFIMA processes are used). When the time-to-maturity exceeds 40 years, the difference is more than 10 percentage points. The results presented in Figure 8 clearly point to the conclusion that ignoring long memory in mortality differentials would lead to an over-estimation of hedge effectiveness.

To understand the reason behind the phenomenon observed in Figure 8, let us analyze the constituents of the longevity risk faced by the hedger. Under our modeling framework, the hedger faces hedgeable risk that arises from the uncertainty surrounding K _t for t > t ₀ and unhedgeable risk (population basis risk) that arises from the uncertainty surrounding both $k^{( {\rm IT}) }_t$ and $k^{( {\rm EW}) }_t$ for t > t ₀. Figure 9 shows the variances of $k^{( {\rm IT}) }_{t_0 + s}$ and $k^{( {\rm EW}) }_{t_0 + s}$ given ${\cal F}_{t_0}$ for s = 1, …, 50, implied by the ARMA and ARFIMA processes. In line with the results presented in section 2, as the forecast horizon s increases, the ARFIMA processes yield larger variances of $k^{( {\rm IT}) }_{t_0 + s}$ and $k^{( {\rm EW}) }_{t_0 + s}$ given ${\cal F}_{t_0}$ compared to the ARMA processes. More specifically, the variances of $k^{( {\rm IT}) }_{t_0 + s}$ and $k^{( {\rm EW}) }_{t_0 + s}$ given ${\cal F}_{t_0}$ implied by the ARMA processes (which incorporate short-term memory only) converge to their respective constant levels fairly quickly, while those implied by the ARFIMA processes (which incorporate both short- and long-term memory) grow at slowly decreasing rates. In effect, when $\{ k^{( {\rm IT}) }_t\}$ and $\{ k_t^{( {\rm EW}) }\}$ are modeled by ARFIMA processes (which incorporate the empirical fact that long memory exists in $\{ k^{( {\rm IT}) }_t\}$ and $\{ k_t^{( {\rm EW}) }\}$) instead of ARMA processes, the proportion of hedgeable risk relative to total risk becomes smaller and so does hedge effectiveness.

Figure 9. The variance of $k^{( {\rm EW}) }_{t_0 + s} (\hbox{left panel}) \hbox{and} k^{({\rm IT}) }_{t_0 + s} \; (\hbox{right panel})$ given ${\cal F}_{t_0}$ for s = 1, …, 50 when $\{ k^{( {\rm EW}) }_t\} \hbox { and } \; \{ k^{( {\rm IT}) }_t\}$ are respectively modeled by an ARMA process (solid line) and an ARFIMA process (dot-dashed line).

4.2. Case study II

Case study II is based on the same assumptions as those made in case study I, except the following:

• • The annuitant's mortality experience is identical to that of NL. The hedge is still subject to population basis risk, as the reference population of the S-forward is EW.

• • When calibrating the hedge, the hedger assumes that {K _t} follows a random walk with drift, and $\{ k_t^{( {\rm EW}) }\}$ and $\{ k_t^{( {\rm NL}) }\}$ may follow either ARFIMA or ARMA processes depending on whether long memory is ignored. The hedge may use a delta-neutral hedge or variance-minimizing hedge. These options produce four hedging scenarios in total.

• • We use HE defined in Equation (12) to evaluate the performance of each hedge. For all four hedging scenarios, the evaluation model is based on ARFIMA processes for $\{ k_t^{( {\rm EW}) }\}$ and $\{ k_t^{( {\rm NL}) }\}$. In other words, we are assuming that long memory exists in reality, but the hedger may or may not incorporate this property when calibrating the hedge.

Our goal is to investigate how delta-neutral and variance-minimizing hedges may underperform if long memory exists in reality but is not accounted for in the hedges.

Figure 10 shows the resulting values of HE for S-forward times-to-maturity ranging from 25 to 45 years. For the variance-minimizing hedge, hedge ratios calculated using the ARMA model assumption lead to materially smaller HE values compared to those computed using the ARFIMA model assumption. However, interestingly, for the delta-neutral hedge, the values of HE produced by hedge ratios computed using the ARMA and ARFIMA model assumptions are highly similar. These results suggest that in reality when long memory in mortality differentials exists, calculating hedge ratios using short memory processes like ARMA would lead to a significant underperformance if the calibration method is variance minimization, but not if the calibration method is delta neutralization.

Figure 10. Values of HE produced by delta-neutral hedges (left panel) and variance-minimizing hedges (right panel), for S-forward times-to-maturity ranging from 25 to 45 years, when $\{ k_t^{( {\rm EW}) }\}$ and $\{ k_t^{( {\rm NL}) }\}$ are modeled by ARMA processes (solid lines) and ARFIMA processes (dot-dashed lines).

To obtain further insights, let us examine the hedge ratios calculated in each of the four hedging scenarios (Figure 11). For the variance-minimizing hedge, hedge ratios computed using the ARMA model assumption are consistently smaller than those calculated using the ARFIMA model assumption; as a smaller than optimal notional amount of S-forward is used, the ARMA model assumption leads to underperformance. For the delta-neutral hedge, the hedge ratios are almost unaffected by the model assumption for $\{ k_t^{( {\rm EW}) }\}$ and $\{ k_t^{( {\rm NL}) }\}$; hence, the hedge does not underperform even when the ARMA model assumption is used.

Figure 11. Hedge ratios for delta-neutral hedges (left panel) and variance-minimizing hedges (right panel), for S-forward times-to-maturity ranging from 25 to 45 years, when $\{ k_t^{( {\rm EW}) }\}$ and $\{ k_t^{( {\rm NL}) }\}$ are modeled by ARMA processes (solid lines) and ARFIMA processes (dot-dashed lines).

We now explain why the hedge ratio for the variance-minimizing hedge is sensitive to the model assumption for $\{ k_t^{( {\rm EW}) }\}$ and $\{ k_t^{( {\rm NL}) }\}$, but that for the delta-neutral hedge is not. For the variance-minimizing hedge, the hedge ratio is calculated using Equation (11), of which both the numerator and denominator are highly sensitive to the model assumption because, as argued earlier, it affects the mix of hedgeable risk (the uncertainty surrounding K _t for t > t ₀) and unhedgeable risk (the uncertainty surrounding $k_t^{( {\rm EW}) }$ and $k_t^{( {\rm NL}) }$ for t > t ₀). For the delta-neutral hedge, the hedge ratio is obtained from Equation (10), which can be expanded to the following:

$$u^{( {\rm NL}, {\rm EW}) }_\Delta = {\Delta^{( {\rm NL}) }_L\over \Delta^{( {\rm EW}) }_H} = {{\partial \over \partial K_{t_0}} \hbox{E} \left[{\cal L}^{( {\rm NL}) } \big\vert {\cal F}_{t_0} \right]\over {\partial \over \partial K_{t_0}} \hbox{E} \left[{\cal H}^{( {\rm EW}) } \big\vert {\cal F}_{t_0} \right]}.$$

Although the equation above involves $k_t^{( {\rm EW}) }$ and $k_t^{( {\rm NL}) }$ for t > t ₀ through the two expectations, it depends a lot more heavily on the rate of change of the two expectations relative to changes in $K_{t_0}$. As a result, $u^{( {\rm NL}, {\rm EW}) }_\Delta$ is quite insensitive to the model assumption for $\{ k_t^{( {\rm EW}) }\}$ and $\{ k_t^{( {\rm NL}) }\}$.