Local hedging of variable annuities in the presence of basis risk, tutoriel & guide de travaux pratiques en pdf.

Variable annuities hedging mechanics

This section outlines the mathematical model representing cash flows involved during the hedging of variable annuity guarantees by insurers.

Cash flows to the insurer

Consider a discrete set of monthly time steps T = {0, . . . , T } and a probability space (⌦, FT , P) equipped with a filtration F = {Ft}t2T . The insurance company issues variable annuities to policyholders at time t = 0 and hedges risk pertaining to these contracts. Without loss of generality, a Guaranteed Minimal Maturity Benefit (GMMB) policy is considered.
Cash flows
The policy account is invested in a mutual fund whose value is a F-adapted process denoted by F = {Ft}t2T . Fees are periodically charged to policyholders and withdrawn from the policy account. Fees apply to all policyholders alive at the beginning of the period, but are only charged at the end of the period. Although fees are sometimes charged at the beginning of the period in practice, the impact of this fees timing assumption is very limited. Indeed, the discount factor applying to a single monthly time period is very close to 1. The total fee rate charged to policyholders is denoted by !tot. For a policyholder active at time t 1, the fees charged at time t are given by !totAt−1 FFtt 1 , where At−1 is the post-fee policy account value at time t 1. The dynamics of A = {At}t2T is thus given by
At+1 = At(1 !tot) Ft+1 , t 2 {0, . . . , T 1}. (2.2.1)
Ft  ) At A0 (1 !tot)t, t 2 T . (2.2.2) = Ft  F0
A policyholder who dies during the period (t, t + 1] receives the amount At+1 at time t + 1. Moreover, at any time before maturity, the policyholder has the possibility to withdraw its investment from the variable annuity, which is called a lapse. In practice, insurers can diversify away a large proportion of idiosyncratic mortality and lapse risks by insuring a large number of policyholders. In the current framework, it is assumed that these risks can be fully diversified in this manner and are thus deterministic. The constant lapse rate on each period is denoted by b. Furthermore, tpx is defined as the probability that a policyholder aged x months at time 0 survives t months. The proportion of policies that are still active at time t is thus given by `t = (1 b)ttpx, t 2 T . (2.2.3)
Surrender charges are not considered for simplicity, and therefore a policyholder who lapses during the period (t, t + 1] receives the amount At+1 at time t + 1. Such a simplification is acceptable in our framework due to the constant lapse rate assumption. However, extensions embedding dynamic lapses would definitely require the inclusion of surrender charges since such charges have a significant impact on policyholder surrender incentives as shown for instance by MacKay et al. (2017).
For a GMMB contract, in the case of a policyholder lapse or death, the benefit provided to the policyholder is fully funded by its policy account ; the insurer does not incur an outflow in this case. However, due to the guarantee, the insurer needs to pay the difference between the benefit and the policy account value at maturity if the policyholder is alive and the policy remains in-force until maturity. The benefit in excess of the account value paid to a GMMB policyholder whose policy is still active at the maturity T is given by max(0, K AT ), where the guaranteed amount K is considered to be deterministic for simplicity.
Only a portion of fees collected from policyholders is allocated to the hedging portfolio for the guarantee, as the rest is allocated to profits and expenses. The fee rate which relates to fees allocated to hedging is denoted by !opt. Hence, if the policyholder is active at time t 1, the amount ! A Ft = !opt A is received at time t and used by the insurance company for opt t−1 Ft 1 1−!tot t  hedging. Therefore, the net cash outflow for the insurer at time t is given by  CFt = !opt At`t−1 + 1{t=T } max(0, K AT )`T , t 2 {1, . . . , T }, (2.2.4)
1 !tot
with CF0 = 0 as no immediate cash flows are involved at time t = 0. Defining  ˜ KF0 A0 t (2.2.5)
K ⌘ , t ⌘ (1 !tot) `t, t 2 T ,
A0(1 !tot)T F0
PTT G0 j=1 j 1
the relationship (2.2.2) can be used to express the cash flows directly in terms of the fund’s value :
˜ , t 2 {1, . . . , T }. (2.2.6)
CFt = !opt t−1Ft + 1{t=T } T max 0, K FT
Pricing
Since variable annuity option liabilities are not openly traded in markets, their fair value must be modeled. A possibility would be to use a value that is endogenous to the hedging strategy, such as the value that would allow optimizing the hedge according to some predefined criterion. Such price could be obtained through the quadratic global hedging approach ; see Rémillard and Rubenthaler (2013) for a general framework and Rémillard et al. (2017) for the Gaussian regime-switching specialization. However, prices are not endogenized to the hedging strategy in practice since insurance companies already have a pricing model which they use to determine the appropriate level of management expenses (MER) charged to clients.
The fair value of liabilities is obtained through risk-neutral valuation using an equivalent martingale measure Q. We consider a risk-free asset whose price at time t is denoted Bt = ert where r is the periodic risk-free rate. Since the discount price process {Ft/Bt}t2T is a martingale under Q, it is straightforward to show that the time-t GMMB contract value is T 5 T
⇧t = Bt EQ 4 X CFj jX j−1 + 1{t<T } T Gt,t 2 T , (2.2.7)
Ft = !optFt
j=t+1 Bj
=t+1
where Gt ⌘ Bt EQ » max ˜ F #.
K T
0, Ft (2.2.8)
BT
Under this convention, the price ⇧t excludes the cash flow CFt from the pricing. In particular, ⇧T = 0. The fee rate !opt is assumed to be a fair fee rate, i.e., the amount of fees which leads to a null initial value for the guarantee. Setting ⇧0 = 0 in (2.2.7) leads to !opt = F0 .
Note that if the profitability provided by the fair fee rate !opt is deemed inadequate, the insurer might decide to adjust the total fee rate !tot provided that it remains competitive. Thus !tot and !opt would be related through a complex non-linear optimization in this case. However, we do not investigate such mechanisms in the current work since the pricing parameter !tot is assumed to be given.
Hedging
To mitigate risks embedded in guarantees provided by variables annuities, insurers perform a cross hedge based on a different hedging asset S, which creates basis risk since the assets F and S are not perfectly correlated. The insurer sets up a hedging portfolio taking positions in two assets : the risk-free asset B = {Bt }t2T and a risky equity futures contract S = {St}t2T , where St denotes the futures price at time t. The use of futures as hedging instruments is consistent with insurers practices, see for instance Chopra et al. (2009), iA Financial Group (2016) and Manulife Financial Corporation (2016). It is justified by the impossibility of taking short positions on many index funds and by the high liquidity of index futures. The number of long positions within the hedging portfolio during the time interval (t, t + 1] are respectively denoted by ✓t(+1B) and ✓t(+1S), with the convention ✓0(B) = ✓0(S) = 0. The insurer performs periodic injections or withdrawals of liquidities from the hedging portfolio at each time step. The injection at time t is denoted by It (negative amounts correspond to withdrawals). Define Vt✓− and Vt✓+ as the value of the hedging portfolio respectively before and after the injection It and the cash flow CFt at time t. This leads to It = Vt✓+ Vt✓− + CFt, t 2 T . (2.2.9)
To ensure the hedging portfolio value tracks the guarantee value, at all steps t 2 T the cash flow injection (or withdrawal) It is performed such that the post-injection portfolio value is equal to the guarantee value : Vt✓+ = ⇧t,t 2 T . (2.2.10)
In particular, the time 0 injection is nil, I0 = 0, because V0✓− ⌘ 0, CF0 = 0 and ⇧0 = 0.
After the post-injection portfolio value Vt✓+ = ⇧t is observed, the hedging portfolio is rebalanced. A new futures position ✓t(+1S) is decided based on the selected hedging strategy, and then the whole portfolio value is invested at the risk-free rate ; entering positions on futures does not involve immediate cash flows besides margin requirements, and liquidities deposited inside the margin are assumed to accrue at the risk-free rate : ✓t(+1B) = Vt✓+ = ⇧t . (2.2.11) Bt Bt
The new pre-injection portfolio value at time t + 1 is then obtained by summing the amount accrued at risk-free rate and profits/losses from futures positions : V ✓ = ✓(B)Bt+1 + ✓(S) (St+1 St). (2.2.12) (t+1)− t+1 t+1
The following proposition proven in Appendix 2.A.1 gives an explicit expression for injections.
Proposition 2.2.1. For t 2 {0, . . . , T 1}, define F t ⌘ Ft+1
G t = Gt+1 Gt. Cash flow injections are given by
I = ⇧ (1 er) F ! jX j−1 + G
t+1 t opt T t =t+1
Ft, S t ⌘ St+1 St, and
✓t(+1S)S t. (2.2.13)
Remark 2.2.1. The no-hedging (or unhedged) injection Itnh+1 is defined as the injection value that would occur with ✓t(+1S) = 0. It follows from Proposition 2.2.1 that It+1 = Itnh+1 ✓t(+1S)S t.

Capital requirements

Insurers must hold reserves and capital to meet future variable annuity guarantee liabilities. In Canada, insurers issuing segregated fund policies (which are the Canadian equivalent of variable annuities) are required by the Office of the Superintendent of Financial Institutions (OSFI) to hold a Total Gross Capital Required (TGCR) at time t = 0 represented by 2  » T #.
TGCR = CVaR0P.95 Xt
e−rtIt (2.2.14) =1
The CVaR risk measure is defined rigorously in Rockafellar and Uryasev (2002). For a continuous random variable X, the CVaR↵[X] can be interpreted as the average of the worse 100(1 ↵)% scenarios. Note that U.S. recommendations for determining capital requirements for variable annuities are also based on the CVaR risk measure. 3

Selection of the hedging strategy

The current paper’s approach for the selection of the hedging strategy ✓(S) = ✓t(S) t2T is to use
a local criterion based on risk measures to optimize the risk. An Ft risk measure is a mapping Rt : XFT ! XFt , where XG is the set of G-measurable random variables for some sigma-algebra G. The number of futures positions in the hedging portfolio is chosen by minimizing the risk related to the next cash injection :
✓t(+1S)⇤ = arg min Rt(It+1) , t 2 {0, . . . , T 1}, (2.2.15)
✓t(+1S)
for a given dynamic risk measure {Rt}Tt=0−1, where Rt is an Ft risk measure for each t. Local procedures in hedging were pioneered by Ederington (1979) who uses Rt(•) = VarP[•|Ft]. However, an important drawback associated with the variance is that it focuses purely on risk in general, penalizing upside risk and failing to incorporate expected costs in the tradeoff. Rockafellar and Uryasev (2000) use the CVaR measure to optimize their hedge which allows reducing the magnitude and frequency of extreme losses. A plethora of other classes of risk measures that were developed in the literature could also be considered, for instance coherent risk measures (Artzner et al., 1999), deviation measures (Rockafellar et al., 2002) and distortion measures (Wang, 2000).
2. See Section 11 of the instruction guide by the OSFI on the “Use of Internal Models for Determining Required Capital for Segregated Fund Risks » which instructs using CVaRP0.95 to determine the TGCR for segregated funds. See also Chapter 6 of the “Capital Adequacy Guideline » by the Autorité des Marchés Financiers for the Province of Québec.
3. See page 11 of the report “The Application of C-3 Phase II and Actuarial Guideline XLIII » by the American Academy of Actuaries which recommends using the CVaRP0.90 to determine the TGCR for variable annuities.

A model involving regime-switching equity risk

This section presents the regime-switching price dynamics model for risky assets S and F .

Market model

Over a long period of time, markets go through various periods of either prosperity or turbulence. To model such dynamics, regime-switching processes have become very popular in the actuarial literature, see for instance Hardy (2001) and Hardy (2003). In these models, the overall state of the market is represented by the state of a Markov chain. Asset return distributions are then presumed to be a function of the current market state.
A regime process h = {ht }t2T characterizes the state of the market, where ht 2 {1, 2} can only take two possible values. Regimes ht are latent variables, i.e., they are not directly observable. The following regime-switching dynamics is assumed for the risky asset prices :
(F ) Ft+1 (F ) (F ) (F ) (S) St+1 (S) (S) (S)
Rt+1 ⌘ log✓ ◆ = µht + ht zt+1, Rt+1 ⌘ log✓ ◆ = µht + ht zt+1,
Ft St
zt+1 zt(+1F) N2 0 1 ⇢ht , (2.3.1)
⌘ « zt(+1S)# ⇠  » , 1 #!
0#  » ⇢ht
where N2(µ, ⌃) is the bivariate normal distribution with mean vector µ and covariance matrix ⌃, z = {zt}t2T is a strong standardized Gaussian bivariate white noise, and the remaining parameters are constants to be estimated. Under this model, asset prices evolve according to Ft+1 = Fteµh(Ft )+ h(Ft )zt(+1F) , St+1 = Steµh(St )+ h(St)zt(+1S) . (2.3.2)
The market information at time t is Ft ⌘ Su, Fu : u = 0, . . . , t . Following the lines of François et al. (2014), a full information filtration G⌘{G t}t2T , where Gt ⌘ Ft _ hu : u = 0, . . . , t is introduced. The regime process h is assumed to have the Markov property with respect to G, i.e., for some transition matrix
P =  » P1,1 P1,2 #, (2.3.3)
P2,1 P2,2
with P1,2 = 1 P1,1 and P2,2 = 1 P2,1, the following relationship holds for i, j 2 {1, 2} :
P ht+1 = j ht = i, {hu}ut−=01, {(Fu, Su)}ut=0 = P ht+1 = j ht = i = Pi,j. (2.3.4)
As in François et al. (2014), the regime mass functions given partial information Ft are defined as follows :
⌘i,tP ⌘ P(ht = i|Ft), i 2 {1, 2}, (2.3.5)
and can be computed recursively through
2 (F ) (S) ⌘j,tPPj,i
j=1 µj ,⌃j Rt+1, Rt+1
⌘i,tP +1 = P 2 ⇣ (F ) (S⌘) ⌘ , i 2 { 1, 2 } , (2.3.6)
P j=1 µj ,⌃j⇣ Rt+1, Rt+1 ⌘j,tP
where µj ,⌃j denotes the bivariate Gaussian probability density function with mean vector and covariance matrix given by
µj(F ) (F ) 2 (F) (S)
µj ⌘  » #, ⌃j ⌘ 2 ⇣ j ⌘(S) ⇢j j j 3 (2.3.7)
µ(S) (F ) ⇣ (S) ⌘ 2 .
j 4 ⇢j j j j 5
The following result deriving from (2.3.2) is useful to develop offline schemes for computing the local hedging strategies.
Proposition 2.3.1. The Ft-conditional distribution of Ft , St depends only on ⌘1P,t under Ft+1 St+1
the physical measure P.
Sojourn time conditional probabilities are quite useful to obtain analytical option pricing formulas (see, e.g., Hardy, 2001). For t 2 {0, . . . , T 1}, ⌧ 2 {0, . . . , T t}, and i 2 {1, 2}, we define the following :
T −1
X (2.3.8)
Ht,⌧,i ⌘ P Yt = ⌧ ht = i ,Yt ⌘ 1{hu=1},
u=t
where Yt is the sojourn time in regime 1 between time t and time T .
A recursive algorithm is available for computing them. Initialize with HT −1,1,1 = HT −1,0,2 = 1 and HT −1,j,1 = HT −1,k,2 = 0 for all j 6= 1 and all k 6= 0. Starting at t = T 2, the following backward induction formulas from Hardy (2001) are used :
X Xi
Ht,k,1 = P1,i Ht+1,k−1,i, Ht,k,2 = P2,i Ht+1,k,i.
i=1 =1

Valuing the guarantee

The above market model is incomplete, and therefore there exists an infinite number of risk-neutral measures. For analytical tractability, the chosen Q is such that the dynamics remain a regime-switching model of the same form and with the same transition matrix, but with the drift parameters h (F ) (S) i replaced by r 1 ⇣ (F ) ⌘ 2 1 ⇣ (S) ⌘ 2 for j 2 {1, 2}. Note µj , µj j , j
that the risk-free rate r does not appear in the risk-neutral drift of S since it is a futures contract. The usual Girsanov-type change of measures could be applied to show the existence of such a measure, see for instance Elliott et al. (2005) for analogous work in continuous-time.
An analytical formula for the price Gt, see (2.2.8), of the option embedded within the guarantee can be obtained following the lines of Hardy (2001). However, the option price in the latter paper depends on the regime currently prevailing in the economy. In the current work, regimes are unobservable and the option price is therefore a weighted average of the prices associated with each regime where weights are the respective risk-neutral probabilities of currently being in each regime, namely ⌘i,tQ ⌘ Q ht = i|Ft.