Hansj¨org Furrer Market-consistent Actuarial Valuation ETH...

Guaranteed Annuity Options

Hansjorg Furrer

Market-consistent Actuarial Valuation

ETH Zurich, Fruhjahrssemester 2008

Guaranteed Annuity Options

Contents

A. Guaranteed Annuity Options

B. Valuation and Risk Measurement

Guaranteed Annuity Options 1

A. Guaranteed Annuity Options

Course material

• Slides

• Boyle, P. and Hardy, M. (2003). Guaranteed Annuity Options. ASTIN Bulletin,Vol. 33, No. 2, pp. 125-152.

• Pelsser, A. (2003). Pricing and Hedging Guaranteed Annuity Options viaStatic Option Replication. Insurance: Mathematics and Economics. Vol. 33,283-296.

• The above documents can be downloaded from

www.math.ethz.ch/~hjfurrer/teaching/


An Introduction into GAO

• Definition:

Under a guaranteed annuity option, the insurer guarantees to convert thepolicyholder’s accumulated funds into a life annuity at a fixed rate g at thepolicy maturity date T

• GAOs provide a minimum return guarantee:

- the policyholder has the right to convert the accumulated funds into a lifeannuity at the better of the market rate prevailing at maturity and theguaranteed rate

- if the annuity rates under the guarantee exceed the market annuity rates, thena rational policyholder will exercise the option. In that case, the insurer mustcover the difference


Origins

• When many of these guarantees were written in the UK in the 1970ies and1980ies

- long-term interest rates were high

- mortality tables did not include an explicit allowance for future mortalityimprovements (longevity)

• Ever since, however, long-term interest rates declined and mortality improvedsignificantly for lives on which these policies were sold. . .

• To summarize, the guarantee corresponds to a put option on interest rates:

- when interest rates rise, the annuity amount per 1000 fund value increases- when interest rates fall, the annuity amount per 1000 fund value decreases


Effect of Mortality Improvement

• The price of the guarantee depends on the mortality:

- 13-year annuity certain corresponds to an interest rate of 5.7% since

1000 =111

1 + 0.057+

111(1 + 0.057)2

+ · · ·+ 111(1 + 0.057)13

- 16-year annuity-certain requires an interest rate of 7.72%:

1000 =111

1 + 0.0772+

111(1 + 0.0772)2

+ · · ·+ 111(1 + 0.0772)16

I when mortality improves, the interest rate at which the guarantee becomeseffective increases


Setting the Scene

• S = {S(t) : 0 ≤ t ≤ T} market value of the accumulated funds

• B = {B(t) : 0 ≤ t ≤ T} with B(t) = exp{∫ t

0ru du} money market account and

{rt : 0 ≤ t ≤ T} instantaneous short rate process

• P (t, T ) time-t price of a zero-coupon bond with maturity T , t ≤ T . Hence,P (t, T ) = EQ[B(t)/B(T )|Ft]

• ax(T ) : market value of an immediate annuity of 1 p.a. (payable in arrear) to alife aged x at T :

ax(T ) =∞∑

k=1

kpx P (T, T + k), (1)

where kpx denotes the conditional probability that a person having attainedage x will survive k years (k-year survival probability)


The Nature of GAO

• g : Conversion rate. Determines the guaranteed annuity payment per annum, e.g.if S(T ) = 1000, then g = 9 implies an annuity payment of 1000/9 (≈ 111) p.a.

• Y (T ) : payoff from exercising the option at maturity T (= value of the guaranteeat maturity T ):

Y (T ) =

0, if S(T )

g ≤ S(T )a65(T )(

S(T )g

)a65(T )− S(T ), if S(T )

g > S(T )a65(T ) .

• Conditional on the survival of the policyholder up to time T , one thus has:

Y (T ) = S(T )(

a65(T )g

− 1)+

. (2)


Valuing the GAO

Assumptions:

• single-premium payments, no expenses

• financial risk is independent from biometric risk

• in a first step, it is even assumed that S and {rt : 0 ≤ t ≤ T} are independent

• there exists an equivalent martingale measure Q, i.e. an arbitrage-free economy(discounted market prices of tradable securities are Q-martingales)

• policyholders behave rationally, i.e. policyholders select the highest annuitypayout


Change of Measure Technique

• For valuing the GAO, a switch from the spot martingale measure Q to theT -forward measure QT turns out to be appropriate.

• Let X be a contingent claim that settles at T . Then

Notation Numeraire Pricing formula State price density

Spot

martingale

measure

Q B(t) πt(X) = B(t) EQ

[X

B(T )

∣∣Ft

]dQdP

∣∣∣Ft

= E[ζ|Ft]

T -forward

measure

QT P (t, T ) πt(X) = P (t, T ) EQT[X|Ft]

dQTdQ

∣∣∣Ft

= P (t,T )P (0,T )B(t)


Main Steps (1/2)

• Let X = Y (T ) =(

S(T )g

)(ax(T )− g)+

• It follows from the above assumptions that the time-t value of the GAO is givenby:

πt(X) = P (t, T ) EQT

[S(T )|Ft

]EQT

[(ax(T )− g

)+∣∣Ft

]T−tpx−(T−t)

g

By the martingale condition, we have that P (t, T )EQT[S(T )|Ft] = S(t). Hence,

πt(X) = S(t) EQT

[(ax(T )− g

)+|Ft

]T−tpx−(T−t)

g(3)


Main Steps (2/2)

• Using the definition of ax(T ) from (1), one obtains

πt(X) = S(t) EQT

[( J∑k=1

kpx P (T, T + k)− g)+∣∣∣Ft

]T−tpx−(T−t)

g(4)

• The expression inside the expectation in (4) corresponds to a call option on acoupon paying bond. The coupon payments at the time instants T + k are kpx.

• Jamshidian [3] showed that if the short rate follows a one-factor process, thenthe option price on a coupon paying bond equals the price of a portfolio ofoptions on the individual zero-coupon bonds (Brigo and Mercurio [2], p. 68):

CBO(t, T, τττ , ccc,K) =n∑

i=1

ci ZBO(t, T, Ti,Ki) (5)


Applying the Hull-White Short Rate Model

• We now specify the term structure of interest rates via the following short ratedynamics (Hull-White one-factor model):

drt = κ(θ(t)− rt

)dt + σ dW (t) (t ≥ 0)

• In the Hull-White context, bond options can be calculated explicitly, see forinstance Brigo and Mercurio [2], p. 65.

• Example: European call option with strike K, maturity S written on a zero-bondmaturing at time T > S:

ZBC(t, S, T,K) = P (t, T ) Φ(h)−K P (t, S) Φ(h− σp) (6)


with

σp =σ

κ

√1− exp{−2κ(S − t)}

2κ

(1− e−κ(S−t)

), h =

1σp

log( P (t, T )P (t, S)K

)+

σp

2

• Combining (4), (5) and (6), one obtains an explicit formula for the price of theGAO:

πt(X) = S(t)∑J

k=1 kpxZBO(t, T, T + k,Kk)P (t, T )

T−tpx−(T−t)

g(7)


Discussion

• the market value for the GAO is proportional to S(t)

• the simple pricing formula (7) relies on strong assumptions such as

- equity returns are independent of interest rates

- term structure of interest rates is given by a one-factor Gaussian short ratemodel

• Formula (7) can be generalized by assuming that S(T ) and P (T, T + k) arejointly lognormally distributed. In that case, a closed-form solution for the valueof the GAO can still be derived.


The Need for Multi-Factor Models

• Example: Hull-White two-factor model:

drt = κ(θ(t) + u(t)− rt

)dt + σ1 dW1(t)

du(t) = −b u(t) dt + σ2 dW2(t)

with u(0) = 0 and dW1(t)dW2(t) = ρdt.

• Note: the value of a swaption depends on the joint distribution of the forwardrates (F (t;T0, T1), F (t;T1, T2), . . . , F (t;Tn−1, Tn)). The payoff can thus not beadditively separated as in the case of e.g. a cap

• Later we will show that the payoff of GAOs can be (statically) replicated by aportfolio of receiver swaptions


• Correlation among the forward rates has an impact on the contract value

• Multi-factor models allow for more general correlation patterns than one-factormodels

• Thus: simple one-factor models usually give reasonable prices for instruments,but good hedging schemes will assume many factors


Interest Rate Swaps (IRS)

• Definition (Interest rate swap): contract that exchanges fixed payments forfloating payments, starting at a future time instant

• Tenor structure:

reset dates: Tα, Tα+1, . . . , Tβ−1

payment dates: Tα+1, . . . , Tβ−1, Tβ

• fixed-leg payments: N τi K (N : notional amount, τi: year-fraction from Ti−1 to Ti)

• floating-leg payments: N τi F (Ti−1;Ti−1, Ti)


Value of a (payer) IRS

V (t) = EQ

[β∑

i=α+1

P (t, Ti) τi

(F (Ti−1, Ti)−K

)∣∣∣Ft

]= . . .

=β∑

i=α+1

(P (t, Ti−1)− (1 + τi K)P (t, Ti)

)

• Forward swap rate: value of the fixed-leg rate K that makes the present valueof the contract equal to zero:

Sα,β(t) =P (t, Tα)− P (t, Tβ)∑β

i=α+1 τi P (t, Ti)


Swap options, Swaptions

• Definition: A European payer swap option where the holder has the right to payfixed and receive floating, is an option on the swap rate Sα,β(t). A Europeanreceiver swap option where the holder has the right to pay floating and receivefix, is an option on the swap rate Sα,β(t).

• The swaption maturity often coincides with the first reset date of the underlyingIRS

• Example: receiver swaption provides payments of the form(K − Sα,β(Tα)

)+.

If K = 7% and Sα,β(Tα) = 5%, it is optimal to exercise the option and receivefixed payments of Sα,β(Tα) + (K − Sα,β(Tα))+ = K

• By entering a receiver swaption, the holder protects itself against the risk thatinterest rates will have fallen when the swaption matures.


IRS and Swaptions in a Nutshell

Type Discounted payoff at Tα Price (time-t value)

Payer IRS

β∑i=α+1

P (Tα, Ti) τi(F (Ti−1, Ti)− K

) β∑i=α+1

(P (t, Ti−1)− (1 + τi K)P (t, Ti)

)=(

Sα,β(Tα)− K)Aα,β(Tα)

PayerSwaption

( β∑i=α+1

P (Tα, Ti) τi(F (Ti−1, Ti)− K

))+Aα,β(t) EQA

[(Sα,β(Tα)− K

)+∣∣Ft

]=(

Sα,β(Tα)− K)+

Aα,β(Tα)

Receiver IRS

β∑i=α+1

P (Tα, Ti) τi(K − F (Ti−1, Ti)

) β∑i=α+1

(−P (t, Ti−1) + (1 + τi K)P (t, Ti)

)=(

K − Sα,β(Tα))Aα,β(Tα)

ReceiverSwaption

( β∑i=α+1

P (Tα, Ti) τi(K − F (Ti−1, Ti)

))+Aα,β(t) EQA

[(K − Sα,β(Tα)

)+∣∣Ft

]=(

K − Sα,β(Tα))+

Aα,β(Tα)


Hedging the Interest Rate Risk of a GAO

• The quantity Aα,β(t) is given by∑β

i=α+1 τi P (t, Ti) and defines the change ofmeasure from the spot martingale measure Q to the measure QA:

dQA

dQ=

Aα,β(Tβ)/Aα,β(0)B(Tβ)/B(0)

=Aα,β(Tβ)

Aα,β(0)B(Tβ).

• Recall that the GAO gives the right to obtain a series of cash payments npxg atdifferent dates T1, T2, . . . . Hence, the interest rate exposure in a GAO is similarto that under a swaption.

• Pelsser [4] advocates the usage of long-dated receiver swaptions for dealing withthe interest rate risk under a GAO (static replicating portfolio approach)

• Price of the GAO ≡ value of a portfolio of long-dated receiver swaptions


Static Replicating Portfolio

• Recall that in the case of GAOs the expresssion

(a(T )

g− 1)+

=

(J∑

k=1

kpx

gP (T, T + k)− 1

)+

gives the right to receive a series of cash payments (kpx/g) for a price of 1

• Cash flows from GAO are gradually decreasing over time (due to the decreasingsurvival probabilities), whereas cash flows associated with an N -year swap areconstant over time

• Idea: Combine positions in receiver swap contracts all starting at time T , butwith different maturities T + k


Construction of the Hedge Portfolio

• Aim: determine the amount to be invested in each swap

• Let ω be the limiting age of the mortality table (e.g. ω = 120)

• At time T + (ω − x):

- cash flow to be replicated: (ω−xpx/g)- cash flow of a swap with fixed leg Kω−x and length ω − x : 1 + Kω−x

- amount Hω−x to be invested at time t:

Hω−x := ω−xpx

g(1 + Kω−x)(8)


• Note: equation (8) can be rewritten as

Hω−xKω−x = ω−xpx

g−Hω−x (9)

• At time T + (ω − x)− 1:

- cash flow to be replicated: (ω−x−1px/g)- cash flow from swap with fixed leg Kω−x and length ω − x : Kω−x

- cash flow from swap with fixed leg Kω−x−1 and length ω−x−1 : 1+Kω−x−1

- amount Hω−x−1 to be invested at time t:

Hω−x−1 :=

(ω−x−1px − ω−xpx

)/g + Hω−x

1 + Kω−x−1(10)

• Observe that Hω−xKω−x + Hω−x−1(1 + Kω−x−1) = ω−x−1px/g


I This yields a recursive relation for the amounts to be invested in swaps withtenor length n:

Hn =

(npx − n+1px

)/g + Hn+1

1 + Kn


Price of the GAO

• With the portfolio of swaps

ω−x∑n=1

HnV swap(T,Kn)

all cash flows of the GAO can be replicated

• Value of the GAO:(ω−x∑n=1

HnV swap(T,Kn)

)+

≤ω−x∑n=1

Hn (V swap(T,Kn))+ =ω−x∑n=1

HnV swapt(T,Kn)


Discussion of the Static Replicating PortfolioApproach

• Pros and cons:

+ no need for dynamic hedging (no further buying and selling until maturity)

+ based on the “right type of interest rate options”

+ swap market is more liquid than bond market

+ cheaper and better protection than reserving (reserving at 99%-level may beinsufficient)

− hedge against the interest rate risk only (hedging mortality risk by sellingmore life insurance?)

− in a period of rising stock returns, insurer must keep purchasing swaptions


B. Valuation and Risk Measurement

Pricing Derivative Securities

• Consider an economy of d + 1 assets (S0, S1, . . . , Sd)

• Trading strategy: H = (H0,H1, . . . ,Hd) with Hi(t) denoting the number ofunits held of the ith asset at time t

• Value of the portfolio at time t:

V (t;H) =d∑

i=0

Hi(t) Si(t)


• The strategy H is self-financing if

V (t)− V (0) =d∑

i=0

∫ t

0

Hi(u) dSi(u)

• When pricing a derivative, the drift parameters µi in the dynamics of Si donot appear: one does not need to know anything about an investor’s attitudetowards risk

• Rationale: risk preferences are irrelevant because contingent claims can bereplicated by trading in the underlying assets

• Price of the derivative: minimal investment to implement the trading strategy


The Role of the Measures P and Q

Real-world measure P Risk-neutral measure Q

asset returns vary by asset class the rate of return on any risky assetis the same as the risk-free rate

the measure P describes theempirical dynamics of asset prices

in EQ[ · ]-expectation, the riskyassets behave like the money marketaccount


Risk Measurement

Risk measurement poses the question:

How does the portfolio value V change in response to changes in the underlyingrisk factors?

• Z = (Z1, Z2, . . . , Zm)′ vector of risk factors

• ∆Z = Z(t + ∆t)− Z(t) : change in Z over ∆t

• Portfolio loss:L = V (t;Z)− V (t + ∆t;Z(t) + ∆Z)


Calculation of the loss distribution function FL

If we were to determine the loss distribution function FL by means of Monte Carlosimulation, we would have to proceed as follows:

• For each of n replications,

(i) generate a scenario under P, i.e. a vector of risk factor changes ∆Zi

(ii) re-value the portfolio V (t + ∆t;Z(t) + ∆Zi) at time t + ∆t under Q, giventhe outcome of ∆Zi

(iii) compute the loss Li = V (t;Z)− V (t + ∆t;Z(t) + ∆Zi)

• Estimate P[L ≤ x] using

1n

n∑i=1

1{Li≤x}


Estimation of a conditional expectation

The bottleneck in the above recipe is the portfolio revaluation of step (ii). Thismeans computing a conditional expectation (or an estimate thereof)


References

[1] Boyle, P. and Hardy, M. (2003). Guaranteed Annuity Options. ASTINBulletin, Vol. 33, No. 2, 125-152.

[2] Brigo, D. and Mercurio, F. (2001). Interest Rate Models. Theory andPractice. Springer, Berlin.

[3] Jamshidian, F. (1989). An exact bond option formula. Journal of Finance,44, 205-209.

[4] Pelsser, A. (2003). Pricing and Hedging Guaranteed Annuity Options viaStatic Option Replication. Insurance: Mathematics and Economics. Vol. 33,283-296.

[5] Wilkie, A. D., Waters, H. R., and Yang, S. (2003). Reserving, Pricing andHedging for Policies with Guaranteed Annuity Options. British ActuarialJournal, Vol. 9, No II, 263-425.


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