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Helmholtzstraße 22 D-89081 Ulm

phone +49 (0) 731/50-31230 fax +49 (0) 731/50-31239

Analysis of Embedded Options in Individual Pension Schemes in Germany

Presentation at the 7th annual APRIA conference, Bangkok, Thailand, July 20-23, 2003 Alexander Kling Jochen Russ Hato Schmeiser

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Agenda

Government subsidized pension products in Germany ❙ Problems of the state pension system

❙ Different products offered

Important criteria according to §1 AltZertGProducts offered by mutual funds

Model Framework

Evaluation of embedded options in these products ❙ Money-back guarantee

❘ Without paid-up option ❘ With paid-up option

Numerical results

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Government subsidized pension products in Germany

Problems of the state pension system ❙ Pay-as-you-go

❙ Changing age structure

❙ In 2030: 1 premium payer for 2 annuitants

Reduction of the national old-age pension and

encouragement of individual pension schemes

Government aid for private pension contracts ❙ Tax relief and direct subsidies to the premiums ❚ Products

❙ Traditional life insurance contracts ❙ Unit-linked life insurance contracts ❙ Bank accounts

❙ Products offered by mutual funds

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Important criteria according to §1 AltZertG

11 criteria settled in §1 AltZertG, the most important are:

Regular premium payment

End of the accumulation phase not before age 60

Annuity beginning at the end of the accumulation phase

Requirement for the seller to disclose all fees included in the

product

Money-back guarantee

Paid-up option for the client (without losing the money-back

guarantee)

Bank accounts and life insurance contracts contain

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Products offered by mutual funds

Currently 8 sellers on the market: Activest, ADIG, BHW,

Deka, dit, DWS, HANSA, Union Investment

Monthly or annual premiums which are invested in funds

containing stocks and bonds

Declining portion of stocks with time going to maturityNo charges for the money-back guarantee

Current regulations allow the sellers to give money-back

guarantees without enforcing hedging strategies

No adequate reserves required

No compensation between the funds of different clientsUncovered risks

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Model for the contract

Annual premium P, term to maturity T years

Only one underlying St containing stocks and bonds

St follows a geometric Brownian motion with time-dependent

volatility σ(t), r(t) deterministic

Value of the assets at time t (before paying the (t+1)st premium)

is given by

Value of the assets at maturity ❙ If paying all the premiums

❙ If exercising the paid-up option at time

− = −

=

=

1 0 T T T T

S

S

P

V

V

ν ν

− = −

=

1 0 t t t

S

S

P

V

ν ν

− = −

=

=

1 0 * τ ν ν τ τ τ

S

S

P

S

S

V

V

T T T

τ

=

t

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Evaluation – without paid-up option

Payoff at T

❙ Value of the assets at T + put option

Value of the money-back guarantee

❙ No closed form solutions exist

[

]

=

Π

E

e

− 0 ( )

TP

V

+

F

0 T ds s r Q T

[

]

+

+

=

=

T T T T

V

TP

V

TP

V

L

max{

;

}

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Evaluation – with paid-up option

Paid-up option can be exercised at time t=1,...,T-1

(Bermuda-style option)

Exercising the paid-up option at time

Price of the option if following an optimal admissible

strategy

Price and optimal strategy calculated by

Monte-Carlo-Simulation following (Douady 02)

[

]

+

+

=

=

τ τ τ τ *

τ

*

τ

* *

max{

;

}

T T T T

V

P

V

P

V

L

τ

=

t

[

]

=

Π

(1)

sup

E

e

0 ( )

P

V

* +

F

0 T ds s r Q S T τ τ

τ

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Evaluation – with paid-up option - additional calculus

2 contracts

Exercising the paid-up option at time and entering a

new contract at the same time

❙ Price of the option if following an optimal admissible strategy

where the value of the assets of the second contract is given by

− = = 1 * T T T S S P W τ ν ν τ

τ

=

t

[

] [

]

(

)

      − − + − ∫ = Π(2) sup E e0 ( ) P V * + (T )P W * + F0 T T ds s r Q S T τ τ τ τ τ

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Evaluation – with paid-up option - additional calculus

T contracts

Exercising the paid-up option annually ❙ Payoff at T

❙ Value of the assets + sum of forward-start performance options ➨ Closed form solution exists

Value of the money-back guarantee

− = + − = +       − + =               − + = 1 0 1 0 ) ( T T T T T T T S S P P V S S P P S S P L ν ν ν ν ν

(

)

− = − − = − ⋅ ⋅ ∫ = Π = Π 1 0 ) ( 1 0 ) ( ) (T T T T 0r s ds 1,1, S e P p T ν ν ν ν

ν

ν

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Numerical Results

Input

❙ 4 contracts

❙ r(t)=5% const.

❙ 6 different volatility structures

35 1,200 € Contract 4 20 2,100 € Contract 3 10 4,200 € Contract 2 5 8,400 € Contract 1 Time to maturity Annual premium 5% 7% 9% 11% 13% 15% 15% 15% 20% Underlying 6 5% 5% 5% 5% 5% 10% 15% 20% 20% Underlying 5 10% 15% 15% 20% 20% 20% 20% 20% 20% Underlying 4 10% 10% 10% 10% 10% 10% 10% 10% 10% Underlying 3 15% 15% 15% 15% 15% 15% 15% 15% 15% Underlying 2 20% 20% 20% 20% 20% 20% 20% 20% 20% Underlying 1 1 2 3 4 5 10-6 15-11 20-16 35-21 Time to maturity

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Numerical Results

629.27 € 433.16 € 465.46 € 425.04 € Underlying 3 1,527.22 € 1,184.35 € 1,275.18 € 1,172.46 € Underlying 2 2,548.72 € 2,093.57 € 2,251.86 € 2,080.33 € Underlying 1 Contract 1 Π (T) S Π ) 1 ( K Π (1) S Π >6% Smaller guarantee level

Value of the options decreasing with decreasing volatilityHighest value in absolute terms occurs with contract 1 and

underlying 1, here

TP

T

S

>

Π

( )

6

%

The value of the pure money-back guarantee is always

increased by the paid-up option even if the “guarantee level” is smaller

For all contracts and underlyings, the value is the highest

when the client chooses to exercise the paid-up option annually

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Numerical Results

49.06 € 4.83 € 5.15 € 4.41 € Underlying 3 250.81 € 110.17 € 119.74 € 106.43 € Underlying 2 639.46 € 392.19 € 432.03 € 381.35 € Underlying 1 Contract 4

Π

Π

(K1)

Π

S(1)

Π

(ST)

The value of the money-back guarantee with paid-up option

if exercising annually can be more than twice as high as the value of the pure money-back guarantee

The value of the pure paid-up option can be higher than the

value of the money-back guarantee without paid-up option

Value of the options decreasing with increasing term to

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Numerical Results

1,360.55 € 1,166.11 € 1,293.76 € 1,008.44 € 1,135.92 € 956.79 € Underlying 4 2,548.72 € 2,339.03 € 2,490.54 € 2,093.57 € 2,251.86 € 2,080.33 € Underlying 1 Contract 1 Π Π(K1) Π(S1) Π(K2) Π(S2) ΠS(T)

The value of the pure money-back guarantee decreases

strongly when we have a decreasing volatility structure

The additional value resulting from the paid-up option

decreases more slowly or even increases

403.76 € 209.32 € 336.97 € 51.65 € 179,13 € 956.79 € Underlying 4 468.39 € 258.70 € 410.21 € 13.24 € 171.53 € 2,080.33 € Underlying 1 Contract 1 Π Π(K1) − Π ΠS(1) − Π Π(K2) − Π Π(S2) − Π Π(TS ) −Π

Reducing volatility as maturity approaches is suitable to reduce

the risk resulting from the pure money-back guarantee but

much less suitable to reduce the risk resulting from the paid-up option

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Summary and outlook

There are embedded options in government subsidized

pension products in Germany

In mutual funds, these embedded options are usually given

away for free

The pure money-back guarantee can have a significant value.The paid-up option that has to be offered by law, has a

significant additional value

Reducing volatility as time to maturity decreases is suitable

to reduce the risk resulting from the pure money-back

guarantee but much less suitable to reduce the risk resulting from the paid-up option

The current law allows the client to speculate against the

provider

The question of fair values of embedded options will become

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ifa

Helmholtzstraße 22 D-89081 Ulm phone +49 (0) 731/50-31230 fax +49 (0) 731/50-31239 email ifa@ifa-ulm.de

Analysis of Embedded Options in Individual Pension Schemes in Germany

Alexander Kling Jochen Russ Hato Schmeiser

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