Valuation and hedge of the Guaranteed Equity-Linked Product ‘FlexInvest’

Valuation and Hedge of the Guaranteed Equity-Linked

Product ‘FlexInvest’

Jan-Willem Roossink 9914862 Financial Econometrics August 2006

1st _{supervisor University of Amsterdam: Prof. Dr. H.P. Boswijk}

2nd _{supervisor University of Amsterdam: Dr. R. Ramer}

Supervisor Zanders: Drs. A.G. van Grootheest

Zanders Brinklaan 134 1404 GV Bussum The Netherlands Telephone: +31 35 692 89 89 Facsimile: +31 35 692 89 99

TABLE OF CONTENTS

INTRODUCTION

4

1.1 Purpose of this Paper ... 4

1.2 Approach ... 4

1.3 Structure... 5

COMPANY and PRODUCT

6

2.1 DBV Levensverzekeringsmaatschappij NV ... 6 2.2 FlexInvest ... 6 2.2.1 FlexInvest 1999 ... 8 2.2.2 FlexInvest 2003 ... 8 2.2.3 FlexInvest 2006 ... 9

ANALYTICAL VALUATION

10

3.1 FlexInvest 1999 and 2003 ... 10 3.2 FlexInvest 2006 ... 12

VALUATION by SIMULATION

14

4.1 Stock Price Behavior: Geometric Brownian Motion ... 14

4.2 Rate of Return and Volatility of FlexInvest... 15

4.3 The Model ... 16

4.4 Statistics ... 17

4.4.1 Standard scenario – Based on Historical Price Development... 18

4.4.2 Scenario – Annual Premium Payments... 19

4.4.3 Scenario – Term of Contract equal to 10 Years ... 19

4.4.4 Scenario – FlexInvest Portfolio 70% Stocks / 30% Bonds ... 21

4.4.5 Scenario – Assumed Return on Stocks of 7%... 23

4.5 Comparison with Analytical Pricing Formula ... 24

HEDGE STRATEGY

27

5.1 Risks of FlexInvest... 27

5.2 Perfect Hedging... 27

5.3 Correlation Stocks FlexInvest and Dow Jones Eurostoxx 50 ... 31

5.4 The Model ... 32

5.4.1 Rebalancing... 32

5.4.2 Correlated Asset Prices... 32

5.4.3 Put Options ... 33

5.5 Statistics ... 34

5.5.1 Standard scenario – Based on Historical Price Development... 34

5.5.2 Scenario – Annual Premium Payments... 36

5.5.3 Scenario – Term of Contract equal to 10 Years ... 37

5.5.4 Scenario – FlexInvest Portfolio 70% Stocks / 30% Bonds ... 37

5.5.5 Scenario – Assumed Return on Stocks of 7%... 38

5.6 Conclusion... 39

RECOMMENDATIONS for FURTHER RESEARCH

41

SUMMARY and CONLUSION

42

REFERENCES

44

Appendix A - Winstdelingsvoorwaarden Flexinvest-Rendement Appendix B - Guidebook to model: FlexInvest Valuation Appendix C - Statistics Valuation by Simulation

Appendix D - Guidebook to model: FlexInvest Hedge Strategy Appendix E - Statistics Hedge Strategy

INTRODUCTION

1.1

Purpose of this Paper

Guaranteed equity-linked products are becoming more and more popular. The common feature of these products is that the client receives a certain minimum amount or percentage as return on its investment. Depending on the products’ conditions the single or periodic premium is invested in bonds or stocks or a combination of both. This makes it possible for the client to have a return of more than the guaranteed return at the end of the contract.

DBV is an insurance company and one of the suppliers of such an equity-linked guarantee product: FlexInvest. FlexInvest has several complicating characteristics, including a ‘click principle’, long term contracts and compensation for losses made in preceding years. If the returns on the investment portfolio are lower than the return guaranteed to the client then DBV suffers a loss. Due to bad performing stock markets and historically low interest rates in recent years, DBV had difficulties to generate returns on their investment portfolio which are sufficient to foresee in the given guaranteed return to the client. DBV currently has problems identifying the value and risks of FlexInvest. The objectives of this paper are: • Valuation of the product FlexInvest.

• Hedging the investment risk involved with FlexInvest.

1.2 Approach

This paper values the product FlexInvest and investigates the performance of a hedge strategy for the investment risk. The valuation is done analytically and by means of a stochastic simulation model. The analytical formulas are not exact because they cannot incorporate all conditions of the product. The valuation by means of a stochastic simulation model assumes a Black-Scholes world such that stock prices follow a geometric Brownian motion; the simulation model has been made flexible such that several scenarios can simply be calculated. Furthermore a hedge strategy for the investment risk is introduced and it is investigated what the performance of this strategy is. The main focus is not to eliminate all risk, but find acceptable lower bounds and minimum losses for the company at reasonable costs; the less risk the higher the costs. It will be examined how the performance of the hedge strategy that hedges the investment risk by buying put options on the Dow Jones Eurostoxx 50 is. This strategy works well, though because the profit / loss for

DBV is a non-linear function of the returns of the stocks, it is difficult to determine the amount of put options to be bought.

1.3 Structure

The next chapter describes in detail the company and product conditions. After this, in chapter 3, closed-form pricing formulas are derived for certain types of the product in a Black-Scholes environment. Chapter 4 values the product in several scenarios by means of a stochastic simulation model. Succeeding this in chapter 5 a hedge strategy is implemented in the simulation model and its performance results are presented. Based on these results recommendations for further research are given in chapter 6. Finally a summary and conclusion are given in chapter 7.

COMPANY and PRODUCT

This chapter gives a short description of DBV, the company which sells the product investigated in this paper: FlexInvest. After this the conditions of FlexInvest itself are described and the contract definitions are given.

2.1

DBV Levensverzekeringsmaatschappij NV

DBV Holding NV is part of Winterthur Group and is specialized in insurances, pensions and mortgages and offers a wide range of products both to corporate and non-corporate clients. Their products vary from insurances with a guaranteed return to innovative combinations of insurance and investment. DBV Holding consists of DBV Levensverzekeringsmaatschappij NV and DBV Schadeverzekeringsmaatschappij NV. The first mentioned develops flexible solutions for life insurances, pensions and mortgages. The last mentioned is focused on insurances as car-, fire- en liability insurance. As from now on DBV will refer to DBV Levensverzekeringsmaatschappij NV.

2.2 FlexInvest

This paper focuses on the product FlexInvest. This product can be described as an equity-linked insurance policy with a guaranteed component. With FlexInvest DBV wants to foresee in the clients’ need to combine saving and investing. FlexInvest reassures a fully guaranteed return at a by the clients’ chosen date and contains the possibility to gain extra investment return. The client has the choice to make a single investment or to use premium payments. With premium payments the client also has the choice to make these payments monthly, quarterly, half-yearly or yearly. The premiums that are to be invested are deposited in an investment pool of equities, bonds, deposits and other financial instruments. Investment experts at MeesPierson manage these investments.

FlexInvest is subject to profit sharing conditions to generate turnover for DBV. These conditions have been changed twice: in 2003 and in 2006 what causes the existence of a first, second and third version of FlexInvest.

To be able to control the risks involved with FlexInvest it first has to be investigated which economic factors influence the value development of the product. The most important risks on the product are investment-, interest rate-, death-, buy off- and costs risk. These last three risks can be calculated using

historical data and do not change significantly at short notice. Investment and interest rate risk are both future related risks. This paper will focus only on those two risks.

DBV guarantees an annual return on the value of the FlexInvest portfolio. For some versions of FlexInvest, any net investment yield in excess of this guaranteed return per annum is frozen on the customers’ behalf and is credited to their policies. This says FlexInvest works by means of the ‘click principle’: yearly realized positive returns stay in possession of the client even though a loss is made in the succeeding year. If the annual return on FlexInvest is less than the guaranteed return, the loss will be remembered and can be compensated in the succeeding years. On which versions of FlexInvest this ‘click principle’ is applicable will be showed later in this chapter. See figure 2.1 for an example of the payoff during several years with a guaranteed return of 3%.

Figure 2.1 An example of the ‘click principle’ and compensation for losses made in preceding years of FlexInvest.

A premature ending of the contract (buy off, conversion or capital transfer) is possible though a penalty clause is applicable. This clause means that 90% of the gained investment profit at the moment of ending, provided positive, is assigned to the client.

FlexInvest is the successor of VarInvest. DBV has stopped to sell VarInvest, though due to long-term contracts, which can take up to 30 years, it will take a long time before the whole product is ended. With VarInvest the client had the choice to invest in a risk averting pool or a more risky pool. The more risky pool (A) invests relatively more in stocks than in bonds; the risk averting pool (B) invests relatively more in bonds than in stocks. With FlexInvest DBV has taken the clients’ possibility to choose an investment pool; FlexInvest is invested in a fixed combination of stocks and bonds. For an overview of the investment pools of VarInvest and FlexInvest see figure 2.2.

Figure 2.2 An overview of the investment combination of stocks and bonds for VarInvest and FlexInvest. To be able to describe the payoff we fix some notation. Let

S

_t be the price of FlexInvest at time t. This should be thought of as a combination of stock and bond prices of which the FlexInvest portfolio exists. Let the start of the contract be at

0

0

=

t

and let

t

_i

,

i

=

1

,...,

n

−

1

be the time points at which a premium

P

_i is possibly paid by the client. Furthermore, let

T

=

t

_n be the expiry date of the contract and K the guaranteed return.

2.2.1 FlexInvest 1999

The first set of conditions, which have been used after the introduction of FlexInvest in 1999, state that the guaranteed annual return on FlexInvest is 3%. If the annual return is higher than 3% a percentage of 15 base points of the difference between the annual return and 3% is calculated as provision for DBV with a maximum of 75 base points. To be able to give an expression for the client’s payoff the net return NRt has to be defined, this is equal to the annual return Rt

minus the profit margin Mt for DBV. The profit margin Mt at time t is defined as:

[3%< <8%]

0

.

15

(

−

)

+

[ ≥8%]

0

.

75

=

t t t R R t

I

R

K

I

M

(2.1) where

( )

x

=

I

A 1 if

x

∈

A

(2.2) 0 if

x

∉

A

Now the net return can be written as:

t t t t t t t

M

S

S

S

M

R

NR

=

−

=

−

−1

−

(2.3)

Also the cumulated loss CLt at time t has to be defined which is equal to the

minimum of zero and the sum of losses and profits over passed time. This gives:

(

0

,

1

)

min

−

+

₋

=

t t

t

NR

K

CL

Now for the payoff Pt of FlexInvest 1999 at time t:

(

NR

CL

K

)

P

_t

=

max

_t

+

_t−1

,

(2.5)

2.2.2 FlexInvest 2003

The second set of conditions, which have been introduced in 2003, state that the guaranteed annual return on FlexInvest is 3%. A fixed percentage of 50 base points, raised by a percentage of 10 base points of the difference between the annual return and 4% if the annual return is higher than 4%, is calculated as provision for DBV with a maximum of 90 base points. For this second set of conditions the client’s payoff Pt at time t can be written the same as equation (2.5)

but where the profit margin Mt at time t is equal to:

[ ]

0

.

10

(

)

[ ]

0

.

40

50

.

+

_{4%< <8%}

−

+

_≥8%

=

t t t R R t

I

R

K

I

M

(2.6)

2.2.3 FlexInvest 2006

The third set of conditions, introduced in 2006, state that the guaranteed annual return is 2% for contracts no longer than 20 years, 2.5% for contracts between 20 and 25 years and 3% for contracts longer than 25 years. A fixed percentage of 50 base points, raised by a percentage of 10 base points of the difference between the annual return and 3% if the annual return is higher than 3% plus a percentage of 40 base points of the difference between the annual return and 10% if the annual return is higher than 10%, is calculated as provision for DBV. The profit margin Mt

at time t can be written as:

[ ]

0

.

10

(

)

[ ]

0

.

40

(

10

%

)

50

.

+

_3%

−

+

_10%

−

=

_{R> t R≥ t} t

I

R

K

I

R

M

t t (2.7)

This is the first version in which the ‘click principle’ is not applicable anymore. This means that only at the end of the contract, instead of every year, there has to be checked whether the real return is higher than the guaranteed return. Now for the payoff Pt of FlexInvest 2006 at time t:

(

)

(

)

_⎥

⎦

⎤

⎢

⎣

⎡

+

+

=

∏

= t i i t t

K

NR

P

1

1

,

1

max

(2.8)

ANALYTICAL VALUATION

This chapter investigates the possibility of an analytical calculation of the value of the product FlexInvest. It is important to notice that the ‘click principle’ makes a big difference when calculating the value of the product. This means the calculations for the products FlexInvest with and without ‘click principle’ are treated separately. First it will be shown that the analytical value of FlexInvest 1999 and FlexInvest 2003 are only possible to find if compensation for losses made in preceding years is not incorporated. After this the analytic calculation of the value of FlexInvest 2006 is presented.

3.1

FlexInvest 1999 and 2003

The contract will have a term of T years and an annual premium payment P at the beginning of the first b years. Note that if b=1 the contract is a single premium payment contract. Like before K is the return guaranteed to the client. This valuation only looks at the investment part of the product; this means for example management fees and death risk will not be accounted for. From Harrison, Kreps (1979) and Harrison, Pliska (1981) we know that the value of an investment

V

_t at time t is given by:

T

t

t

V

e

E

V

_T ds s r Q t T t

≤

≤

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛ ∫

=

−

,

0

) ( (3.1)

Equation (3.1) shows that the value of the investment at time T,

V

_T is discounted with the continuously compounded interest rate r under the risk-neutral probability measure Q. Filling in the payoff function of FlexInvest version 1999 and 2003 of equation (2.5) the value at time T can be written as:

(

)

(

)

∑∏

= = −

+

+

=

b i T i j j j T

P

NR

CL

K

V

1 1

,

max

1

(3.2)

where

NR

_j respectively

CL

_j−1 are equal to equation (2.3) respectively (2.4). In words we can describe equation (3.2) as that DBV guarantees for all premiums of the first b years a minimal annual return of K%. The trickiest part of this equation

to solve is compensation for losses made in preceding years. After a lot of research we conclude that it is not possible to find a solution for equation (3.2). Though to give a feeling what the solution could be it is possible to give a maximum value of the solution for equation (3.2). To do this we have to make the assumptions that there is no margin for DBV and no compensation for losses made in preceding years. Given these assumptions the value at time T can be written as:

∑∏

= = − −

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

+

=

b i T i j j j j T

K

S

S

S

P

V

1 1 1

,

max

1

(3.3)

and this can be rearranged into:

(

)

∑∏

= = − −

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

−

+

+

=

b i T i j j j j T

S

S

K

S

K

P

V

1 1 1

0

),

1

(

max

1

1

(3.4)

from which it can be seen that

max

(

S

_j

−

S

_j−1

(

1

+

K

),

0

)

is the payoff function of a call option on a stock starting at

t

= j

−

1

, maturing at

t

=

j

and with a strike of

)

1

(

1

K

S

j−

+

. Actually this is the same payoff function as for FlexInvest only without compensation for losses made in preceding years.

Black and Scholes (1973) have derived a mathematical formula for the theoretical value of European call (and put) stock options that can be derived from the assumptions of the Black-Scholes model. The formula for the price of a call on a stock currently trading at price

S

_j−1 where the option has a strike price of

)

1

(

1

K

S

_j−

+

and time to maturity of T=1 years is:

(

S

₁

,

S

₁

(

1

K

),

1

)

S

₁

N

(

d

₁

)

S

₁

(

1

K

)

e

N

(

d

₂

)

C

_{j− j−}

+

=

_j−

−

_j−

+

−r (3.5) where

σ

σ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

+

⎟

⎠

⎞

⎜

⎝

⎛

+

=

1

2

1

ln

2 1

r

K

d

(3.6)

σ

−

=

1 2

d

d

(3.7)

and r is the risk free interest rate,

σ

the stock volatility and

N

( )

⋅

the standard normal cumulative distribution function.

When filling in the above results of Black and Scholes into equation (3.4) and discounting to time T=0 the payoff function at time 0 can be written as:

(

)

(

)

∑∏

= = − −

₊

₊

₋

₊

=

b i T i j r rT

d

N

e

K

d

N

K

P

e

V

1 2 1 0

1

(

)

(

1

)

(

)

(3.8)

where

d

₁ respectively

d

₂ are equal to equation (3.6) respectively (3.7).

To get a feeling for this result figure 3.1 reflects payoffs for different terms of the contract FlexInvest 1999 / 2003 with a single premium of 1.000 when the risk free rate is 4.5% and the volatility is 10%.

Figure 3.1 Payoffs for different terms of the contract FlexInvest 1999 / 2003 when

r

=

4

.

5

%

and

%

10

=

σ

.

We conclude that DBV always suffers a loss on FlexInvest 1999 / 2003 given the assumptions since for all terms of the contract it sells the contract for 1.000 which is less than the value of the contract.

3.2 FlexInvest

2006

The same assumptions and notation will be used as in the section before. This means that the contract will have a term of T years and an annual premium payment P at the beginning of the first b years; K is the return guaranteed to the client. For FlexInvest with the profit sharing conditions of 2006 the value at time T is given by:

(

)

∑

= − − +

⎥

⎦

⎤

⎢

⎣

⎡

+

=

b i i T i T T

S

S

K

P

V

1 1 1

,

1

max

(3.9)

This can be rearranged into:

(

)

(

(

)

)

∑

= − + − − − +

⎥

⎦

⎤

⎢

⎣

⎡

+

−

+

+

=

b i i T i T i i T T

S

S

K

S

K

P

V

1 1 1 1 1

1

,

0

max

1

1

(3.10)

from which it can be seen that

(

T i

)

i

T

S

K

S

−

₋₁

(

1

+

)

+1−

,

0

max

is the payoff function of a call option on a stock starting at

t

= i

−

1

, maturing at

t

=

T

and with a strike of

i T

j

K

S

₋₁

(

1

+

)

+1− . Again using the Black-Scholes option pricing formula for this call in combination with equation (3.10) and discounting to time T=0 the value of FlexInvest 2006 at time 0 can be written as:

(

)

(

( ) (

)

( )

)

[

]

∑

= − + − − + − + −

₊

₊

₋

₊

=

b i i T r i T i T rT

d

N

e

K

d

N

K

P

e

V

1 2 ) 1 ( 1 1 1 0

1

1

(3.11) where

(

)

T

T

r

K

d

i T

σ

σ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

=

− +

2

1

1

ln

2 1 1 (3.12)

T

d

d

₂

=

₁

−

σ

(3.13)

To get a feeling for this result figure 3.2 reflects payoffs for different terms of the contract FlexInvest 2006 with a single premium of 1.000 when the risk free rate is 4.5% and the volatility is 10%.

Figure 3.2 Payoffs for different terms of the contract FlexInvest 2006 when

r

=

4

.

5

%

and

σ

=

10

%

. We conclude that, given the assumptions, for short terms DBV suffers a loss and for longer terms DBV makes a profit on FlexInvest 2006. The explanation for this is that the probability of a negative return on a long term is less than the probability of a negative return on a short term.

VALUATION BY SIMULATION

This chapter constructs a simulation model to find the value of the guarantee given in the different FlexInvest products. Outputs such as the value of the given guarantee and the possible profit / loss for DBV of different scenarios are presented. These scenarios are based on historical price developments and managerial preferences. The model developed in this chapter acts as a starting point for the hedging strategy that will be developed in the following chapter. Firstly it is described how the future returns for FlexInvest are simulated using geometric Brownian motion. After this it is shown how the historical volatility of FlexInvest has been calculated which is needed as input for simulating future returns. Thirdly the calculations which are made within the model are briefly described. After this outputs and other statistics with comments on them are presented. Then these outputs are compared with the values given by the analytical pricing formulas from chapter 3. This chapter will end with an overview and conclusion of given statistics.

4.1

Stock Price Behavior: Geometric Brownian Motion

To run the model simulated future returns of the FlexInvest portfolio are needed. To generate these realized returns the geometric Brownian motion stochastic differential equation will be used. This is often used for generating asset returns and widely accepted in modern financial literature (Muldowney and Wojdowski). The input that is needed to generate the portfolio prices are:

•

μ

: expected rate of return of the portfolio •

σ

: volatility of the portfolio

Later in this chapter it is explained how this expected rate of return and volatility have been calculated to come to the standard scenario. The discrete time version of the model is:

t t t t t

e

S

S

δ σε δ σ μ δ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − −

=

2 2 ,

ε

_t~N(0,1) (4.1)

where

ε

_t is a random drawing from a standardized normal distribution and δ

−

−

_t

t

S

S

is the change in the stock price S in time interval

δ

t

. This model exhibits the Markov property which says that previous prices are not relevant for predicting future prices, given knowledge of the current price.

Note that in this model stochastic volatility of the FlexInvest portfolio is not taken into account. This has not been done because the only interesting value of the portfolio return is after a whole year. When considering yearly returns we think the effect of stochastic volatility is marginal.

4.2

Rate of Return and Volatility of FlexInvest

The portfolio of FlexInvest currently consists of 40 stocks Xi and 8 bonds Bj and the

objective is to have a fixed proportion of 40% stocks and 60% bonds. Bonds have lower risk but also lower rate of return than stocks. This means a combination of stocks and bonds has risk and return higher than only bonds but lower than only stocks. Historical prices have been collected from all the stocks from the last 1829 trading days; that is as from March 26, 1999 until May 16, 2006. Using the historical prices it is easy to obtain the daily returns Rt at time t. Note that bonds

are valued at their face value. This means the constant daily returns are derived from the coupon interest and therefore have a volatility of zero. With these daily returns of every stock and bond in the FlexInvest portfolio the daily returns of the FlexInvest portfolio can be calculated. To calculate this the weight

ω

_i of each stock

Xi has to be known. This weight is the market value of the stock proportionally to

the total market value of all stocks in the portfolio 1)_:

∑

=

=

₄₀ 1 i X X X X i i t i t i t i t

N

P

N

P

ω

(4.2) where i t X

P

is the price of stock Xi and i

t

X

N

the number of shares of stock Xi at

time t. Let B t

R

be the weighted average rate of return on bonds, then the daily return at time t of FlexInvest is:

B t X t i i Flex t

R

R

R

₄₀

_%

_*

_*

i

₆₀

_%

_*

40 1

+

=

∑

=

ω

(4.3) Based on these daily returns of the FlexInvest portfolio the historical mean

∑

=

=

N i Flex i Flex

R

N

R

1 _

₁

and volatility

∑

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

=

N i Flex Flex i

R

R

N

1 2 _

1

σ

of the portfolio can be calculated in the usual way. The volatility

σ

of the portfolio is equal to 8.50% per year and the historical rate of return

μ

of the portfolio is equal to 4.23% per year.

4.3 The

Model

The model starts by following the calculations as given in the second chapter. This means that firstly the margin for DBV, net return and possible profits and losses based on the realized return in percentages are calculated. This chapter has explained before that the realized returns are obtained from geometric Brownian motion; the Box-Muller transformation (see Carter) has been used to produce a set of variables with a Gaussian, that is normal distribution. From the calculated percentages each year the corresponding nominal value of loss, profit, cumulated loss and cumulated margin for DBV are computed to end up with the following figures: clients return with guarantee, clients return without guarantee, cumulated loss and cumulated margin DBV. From these figures it can be seen how much the client’s total return and DBV’s total profit or loss is at the end of the contract and what the value of the guarantee is.

To come to the present value of these figures we choose the average rate of return as discount factor. Though in financial literature it is common to use the risk free rate as discount factor (Cox, Ross and Rubinstein) for pricing options, we think in this case the average rate of return as discount factor provides more practical values of the guaranteed option in FlexInvest.

Simulating the procedure above many times and taking the average gives in the long run the theoretical value of FlexInvest. With the simulation output the percentage of contracts in which DBV has suffered a loss and upper and lower percentiles can be seen. Appendix B gives an exact description of the calculations made in the model.

4.4 Statistics

This section presents and analyzes outcomes of interesting simulations. For each series of 60.000 simulations it will be shown how much the investment is worth at the end of the contract and how this amount is distributed among the client and DBV. This way it can be seen how much the profit or loss for DBV is, how much the value of the given guaranteed return is, how much the minimum and maximum loss and profit for DBV is and how much the upper and lower percentile bounds are. This will be done for every one of three types of the FlexInvest product and also for FlexInvest 1999 with a guarantee of 4%. The statistics will show averages discounted at the average rate of return of the FlexInvest portfolio; not discounted averages are given in appendix C.

Return client – with guarantee shows what the client will receive on average at the

end of the contract with DBV giving the guaranteed annual return; Return client –

without guarantee shows what the client will receive on average at the end of the

contract with DBV not giving the guaranteed annual return. The difference between these two is equal to the average value of the guarantee given by DBV.

Furthermore the statistics show what the average loss of DBV is at the end of the contract; a loss occurs if investment results are worse than the annual guarantee DBV has to pay to the client. In the same way the average margin for DBV is calculated. Summing above two results gives the total result of DBV at the end of the contract; when positive DBV has made a profit, when negative DBV has made a loss. With the results of the simulations the maximum, minimum, median and upper and lower bounds of the result of DBV can be calculated. Also the percentage in how many contracts DBV has had a negative result is given: DBV Result - % not

covered.

The Return total is equal to the Return client – with guarantee plus the DBV result. Suppose an initial amount is invested at the beginning of the contract. This equality says the amount at the end of the contract is distributed among the client and DBV.

4.4.1 Standard scenario – Based on Historical Price Development

In figure 4.1 the output with volatility and expected rate of return based on historical price developments is presented (

σ

=8.50% per year and

μ

=4.23% per year). The contract has a single premium of 1.000 and a term of 25 years.

Figure 4.1 Statistics with volatility and expected rate of return based on historical price developments, that is 8.5% respectively 4.23%. The contract is single premium and runs for 25 years.

When looking at the statistics it immediately attracts attention that FlexInvest 2006 has the first conditions that cause an average profit for DBV. Also the DBV Result of 2006 versus 2003 improves much more than 2003 versus 1999 3%. This is partly because of higher margins but also because of less loss for DBV. The less loss can be explained by the fact that the ‘click principle’ is no longer applicable in 2006. The higher margins are mainly caused by the fact that when high returns (>10%) on the investment are made DBV takes 0.4% per percentage point above 10% as a margin.

Furthermore we see the Value of Guarantee increases from version 1999 3% to version 2003, even though the total DBV Result decreases. This means the loss for DBV is higher, but the margin DBV charges is relatively more higher which causes the result also to be higher. This loss is higher because of the standard margin of 0.5% DBV takes; it is more likely to make a loss.

The fact that version 2006 versus version 2003 performs much better than version 2003 versus version 1999 3% can also be seen in the percentage of covered results; with the 2006 conditions DBV has a negative result in 29% of all contracts whereas with the conditions of 2003 DBV has a negative result in 57% of all contracts.

When looking at the maxima, minima and upper and lower bounds of the DBV result it is noticeable that the change in conditions from version 1999 to version 2003 mainly improved the upside potential whereas the change in conditions from version 2003 to version 2006 improves upside potential but also downside risk. From the differences between the average and median of the DBV result from version 1999 to version 2003 it is concluded that the distribution of the result of DBV is skewed; the values are not equally distributed about the mean. In this case the distribution has a negative skew: there are a small number of (very) negative results for DBV.

4.4.2 Scenario – Annual Premium Payments

In this scenario the volatility and expected rate of return are kept the same as in the standard scenario. The contract has a term of 25 years but an annual premium of 1000 instead of a single premium. In this scenario the same conclusions can be drawn as in the standard scenario; for this reason the statistics are not presented.

4.4.3 Scenario – Term of Contract equal to 10 Years

In figure 4.2 the output with volatility and expected rate of return based on historical price developments is presented (

σ

=8.50% per year and

μ

=4.23% per year). This is a deviation on the standard scenario because the contract has a single premium of 1.000 but a term of 10 years instead of 25 years.

Figure 4.2 Statistics with volatility and expected rate of return based on historical price developments, that is 8.5% respectively 4.23%. The contract is single premium and runs for 10 years.

When looking at the statistics we first notice that the value of the guarantee is much lower than in the standard scenario. This is straightforward because the term of the contract is less than half the term of the standard scenario. Actually all statistics are in proportion of 10 years / 25 years except for the average of the DBV result of FlexInvest 1999 3%. This is slightly higher with a term of 25 years than with a term of 10 years though it would be expected to be much higher. Remarkable is that the result of DBV for version 2003 becomes more negative compared to the standard scenario whereas the results of DBV for versions 1999 become less negative.

Again the conditions of FlexInvest 2006 cause the presence of a single positive DBV result. The difference between version 2006 versus version 2003 and version 2003 versus version 1999 3% is present, though not as obvious as in the standard scenario. Similar as in the standard scenario is that the value of the guarantee of version 1999 3% versus version 2003 increases, even though the result of DBV decreases; the reason for this has been explained in the standard scenario.

Remarkable is the fact that the maximum of the DBV result of version 1999 3% respectively 4% are equal. This happens if there is no loss for DBV and because of the reason that the margins for 1999 3% respectively 4% are the same, the DBV

result is also the same. In the standard scenario this is not possible because of the long term of the contract.

Also remarkable is the fact that the percentage of contracts in which DBV is not covered is more or less the same as in the standard scenario. This means that it looks like the probability for DBV to suffer a loss is not depending on the term of the contract.

4.4.4 Scenario – FlexInvest Portfolio 70% Stocks / 30% Bonds

In figure 4.3 the output with volatility and expected rate of return based on historical price developments is presented but where the combination of stocks respectively bonds have changed to 70% respectively 30% (

σ

=14.87% per year and

μ

=3.90% per year). The contract has a single premium of 1.000 and a term of 25 years.

Actually a change in the combination of stocks and bonds is equal to a change in volatility and average rate of return. Changing the FlexInvest portfolio mix from 40% stocks respectively 60% bonds to 70% stocks respectively 30% bonds causes the volatility respectively the average rate of return of the FlexInvest portfolio to change from 8.5% respectively 4.23% to 14.87% respectively 3.90%. It is straightforward that a higher volatility and a lower average rate of return is not in line with economic theory and makes an investment less worth, though it is interesting to see what this does to the FlexInvest portfolio.

Figure 4.3 Statistics with volatility and expected rate of return based on a combination of 70% stocks and 30% bonds in the FlexInvest portfolio, that is 14.87% respectively 3.90%. The contract is single premium and runs for 25 years.

The values of the guarantee are in each version of FlexInvest more than twice than what they are in the standard scenario. In all but the version of 2006 this can be explained by the higher Return client – with guarantee that in its turn can be explained by the higher volatility in combination with the ‘click principle’. For the 2006 version the doubling of the value of the guarantee is mostly accounted for by a decrease in Return client – without guarantee. This can be explained by the lower expected rate of return. Because the ‘click principle’ is no longer applicable, the higher volatility doesn’t cause higher returns with guarantee like in the other versions.

The average losses for DBV also double compared to the standard scenario. Remarkable is that the margins stay more or less equal except for the 2006 version. In this version the average margin for DBV is much higher than in the standard scenario. We can explain this by the occurrence of relatively more returns higher than 10% (because of higher volatility) that causes high margins for DBV. The 95% upper bound for the versions 1999 and 2003 remain almost equal as in the standard scenario; due to the higher volatility the distribution becomes more stretched but in combination with the lower rate of return the 95% upper bound remains equal.

Looking at all extremes and upper and lower bounds we see the distribution of the DBV result has a lower mean and becomes more flat, that is larger extremes. Noticeable is the large difference between the median and average of the 2006 version. This means the distribution of the DBV result has a positive skew; there are a small number of (very) positive results for DBV.

Comparing the percentages in which DBV is not covered with the standard scenario we see DBV more frequently makes a loss. An explanation for this is because of lower average rate of return there is less return so less margin and higher probability on a loss and because of the ‘click principle’ in combination with the higher volatility.

4.4.5 Scenario – Assumed Return on Stocks of 7%

In figure 4.4 the output is presented of the volatility and expected rate of return when there has been made the assumption that the stocks of the FlexInvest portfolio will have an average rate of return of 7% and the total FlexInvest portfolio will have a volatility of 10%. This is equal to the FlexInvest portfolio having a volatility of 10.0% per year and an average rate of return of 5.60% per year. The contract has a single premium of 1.000 and a term of 25 years.

Figure 4.4 Statistics with volatility and expected rate of return based on the assumption that the stocks of the FlexInvest portfolio will have an expected rate of return of 7%. This results in the portfolio having a volatility of 10.0% and an average rate of return of 5.6%. The contract is single premium and runs for 25 years.

The average value of the guarantee is lower than in the standard scenario. This is explained because of the higher average rate of return it occurs less frequently for DBV not to make a return higher than the guaranteed return and thus there is less loss.

Now the conditions of FlexInvest version 2006 and version 2003 both cause the presence of a positive DBV result. The difference between version 2006 versus version 2003 and version 2003 versus version 1999 3% is present and even more obvious as in the standard scenario. The same as in the standard scenario is that the value of the guarantee of version 1999 3% versus version 2003 increases, even though the result decreases; the reason for this has been explained in the standard scenario.

Furthermore, like in the case with an annual premium payment and in the case with a term of 10 years, the maximum of the DBV result of version 1999 3% and 4% is the same. If there is no loss for DBV, the margins are the same for both FlexInvest conditions and thus too are the results.

It can be seen that in the scenario with a higher average rate of return almost all statistics improve, though the improvements made by the 2006 conditions are relatively much larger. This is true for the average results, extremes and the percentage of contracts in which DBV is not covered. Only the minima for all versions have become more negative because of the higher volatility.

4.5

Comparison with Analytical Pricing Formula

This section compares the results of the simulation model with the values given by the pricing formulas as calculated in chapter 3. Notice that in the analytical pricing formula the calculation has been made with the risk-neutral probability measure and in the simulation model with the physical probability measure. Under the risk-neutral probability measure today’s arbitrage-free price of the guarantee is equal to the discounted expected value (under this risk-neutral measure) of the future payoff of the guarantee. This measure is called risk-neutral because all financial assets in the economy have the same expected rate of return, no matter how much the amount of risk in the asset. In contrast, the physical probability measure is the actual probability measure of prices where an asset with high risk has a higher expected rate of return than an asset with low risk.

To be able to compare both values the simulations have been repeated with the risk-neutral probability measure. In figure 4.5 a table is presented with values both derived from simulation and the analytical pricing formulas; all values have been discounted to time t=0.

Figure 4.5 Table with values under the risk-neutral probability measure both derived from simulation and the analytical pricing formulas for T=25 and T=10 years.

It is clear that for the versions 1999 and 2003 there are big differences between the theoretical value and the value by simulation. These differences can be explained because compensation for losses made in preceding years and margins taken by DBV have not been incorporated in the analytical formulas.

For the version of 2006 it can be seen that the theoretical values are close to the values by simulation, though small differences remain. Again compensation for losses made in preceding years and margins taken by DBV have not been incorporated in the analytical formula, though it can be concluded these have much less impact on the return of the client than as in the versions 1999 and 2003.

4.6 Conclusion

This section gives an overview of the results of this chapter. In all scenarios it can be seen that:

• The 2006 version is the first version with positive result for DBV, except in the scenario where the expected rate of return is 7%. Then the 2003 version also has a positive DBV result.

• When changing from version 1999 3% to version 2003 there is more loss but relatively more margin: the result of DBV improves changing from version 1999 3% to version 2003.

• The improvements (higher lower bounds, minima, mean and percentages of covered contracts) for DBV when changing from version 2003 to version 2006 are bigger than when changing from version 1999 3% to version 2003; in the scenario of an average rate of return of 7% this is more clear while in the scenario of a contract term of 10 years this is less clear. This relative improvement is mainly caused by higher margins for DBV in the 2006 version.

• Moving along the versions of FlexInvest, starting at version 1999 4%, the percentage of contracts in which DBV is covered increases. For the scenario of 7% expected rate of return the percentages are relatively very high compared to the standard scenario.

• Changing from version 1999 3% to version 2003 mainly improves upside potential whereas changing from version 2003 to version 2006 also improves downside risk.

Overall the two most important conclusions are:

• In the scenario of 70% stocks and 30% bonds, the higher volatility is causing more extreme values of the DBV result, more losses and more uncovered contracts for DBV as a result of the ‘click principle’. In other words, because of the ‘click principle’ it is dangerous for DBV when volatility increases.

• The conditions of the 2006 version are much better for DBV (when looking at lower bounds, minima, mean and percentage of covered contracts) than the previous versions of FlexInvest.

Comparing the results of the analytical pricing formulas with the simulation results it is seen that for the versions 1999 and 2003 big differences remain; the performance of the analytical pricing formula for the version 2006 is accurate. In all formulas compensation for losses made in preceding years and margins taken by DBV are not been incorporated, it can be concluded these contract conditions have much less impact on the return of the client in version 2006 than as in the versions 1999 and 2003.

HEDGE STRATEGY

This chapter presents a hedge strategy. Its performance is measured in a simulation model similar like that of the previous chapter. First it will be explained which risks involved with FlexInvest will be hedged. Secondly the theoretical framework for perfect hedging is given. After that it is shown that the Dow Jones Eurostoxx 50 index is highly correlated with the stocks in the FlexInvest portfolio and thus derivatives on this index seem to be good instruments to use in the hedge strategy. Next the model that has been built to measure the performance of the hedge strategy and the assumptions for this model are explained in summary. Then some scenarios are presented and it is shown how the performance of the hedge strategy is in these different scenarios. Finally an overview of the results of this chapter is given.

5.1

Risks of FlexInvest

As it has been explained before, there are more risks on the FlexInvest portfolio then there have been considered in this paper; such risks are death-, buy off-, investment and interest rate risk. So far the FlexInvest portfolio has been considered to follow a single geometric Brownian motion, though for the construction of a hedge strategy it is necessary to look at the evolution of stock and bond prices separately.

The model that is constructed in this chapter will simulate future FlexInvest stock prices by geometric Brownian motion; bond prices will be kept constant. We keep the average rate of return on bonds constant because of two reasons. Firstly we think the interest rate risk is lower than the investment risk. This is based on the long-term bond contracts in the present portfolio, present low interest rates (historically seen) and relatively ‘easy’ hedge instruments (see chapter 6). Secondly the results are clearer if we use interest rate scenarios rather than an interest rate model; it allows us to change only one parameter at a time. The hedge strategy will not concern the interest rate risk; though being interesting it leads to a too complex situation which is beyond the scope of this paper. It remains to find a hedge strategy for all stocks present in the FlexInvest portfolio.

5.2 Perfect

Hedging

The option pricing theory developed by Black and Scholes relays on the arbitrage argument by which investors can use a replicating portfolio of a position in the

stock portfolio and a position in bonds to exactly reproduce the payoff of any option. Note that for a perfect hedge, that is to eliminate all the risk from the total position, the weights of the replicating portfolio have to be adjusted continuously. The key insight is that the price behavior of an option is very similar to a portfolio of the underlying stocks and bonds that is revised in a particular way over time. As the value of the stock portfolio increases, bonds are sold and the position in the stock portfolio is increased. As the value of the stock portfolio declines, the position in the stock portfolio is decreased and bonds are purchased.

Any payoff P(St, t) can be replicated by means of a self-financing trading strategy.

A trading strategy is described by a pair of weights

(

ψ

_t

,

φ

_t

)

, where

ψ

_t is the number of shares and

φ

_t is the number of bonds held in a portfolio

Π

at time t. The value of the portfolio at time t is equal to:

t t t t t

=

ψ

S

+

φ

B

Π

(5.1) A self-financing strategy is defined by a pair of predictable processes

(

ψ

_t

,

φ

_t

)

satisfying: 1.

∫

+

∫

<

∞

T T t t

dt

0 0 2

φ

ψ

(5.2) 2.

+

=

+

+

∫

+

∫

∀

∈

[ ]

t t u u u u t t t t

S

B

S

B

dS

dB

t

T

0 0 0 0 0 0

φ

ψ

φ

0

,

ψ

φ

ψ

(5.3)

(see Etheridge, page 113). The self-financing property can be explained in that rebalancing the portfolio cannot involve any extra input of cash; the purchase of more stocks must be funded by the sale of bonds and vice versa. In discrete time symbols: t t t t t t t t

S

+

φ

B

=

ψ

+1

S

+

φ

+1

B

ψ

(5.4) The payoff P(St, t) can be replicated by the self-financing strategy:

( )

S

t

S

P

t t

,

∂

∂

=

ψ

and

φ

_t

=

B

_t−1

(

P

( )

S

_t

,

t

−

S

_t

ψ

_t

)

(5.5) Now assume that there is no compensation for losses made in preceding years and no margin for DBV, then the return of DBV, the client and the stock portfolio for one year can be represented as in figure 5.1.

-15% -12% -9% -6% -3% 0% 3% 6% 9% 12% 15% -10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

Return Stocks Portfolio

RETURN DBV RETURN CLIENT RETURN PORTFOLIO

Figure 5.1 Payoff diagram of the return of DBV, the client and the stock portfolio for one year when it is assumed that there is no compensation for losses made in preceding years, no margin for DBV and a guaranteed return of 3%.

From figure 5.1 it can be seen that DBV can perfectly hedge its investment risk, given the assumptions, with a long position in a put option on the stock portfolio with a strike price of the guaranteed return and a maturity of one year. Figure 5.2 reflects the result of this strategy.

-15% -12% -9% -6% -3% 0% 3% 6% 9% 12% 15% -10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

Return Stocks Portfolio

RETURN DBV RETURN PUT RETURN PUT + DBV

Figure 5.2 Payoff diagram of the return of DBV, a put option and the sum of these for one year when it is assumed that there is no compensation for losses made in preceding years, no margin for DBV and a guaranteed return of 3%.

Given that the payoff of a put option is equal to:

(

S

T

K

)

(

K

S

T

)

P

,

=

max

0

,

−

(5.6) and continuously applying the replicating strategy of equation (5.4) the investment risk for DBV is perfectly hedged.

Because the stock portfolio of FlexInvest consists of 40 stocks it is inconvenient to apply the hedge strategy above; this implies trading in all these stocks continuously. Also continuous trading in reality is not possible and would lead to extreme transaction costs (Leland, 1985). Because of these reasons this chapter continues with a more realistic hedge strategy of trading only once a year in put options on an index highly correlated with the stocks of FlexInvest.

Later on it is shown that in reality this strategy also has a disadvantage: sometimes the costs of the hedge strategy are much more than the proceeds. Reason for this is that we have assumed no compensation for losses made in preceding years and no margin for DBV.

5.3

Correlation Stocks FlexInvest and Dow Jones Eurostoxx 50

This section shows that the correlation between the Dow Jones Eurostoxx 50 (DJ Euro50) and the stocks in the FlexInvest portfolio is very high. This is expected because the benchmark for the stocks in the FlexInvest portfolio is the DJ Euro50. The Dow Jones Eurostoxx 50 is the average of stock prices of the 50 leading companies of the countries participating in the Euro. The DJ Euro50 is an important index of Europe and a reliable estimator of stock price developments in the Euro zone.

The correlation coefficient is a measure for the correlation between two stochastic variables and can be written as:

( )

_{( ) ( )}

y

x

y

x

y

x

σ

σ

ρ

,

=

cov(

,

)

(5.7)

where

σ

2

( )

a

is the variance of a.

The prices and thus returns of stocks in the FlexInvest portfolio are calculated as has been explained in the previous chapter. Likewise the prices and thus returns of the DJ Euro50 have been collected from the last 1829 trading days. Now with these two time series it is straightforward to calculate the correlation; this is equal to 97.4%.

Because of this high correlation and because the benchmark for DBV’s investments is the DJ Euro50, it seems logical to hedge the stocks in the FlexInvest portfolio with derivatives on the DJ Euro50. The hedge strategy that will be examined in this model is that at the beginning of each year put options on the DJ Euro50 will be bought with a strike price of the given guarantee and a maturity of one year. If the realized return of DBV is lower than the return guaranteed to the client (equal to the strike of the put option), DBV can exercise the put option and compensates its loss. If the realized return of DBV is higher than the return guaranteed to the client, the option expires worthless. By cumulating the prices and payoffs of the put option it is possible to see at the end of the contract whether the hedge strategy was successful or not.

5.4 The

Model

The model has been build up from the model used in the previous chapter. To calculate the clients return with guarantee, clients return without guarantee, cumulated loss and cumulated margin DBV exactly the same calculations have been used. One calculation has been added for the calculations of the hedge strategy. The calculation in which previously the FlexInvest portfolio price path was simulated has now been changed to simulate prices of FlexInvest stocks, FlexInvest bonds, the FlexInvest portfolio and the DJ Euro50. Simulating many times and taking the average gives in the long run the theoretical value of FlexInvest; with the simulation output the percentage of contracts in which DBV has suffered a loss and upper and lower bound percentiles can be seen. It is possible now to compare the distribution of the result of DBV with and without the hedge strategy. Appendix D gives a detailed description of the calculations made in the model.

5.4.1 Rebalancing

An important difference with the model of the previous chapter is that because stock and bond prices for the FlexInvest portfolio are simulated separately, it has become necessary to follow the percentages of stocks and bonds in the portfolio. These preferred percentages, and upper and lower bound, are defined in the contract. The asset manager of DBV controls these percentages mainly by investing new premiums in either bonds or stocks. In the model, if the upper or lower bound of the percentage of stocks or bonds is exceeded, the proportion of stocks and bonds will be reset to the preferred percentage.

5.4.2 Correlated Asset Prices

In this model the prices for FlexInvest stocks are simulated. It is required to simulate DJ Euro50 prices with a correlation as calculated above (97.4%). Like in the previous chapter a set of variables with a Gaussian distribution is produced by using the Box-Muller transformation. To correlate the DJ Euro50 with the stocks of FlexInvest the Cholesky decomposition is applied (Rapisarda, Brigo and Mercurio). This results in that if

δ

₁ and

δ

₂ are independent standardized normal variables then: 1 1

δ

ε

=

(5.8) and 2 2 1 2

δ

ρ

δ

1

ρ

ε

=

+

−

(5.9)

are correlated with correlation coefficient

ρ

. Implementing these two variables in the geometric Brownian motion the result is that both asset prices are correlated with correlation coefficient

ρ

.

5.4.3 Put Options

The number of put options that will be bought is equal to a percentage of the proportion of the FlexInvest stock price and the DJ Euro50 stock price. Note that when taking this percentage at 100% the costs of the hedge strategy will be much more than the proceeds; this way the lower bound and minimum become even more negative. When taking this percentage low (say 10%) the costs of the hedge strategy will not be much but the proceeds will not be much also. Figure 5.3 reflects the results of several percentages when there is a constant return on stocks of -5% and +5% for 25 years; the result put is equal to the proceeds minus the costs of the put option.

-600 -450 -300 -150 0 150 300 450 1 3 5 7 9 11 13 15 17 19 21 23 25 Year Stock Price Result Put - 100% Result Put - 50% Result Put - 25% Result Put - 10% -2,000 -1,500 -1,000 -500 0 500 1,000 1,500 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Year

Figure 5.3 Result of several percentages when there is a constant return on stocks of -5% (left) and 5% (right). Result put is equal to the proceeds minus the costs of the put option.

The movement in the result of DBV is not linear dependent on the return of the DJ Euro50 (or the return of FlexInvest stocks) because of compensation for losses made in preceding years and a non-linear profit sharing function. This makes it difficult to determine the ‘best’ percentage; it depends on the risk preferences of DBV which percentage is bought. In the remaining part of this chapter the calculations are done with a percentage of 50% of the proportion of the FlexInvest stock price and the DJ Euro50 stock price.

The price of the put option in the model will be determined with the Black-Scholes option pricing formula. When a stock is currently trading at price S, the exercise price is K and time to maturity is T years this price is given by:

( )

S

,

T

Ke

N

(

d

₂

)

SN

(

d

₁

)

P

=

−rt

−

−

−

(5.10)

Here

N

( )

⋅

is again the standard normal cumulative distribution function and:

T

T

r

K

S

d

σ

σ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

+

⎟

⎠

⎞

⎜

⎝

⎛

=

2

ln

2 1 (5.11)

T

d

d

₂

=

₁

−

σ

(5.12)

The strike price of the put option is the price at which the stock can be sold at the end of the contract. Comparing simulations done under equal circumstances but changing the strike it can be seen that when increasing the strike price a higher price for the put option has to be paid and the lower bound and minimum increases. Disadvantage of a higher strike price is that the upper bound and maximum decreases. In the remaining part of this chapter the calculations are done with a strike price of 3% higher than the value at the beginning of the contract.

5.5 Statistics

This section presents and comments outcomes of interesting scenarios. Similar statistics like that of the previous chapter are shown with and without the performance of the hedge strategy. Again this will be done for every version of the FlexInvest product. The statistics will show averages discounted at the average rate of return of the FlexInvest portfolio; not discounted averages are given in appendix E. The standard scenario and deviations on this standard scenario that will be considered are the same as in the preceding chapter. The differences in statistics between all FlexInvest versions have been analyzed in chapter 4; this chapter only analyzes the figures concerning the hedge strategy. All other figures are in line with the statistics from chapter 4 though because of rebalancing these figures are not exactly the same.

5.5.1 Standard scenario – Based on Historical Price Development

In figure 5.3 the output with volatility and expected rate of return based on historical price developments is presented (

σ

=8.50% per year and

μ

=4.23% per year). The contract has a single premium of 1.000 and a term of 25 years.

Figure 5.3 Statistics with volatility and expected rate of return based on historical price developments, that is 8.5% respectively 4.23%. The contract is single premium and runs for 25 years.

Looking at the average result of DBV it can be seen this result increases when comparing with and without hedge strategy; for all versions the real costs of the hedge strategy are the same (equal to 20). This increase is caused by a lower loss for DBV and not expected because it is expected that DBV has to pay a positive amount for a hedge strategy. This can be explained by the fact that with 4.671% the risk free rate is higher than the expected rate of return of 3.56% on stocks FlexInvest (and 3.71% on DJ Euro50); would the risk free rate be lower then the hedge strategy will cost a positive amount, this can be seen later on in this chapter.

Furthermore it is interesting to see what happens to the distribution of the result of DBV. It can be seen that for all versions the maximum and 95% upper bound decrease and the minimum and 5% lower bound increase. This can be explained in that the hedge strategy always has positive costs, so lower maximum and 95% upper bound, but provides compensation for negative results for DBV, so higher minimum and 5% lower bound. Moving along the versions, starting at 1999 4%, it can be seen the decrease of maximum and upper bound become less and increase in minimum and lower bound become more; stated otherwise the hedge strategy performs better moving along all versions where the 2006 version performs best. Remarkable is that all versions but 1999 4% have a lower percentage of a not covered result for DBV with hedge than without hedge. Not good is the lower minimum of result of DBV in the 1999 3% version; where the hedge strategy should compensate negative results for DBV in this simulation both the DBV result and the hedge strategy were negative. We conclude that overall the hedge strategy is performing very well.