analytic solution for return of premium and rollup guaranteed minimum death benefit ... · 2018. 5....

ANALYTIC SOLUTION FOR RETURN OF PREMIUM AND ROLLUPGUARANTEED MINIMUM DEATH BENEFIT OPTIONS

UNDER SOME SIMPLE MORTALITY LAWS

BY

ERIC R. ULM1

ABSTRACT

Much attention has been focused recently on the issue of valuing guaranteedminimum death benefits embedded in annuity contracts. These benefits resemblea sequence of put options and their value should obey a differential equationsimilar to the Black-Scholes equation for simple put options. This paper derivesa number of analytic solutions to this equation for a number of simple mor-tality laws.

KEYWORDS

Guaranteed Annuity Options, Black-Scholes Equation.

1. INTRODUCTION

The valuation of Guaranteed Minimum Death Benefit (GMDB) riders embed-ded in variable annuity contracts has been a hot topic in recent years. In themain contract, a premium is paid that accumulates a sum of money on a taxdeferred basis which can be converted to a payout annuity upon retirement.The initial premium can be invested in a number of different types of funds,usually including both a fixed fund earning a guaranteed rate of interest anda variable fund invested in equities.

In order to differentiate their products, variable annuity issuers have addeda number of different types of riders, including GMDBs. These riders guaran-tee some minimum benefit at death, which is frequently the return of premiumspaid and sometimes includes the accumulation of those premiums at a fixedinterest rate. This situation is commonly called a “rollup” GMDB. Because

Astin Bulletin 38(2), 543-563. doi: 10.2143/AST.38.2.2033353 © 2008 by Astin Bulletin. All rights reserved.

1 Eric Ulm, FSA, is an Assistant Professor in the Department of Risk Management and Insurance,J. Mack Robinson College of Business, Georgia State University, Atlanta, GA. Phone: 404-413-7485.E-mail: [email protected]. Some work performed while at the University of Central Florida,Orlando, FL.

the underlying assets are invested in stock funds, GMDB riders resemble asequence of European put options, with the put value on a given date beingmultiplied by the probability the annuitant dies that date and has not yet lapsedhis policy. The integral that determines the value of the option can be writtenas follows (see, for example, Hardy (2003)),

a

x t+

,

, ,

f S t

Xe e e N S w Se N d S w t w p dwmpt pw rw qw

w2 1

0

=

- - - +

3- - ( )td#

]] ] ]

gg g g6 6@ @% / (1)

where X is the strike at issue, p is the roll-up rate, r is the risk-free rate, S isthe stock level today, q is the level of asset fees, x is the age at issue and t isthe years since issue. Also:

, ;

,

ln

ln

d S ww

XeS r q p w

d S ww

XeS r q p w

s

s

s

s

2

2

pt

pt

1

2

2

2

=

+ - - +

=

+ - - -

J

L

KK

J

L

KK

]d

]d

N

P

OO

N

P

OO

gn

gn

(2)

A number of papers address issues specific to GMDB pricing, includingMilevsky and Posner (2001) and Milevsky and Salisbury (2001), which derivesa differential equation that must be satisfied by the GMDB value. A numberof solutions have been found assuming constant mortality and lapse for returnof premium GMDBs, both in that paper and by Ulm (2006). This paper extendsthose results to roll-up GMDBs. In addition, solutions are found for terminsurance and endowment insurance under constant force of mortality. Finally,a solution is found if mortality follows DeMoivre’s Law.

2. THE DIFFERENTIAL EQUATION TO BE SOLVED

Assume the existence of a deferred annuity with a variable account. The annu-ity has a GMDB rider that guarantees a return of premium with p% interestcompounded continuously upon the death of the annuitant. This could bemodeled as the sum of a continuous sequence of European put options (see,for example, Hardy (2003)). The weight at a given option duration would beequal to the instantaneous probability of death at that moment. The value ofthe strike at that moment would be Xept, where X is the value of the initial pre-mium paid. The equation that must be satisfied by the value of the GMDB,fa(S, t ), is:

544 E.R. ULM

a a

a

2

, ,

t r q S S SS

r t S t t Max Xe S

s

l

21

0

a

x xpt

22

2

22

22

2

2+ - +

= + + - -m m

f f f

f

^] ] ] a

hg g g k6 6@ @

(3)

which is derived in Milevsky and Salisbury (2001), or Ulm (2006) assuming thatthe stock fund follows a geometric Brownian motion with standard devia-tion s . fa(S, t) is the value of the option if the individual is alive and has notlapsed his policy. q represents the asset fees that are taken out of the account.µx (t) is the force of mortality for a person age (x) and l (S, t) is the lapse rate,which could in theory depend on both the stock level and time. In the usualBlack-Scholes differential equation, the hedge must grow at the risk free rate.In deriving Equation (3), it is the expected value of the option that must behedged, which consists of both the live value of the option and the expectedvalue of any benefits paid out on death or surrender. This accounts for thelarger factor multiplying the value of fa(S, t) on the right hand side of Equa-tion (3), as well as the additional subtracted term. Equation (3) can be derivedby constructing a riskless portfolio of GMDB riders and underying stockwhich must earn the risk-free rate, as in the derivation of the standard Black-Scholes equation. Alternatively, the integral in equation (1) can be shown tosatisfy equation (3).

2.1. General Solution to Equation (3).

Assuming a constant lapse rate l(S,t)∞∞=∞∞l, but without making any assumptionson µx(t), it can be shown that

a , ,S t Xe A t SB t H Xe S Xe f w yt tpt pt ay= - - +f ] ] ] a ] ^g g g k g h9 C (4)

with the following definitions: lny XS

= ` j and tT ts2

2

=-] g are dimensionless vari-

ables. A(t) and B (t) are Ax+ t (single premium for a fully continuous whole lifepolicy on an x + t year old) at force of interest r + l – p and q + l respectively.H(x) is the Heaviside step function.

ar q

s21

= --

2

^ h(5)

sf f e et t( ) ( )s dsk t t m

0 t

t2

02=

- - -0 #] ^g h , t0 and f (t0) arbitrary (6)

And

ROLLUP GUARANTEED MINIMUM DEATH BENEFIT OPTIONS 545

2

2

2

2

2

2

,a

a

w y e e f sA s

se

ds

e e f sB s

se

ds

ee f s

A sy pT s

se

ds

ee f s

B sy pT s

se

ds

tpb

t

pb

t

pb

t

pb

t

4

4

4

4

1

( ) ( )

( ) ( )

( )( )

/

( )( )

/

a a

a a

aa

aa

pT sy pT s s

pT sy pT s s

pTs

y pT s s

pTs

y pT s s

bt b t

bt b t

bt b t

bt b t

1 1

0

4

1 1

0

4

11

03

4

11

03

4

=+

-

-+ -

-

+ - --

- - --

- -- - - -

- -- - - -

--

- - - -

--

- - - -

#

#

#

#

^ ]]] ]

]]] ]

]] ^ ]] ]

]] ^ ]] ]

h ggg g

ggg g

gg h gg g

gg h gg g

5

5

6 5

6 5

?

?

@ ?

@ ?

(7)

where T is either an endowment age, or the age at which the force of mortality

becomes infinite, and b =s

p22

. The details of the derivation are in Appendix A.

3. SOLUTION TO EQUATION (3) UNDER SPECIFIC MORTALITY LAWS

3.1. The Value of a Roll-Up GMDB under De Moivre’s Law Mortality

The general procedure for determining the value of the rollup is to find A(t),B(t) and f(t), substitute them into Equation (7), and then compute the integrals.

For De Moivre’s Law, t T tm1

=-] g so tm t

s2=

2] g .

also:

sA r p etl ts

2 1 t( )r pl2

=+ -

--

+ -

2

2] ^g h < F (8)

sB q etl t

s2 1 t

( )q l2

=+

--

+

2

2] ^g h < F (9)

also, from Equation (6) with the arbitrary constant t0 chosen to satisfy t0 ekt0 = 1,and f (t0) = 1:

fe

t t

kt

=-J

L

KK] N

P

OOg (10)

Substituting Equations (8), (9) and (10) into Equation (7) yields the followingset of integrals for w (y,t):

546 E.R. ULM

We will use the following identities:


2

2

2

2

2

2

2

2

2

2

2

2

, a

a

w y e r p se e

ds

se e

ds

e q se e

ds

se e

ds

e r p se e

y pT s ds

se e

y pT s ds

e q se e

y pT s ds

se e

y pT s ds

t bl

sp t

p t

bl

sp t

p t

ls

p tb

p tb

ls

p tb

p tb

2 4

4

2 4

4

2 4

4

2 4

4

1

( )( )

( )( )

( ) ( )

( )/

( )

/

( )/

( )

/

( ) ( )

aa

a a

aa

a a

aa

a a

aa

a a

pTs y pT s s

s y pT s s

pTs y pT s s

s y pT s s

pTs y pT s s

s y pT s s

pTs y pT s s

s y pT s s

k b b tt

b b tt

k b b tt

b b tt

k b b tt

b b tt

k b b tt

b b tt

11 4

0

4

0

11 4

0

1 1 4

0

13

1 4

0

3

4

0

13

1 4

0

3

1 1 4

0

2

2

2

2

= ++ - -

--

- + -+ -

--

++ - -

- -

--

- -

-+ -

- -

--

- -

-+ - - - - -

+ - - - -

-+ - - - - -

- + - - - - -

-+ - - - - -

+ - - - -

-+ - - - - -

- + - - - - -

2

2

2

2

#

#

#

#

#

#

#

#

^ ^ ^ ]] ]

]] ]

^ ^ ]] ]

]] ]

^ ]] ] ^

]] ] ^

^ ]] ] ^

]] ] ^

h h h gg g

gg g

h h gg g

gg g

h gg g h

gg g h

h gg g h

gg g h

R

T

SSS

R

T

SSS

R

T

SSS

R

T

SSS

5 5

7 5

5 5

7 5

5 5 67 5 6

5 5 67 5 6

V

X

WWW

V

X

WWW

V

X

WWW

V

X

WWW

? ?

A ?

? ?

A ?

? ? @A ? @

? ? @A ? @

2

t

se e

ds

e e

a

ee N

y pTa

e Ny pT

a

p t

b tbt b

t

tbt b

t

4

2 4

22 4

22 4

as y pT s s

y T

aa y pT

a y pT

b tt

b

bt

bt

4

0

2

2 2

2

pb b b

b

2 2

2

4

2

4

2

-

=

+

-- +

- +

- -- +

+ +

- - - -

+- + - +

- + - +

-

#

J

L

KK

J

L

KKK

J

L

KK

J

L

KKK

]] ]

d N

P

OO

N

P

OOO

N

P

OO

N

P

OOO

gg g

n R

T

SSSS

R

T

SSSS

5

V

X

WWWW

V

X

WWWW

?

*

4(12)

(11)

and

This leads, after much algebra, to the following solution for fa(S, t):

REGION I. S > Xept

m m

a ,S t T tC

XeXe

S N T tC

XeXe

S N

q T tSe N d r p T t

Xe e N d

z z

l l

( ) ( ) ( ) ( )

ptpt

ptpt

q T t pt r p T tl l

11

22

1 2

1 2

=-

- +-

-

++ -

- -+ - -

-- + - - + - -

f ] ] d ^ ] d ^

^ ] ^ ^ ] ^g g n h g n h

h g h h g hand

REGION II. S < Xept

m m

a ,S t r p T t

Xe e

q T t

S e

T tC

XeXe

S N T tC

XeXe

S N

r p T tXe e N d q T t

Se N d

l l

z z

l l

1 1( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

pt r p T t q T t

ptpt

ptpt

pt r p T t q T t

l l

l l

11

22

2 1

1 2

=+ - -

--

+ -

-

--

--

++ - -

-+ -

- + - - - + -

- + - - - + -

f ] ^ ]`

^ ]`

] d ^ ] d ^

^ ] ^ ^ ] ^

g h gj

h gj

g n h g n h

h g h h g h(14)

with definitions:

548 E.R. ULM

2

2

t

t

se e

y pT s ds

e ey pT

e y pTe N

y pTa

e Ny pT

a

e e

a

ee N

y pTa

e Ny pT

a

p tb

btbt

tbt b

t

tbt b

t

b

b tbt b

t

tbt b

t

4

2 22 4

22 4

22 4

22 4

22 4

/

as y pT s s

y T

aa y pT

a y pT

y T

aa y pT

a y pT

b tt

b

bt

bt

b

bt

bt

3

4

0

2 2

2

2

2 2

2

p

p

b b b

b

b b b

b

2 2

2

4

2

4

2

2 2

2

4

2

4

2

-- -

=- +

- +-

- +- +

+ -- +

+ +

-

+

-- +

- +

- -- +

+ +

- - - -

+- + - +

+ - +

+- + - +

+ - +

-

-

#

J

L

KK

J

L

KKK

J

L

KK

J

L

KKK

J

L

KK

J

L

KKK

J

L

KK

J

L

KKK

]] ] ^

d ^

d

N

P

OO

N

P

OOO

N

P

OO

N

P

OOO

N

P

OO

N

P

OOO

N

P

OO

N

P

OOO

gg g hn h

n

R

T

SSSS

R

T

SSSS

R

T

SSSS

R

T

SSSS

5 6V

X

WWWW

V

X

WWWW

V

X

WWWW

V

X

WWWW

? @Z

[

\

]]

]

Z

[

\

]]

]

_

`

a

bb

b

_

`

a

bb

b

(13)

(15)

;

;

; ;

; ; ; ;

;

;

ln

ln

ln ln

dT t

XeS r q p T t

dT t

XeS r q p T t

T tXe

S K T t

T tXe

S K T t

m A K m A K Ar q p

K Ar p

C q KA

r p KA

C r p KA

q KA

s

s

s

s

zs

sz

s

s

s s

l

l l

l l

2

2

21 2

21 1 1 2

1 1

21 1 2

1 1 1

pt

pt

pt pt

1

2

1

2

2

2

1 22

1

2

=-

+ - - + -

=-

+ - - - -

=-

+ -

=-

- -

= + = - = -- -

= ++ -

=+

-- -

+ --

=+ -

+ -+

+-

2

2

2 2

J

LKK

J

LKK

J

LKK

J

LKK

]]

]]

]d ]

]d ]

^ ^

^ ^^ ^

N

POO

N

POO

N

POO

N

POO

gg

gg

gn g

gn g

h h

h hh h

< << <

F FF F

The existence of distinct solution in the two regions is due to the absolutevalues in equation (13) combined with careful evaluation of expressions of the

form A2 in the later stages of the derivation.

3.2. The Value of a Roll-Up GMDB under Constant Force of Mortality withEndowment

We now turn our attention to solution of Equation (3) for constant mortality.The solution for p = l = 0 was found in Milevksy and Salisbury (2001) and thesolution for p = 0; l ! 0 is in Ulm (2006). We now find the answer for p ! 0.We will begin by finding the value assuming an endowment age correspond-ing to an endowment time T. At this age, a surviving individual receives hisfull GMDB. We will finally allow T to go to infinity.

We will follow a procedure analogous to that in section 3.1. We must find A(t),B(t) and f(t), substitute them into Equation (7), and then compute the integrals.In this case, m(t) = m and m(t) = m. Including the effect of the endowment,

A r p r pr p

et m lm

m ll t

( )r p

s

m l2

=+ + -

++ + -

+ - -+ + -

2] dg n (16)

and


B q qq

et m lm

m ll t

( )q

s

m l2

=+ +

++ +

+ -+ +

2] dg n (17)

also, from Equation (6) with f (t0) = 1

-sf e e et ( ) ( )dsk t t m k g tt

t

0 22

0= =- - - +#] g (18)

where

gs

m2=

2(19)

Substituting Equations (16), (17) and (18) into Equation (7) yields the followingset of integrals for w(y, t):

550 E.R. ULM

2

2

2

2

2

2

2

2

2

2

2

2

, a

a

w y e r p se e

ds

r pr p

se e

ds

e q se e

ds

qq

se e

ds

e r p se e

y pT s ds

r pr p

se e

y pT s ds

e q se e

y pT s ds

qq

se e

y pT s ds

t b m lm

p t

m ll

p t

b m lm

p t

m ll

p t

m lm

p tb

m ll

p tb

m lm

p tb

m ll

p tb

4

4

4

4

4

4

4

4

1

( )( )

( )( )

( ) ( )

( )/

( )

/

( )/

( )

/

( ) ( )

aa

a a

aa

a a

aa

a a

aa

a a

pTs y pT s s

s y pT s s

pTs y pT s s

s y pT s s

pTs y pT s s

s y pT s s

pTs y pT s s

s y pT s s

k g b b tt

b b tt

k g b b tt

b b tt

k g b b tt

b b tt

k g b b tt

b b tt

11 4

0

4

0

11 4

0

1 1 4

0

13

1 4

0

3

4

0

13

1 4

0

3

1 1 4

0

2

2

2

2

= ++ + - -

++ + -

+ -

-

- + -+ + -

++ +

+

-

++ + - -

- -

++ + -

+ -

-- -

-+ + -

- -

++ +

+

-- -

-+ + - - - - -

+ - - - -

-+ + - - - - -

- + - - - - -

-+ + - - - - -

+ - - - -

-+ + - - - - -

- + - - - - -

#

#

#

#

#

#

#

#

^ ^ ]] ]

]] ]

^ ]] ]

]] ]

]] ] ^

]] ] ^

]] ] ^

]] ] ^

h h gg g

gg g

h gg g

gg g

gg g h

gg g h

gg g h

gg g h

R

T

SSS

R

T

SSS

R

T

SSS

R

T

SSS

5 5

7 5

5 5

7 5

5 5 67 5 6

5 5 67 5 6

V

X

WWW

V

X

WWW

V

X

WWW

V

X

WWW

? ?

A ?

? ?

A ?

? ? @A ? @

? ? @A ? @

Applying Equations (12) and (13) and working through the algebra gives thefollowing results for fa(S, t).

(20)

REGION I. S > Xept

m m

a ,S t XeXe

S N XeXe

S N

r pr p

Xe e N d

qq

Se N d

z z

m ll

m ll

( ) ( )

( ) ( )

ptpt

ptpt

pt r p T t

q T t

m l

m l

1 1 2 2

2

1

1 2

= - + -

++ + -

+ --

-+ +

+-

- + + - -

- + + -

C Cf ] d ^ d ^^ ^^ ^

g n h n hh h

h hand

REGION II. S < Xept

m m

a ,S tr p

Xeq

S

C XeXe

S N C XeXe

S N

r pr p

Xe e N d

qq

Se N d

m lm

m lm

z z

m ll

m ll

( ) ( )

( ) ( )

pt

ptpt

ptpt

pt r p T t

q T t

m l

m l

1 1 2 2

2

1

1 2

=+ + -

-+ +

- -

++ + -

+ --

-+ +

+-

- + + - -

- + + -

f ] ^ ^d ^ d ^

^ ^

^ ^

g h hn h n h

h h

h h

(21)

The parameters are defined as in Equation (15), with the exceptions:

;

;

K Ar p

Cq K

Ar p K

A

Cr p K

Aq K

A

s

m l

m lm

m lm

m lm

m lm

2

21 1 2 1

2 1 2 1 1

2

1

2

= ++ + -

=+ +

-- -

+ + --

=+ + -

+ -+ +

+-

2

^

^ ^^ ^

h

h hh h

< << <

F FF F

(22)

The value of the endowment alone is given by the value of a put option attime T multiplied by the probability that the person has not lapsed and is aliveto receive the endowment. This value is therefore:

Xe e N d Se N d( ) ( ) ( ) ( )pt r p T t q T tm l m l2 1- - -

- + + - - - + + -^ ^h h (23)

in both ranges. The term insurance alone is the difference between Equations (21)and (23).


The value of a rollup GMDB under constant force of mortality at all agescan be found from Equation (21) by allowing the endowment age T " �. Thisgives the following equations:

m

a> ,S Xe S t XeXe

Spt ptpt2

2

= Cf ] dg nm

a< ,S Xe S tr p

Xeq

S C XeXe

Sm l

mm l

mpt pt ptpt1

1

=+ + -

-+ +

-f ] ^ ^ dg h h n (24)

These equations collapse to those shown in Milevsky and Salisbury (2001) forconstant force of mortality and l = p = 0.

4. NUMERICAL COMPARISONS

We will compare the values that we obtain from these derived formulas withnumeric integration of Equation (1). Two features of Equation (1) are worthmentioning. First, the interest rate r only appears in the combination r – p.This implies that Equations (14), (21) and (24) should not involve r except in thecombination r – p and this turns out to be true. Second, the strike X only appearsin the combination Xept, i.e. the strike at time t. This implies that only that com-bination should appear in Equations (14), (21) and (24), which is also true.

We will now compare the results of Equations (14), (21) and (24) with theresults one would get from Equation (1). These results should be identical, andwe include them here as a check that the analytic solution is correct.

For De Moivre’s Law, Equation (1) becomes:

a , , ,S t Xe e e N S w Se N S w T te dwpt pw rw qw

T t wl

2 1

0

= - - --

- -- -

d df #] ] ]g g g6 6@ @$ . (25)

Equation (25) must be evaluated using numeric techniques (although the pro-cedure described in the preceding sections could be viewed as a very compli-cated way of finding an analytic solution to the integral). Table 1 gives somevalues of Equation (14) and Equation (25) for various values of X, r, p, q, s,l, S and T – t.

For constant force with endowment, Equation (1) becomes:

(26)

a , , ,

,

,

S t Xe e e N S w Se N S w e dw

Xe e e N S T t

Se N S T t e

m ( )

( ) ( )

( ) ( ) ( )

pt pw rw qwT t

w

pt p T t r T t

q T t T t

m l

m l

2 1

0

2

1

= - - -

+ - -

- - -

- --

- +

- - -

- - - + -

d d

d

d

f #] ] ]]

]

g g gg

g

6 66

6

@ @@

@

%%

/

/

552 E.R. ULM

Again, Equation (26) is evaluated using numeric integration. Table 2 gives somevalues of Equation (21) and Equation (26) for various values of X, m, r, p, q,s, l, S and T – t.

Finally, for constant force, Equation (1) becomes:

a , , ,S t Xe e e N S w Se N S w e dwm ( )pt pw rw qw wm l2 1

0

= - - -

3- - - +d df #] ] ]g g g6 6@ @$ .

(27)

Table 3 gives some values of Equation (24) and Equation (27), also evaluatedby numeric integration, for various values of X, m, r, p, q, s, l and S. In all threetables, the differences are mostly due to the discreteness of the numerical inte-gration, and the closed form answers are actually the more accurate values.


TABLE 1

COMPARISON OF CLOSED FORM AND INTEGRATION FOR DE MOIVRE’S LAW MORTALITY

Strike r p q s l S T – t Closed Form Integration

1 8% 3% 1% 20% 2% 0.5 20 0.21424733 0.214247301 8% 3% 1% 20% 2% 1 20 0.05555276 0.055552751 8% 3% 1% 20% 2% 2 20 0.00671562 0.006715651 5% 3% 1% 20% 2% 1 20 0.12664668 0.126646691 8% 1% 1% 20% 2% 1 20 0.03202345 0.032023461 8% 3% 4% 20% 2% 1 20 0.09328518 0.093285181 8% 3% 1% 40% 2% 1 20 0.16380654 0.163806581 8% 3% 1% 20% 10% 1 20 0.02982873 0.029828751 8% 3% 1% 20% 2% 1 40 0.03611639 0.03611640

TABLE 2.

COMPARISON OF CLOSED FORM AND INTEGRATION FOR CONSTANT FORCE OF MORTALITY

WITH ENDOWMENT

Strike m r p q s l S T – t Closed Form Integration

1 2% 8% 3% 1% 20% 2% 0.5 20 0.12626180 0.126261821 2% 8% 3% 1% 20% 2% 1 20 0.03977509 0.039775111 2% 8% 3% 1% 20% 2% 2 20 0.00779967 0.007799521 6% 8% 3% 1% 20% 2% 1 20 0.05063465 0.050634661 2% 5% 3% 1% 20% 2% 1 20 0.12604128 0.126041241 2% 8% 1% 1% 20% 2% 1 20 0.01810033 0.018100311 2% 8% 3% 4% 20% 2% 1 20 0.07744060 0.077440571 2% 8% 3% 1% 40% 2% 1 20 0.13253453 0.132534601 2% 8% 3% 1% 20% 10% 1 20 0.01472027 0.014720291 2% 8% 3% 1% 20% 2% 1 40 0.02587016 0.02587016

5. APPLICABILITY OF THE ANALYTIC FORMULAS

Once the formulas in equations (14), (22) and (24) have been found, it isimportant to know under what circumstances they can be used in practicalcalculations. In the following sections, we address some of the primary issuesassociated with the use of these formulas.

5.1. Applicability of the Mortality Laws

One of the primary drawbacks of the derived formulas is the reliance on sim-plistic mortality laws in their derivation. We therefore tested the formulasagainst more realistic mortality represented by the 2000 Group Annuity Mor-tality table.

5.1.1. Tests of the Constant Force of Mortality Formulas

We will begin by comparing results of equation (24) with results from the 2000Group Annuity Mortality table. The initial parameters for r, p, q, s and l usedare those in the first row of Table 3. The value of S is allowed to vary from 0to 2 in steps of 0.1 and the value of m is age dependent and is set such thatthe value from equation (24) and the value found by integrating equation (1)for the realistic mortality are equal at S = 0. The results at ages 40 and 65 areshown in Table 4. The values at age 65 are in quite good agreement with aconstant force approximation. However, the values at age 40 show significantdeviations. In particular, although the values at S = 0 have been set equal, theslopes have not been, and the slopes are matched much more closely at age 65than at age 40. This suggests adjusting two parameters in order to match boththe value and the slope at S = 0. The most reasonable parameters to adjust arethe mortality and lapse rates, which now become age dependent.

554 E.R. ULM

TABLE 3.

COMPARISON OF CLOSED FORM AND INTEGRATION FOR CONSTANT FORCE OF MORTALITY AT ALL AGES.

Strike m r p q s l S Closed Form Integration

1 2% 8% 3% 1% 20% 2% 0.5 0.08498966 0.084989541 2% 8% 3% 1% 20% 2% 1 0.02326991 0.023269941 2% 8% 3% 1% 20% 2% 2 0.00363245 0.003632411 6% 8% 3% 1% 20% 2% 1 0.04547443 0.045474541 2% 5% 3% 1% 20% 2% 1 0.07619048 0.076189251 2% 8% 1% 1% 20% 2% 1 0.01214452 0.012144541 2% 8% 3% 4% 20% 2% 1 0.04300611 0.043006161 2% 8% 3% 1% 40% 2% 1 0.07425119 0.074251301 2% 8% 3% 1% 20% 10% 1 0.01096458 0.01096464

The value of the GMDB option at S = 0 can be found by substitution intoequation (1). It turns out that

a x, Xe e e w p dw r pm m lm

0 0 pw rw wx w

l

0

= =+ + -

3- -f #] ]g g (28)

The slope a

SD22

=f

at S = 0 can be found by differentiating the inside of the inte-

gral in equation (1) and then substituting S = 0. So,


TABLE 4.

COMPARISON OF CONSTANT FORCE EQUATION AND 2000 GROUP ANNUITY MORTALITY AT AGE 40 AND 65

Strike m Age r p q s l S Constant Force 2000GAM

1 0.86% 40 8% 3% 1% 20% 2% 0 0.10912893 0.109128931 0.86% 40 8% 3% 1% 20% 2% 0.1 0.09046434 0.084113241 0.86% 40 8% 3% 1% 20% 2% 0.2 0.07506622 0.067769891 0.86% 40 8% 3% 1% 20% 2% 0.3 0.06188460 0.055484951 0.86% 40 8% 3% 1% 20% 2% 0.4 0.05052989 0.045651571 0.86% 40 8% 3% 1% 20% 2% 0.5 0.04077308 0.037479351 0.86% 40 8% 3% 1% 20% 2% 0.6 0.03245779 0.030509811 0.86% 40 8% 3% 1% 20% 2% 0.7 0.02546816 0.024449591 0.86% 40 8% 3% 1% 20% 2% 0.8 0.01971373 0.019098271 0.86% 40 8% 3% 1% 20% 2% 0.9 0.01512125 0.014312461 0.86% 40 8% 3% 1% 20% 2% 1 0.01162979 0.009994991 0.86% 40 8% 3% 1% 20% 2% 1.2 0.00731340 0.007409591 0.86% 40 8% 3% 1% 20% 2% 1.4 0.00494076 0.005702001 0.86% 40 8% 3% 1% 20% 2% 1.6 0.00351763 0.004513311 0.86% 40 8% 3% 1% 20% 2% 1.8 0.00260681 0.003653481 0.86% 40 8% 3% 1% 20% 2% 2 0.00199385 0.003012061 4.72% 65 8% 3% 1% 20% 2% 0 0.40288709 0.402887091 4.72% 65 8% 3% 1% 20% 2% 0.1 0.34438793 0.341475261 4.72% 65 8% 3% 1% 20% 2% 0.2 0.29099806 0.287082881 4.72% 65 8% 3% 1% 20% 2% 0.3 0.24260656 0.239673091 4.72% 65 8% 3% 1% 20% 2% 0.4 0.19915553 0.198228011 4.72% 65 8% 3% 1% 20% 2% 0.5 0.16060496 0.161791811 4.72% 65 8% 3% 1% 20% 2% 0.6 0.12692423 0.129572521 4.72% 65 8% 3% 1% 20% 2% 0.7 0.09808852 0.100927631 4.72% 65 8% 3% 1% 20% 2% 0.8 0.07407701 0.075335361 4.72% 65 8% 3% 1% 20% 2% 0.9 0.05487173 0.052369431 4.72% 65 8% 3% 1% 20% 2% 1 0.04045691 0.031717021 4.72% 65 8% 3% 1% 20% 2% 1.2 0.02353173 0.021164121 4.72% 65 8% 3% 1% 20% 2% 1.4 0.01488257 0.014829261 4.72% 65 8% 3% 1% 20% 2% 1.6 0.01000730 0.010781031 4.72% 65 8% 3% 1% 20% 2% 1.8 0.00705154 0.008072611 4.72% 65 8% 3% 1% 20% 2% 2 0.00515568 0.00619253

ax, S e e w p dw qm m l

mD 0 0

S

qw wx w

l

0 022

= = - = -+ +

3

=

- -f #] ]g g (29)

The two equations for m and l solve for:

a

a a

, ,, , , ,r p q

l DD D

0 0 0 01 0 0 0 0 1 0 0 0 0

=- -

- + + -

ff f] ]^ ] ] ] ]g gh g g g g6 6@ @

(30)

556 E.R. ULM

TABLE 5

COMPARISON OF CONSTANT FORCE EQUATION AND 2000 GROUP ANNUITY MORTALITY AT AGE 40 AND 65WITH MATCHED SLOPES

Strike m Age r p q s l Adj l S Closed Form 2000GAM

1 0.65% 40 8% 3% 1% 20% 2% 0.33% 0 0.10912893 0.109128931 0.65% 40 8% 3% 1% 20% 2% 0.33% 0.1 0.08783152 0.084113241 0.65% 40 8% 3% 1% 20% 2% 0.33% 0.2 0.07190695 0.067769891 0.65% 40 8% 3% 1% 20% 2% 0.33% 0.3 0.05885151 0.055484951 0.65% 40 8% 3% 1% 20% 2% 0.33% 0.4 0.04792422 0.045651571 0.65% 40 8% 3% 1% 20% 2% 0.33% 0.5 0.03872864 0.037479351 0.65% 40 8% 3% 1% 20% 2% 0.33% 0.6 0.03101132 0.030509811 0.65% 40 8% 3% 1% 20% 2% 0.33% 0.7 0.02459377 0.024449591 0.65% 40 8% 3% 1% 20% 2% 0.33% 0.8 0.01934238 0.019098271 0.65% 40 8% 3% 1% 20% 2% 0.33% 0.9 0.01515273 0.014312461 0.65% 40 8% 3% 1% 20% 2% 0.33% 1 0.01194058 0.009994991 0.65% 40 8% 3% 1% 20% 2% 0.33% 1.2 0.00784918 0.007409591 0.65% 40 8% 3% 1% 20% 2% 0.33% 1.4 0.00550523 0.005702001 0.65% 40 8% 3% 1% 20% 2% 0.33% 1.6 0.00404886 0.004513311 0.65% 40 8% 3% 1% 20% 2% 0.33% 1.8 0.00308765 0.003653481 0.65% 40 8% 3% 1% 20% 2% 0.33% 2 0.00242291 0.003012061 4.41% 65 8% 3% 1% 20% 2% 1.53% 0 0.40288709 0.402887091 4.41% 65 8% 3% 1% 20% 2% 1.53% 0.1 0.34287480 0.341475261 4.41% 65 8% 3% 1% 20% 2% 1.53% 0.2 0.28886389 0.287082881 4.41% 65 8% 3% 1% 20% 2% 1.53% 0.3 0.24035511 0.239673091 4.41% 65 8% 3% 1% 20% 2% 1.53% 0.4 0.19710314 0.198228011 4.41% 65 8% 3% 1% 20% 2% 1.53% 0.5 0.15894350 0.161791811 4.41% 65 8% 3% 1% 20% 2% 1.53% 0.6 0.12575293 0.129572521 4.41% 65 8% 3% 1% 20% 2% 1.53% 0.7 0.09743329 0.100927631 4.41% 65 8% 3% 1% 20% 2% 1.53% 0.8 0.07390323 0.075335361 4.41% 65 8% 3% 1% 20% 2% 1.53% 0.9 0.05509348 0.052369431 4.41% 65 8% 3% 1% 20% 2% 1.53% 1 0.04094380 0.031717021 4.41% 65 8% 3% 1% 20% 2% 1.53% 1.2 0.02416882 0.021164121 4.41% 65 8% 3% 1% 20% 2% 1.53% 1.4 0.01547731 0.014829261 4.41% 65 8% 3% 1% 20% 2% 1.53% 1.6 0.01052026 0.010781031 4.41% 65 8% 3% 1% 20% 2% 1.53% 1.8 0.00748396 0.008072611 4.41% 65 8% 3% 1% 20% 2% 1.53% 2 0.00551868 0.00619253

and

,,

qm l DD

1 0 00 0

= ++-^ ]]h gg< F (31)

m and l now both depend on the issue age of the policy. The results withmatched slopes are given in Table 5. The agreement is quite good. The mostsubstantial deviations occur around S = 1, which suggests that the formula isof more use in valuation than in pricing.


TABLE 6.

COMPARISON OF DEMOIVRE’S LAW EQUATION AND 2000 GROUP ANNUITY MORTALITY AT AGE 40 AND 65WITH MATCHED SLOPES

Strike Age r p q s l Adj l T – t S DeMoivre’s Law 2000GAM

1 40 8% 3% 1% 20% 2% 0.82% 157.33 0 0.10912893 0.1091291 40 8% 3% 1% 20% 2% 0.82% 157.33 0.1 0.08753263 0.0841131 40 8% 3% 1% 20% 2% 0.82% 157.33 0.2 0.07156195 0.0677701 40 8% 3% 1% 20% 2% 0.82% 157.33 0.3 0.05852753 0.0554851 40 8% 3% 1% 20% 2% 0.82% 157.33 0.4 0.04764948 0.0456521 40 8% 3% 1% 20% 2% 0.82% 157.33 0.5 0.03851424 0.0374791 40 8% 3% 1% 20% 2% 0.82% 157.33 0.6 0.03085899 0.0305101 40 8% 3% 1% 20% 2% 0.82% 157.33 0.7 0.02449955 0.0244501 40 8% 3% 1% 20% 2% 0.82% 157.33 0.8 0.01929858 0.0190981 40 8% 3% 1% 20% 2% 0.82% 157.33 0.9 0.01514912 0.0143121 40 8% 3% 1% 20% 2% 0.82% 157.33 1 0.01196512 0.0099951 40 8% 3% 1% 20% 2% 0.82% 157.33 1.2 0.00789809 0.0074101 40 8% 3% 1% 20% 2% 0.82% 157.33 1.4 0.00555902 0.0057021 40 8% 3% 1% 20% 2% 0.82% 157.33 1.6 0.00410086 0.0045131 40 8% 3% 1% 20% 2% 0.82% 157.33 1.8 0.00313569 0.0036531 40 8% 3% 1% 20% 2% 0.82% 157.33 2 0.00246650 0.0030121 65 8% 3% 1% 20% 2% 2.24% 30.54 0 0.40288709 0.4028871 65 8% 3% 1% 20% 2% 2.24% 30.54 0.1 0.34083990 0.3414751 65 8% 3% 1% 20% 2% 2.24% 30.54 0.2 0.28468545 0.2870831 65 8% 3% 1% 20% 2% 2.24% 30.54 0.3 0.23519353 0.2396731 65 8% 3% 1% 20% 2% 2.24% 30.54 0.4 0.19201047 0.1982281 65 8% 3% 1% 20% 2% 2.24% 30.54 0.5 0.15467650 0.1617921 65 8% 3% 1% 20% 2% 2.24% 30.54 0.6 0.12276945 0.1295731 65 8% 3% 1% 20% 2% 2.24% 30.54 0.7 0.09592391 0.1009281 65 8% 3% 1% 20% 2% 2.24% 30.54 0.8 0.07382662 0.0753351 65 8% 3% 1% 20% 2% 2.24% 30.54 0.9 0.05620810 0.0523691 65 8% 3% 1% 20% 2% 2.24% 30.54 1 0.04283475 0.0317171 65 8% 3% 1% 20% 2% 2.24% 30.54 1.2 0.02636432 0.0211641 65 8% 3% 1% 20% 2% 2.24% 30.54 1.4 0.01738788 0.0148291 65 8% 3% 1% 20% 2% 2.24% 30.54 1.6 0.01206547 0.0107811 65 8% 3% 1% 20% 2% 2.24% 30.54 1.8 0.00870532 0.0080731 65 8% 3% 1% 20% 2% 2.24% 30.54 2 0.00647835 0.006193

We now compare the results of equation (14) with the 2000 GAM table.We expect greater agreement in this case because DeMoivre’s Law is a betterapproximation of human mortality in the sense that mortality rates rise withage and there is a maximum age as well. Again, the initial parameters for r, p,q, s and l used are those in the first row of Table 3. The value of S is allowedto vary from 0 to 2 in steps of 0.1. Finally, we vary both T – t and l to matchboth the values and slopes at S = 0 and the results are presented in Table 6.

5.2. Stochastic Interest Rates

Equations (1) and (3) are both based on the idea that yield curves are flat anddo not change over time. If the yield curve is not flat, but future interest ratesare completely determined by today’s forward rates, equation (1) is changedonly marginally to:

x t+

a ,

, ,

S t

Xe e e N S w Se N S w t w p dwm ( )pt pw r w qww

t2 1

0

w t

=

- - - +

3- -d d

f

#

]] ] ]

gg g g6 6@ @$ . (32)

where wrt represents the forward rate of interest for cash flows invested at time tand maturing at time t + w. We will now check equation (24) against a numer-ical evaluation of equation (32) for a non-level yield curve. The initial para-meters for p, q, s and l used are those in the first row of Table 3, and thestarting yield curve is t r0 = .08 – .04e– t /10. We will set the rate r = 7.14% inequation (24) to be the one that sets the two equations equal at S = 0. Theagreement is quite good, with a maximum deviation of 0.0034. Checking equa-tion (14) against equation (32) under similar conditions yields similarly goodagreement.

Values of put options when interest rates are stochastic were first consideredby Merton (1973). Rabinovitch (1989) derives option prices for the specific casewhere interest rates follow a constant volatility mean reverting process. This isunlikely to have a large effect, as Kim and Kunitomo (1999) claim that “the trendof interest rate plays a crucial role in determining the stock option values. Theeffects of the volatility of interest rate process as well as the correlation betweenthe stock price and interest rate are of secondary importance”. We have alreadyexamined the effect of interest rate trends and shown it to be relatively unim-portant. Stochastic hazard rates have a similarly small effect if the mortalityyield curve implied in market prices of annuities is used in the formulas.

It may seem at this point that fitting to numerical evaluations do not resultin any gain from using the closed form solutions. The advantage comes whena model needs to rapidly compute a large number of option prices. In this case,the parameters in the closed form can be precomputed and used repeatedly,which could result in a considerable time savings.

558 E.R. ULM

FIGURE 1: Sensitivity of the Constant Force Equation to its Parameters.The Base Values are m = 0.0086, r = 5%, p = 3%, q = 1%, s = 20% and l = 10%.

5.3. Sensitivity of Results to Parameter Values

We now show the sensitivity of the equations to their parameters for a reason-able set of starting values. We used the constant force of mortality formula inEquation (24) and took numerical derivatives with respect to six parameters.The results are shown in figure 1. The graph shows the change in the value ofthe option for a 1% change in the parameter values for various market levels.The value of the option increases with an increase in most parameter values,but decreases with an increase in lapse rates or the risk-free rate. Therefore, weplot the absolute values of these derivatives to facilitate comparison.

When the GMDB is in-the-money, the option value is most sensitive tochanges in mortality and lapse rates. In addition, the lapse rate is the mostdifficult parameter to estimate. When the GMDB is out-of-the-money, thevolatility and the risk-free rate become more important than the lapse andmortality rate. However, these economic parameters can be estimated muchmore easily than the lapse rate. The risk-free rate is available from the currentyield curve, and the volatility can be inferred from traded option values. It appearsthat good estimates of the lapse rate, while difficult, are necessary for accuratevaluation of GMDB riders. Next, we use Table 4 and Figure 1 to compare theerrors introduced by using a simplified mortality law to the errors introduced byuncertainty in parameter values. The errors in Table 4 are of the order of 0.0001to 0.001. This is the same general range of 1-10% errors in parameter values.That is, using a lapse rate of 10% in valuation when the experience lapse rateis 11% (a 10% change) is at least as important as the error introduced by usinga simplified mortality law.


0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0 0.5 1 1.5 2

Stock Level S

Cha

nge

in V

alue

m

- l

- r

s

p

q

6. CONCLUSIONS

In this paper, we have solved the differential equation that must be satisfied fora rollup GMDB attached to a variable annuity contract for two well-knownmortality laws. Solutions have been found previously only in the case of con-stant force of mortality and rollup interest rate p = 0%. This paper extends theresults to positive p, as well as providing a solution for term insurance andendowment insurance under constant force of mortality. In addition, solutionsfor the GMDB value under DeMoivre’s law mortality are derived. The answerscompare well with past work, as well as answers obtained from commonlyused numerical integration methods.

APPENDIX A. DERIVATION OF EQUATIONS (4)-(6)

Equation (3) can be rewritten as:

aa

a at r t r q S S S

S

t Xe S H Xe S

m l s

m

21

x

xpt pt

22

2

22

22

2

2- + + + - + =

- - -

2ff

f f] ^] `

g hg j

66 9

@@ C

(A1)

where H (x) is the Heaviside step function, whose derivative is the Dirac deltafunction d (x). Now, I will assume fa(S, t) is of the form:

a , ,S t Xe A t SB t H Xe S C S tpt pt= - - +f ] ] ] ` ]g g g j g9 C (A2)

Substituting Equation (A2) into Equation (A1) gives:

560 E.R. ULM

Xe A t r t p Xe A t SB t q t SB t H Xe S

Xe SA t r q r q S B t pX e A t pXe SB t Xe S

Xe S A t S B t Xe S

tC

r t C r q S SC

SSC

t Xe t S H Xe S

m l m l

s d

s s d

m l s m m

� �

�

21

pt pt pt

pt pt pt

pt pt

x xpt

xpt

2 2 2

21 2

21 3

22

2

22

22

2

- + + - + - + + + -

+ - - + - + + - -

+ - -

+ - + + + - + = - + -

2

2 2

2

pt

2

] ]^ ] ] ]^ ] a] ^ a ] ] ] a

] ] a] ^ ] ] a

g g h g g g h g kg h k g g g k

g g kg h g g k

6

6 9

@

@ C

%%%

//

/

9 C

The heaviside functions can be canceled and the right hand side of equation(A3) can be set equal to 0 if

A�(t) – (r + m(t) + l – p)A(t) = – m(t) (A4)

(A3)

and

B�(t) – (q + m(t) + l)B(t) = – m(t) (A5)

Bowers et al page 125 (1997) states that

dxd Ax – (d + m(x)) Ax = – m(x) (A6)

where d is the force of interest and Ax is the net single premium for a wholelife policy on a person age x. This implies that A(t) and B(t) are Ax+ t at forceof interest r + l – p and q + l respectively.

C must now obey the following equation:


tC

r t C r q S SC

SSC

r q Xe SA t r q S B t pX e A t pXe SB t Xe S

S B t Xe S A t Xe S

m l s

s d

s s d �

21

x

pt pt pt

pt pt

22

2

2 2 2

21 3

21 2

22

22

2- + + + - +

= - - - + - + -

+ - -

2

2

2 2

pt

2] ^^ ] a ] ] ] a

] ] a

g hh g k g g g k

g g k

6 @%%

// (A7)

The left hand side is similar to Equation (3) but the right hand side now has sourcesrelated to the delta function and its derivatives. We now make Equation (A7)dimensionless. The derivation follows similar lines to that found in Wilmott et al.(1995). We pick dimensionless variables y = ln X

S` j and t = ( )T Ts2-2

. T is cur-rently an arbitrary parameter as, unlike the case for the vanilla Europeanoptions, there is no expiration date to the GMDB option. Later, we willsee that T represents the age at which the GMDB must be exercised. For a

mortality function such as de Moivre’s law with a built in maximum age,T represents the time remaining until that age is reached. For a mortalityfunction without a maximum age, T can be viewed as the time until theGMDB is no longer a death benefit, but an endowment benefit. If there is noendowment age, we will let T " � as the final step. We will assume C(S, t) =Xeay f (t)w(y,t):

We chose a to be ( )r q

s21

--

2 , define b to be p

s

22 , and assume that f (t) satisfies:

afr

fts

m t lt�

20x 2

++ +

+ =2] ]^ ]g g h g= G (A8)

Being careful with the delta functions (see Barton (1989)), this leads to thefollowing equation for w(y,t):

( ) ( )a awyw e e e f

Be f

A

e e e fA

e fB

y pT

e e e fB

e fA

y pT

t tt

tt

b tt

b tt

d bt

tt

tt

d bt�

3 2 1 2( ) ( )

( )

( ) ( )

a a

a

a a

y pT y

ay pT y

y pT y

bt

bt

bt

2

22 1

1

2 1

22

2- = - - -

+ - - -

+ - - -

- - -

- - -

- - -

2 ]] ]]

]] ]] ^]] ]] ^

gg gg

gg gg hgg gg h

66

@@

(

(22

(A9)

The ordinary differential equation for f(t) is solved by:

-sf f e et t ( ) )(s dsk t t m

0t

t

0 22

0=- - #] ^g h (A10)

where ark

sl2 2

=+

+2

] g , t0 and f(t0) are arbitrary constants, and m(s) is the

functional form of m(t) not m(t).Equation (A9) is a diffusion equation with sources. The solution is:

562 E.R. ULM

, ( )

( )

a

a

w ys

ee e e pT s d ds

se e pT s d ds

s

ee e e pT s d ds

se e pT s d ds

s

ee e e pT s d ds

se e pT s d ds

tp t

d z b z

p td z b z

p tb d z b z

p tb d z b z

p td z b z

p td z b z

�

�

43 2

42

4

4

4

4

1

( )( )

( ) ( ) ( )

( )( )

( ) ( ) ( )

( )( )

( ) ( )

( )( )

( ) ( ) ( )

( )( )

( ) ( ) ( )

( )( )

( ) ( ) ( )

a

a

a

a

a

a

sf s

B spT y s

f sA s

y s

sf sA s

pT y s

f sB s

y s

sf s

B spT y s

f sA s

y s

btz z t

tz z t

btz z t

tz z t

btz z t

tz z t

0

2 4

0

1 4

0

4

0

1 4

0

2 4

0

1 4

2

2

2

2

2

2

=-

- - -

+-

- -

+-

- -

--

- -

+-

- -

--

- -

-

3

3

3

3

3

3

3

3

3

3

3

3

-

- - - - -

-

- - - -

-

-

- - - -

-

- - - -

-

- - - - -

-

- - - -

##

##

##

##

##

##

^ ] ^

] ^

] ^

] ^

] ^

] ^

h g h

g h

g h

g h

g h

g h

6

6

6

6

6

6

@

@

@

@

@

@

assuming w(y,0) = 0. This is the same condition as

a ,S T Xe A T SB T H Xe SpT pT= - -f ] ] ] `g g g j9 C

implying that T is either the age at which mortality is 100%, or is an endowmentage that can be allowed to increase to infinity.

(A11)

Integrating over the delta functions gives:

(A12)

2

2

2

2

,a

a

w y e e f sA s

se ds

e e f sB s

se ds

e e f sA s

y pT ss

e ds

e e f sB s

y pT ss

e dss

tpb

t

pb

t

pb

t

pb

t

4

4

4

4

1

( ) ( )( )

( ) ( )( )

( )( )

/

( )

( )( )

/

( )

a a

a a

aa

aa

pT sy pT s s

pT sy pT s s

pTs

y pT s s

pTs

y pT s s

bt b t

bt b t

bt b t

bt b t

1 1

0

4

1 1

0

4

11

03 2

4

11

03 2

4

=+

-

-+ -

-

+ - --

- - --

- -- - - -

- -- - - -

--

- - - -

--

- - - -

#

#

#

#

^ ]]]

]]]

]] ^ ]]

]] ^ ]]

h ggg

ggg

gg h gg

gg h gg

5

5

6 5

6 5

?

?

@ ?

@ ?

REFERENCES

BARTON, G. (1989) Elements of Green’s Functions and Propagation, New York: Oxford Univer-sity Press.

BOWERS, N.L., GERBER, H.U., HICKMAN, J.C., JONES, D.A. and NESBITT, C.J. (1997) ActuarialMathematics, Schaumburg, Illinois: The Society of Actuaries.

HARDY, M. (2003) Investment Guarantees, Hoboken, N.J.: John Wiley and Sons.KIM, Y.J. and KUNITOMO, N. (1999) Pricing Options under Stochastic Interest Rates: A New

Approach, Asia-Pacific Financial Markets 6, 49-70.MERTON, R.C. (1973) Theory of Rational Option Pricing, Bell Journal of Economics and Man-

agement Science 4(1), 141-183.MILEVSKY, M.A. and POSNER, S.E. (2001) The Titanic Option: Valuation of the Guaranteed Min-

imum Death Benefit in Variable Annuities and Mutual Funds, Journal of Risk and Insurance68(1), 93-128.

MILEVSKY, M.A. and SALISBURY, T.S. (2001) The Real Option to Lapse and the Valuation of Death-Protected Investments, Conference Proceedings of the 11th Annual International AFIR Col-loquium, Sep 2001, p. 537.

RABINOVITCH, R. (1989) Pricing Stock and Bond Options when the Default Free Rate is Sto-chastic, Journal of Financial and Quantitative Analysis 24(4), 447-457.

ULM, E.R. (2006) The Effect of the Real Option to Transfer on the Value of Guaranteed Min-imum Death Benefits, Journal of Risk and Insurance 73(1), 43-69.

WILMOTT, P., HOWISON, S. and DEWYNNE, J. (1995) The Mathematics of Financial Derivatives,Cambridge, England: Cambridge.

ERIC R. ULM

Department of Risk Management and InsuranceJ. Mack Robinson College of BusinessGeorgia State University, Atlanta, GA.Tel.: 404-413-7485.E-mail: [email protected].


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