The Effect of Policyholder Transfer Behavior on the Value of
Guaranteed Minimum Death Benefits
Eric R. Ulm1
Georgia State University
JEL Classification: G13
Keywords:
Abstract
Variable annuity contracts frequently include both Guaranteed Minimum Death
Benefit options and options to transfer funds between fixed and variable accounts. We
model the difference between fixed and variable rates as the primary determinant of
policyholder transfer behavior. We find that people tend to transfer their money into
variable accounts at about 39% of the rate that would be required to maintain constant
percentage rebalancing, but with the opposite sign. If these transfers are not taken into
account, the GMDB options on the variable accounts will be overvalued and over
hedged. Ignoring this effect can have a substantial impact on the size of the futures
portfolio needed to hedge this risk and a non-negligible impact on the earnings of the
variable annuity portfolio.
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: 404-463-4809. E-mail: inseuu@langate.gsu.edu.
1. Introduction
Insurance contracts often contain two types of options, financial options and
“real” options. Over the last ten to twenty years, insurance companies have become more
adept at recognizing and hedging the financial options embedded in their contracts. One
such option is the guaranteed minimum death benefit, or GMDB, embedded in variable
annuity contracts. A GMDB pays the maximum of variable annuity contract value or
initial premium on death. This acts as a put option on the fund with a random exercise
time, the moment of the policyholder’s death. These options have been examined in some
detail in, for instance, Milevsky and Posner (2001), Hardy (2003), Ulm (2004) and Ulm
(2008). The valuation of these options is straightforward using numerical methods, and
insurance companies have been steadily developing and implementing strategies for delta
hedging the options.
Less recognized, however, are the real options available to policyholders and how
they interact with the financial options in the contracts. Milevsky and Salisbury (2001)
have looked at the policyholder option to surrender a contract that is out-of-the-money
and replace it with a new at-the-money contract. Ulm (2006) has looked at the option
policyholders have to transfer money between funds that have different risk profiles.
Both of these papers assume optimal exercise of these real options on the part of
policyholders.
The assumption of optimal exercise is usually considered reasonable in the case of
tradeable options. In this situation, most market participants will be knowledgeable and
exercise optimally, and the other participants will find it more cost effective to sell the
option than to exercise it when they desire to cash out their position. On the other hand,
optimal exercise is not considered reasonable when participants own assets that cannot be
traded. Residential mortgages with prepayment options are one example. A number of
models of mortgage prepayments exist (see, for example, Richard and Roll (1989),
Schwartz and Torous (1989) or Hayre and Rajan (1995)), which do not posit optimal
behavior on the part of the homeowners. Also, insurance companies regularly use nonoptimal policyholder lapse and transfer behavior in their pricing, cash flow testing and
asset-liability management models.
2. The Model of Real Policyholder Transfer Behavior
First, we will address the issue of whether policyholders tend to rebalance their
portfolios to maintain a constant percentage in fixed and variable accounts, as
recommended by financial planners, or they choose an initial allocation and then transfer
rarely, allowing both accounts to grow independently at their separate rates. To answer
this question, we used data supplied by Morningstar Inc. They report quarterly data on
the breakdown of variable annuity assets between equity, fixed, balanced, bonds and
money market funds, as well as quarterly returns on all classes except fixed. The return
on fixed accounts can be inferred from NAIC Annual Statements. We assume that the
fixed account contains the fixed and money market funds and the variable account
contains the equity, bond and balanced funds. This information is available from 1st
Quarter 1998 to 4th Quarter 2006.
If the hypothesis of constant percentage rebalancing were true, you would expect
the percentage in variable funds in one quarter to be the same as in the previous quarter.
We find that the square root of the average squared deviation of this prediction is 0.0235
or that this hypothesis typically mispredicts the variable funds by about 2.35%. To test
the “buy and hold” hypothesis, we assumed that the fund performance agreed with that
supplied by Morningstar Inc. and that there were no transfers. In this case, the square root
of the average squared deviation of this prediction is 0.0112 or this hypothesis typically
mispredicts the variable funds by about 1.12%. Table 1 contains the values of the variable
account percentages and the two predicted percentages. These values are displayed
graphically in Figure 1. It is clear from the graph that the “buy and hold” hypothesis
predicts the account percentages better than the rebalancing hypothesis, especially in
rapidly changing markets as was true in late 2002 and early 2004.
It is also evident from the graph that both hypotheses underpredict the magnitude
of the change in the variable account percentage. In the falling market of late 2002, both
hypotheses predict more variable funds than are actually found, whereas in early 2004,
both hypotheses predict fewer variable funds. This means that, relative to a buy and hold
strategy, policyholders transfer into variable funds when the market is rising, and out of
variable funds when the market is falling. This complements previous data on mutual
fund sales, which have shown ambiguous results. Ippolito (1992) shows using annual
data that investors direct investments toward individual funds that have performed well in
the recent past. Warther (1995) studies the full market mutual fund flows and concludes
that fund investors “do not follow positive feedback strategies”. That is, they do not in
general direct more money to stock funds as a result of those funds performing well.
Edelen and Warner (2001) find mutual fund flow responds to lagged returns with a
positive correlation over 1 day and negative correlations after 2-3 days.
There is still the complication that the base funds from one quarter to the next are
not identical. New accounts have been sold, new premium added and some policies have
lapsed. We will make the simplifying assumption that new premium is distributed
proportionately to existing funds, and that lapses come proportionately from existing
funds.
2.1 The Policyholder Transfer Model.
We will start with a model of variable annuity fund performance. A percentage p
of the funds will be invested in an variable account of size S , itself a stochastic process,
and therefore 1  p will be invested in an account earning a fixed rate g . The fund value
F obeys the following equation:
dF  g (1  p) Fdt  pF
dS
S
(1)
In the case of constant rebalancing, the percentage p is constant. For the buy and
hold case, p obeys the following equation, assuming that S is a differentiable function,
rather than a Brownian motion:
 dS

dp  p(1  p)
 gdt 
 S

(2)
If S follows a geometric Brownian motion process, then
 dS

dp  p(1  p)
 gdt   p 2 (1  p ) 2 dt
 S

(3)
where the second term is present to correct for the drift caused by the quadratic variation
in dS . When Equation 2 is treated as an ordinary differential equation, and Equation 3 is
treated as a stochastic differential equation, both equations solve for
S
S0
p0
p
(4)
S
(1  p0 )e  p0
S0
gt
which is the correct answer in the buy-and-hold case. The derivations of both equations
can be found in Appendix A.
We will assume transfers are proportional to the difference between the
instantaneous difference in fund rates
dS
 gdt . We will also assume that they are
S
proportional to the percentage in the fund being transferred from, and the fund being
transferred into, i.e. proportional to p (1  p ) .
This latter assumption is mathematically useful, as well as having some intuition
behind it. Those policyholders with 0% in equity or 0% in fixed are unlikely to be
making constant transfers. Only those near the middle seem likely to transfer. In addition,
if either fund is at 100%, it is impossible to transfer more money into it. This transfer
assumption prevents this situation from occurring. Finally, the values of p (1  p ) are
between 0.16 and 0.24 when p is in the generally realized ranges from 60% to 80%, so
the effect of this assumption is small. This means, in our model, with S differentiable,
not stochastic:
 dS

 dS

 dS

dp  p(1  p)
 gdt   qp(1  p)
 gdt   p(1  p)(1  q)
 gdt 
 S

 S

 S

and, if S follows a geometric Brownian motion process, then
(5)
1
 dS

1

dp  p(1  p)(1  q)
 gdt   p 2 (1  p) 2 dt  qp(1  p)  2 p  q  p   2 dt (6)
 S

2

2
with the last two terms correcting for the drift from the quadratic variation in S as in
Equation 3. When Equation 5 is treated as an ordinary differential equation, and Equation
6 is treated as a stochastic differential equation, both equations solve for
p
 S
p0 
 S0



(1 q )
 S 
(1  p0 )e g (1 q )t  p 0  
 S0 
(7)
1 q
Now, assuming the stock market follows a standard geometric Brownian motion,
dS
 dt  dZ
S
(8)
We find from Equations 6 and 8:

1
dp  p(1  p)(1  q)    g  p 2  q 
2

 
p  2  dt  p(1  p)(1  q)dZ
 
(9)
2.2 Empirical Determination of the Value of the Transfer Parameter
We will now turn our attention to the determination of the transfer parameter q . It
can be seen from Equation 9 that a value of q  1 sets dp  0 and therefore q  1
corresponds to constant percentage rebalancing (CPR hereafter). A value of q  0
corresponds by design to the buy-and-hold strategy. Positive values of q imply
policyholders transfer into a rising market, and negative values of q implies transfers
into a falling market. Table 2 contains 35 quarterly values of p and returns determined
by the change in values of the Morningstar catagories over the quarter weighted by the
starting percentages. Milevsky and Posner (2001) find an industry average of 115 basis
points of asset fees, so
S
is determined as the returns less 115 bp in asset fees. We can
S0
adjust q in Equation 7 to fit these values. We will begin by transforming equation 7 to a
linear equation, obtaining:
  S
q ln 
  S0

 p(1  p 0 )e gt S 0

  gt   ln 

 (1  p) p 0 S





(10)
Performing a regression with slope q and intercept c gives q  0.39 and c  0.0044
with standard errors of 0.10 and 0.0082. The value of q is significant at the 0.001 level
with a 95% confidence interval 0.19  q  0.60 . The constant is not significant with a
95% confidence interval  0.0121  c  0.0210 , so there is no evidence of a bias to
transfer one way or the other. A plot of the regression is shown in Figure 2.
Figure 3 contains a comparison of the percentage of funds that would be predicted
if q  0 , the buy and hold strategy, and q  0.39 . The raw data can be seen in table 1.
The sum of the squared deviation goes down as you move from right to left across the
table. For the CPR model, the sum of the squared deviations is 0.01932 (off by 2.35% on
average). For the buy and hold model it is 0.004451 (off by 1.12% on average) and for
q  0.39 it is 0.003048 (off by 0.93% on average). A second estimate of the parameters
can be obtained by minimizing the sum of the squared deviations. This produces a value
of q  0.41 , well within the range given by the linear fit. The sum of the squared
deviations is 0.003045, negligibly different from that produced by the best linear fit.
3. GMDB Values under the Constant Percentage Rebalancing and Buy-and-Hold
Assumptions.
It is relatively straightforward to calculate the values of the GMDB option under
the constant percentage rebalancing assumption. The fund value obeys Equation 1 with
p constant. If S follows a Geometric Brownian Motion process, this leads to:
dF  g (1  p)  pFdt  pFdZ
(11)
Since p is constant, this is just a Geometric Brownian Motion with variance p . The
GMDB can be calculated in the usual way from the integral:



f a ( F0 , t )   Xe rw N  d 2 ( F0 , w)  SN  d1 ( F0 , w) w p x()t  x (t  w)dw
(12)
0
where
p 2 2 
F  
w
ln  0    r 
2 
X  
d1 ( F0 , w) 
p w
(13a)
and
p 2 2 
F  
w
ln  0    r 
2 
X  
d 2 ( F0 , w) 
p w
Where
w
(13b)
p x()t includes both mortality and lapse decrements (see Hardy 2003). The values
of d1 and d 2 are calculated with a variance of p .
The calculation of GMDB values under the buy and hold assumption is only
slightly more difficult. At any given time, the fixed fund amount is known to be
(1  p 0 ) F0 e gt . If this value is larger than the strike, the GMDB value will have no
contribution from the put option based on death at time t . On the other hand, if the fixed
fund is smaller than the strike at time t , the contribution to the GMDB at that time can be
converted to a put option at that time with an adjusted strike price J (t ) assuming an
initial stock level of 1, where:
J (t ) 

S 0 X  (1  p0 ) F0 e gt
p0 F0

(14)
The value of the contribution at time t needs be multiplied by
p0 F0
. Equation 14 comes
S0
from the fact that

S 
MaxX  F ,0  Max  X  (1  p0 ) F0 e gt  p0 F0
,0
S0 




 S X  (1  p0 ) F0 e gt

p0 F0
Max  0
 S ,0
S0
p0 F0


(15)
This leads to the following integral for the GMDB Value:
p F
f a ( F0 , t )  0 0
S0
 J (w)e
t0
 rw

N  d 2 ( J ( w), w)  S 0 N  d1 ( J ( w), w) 
0
 w p x()t  x (t  w)dw
(16)
where
 S  
2
ln  0    r 
2
 J ( w)  
d1 ( J ( w), w) 
 w

 w

(17a)
 S0  
2
   r 
ln 
2
 J ( w)  
d 2 ( J ( w), w) 
 w

 w

(17b)
and


X
ln 

F (1  p 0 ) 
t0   0
g
(17c)
The integral in equation 16 is zero if t 0  0 .
The next step is to determining the values of  
f a
to be used in hedging. The
S
calculation of delta in the constant percentage rebalancing case is straightforward. Since
p is constant,  
f a F
F f
f
 p 0 a . The remaining partial derivative, a , can be
F
F S
S 0 F
calculated numerically from equation 12.
The value of  in the buy and hold case is a little more difficult, as p can vary as
S changes. Therefore,  
f a F f a p
F f
p(1  p) f a
, with the

p 0 a 
F S p S
S 0 F
S0
p
remaining partial derivatives calculated numerically from equation 16. It requires fewer
calculations to calculate  
f a
numerically by increasing and decreasing S by some
S
small amount, calculating F  (1  p 0 ) F0 e gt  p 0 F0
S
and p 
S0
p0
S
S0
S
(1  p0 )e  p0
S0
in
gt
both the up and down cases, and then using equation 16 to calculate the value.
3.1 GMDB Values under q  1, q  0 assumptions.
We now tackle the much more difficult problem of calculating the value of the
GMDB option for arbitrary values of the transfer parameter q . There are three basic
methods of calculating GMDB values. The first is by using an analytic formula. Only a
few exact formulas exist (see Ulm 2008), but if the value of the option given death at a
specific time t is known analytically, the problem can be solved through integration, as
in equations 12 and 16. If the probability distribution of the fund values f F ( y) is known
at time t , the GMDB value can be determined by double integration, for example:

f a ( F0 )    e rt Max( X  y,0) f F ( y) t p x( )  x (t )dydt
(18)
0 0
The second method is through simulation. A number of Brownian Motion Paths
can be produced, and the fund value determined from Equations 8 and 9. This is
equivalent to determining the interior integral in Equation 18 stochastically, and then
computing the exterior integral numerically. Unfortunately, this method can take a
considerable amount of time.
The third method would be to solve the differential equation obeyed by the
GMDB:
f a
f
2 fa
1
 rS a   2 S 2
 [r   x (t )   ( S , t )] f a  [  x (t )]Max( X  S ,0)
t
S 2
S 2
(19)
either analytically or numerically. For a derivation of this equation, see Milevsky and
Salisbury (2001).
We implement the first method by finding a good approximation to the
probability density of the fund distribution under the transfer assumptions of the model in
Section 2. The second method is implemented directly, and acts as a check on the first
method. We also derive the differential equation satisfied by the GMDB value under
those transfer assumptions. In general, the differential equation is too difficult to solve
directly.
We first turn our attention to the determination of the fund values F (t ) . It is
shown in Appendix B that, under risk neutral growth assumptions,
1
1 q 1 q q  2  p 2  p dt




S
2 
F (t )  F0 (1  p0 )e r (1 q )t  p0    e 0

 S 0  
t
(20)
Unfortunately, the value is path dependent in the general case, due to the stochastic
integral in the last term. In practice, this isn’t as bad as it at first seems. If q  0 , the
stochastic term disappears and equation 20 reduces to the value of the fund in the buyand-hold case. If q  1 , the integral is non-stochastic as p  p0 . The integral is also
zero when p 0  0 and p0  1 . With so many anchor points, you might expect that
replacing the stochastic integral with the approximation
e
q 2

2
 p
t
0
2

 p dt
e
q 2

2
  p0  p0 dt
t
2
0
q
 e2


 2 p02  p0 t
(21)
would be reasonable. This approximation is exact in the four cases mentioned above. It is
very good when q is small or p 0 is near the extremes. Even when p 0 is not extreme, if
p remains in the range from 60%-80%, as is typical, and   20% as is typical, the
integral would fall in the range e
.0048qt
e
q 2

2
 p
t
2

 p dt
0
 e .0032qt . If qt  7.8 , about 20
years at the fitted value of q  0.39 , the integral will fall between 0.9632 and 0.9753, an
error of about 1.2%. In fact, the difference between the largest and smallest physically
possible values of the integral is only 4.3% after 20 years.
We also show the following approximation:
1
1 q

 S   1 q q2  2  p02  p0 t
r (1 q ) t
F (t )  F0 (1  p0 )e
 p0    e

 S 0  
(22)
agrees very well with simulated results, both in the determination of fund values along
simulated Brownian paths, and in the determination of option values. The error is
certainly substantially less than the error introduced in the determination of the input
parameters q and  . The approximation could be improved somewhat by using a
weighted average of the beginning value of p , that is, p 0 , and the ending value of p ,
which can be determined from S from Equation 7. This increases the complexity of the
calculations considerably, with little gain in the accuracy of the approximation.
Now that the fund value is known as a function of S , we can use the lognormal
density of S to compute the option value from the integral in equation 18. The integral
then becomes:
1
 
f a ( F0 )    e  rt Max ( X  F0 e
0 
q 2

p0 (1 p0 ) t
2


 1 q
( r  )(1 q ) t
r (1 q ) t
 p0 e 2
e (1 q ) t y  ,0)
(1  p 0 )e


2

1
2
e

y2
2
t
p x( )  x (t )dydt
(23)
which can now be evaluated numerically. Figure 4 shows the results of this evaluation for
typical parameters. While there is a substantial difference between the values for the CPR
and buy-and-hold cases, there is little added effect from increasing q above zero.
We now turn our attention to the calculation of the delta values. As before,

f a F f a p
F f
p(1  p)(1  q) f a

p 0 a 
F S p S
S 0 F
S0
p
with the partial derivatives with respect to S obtained from equations 1 and 6. The
remaining partial derivatives can be evaluated numerically from equation 23, by
(24)
increasing and decreasing F and p independently by some small amount. It requires
fewer calculations to simply increase and decrease S directly, calculate the
corresponding F and p from equations 7 and 24, and then calculate the derivative
numerically from equation 25. Figure 5 shows the results of this evaluation for typical
parameters. Figure 6 shows the delta values as a percentage of fund value. The effect on
the delta values is somewhat larger than the effect on the GMDB values themselves,
through the existence of the second term in Equation 24.
We next determine the value of the options through simulation. We ran 1000
simulations of the Brownian process Z with a time step of 0.01 for 100 years. At each
time step, the stock level was determined from:
St  e
 2
 r

2


 t Z


(25)
p was determined from Equation 7, and the fund level F was determined from Equation
1. The integrated results agree very well with the simulated ones as can be seen in Figure
7 and Figure 8.
In addition, Figure 9 shows a comparison of the cumulative probability
distribution of F after 20 years determined through simulation with that obtained from
Equation 22. At this scale the lines are indistinguishable. Figure 10 shows a closer view
of the most relevant area.
Finally, there is the possibility of valuing the GMDB from the differential
equation that the value satisfies. We will start from Equations 9 and 11 and then construct
a riskless portfolio. If f a is any function of the state variables F and p and time, then:
df a ( F , p, t ) 
1 2 fa
f a
f
f
2 fa
1 2 fa 2 
2
dt  a dF  a dp  
dF

dFdp

dp 
2
t
F
p
Fp
2 p 2
 2 F

(26)
We can then substitute Equations 9 and 11 for the differentials dF and
dp into Equation 26. The second order terms only affect the results through the
dZ 2  dt terms. This leads to:
 f
f

1
  f
df a   a  g (1  p)  pF a  p(1  p)(1  q)  g  p 2  q  p  2  a
F
2
  p

 t

2
2
2 fa
2 fa 
1 2 2 2 2 fa
2
2
2
2 (1  q) 
p F

p
(
1

p
)(
1

q
)

F

p
(
1

p
)
dt
2
Fp
2
F 2
p 2 
f
f 

 pF a  p(1  p)(1  q) a  dZ
F
p 

(27)
Then, we set up the portfolio   f a  S to hedge, so
d  df a  dS  df a  Sdt  SdZ
(28)
And then, if we require the dZ term to vanish,
 p
F f a p (1  p )(1  q) f a

S F
S
p
(29)
Since  is riskless, d  rdt . Substituting Equations 29 and 30 into this relationship
and canceling the dt leads to:
f a
f

1
 f
 g (1  p)  rp F a  p(1  p)(1  q) r  g  p 2  q  p  2 a
t
F
p
2



2 fa
1 2 2 2 2 fa
2
2
p F

p
(
1

p
)(
1

q
)

F
2
Fp
F 2
(1  q) 2  2  2 f a
 p (1  p)
 rf a
2
p 2
2
2
(30)
Equation 30 represents the value of the put at given time of death t . An analysis
similar to that in Ulm (2006) for the value of the full GMDB yields:
f a
f

1
  f
 g  (r  g ) p F a  p(1  p)(1  q) r  g  p 2  q  p  2  a
t
F
2
  p


2 fa
1 2 2 2 2 fa
2
2
p F

p
(
1

p
)(
1

q
)

F
2
Fp
F 2
 p 2 (1  p) 2
(1  q) 2  2  2 f a
 r   x (t )   ( F , t ) f a   x (t ) Max ( X  F ,0) (31)
2
p 2
This equation could be solved by working backwards in time in a 3 dimensional
lattice. It can also be made easier by assuming the fund growth rate g is equal to the risk
free rate, and that the forces of mortality and lapse are constant, which eliminates the
f a
t
term. This leads to:
 (r     ) f a  rF

f a
1
 f
 p(1  p)(1  q)  q 2  (1  q) p 2  a
F
2
 p
2 fa
1 2 2 2 2 fa
2
2
p F

p
(
1

p
)(
1

q
)

F
2
Fp
F 2
 p 2 (1  p) 2
(1  q) 2  2  2 f a
  Max ( X  F ,0)
2
p 2
(32)
Equation 32 can be put into a form resembling the heat equation by using the
substitutions:
 F 1 q

 (1  p)
 X 

   ln 
(33a)
1


1
q
 p  


  ln 
 1  p  


(33b)
And
f a ( X , p)  H ( ,  )e
  16( r     ) 
   2

2 
2

(33c)
This leads to:

e (1 q )
 2r (1  q)

q
(
1

q
)

2

1  e (1 q )
 

  (1  q) e
2

 H  2 H

2 


 2



  16( r     ) 
   2

2 
2


Max 1  e  e (1 q )  1,0

(34)
which can be solved numerically, or through Green’s functions and Sturm-Liouville
theory (see Polyanin 2002, pg. 151). In practice, Equation 34 is of limited use, as it is not
analytically solvable, and the integral methods of Equation 23 are faster. In addition, it
was derived for unrealistic constant force of mortality and lapse assumptions. One
advantage of equation 34 is that if the equation is solved on a lattice, many values of
f a ( X , p) would be obtained simultaneously. These values could be stored in memory,
and values at non-lattice points could be obtained through interpolation.
4. Effect of Transfers on Hedging Effectiveness
Insurance companies are in the business of taking on risk. However, they are
much more comfortable with risks that are well understood and independent. Mortality
risk is the most typical example. In these cases, the law of large numbers will allow the
insurance company to make a consistent profit with little actual risk. GMDB riders on
variable annuities, however, are based on market risk, which is not diversifiable. Adding
more GMDBs to an insurance portfolio will only increase the risk in a falling market, as
the GMDB payoffs are not independent. Therefore, insurance companies will often
attempt to transfer the market portion of the risk to another entity, either through
reinsurance or delta hedging with futures contracts. In this section, we will show how the
transfer model should affect hedging strategy, and how serious the consequences of
failing to incorporate transfer behavior into the hedging model are.
The typical hedging program consists of calculating the dependence of the entire
portfolio of GMDB options on the market (i.e., the values of delta) and then buying or
shorting futures contracts of the appropriate size to leave the economic value of the
company unchanged when the market moves. Figure 5 shows the values of delta that
would be calculated under each of the three assumptions (CPR, Buy-and-Hold and
q  0.39 ). In this section, we will examine the consequences if an insurance company
attempts to hedge on either the CPR or Buy-and-Hold model when the q  0.39 model is
actually the correct one.
As can be seen in the figures, delta is smallest for the q  0.39 transfer model.
This implies that if an insurance company is hedging on the basis of the other two
models, it will invest in too many futures contracts, and therefore be over hedged. This
may not be that bad in practice, however. Additional futures contracts will lead to
additional transaction costs, but there are also some benefits to being slightly over
hedged. GMDB options are not the only equity exposure assumed by insurance
companies. Asset fees on variable products are also highly dependent on stock market
performance, and this is often a larger effect than the GMDB option exposure. Being
slightly over hedged will mitigate some of the risk of reduced asset fees, and the
magnitude of the error in delta is not large enough to result in the entire company being
over hedged. Aside from transaction costs, the overvaluing and over hedging of the
GMDB options does not change the long-term earnings of the company. It merely
redistributes the earnings to different time periods.
It can also be seen from the figures that that advantage of going from Constant
Percentage Rebalancing assumptions to Buy-and-Hold assumptions is significant (up to
0.66% of fund value when the fund is at 60% of the strike value). The additional gain
from including transfers is smaller (another 0.22% of fund value, which is approximately
0.39  0.71% ), but could easily still be important. If an insurance company has
$10,000,000,000 in variable annuity funds with at-the-money GMDB riders, the change
in the futures value from a 1% change in the S&P if CPR assumptions are used is
$4,003,000. The change from Buy-and-Hold assumptions is $3,431,000 and the change
from q  0.39 assumptions is $3,226,000. The effect on the hedge from a change in
assumptions could be on the order of $5,000,000 a year, assuming a typical 8% change in
the S&P, and even more in years with particularly large positive or negative equity
moves. Assuming an operating margin of 20 - 50 bp, which is about $20,000,000 $50,000,000 per year in income, the effect of the incorrect hedge assumptions would be a
non-negligible part of the total income of the variable annuity segment, and care should
be taken to insure that the futures position is hedging the actual exposure, including
transfer effects.
5. Conclusions
A model of variable annuity subaccounts where transfers are primarily determined
by recent fund returns is a good fit to empirical data on fund allocation percentages. This
is consistent with previous studies of mutual funds that show that individual cash flows to
mutual funds are positively related to recent fund performance. A model based on these
assumptions shows that people tend to transfer their money into variable accounts at
about 39% of the rate that would be required to maintain constant percentage rebalancing,
but with the opposite sign. If these transfers are not taken into account, the GMDB
options on the variable accounts will be overvalued and over hedged. Ignoring this effect
can have a substantial impact on the size of the futures portfolio needed to hedge this risk
and a non-negligible impact on the earnings of the variable annuity portfolio.
6. Acknowledgements
This research was supported, in part, by a research grant from the
Robinson College of Business, Georgia State University and by a grant from The
Actuarial Foundation. The data on mutual fund percentages and returns was provided by
Morningstar Inc. We would like to thank Conrad Ciccotello, Steven Craighead and Ajay
Subramanian for useful discussions.
References:
Edelen, R.M. and Warner, J.B. 2001. Aggregate Price Effects of Institutional
Trading: A Study of Mutual Fund Flow and Market Returns, Journal of Financial
Economics 59(2):195-220
Hardy, M. 2003. Investment Guarantees, Hoboken, N.J.: John Wiley and Sons.
Hayre, L. and Rajan, A. 1995, Anatomy of Prepayments: The Salomon Brothers
Prepayment Model, Salomon Brothers.
Ippolito, R.A. 1992. Consumer Reaction to Measures of Poor Quality: Evidence
from the Mutual Fund Industry, Journal of Law and Economics 35:45-70.
Milevsky, M.A., and Posner, S.E. 2001. The Titanic Option: Valuation of the
Guaranteed Minimum Death Benefit in Variable Annuities and Mutual Funds, Journal of
Risk and Insurance 68(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 Colloquium, Sep 2001, pg 537
Polyanin, Andrei D. 2002, Handbook of Linear Partial Differential Equations for
Engineers and Scientists, Boca Raton, Fl: Chapman & Hall/CRC.
Richard, S. and Roll, R. 1989, “Prepayments on Fixed Rate Mortgage Backed
Securities”, Journal of Portfolio Management 15(3):73-83
Schwartz, E. and Torous, W. 1989, “Prepayment and the Valuation of Mortgage
Backed Securities”, Journal of Finance 44(2):375-392
Ulm, E.R. 2004. “Rapid Calculation of the Price of Guaranteed Minimum Death
Benefit Ratchet Options Embedded In Annuities”, Journal of Actuarial Practice 11, 169195
Ulm, E.R. 2006. The Effect of the Real Option to Transfer on the Value of
Guaranteed Minimum Death Benefits, Journal of Risk and Insurance 73(1), 43-69
Ulm, E.R. 2008. Analytic Solution for Return of Premium and Rollup Guaranteed
Minimum Death Benefit Options Under Some Simple Mortality Laws, Submitted.
Warther, V.A. 1995. Aggregate Mutual Fund Flows and Security Returns,
Journal of Financial Economics 39(2-3):209-235.
Appendix A. Derivation of Equations 2 and 3, the stochastic equation followed by
the variable fund percentage under a buy and hold strategy.
Neglecting terms of order 2 or higher:


 dS 

pX (t ) 1  

S 


 p
dp  p (t  dt )  p (t ) 

 dS  
 (1  p ) X (t )(1  gdt )  pX (t ) 1   
S 


(A1)
Now, cancel X (t ) , place the terms over a common denominator and subtract:
dp 
dS
 p(1  p) gdt
S
dS
1  (1  p) gdt  p
S
p(1  p)
(A2)
The differentials in the denominator are nearly zero, so:
dp  p (1  p )
dS
 p (1  p ) gdt
S
(A3)
Equation 3 can be derived in several ways. The simplest is to assume
p0
p
S
S0
S
(1  p0 )e  p0
S0
(A4)
gt
and
dp 
p
p
1 2 p 2
dt 
dS 
dS
t
S
2 S 2

2
p
p
1  2 p 2 2 2  p
p
2 2 1  p

dt 
dt 
dS 

S
dZ



S
dS
2
2 

t
S
2 S
2 S 
S
 t
We find:
(A5)
p
  gp(1  p)
t
(A6)
p 1
 p (1  p )
S S
(A7)
and
2 p
2
  2 p 2 (1  p)
2
S
S
(A8)
Substitution into Equation A5 yields:
 dS

dp  p(1  p)
 gdt   p 2 (1  p ) 2 dt
 S

(A9)
Appendix B. Derivation of the equation for the fund value associated with the
transfer model.
We start from Equation 11 under risk neutral growth assumptions:
dF  rFdt  pFdZ
(B1)
Ito’s Lemma implies that:
  2 p2 
dt  pdZ
d ln F   r 
2 

(B2)
Or, integrating,
ln F  ln F0  rt 
2
2
p
2
dt    pdZ
(B3)
The last two terms can be put into a form that can be more easily approximated by
showing that:

2
2
p
2
dt    pdZ 
1
  2 (1 q ) t

 q 2
1
 (1 q ) Z
2
d
ln
(
1

p
)

p
e
e


0
0
1 q 
2


 p
2

 p dt
(B4)
To prove Equation B4, we will use Ito’s Lemma on the differential in the first
integral on the right hand side, that is:
1
  2 (1 q ) t


1
d ln (1  p 0 )  p 0 e 2
e  (1 q ) Z 
1 q


1
  2 (1 q ) t
 1 2
  (1  q) p0 e 2
e (1 q ) Z
1  2


1
  2 (1 q ) t
1 q 
2
(
1

p
)

p
e
e (1 q ) Z
0
0



 dt


1
  2 (1 q ) t

e (1 q ) Z
1   (1  q) p0 e 2

1
  2 (1 q ) t
1 q 
 (1  p0 )  p0 e 2
e (1 q ) Z

 dZ


1
  2 (1 q ) t
 2
2
e (1 q ) Z
1  1  (1  q) p0 e 2

 
1
  2 (1 q ) t
1  q  2 
2
(
1

p
)

p
e
e (1 q ) Z
0
0

2
  12 2 (1 q )t  (1 q ) Z  
 (1  q) p e
e
 

 dt

2 
1
  2 (1 q ) t



2
e  (1 q ) Z  
(1  p 0 )  p 0 e

 
2
2
2
0
(B5)
Now, since from Equation 7 (again, risk neutral growth assumptions),
p
 S
p0 
 S0



(1 q )
 S 
(1  p0 )e r (1 q )t  p0  
 S0 
1 q
p0 e

1
  2 (1 q ) t
 (1 q ) Z
2
e
(1  p0 )  p0 e
(B6)
1
  2 (1 q ) t
 (1 q ) Z
2
e
Equation B5 becomes:
1
  2 (1 q ) t


1
d ln (1  p 0 )  p 0 e 2
e  (1 q ) Z 
1 q



2
2
pdt  pdZ 
 2 (1  q)
2
pdt 
 2 (1  q)
2
p 2 dt
(B7)
Or, with appropriate cancellations:
1
  2 (1 q ) t


1
2 2
q 2 2
d ln (1  p 0 )  p 0 e 2
e  (1 q ) Z   
p dt  pdZ 
p  p dt
1 q
2
2




(B8)

(B9)
This proves Equation B4. Substituting into Equation B3 produces:
ln F  ln F0  rt 
1
  2 (1 q ) t

 q 2
1
 (1 q ) Z
2
d
ln
(
1

p
)

p
e
e


0
0
1 q 
2


 p
2
 p dt
The logarithm in the differential is zero at time zero, so:
ln F  ln F0  rt 
1
  2 (1 q ) t

 q 2
1
ln (1  p 0 )  p 0 e 2
e  (1 q ) Z  
1 q 
2

 p
2

 p dt
(B10)
or
1
1
  2 (1 q ) t

 1 q q   p 2  p dt
F (t )  F0 e rt (1  p0 )  p0 e 2
e (1 q ) Z  e 2


2
(B11)
Moving e rt inside the square brackets yields Equation 20:

 S
F (t )  F0 (1  p0 )e r (1 q )t  p0 

 S0



1 q
1
 1 q q2  2   p 2  p dt
 e 0

t
(B12)
Table 1. Percentage of Variable Annuity Funds held in Variable Accounts
Quarter
6/30/1998
9/30/1998
12/31/1998
3/31/1999
6/30/1999
9/30/1999
12/31/1999
3/31/2000
6/30/2000
9/30/2000
12/31/2000
3/31/2001
6/30/2001
9/30/2001
12/31/2001
3/31/2002
6/30/2002
9/30/2002
12/31/2002
3/31/2003
6/30/2003
9/30/2003
12/31/2003
3/31/2004
6/30/2004
9/30/2004
12/31/2004
3/31/2005
6/30/2005
9/30/2005
12/31/2005
3/31/2006
6/30/2006
9/30/2006
12/31/2006
Var Pct
76.63%
73.97%
78.32%
76.63%
77.68%
76.92%
80.35%
81.20%
80.49%
80.32%
79.94%
76.67%
77.62%
72.88%
75.32%
75.40%
71.43%
66.27%
64.56%
63.20%
68.38%
66.92%
70.06%
72.22%
71.96%
71.44%
74.16%
73.12%
73.78%
74.75%
74.94%
76.71%
76.03%
77.01%
78.93%
q=0.39
Buy and Hold CPR
Prediction Prediction
Prediction
75.95%
72.95%
77.59%
78.49%
78.21%
76.43%
80.60%
81.18%
80.16%
80.32%
78.37%
76.78%
77.73%
73.16%
75.55%
75.10%
72.34%
66.66%
67.59%
63.33%
67.09%
69.19%
69.61%
70.59%
71.87%
71.30%
73.69%
73.24%
73.40%
74.75%
75.07%
76.02%
75.78%
76.46%
78.28%
75.95%
73.95%
76.54%
78.38%
77.71%
76.73%
79.55%
80.89%
80.40%
80.31%
78.87%
77.64%
77.37%
74.41%
74.76%
75.11%
73.16%
67.97%
67.15%
63.61%
65.95%
68.90%
68.80%
70.39%
71.91%
71.43%
73.01%
73.45%
73.26%
74.43%
74.93%
75.67%
75.99%
76.29%
77.88%
76.17%
76.63%
73.97%
78.32%
76.63%
77.68%
76.92%
80.35%
81.20%
80.49%
80.32%
79.94%
76.67%
77.62%
72.88%
75.32%
75.40%
71.43%
66.27%
64.56%
63.20%
68.38%
66.92%
70.06%
72.22%
71.96%
71.44%
74.16%
73.12%
73.78%
74.75%
74.94%
76.71%
76.03%
77.01%
Table 2. Variable Percentages and Variable Returns from Data Supplied by Morningstar
Inc.
Quarter
3/31/1998
6/30/1998
9/30/1998
12/31/1998
3/31/1999
6/30/1999
9/30/1999
12/31/1999
3/31/2000
6/30/2000
9/30/2000
12/31/2000
3/31/2001
6/30/2001
9/30/2001
12/31/2001
3/31/2002
6/30/2002
9/30/2002
12/31/2002
3/31/2003
6/30/2003
9/30/2003
12/31/2003
3/31/2004
6/30/2004
9/30/2004
12/31/2004
3/31/2005
6/30/2005
9/30/2005
12/31/2005
3/31/2006
6/30/2006
9/30/2006
12/31/2006
Variable Variable
Percentage Return
76.17%
76.63%
73.97%
78.32%
76.63%
77.68%
76.92%
80.35%
81.20%
80.49%
80.32%
79.94%
76.67%
77.62%
72.88%
75.32%
75.40%
71.43%
66.27%
64.56%
63.20%
68.38%
66.92%
70.06%
72.22%
71.96%
71.44%
74.16%
73.12%
73.78%
74.75%
74.94%
76.71%
76.03%
77.01%
78.93%
0.37%
-12.02%
16.58%
1.93%
7.98%
-3.80%
18.54%
5.12%
-3.53%
0.41%
-7.05%
-11.50%
5.61%
-14.92%
11.72%
0.24%
-9.80%
-13.89%
5.53%
-2.75%
14.24%
3.82%
10.49%
2.86%
-0.23%
-1.32%
9.58%
-2.35%
2.05%
4.74%
2.26%
5.35%
-2.62%
2.78%
6.44%
Figure 1. Percentage of variable annuity funds held in equity accounts
85%
80%
75%
Var Pct
70%
Buy and
Hold
Prediction
CPR
Prediction
65%
60%
12/31/1997 12/31/1998 12/31/1999 12/30/2000 12/30/2001 12/30/2002 12/30/2003 12/29/2004 12/29/2005
  S
Figure 2. Best fit line to c  q ln 
  S0

 p (1  p 0 )e gt S 0

  gt   ln 

 (1  p ) p 0 S


.


Dependent Variable
0.1
0.05
0
-0.2
-0.1
0
0.1
-0.05
-0.1
-0.15
Variable Rate minus Fixed Rate
0.2
Empirical Values
Regression
Figure 3. Percentage of variable annuity funds held in equity accounts. Comparison of
predictions for buy-and-hold and q=0.39.
85%
80%
75%
Var Pct
70%
Buy and
Hold
Prediction
q=0.39
Prediction
65%
60%
12/31/1997 12/31/1998 12/31/1999 12/30/2000 12/30/2001 12/30/2002 12/30/2003 12/29/2004 12/29/2005
Figure 4. Comparison of GMDB Values for a strike price of 1 with parameters r  5% ,
  20% , p0  0.70 , constant force of mortality   2% and constant force of lapse
  10% .
0.025
GMDB Value
0.02
0.015
CPR, q=-1
0.01
Buy And
Hold, q=0
q=0.39
0.005
0
0.7
0.8
0.9
1
Fund Value
1.1
1.2
1.3
Figure 5. Comparison of GMDB deltas for a strike price of 1 with parameters r  5% ,
  20% , p0  0.70 , constant force of mortality   2% and constant force of lapse
  10% .
0
-0.01 0
0.5
1
1.5
2
-0.02
Delta
-0.03
-0.04
CPR, q=-1
-0.05
-0.06
Buy And
Hold, q=0
q=0.39
-0.07
-0.08
-0.09
-0.1
Fund Value
Figure 6. Comparison of GMDB deltas per fund for a strike price of 1 with parameters
r  5% ,   20% , p0  0.50 , constant force of mortality   2% and constant force of
lapse   10% .
0
0
0.5
1
1.5
2
Delta per Fund
-0.05
-0.1
CPR, q=-1
-0.15
Buy And
Hold, q=0
q=0.39
-0.2
-0.25
Fund Value
Figure 7. Comparison of GMDB Values obtained from simulation and integration for a
strike price of 1 with parameters r  5% ,   20% , p0  0.70 , constant force of
mortality   2% and constant force of lapse   10% .
0.05
GMDB
0.04
0.03
Integral,
q=0
0.02
Simulation
q=0
0.01
0
0.5
1
Fund Value
1.5
Figure 8. Comparison of GMDB Values obtained from simulation and integration for a
strike price of 1 with parameters r  5% ,   20% , p0  0.70 , constant force of
mortality   2% and constant force of lapse   10% .
0.05
GMDB
0.04
0.03
Integral,
q=0.39
0.02
Simulation
q=0.39
0.01
0
0.5
1
Fund Value
1.5
Figure 9. Comparison of GMDB Values obtained from simulation and integration for an
initial fund of 1 with parameters r  5% ,   20% , p0  0.70 after 20 years.
Cumulative Probability
1.2
1
0.8
Simulated
0.6
Approximated
0.4
0.2
0
0
2
4
6
Fund Value
8
10
Figure 10. Comparison of GMDB Values obtained from simulation and integration for an
initial fund of 1 with parameters r  5% ,   20% , p0  0.70 after 20 years at small
fund values.
Cumulative Probability
0.3
0.2
Simulated
Approximated
0.1
0
1.1
1.2
1.3
Fund Value
1.4
1.5