Morningstar - Q2 2020 - 65

n=
Wt

ct =

f = n- qT

t
T

t

=

∑

v= t

n

c
q

P=

1
( 1 + rˉ ) v - t

-j

∑
j =1

EXHIBIT 2

T

v-t
g circumstances.
preferences1 +and
The formula
=
t
+
for g vis:
1
r
ˉ
=t

1 + rˉ
1+

Survival Probability

EOIS = 75%

EOIS = 50%

n
j =1

EOIS = 25%
D ( y; c, T ) =

∑

c
q

∑

0

The EOIS
F-L
ct = cˉ t + t 1 t
t
spending).

c
q

j -f
q

0
Year

10

20

f- j

c
q

y
q

n
q

100%
75

) (1 + )

y f
q

f- j

(1 + )
P ( y; c, ) (1 + )
y
q

c
q

=

P ( y; c, T )

n
q

y f
q

-j

(1 + )
30
P ( y; c, )
y
q

50
25
0

40

n
q

Pc ( y ) =

∑

1- 1
O

-1 , O
1- O1

1

1
of consumption.
c~t+1 - O ~t
1
= Qt+1
ct
1+
When consumption
changes over time, the rate
of
change
depends
on
the EOIS. The EOIS
c~t+1
~t - O
1 , captures
how responsive an investor
parameter,
=
Q
ct
(1 + ) O t+1

( )

we can see exactly
~t - Ohow this works. Consider
Qt+1
~ case when
the
rˉ < so that consumption is
-1
ROt+1 =
1-O
t
declining.E TheQ~higher
the value of , the greater
t
t+1
the rate of decline.
~t 1 - O
Et Qt+1
c~t+1
~
Taking
Longevity
Into Account
=
1 + ROt+1
O
ct
+
(1
)
In "When Will I Go?" (Quant U, February/March
2015), I present the Gompertz function, which
is a formula for the probability of surviving for any

( )
( )

( )

=

f- j

P ( y; c, T )

(1 + )
y
q

y
q

Source: Morningstar.

T
yv
H
=
Whether
consumption
increases, decreases, or
t
(1+ r )v- t
v= t
remains constant depends on whether the
certainty equivalent
is above, below, or the
F + Ht - Lreturn
t
= cˉas
+ t
csame
t
t the subjective discount rate. If rˉ > ,
t
it is worthwhile for the retiree to forgo some
- the
L earlier years to potentially
H in
consumption
xt = 1 + t t
earn a higherFmarket
return. If rˉ < , the
t
preference for earlier consumption over later
~
~t
1 u'(ct+1 )
=soQconsumption
t+1
market
returns,
is higher
+
1
u'(ct)
in the earlier years than in the later years. If rˉ = ,

u(c) = { ln(c), O = 1 c

- qT

y
q

(

)

given number of years. The Gompertz function
has three parameters:
1 Current age.
2 The mode of the distribution of the age

of death.2
3 The dispersion around the mode of the age
of death.
Taking these parameters as part of the retiree's
circumstances, the Kaplan-Blanchett model
takes longevity
into account by calculating the
T
1
= probabilities
survival
of the retiree using the
t
(1+ r )v- t
v= t
Gompertz
formula. Let:

∑

W
cqtt == thet probability of the retiree surviving
v
t
to least
year v, given that the retiree
is
alive
in
year t.
T
1
=
t
( 1 + rˉ ) v - t
E X H I Bv I=Tt 2 includes
a plot of the survival
probabilities for a 2-year-old woman. The other
T
v-t
+ g we
curves
are 1what
call rescheduling factors,
=
t
+
1
r
whichv =indicate
how
much spending to move
ˉ
t
from later years to nearer years because of
1 + rˉ
the
g = lower survival
- 1probabilities in later years. The
+
1
rescheduling factor for a given retiree is:
RFvt = ( qvt )
T

t

=

∑ RF

v

v= t
T

Lt =

∑

v= t

1+ g
1 + rˉ

cˉ v
(1+ r )v- t

v-t

(1 + )

y f
q

(

y

)

Dc ( y ) = y1 1 + q - qf
In the Waring-Siegel model, the retiree is
T
1
=
t
v - t P ( y; c, T )
+
(
1
(y;
c,r )T ) =
D
v= tm
P ( y;probability
c, T )
sense because survival
declines over
Wt horizon. Hence, planned spending
the
planning
ct =
D ( y; c,decline
T)
out intoD the
future should
as well. However,
t
m (y; c, T ) =
y
1+ q

∑

T

to scheduling1 spending. This is why the survival
=
t
v-t
P( 1is+ raised
probability
-rˉ )Dm yto the EOIS power in the
v= t
P
formula for the rescheduling factor. The closer that
EOIS T 01 + g v - t
D
= DV 01
ast shown
by+the
shapes of the curves in E X H I B I T 2
1 =rˉ P 1 + y = P Dm
v= t
for EOIS values of 75%q, 50%, and 25%.
1 + rˉ
g
-1
We= incorporate
the rescheduling factor into the
1+
P ( y; c, T )
spending
rule
by
C (y; c, T ) =including it in the present value
( y; c,
T)
(pricing)
calculation.PWe
have:
RFvt = ( qvt )

∑
∑

∑
∑

1
y

t

*

T
( y; c, T1)+=g
= CRF
v
1 + rˉ
v= t

∑

c
T q

∑

( )(
cˉ

n
j =1

f-j
q

v-t

f- j-1
q

f- j-2

) (1 + )
y
q

Taking
Lt = Needsv PInto
( y; c,Account
T)
(1+ r )v- t
v = t typically have some minimum amount
Retirees

(

)(

- qT

1+ )
( ) Rescheduling
(1 + ) + T (Factor

(

T

2

f- j

y
q

P ( y; c, T ) = P y; c,

v= t

y f
q

∑ (1 + ) + (1 + )

-1

t
= the impatience
called the
1 + g v - parameter,
= RFv
subjective
discount
rate.
1 + rˉ
v= t
=
T the elasticity of intertemporal substitution,
cˉ
Lt = or EOISv . v - t
(1+ r )

-n

) (1 + )

j =1

where:
RFvt = ( qvt )

t

y

n

c
q

Survival Probabilities and Rescheduling FactorsP ( y; c, T ) =

∑

g=

(

+ 1+ q

y

)

- qT - 2

1
T T+ q 1+ q
-
L
F
+
food, housing, and
t other necessities. This is
ct = cˉ t + t P ( y;
c, T )
nondiscretionary
t spending. Let cˉ v denote the
2 v. At a minimum,
P
1
nondiscretionary
spending
in year
T
y-v Dm y + 2 C ( y )
Ht = P
(1+ r )v- t
v= t

∑

Wt =F +HtH+t -Ft Lt
ct = cˉ t + t
t

morningstar.com/lp/magazine

Wt =H (1- +Lr) (Wt -1 - ct-1 )
xt = 1 + t t

5
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Morningstar - Q2 2020

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