Morningstar Magazine - October/November 2017 - 56

Ft =

Strategies

EXHIBIT 3

Evolution of Consumption with Different Levels of the EOIS
r=

r< ,

= 0.1

r< ,

= 0.5

r< ,

= 0.9
50K
USD

40

30

20
25
Age

35

45

55

65

75

85

Source: Morningstar.

2 Whether consumption increases, decreases, or

remains constant depends on whether the market
rate of return (r) is above, below, or the
same as the subjective discount rate ( ). If r> , it
is worthwhile for the investor to forgo some
consumption in the earlier years to earn the
market rate of return because earning the market
rate of return outweighs the preference
for earlier consumption over later consumption.
If r< , the preference for earlier consumption over
later consumption outweighs the benefit of
earning the market rate of return, so consumption
is higher in the earlier years than in the later
years. If r= , the benefit of earning the market
rate of return offsets the preference for earlier
consumption over late consumption, resulting in
a completely flat pattern of consumption.
3 When consumption changes over time (r≠ ), the

rate of change depends on the elasticity of
intertemporal substitution, . Recall that the EOIS
parameter, , captures how responsive an
investor is to changes in intertemporal trade-offs.
Here, we can see exactly how this works. Consider
the case when r< , so that consumption is
declining. The higher the value of , the greater
the rate of decline, as illustrated in E X H I BI T 3 . In
this exhibit, I first show consumption over the
lifecycle when r= , which results in a straight line.
I then show consumption for three values of
(0.1, 0.5, and 1.0) with r= 3% and = 4% so that
consumption declines over time. When = 0.1,

56

Morningstar October/November 2017

t-1

t-1

(1 + r) (Ft-1 - ct-1 ),
t ≥ L +1
F0 = 0 (1 + r) (Ft-1 + y - ct-1 ), t ≤ L +1
Ft =
(1 + r) (Ft-1 - ct-1 ),
t ≥ L +1
Wt = Ht(1+ +Ftr) (Ft-1 + y - ct-1 ), t ≤ L +1
Ft =
(1 + r) (Ft-1 - ct-1 ),
t ≥ L +1
Wt = Ht + Ft
Wt = (1 + r) (Wt -1 - ct-1 )
component to the model, which leads to
Wt = Ht + Ft
the
demand
fort -1
life- insurance
and annuities.
T =
ct-1 )
(1 -+t r) (W
W
t
∑ (1 + r) ct-1 = W0

the pattern of consumption is nearly flat because
the investor is fairly insensitive to the
intertemporal trade-off that the market is offering.
At higher values of , the consumption curve
becomes steeper due to higher levels of investor
responsiveness to intertemporal trade-offs.

t=0
Paul
D. Kaplan, Ph.D., CFA, is director of research with
T
-t
t
t -1 Het-1
Morningstar
Canada.
t-1
0 is a member of the editorial board
t=0
of Morningstar
magazine.
-1
T0
0 -t0

W
∑ (1=+ (1
r) +cr) (W= W- c )

c =∆ W
∑ (1 + r) ct-1 = W0
t=0
-1
c0 = ∆ 0 W
-
Appendix:L 0Statement
of the Model and Its Solution
r + r) - t
y 1∑+ (1
ct-1 maximization problem
cHt0 ==model
The
t = 0is a utility
-1 +
∆ 0 1W0 -
cas0 =follows:
1+ r
ct-1
ct =
- y), if t ≤ L
(1++ r) (Ht-1
Ht = 1
-
if t >-1L
T
max 10,+ r T
-t
ct1 ,c=2 , ... ,cT ∑ (1 +ct-1)- t
ct s.t. ∑ (1 + r) ct
- 1 -1
1+ t = 0
t=0
T
T
max
-t
-t
0
cF10 ,cW=
ct s.t. ∑ (1 + r) ct
=
2 ,0 ... ,cT ∑ (1 + )
-
1
t=0
t=0
-1
T
T
max
=
+∑r)(1(F+t-1 +)- ty - ct-1 ), c t ≤s.t.L +∑1 (1 + r) - t c
(1
0 ... ,c
cF1t ,cW=
,
2
T
t
t
1-
To =get(1the
+t =r)0(1(F+t-1 )-calculate:
ct-1 ),- 1
t ≥ L +t =10
+ (1
r)solution,
= W0
1-
= (1 + r)- -T(1 + )
∆
=
W0t = Ht +-F1t
1- -T
and
(1 + r)- (1 + )
∆0==then:
-1
-c )
Wt = (1 + r) (W
- -T t -1 t-1
∆0 =
-1
T
-t
∑ (1 + r) ct-1 = W0
t=0

As I state in the main text, consumption is given by:

Up Next: the Mortality Component
Individual investors need to make decisions beyond
which investment to buy. These include
decisions regarding consumption and savings, life
insurance and annuities, and how to change
these decisions over time.

In financial economics, lifecycle models provide a
way to address these decisions in a rational
manner. In this issue of Quant U, I present
a barebones version of a lifecycle model to address
questions of savings, investment, and consumption.
From this model, we see (1 ) the central role
that human capital plays in determining the level
of consumption, (2 ) that the optimal pattern of
consumption is smooth and is the same before and
during retirement, and (3 ) the role that market
and investor preference parameters play in
determining (a) whether consumption increases or
decreases over the lifecycle and (b) the rate of
change in consumption, increasing or decreasing.
In this barebones model, I assume that the
year of death is fixed and known in advance. In the
next issue of Quant U, I will introduce a mortality

-1

c0 = ∆ 0 W0
Once the +value- of c0 is determined, the values
1 r
c
coft =
ct for 1t =
+ 1, 2, ...t-1, T can be calculated using the
equation for the evolution of consumption
presented in the main text. From these, the
T
max
evolution
of Tfinancial- twealth and-1 total wealth
can
-t
c1 ,c2 , ... ,cT ∑ (1 + )
ct presented
s.t. ∑ (1 + r) ct
be calculated
using
the
equations
-
1
t=0
t=0
=in the
W main text.
0

= (1 + r)
∆0 =

1-

(1 + )

- -T
-1



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