Morningstar Magazine - February/March 2018 - 46

Strategies

EXHIBIT 3

investment opportunities given by the efficient
frontier, 2) the investor's attitude toward
risk as expressed in the risk tolerance parameter,
and 3) the investor's intertemporal preferences
given by the subjective discount rate
and the elasticity of intertemporal substitution.
shows what the efficient frontier
looks like in the lifecycle model. It is nearly linear
and includes 100% allocation to the riskless
asset. To see which portfolio the investor chooses,
under the assumptions of the model, we can
derive an expected utility function of the form:

Total Wealth and Cash Flows in Lifecycle Model With Annuities and Life Insurance
Expected Return
10%

INDI

EXHIBIT 3

Selected Portfolio

In EX H I B I T 3 , I have drawn an indifference curve.
An indifference curve is made up of all
combinations of SD and ER that result in the same
level of expected utility. The curve in E X H I BI T 3
is a special one in that it is tangent to the efficient
frontier. The point of tangency is the SD / ER
combination that maximizes expected utility
subject to the constraint that ( SD, ER ) is a point on
the efficient frontier. This point represents the
portfolio of risky assets that the investor holds in
the regular account to fund the variable
request and part of lifestyle consumption, as well
as the immediate variable annuities held to fund
the remaining part of lifestyle consumption. The
higher the level of risk tolerance ( ), the further up
the efficient frontier is the point of tangency.
The tangent indifferent curve intersects the ER axis
at the certainty equivalent return. This is the
risk-free rate of return that would give the investor
the same level of expected utility as the optimal
portfolio. Letting ( SD*, ER* ) denote the point of
tangency and CE denote the certainty equivalent
return, we have:
U ( CE, 0 ) = U ( SD*, ER*)

46

Morningstar February/March 2018

4

Certainty Equivalent Return
Assumed Interest Rate

2

SD

U (.,.; )

expected utility
expected return
standard deviation
the risk tolerance parameter
expected utility function, which
is increasing in ER and decreasing
in SD, for a given value of

8

ER

0

6.97%

0%
Standard Deviation

=
=
=
=
=

E
URV
IER
CE C
T FRONT
EFFIC IEN

6

4.38%

EU = U ( ER, SD ; )
Where:
EU
ER
SD

EN
FFER

5

10

15

20

Source: Morningstar.

The certainty equivalent return is one of three
inputs to the assumed interest rate. Because
the AIR is in logarithm form, we convert CE into a
logarithmic return that we denote :
Payout
$3.50
= ln (1 + CE ) Logarithmic Scale
2.50
The other two inputs are the intertemporal
preference parameters, which I introduced in Part I
1.50
of this series in the October/November 2017 issue:
1.00
= the subjective discount rate (here in
logarithmic form)
0.50
h = the
0 elasticity of intertemporal
10
20
substitution
Years in the(EOIS)
Future
The subjective discount rate, , captures the idea
that, all other things being equal, investors
Equity
of Risky
prefer Allocation
consumption
todayAssets
to consumption in the
100%
future. The EOIS parameter, h, captures how
responsive an investor is to changes in
intertemporal trade-offs, such as a rise in interest
75
rates. An investor with a high EOIS would lower
current consumption and raise future consumption
50
in response to the improved trade-off between
consumption today and consumption tomorrow, as
25
compared to an investor with a low EOIS.
0
0% interest rate20is given by: 40
The assumed
Percentage of Portfolio in Risky Assets
=

+ (1 - )

The assumed interest rate is a weighted
average of the subjective discount rate and the
certainty equivalent return, the weight being
the EOIS.
Percentile

95th
th
In this three-part series, I used lifecycle90investment
75th
models to address key questions that investors
50th
th
must answer over their lifetimes. These25include
10th
decisions regarding consumption, savings,
th
life insurance and annuities, and how 5to invest
their portfolios. The models provide rational
answers and show how they change over time.

The Annuity Puzzle

30
40
50
But investors are not completely rational, and
markets do not function perfectly. In the
model, investors make heavy use of immediate
fixed annuities and variable annuities to generate
income. In the real world, few investors do so.

Effective Equity
Allocation %

This gap between theory and practice is
90known
as the annuity puzzle. Perhaps the most80important
lesson is that it would be in investors' 70
own
60
interests to make more use of annuity products to
50
avoid running out of money before death.
40
30
Paul D. Kaplan, Ph.D., CFA, is director of research
20 with
Morningstar Canada. He is a member of the 10
editorial board
of Morningstar magazine.

60

80

100



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