# Morningstar Magazine - February/March 2018 - 45

```5th

1.00
0.50
0
10
Years in the Future

20

30

40

50
Expected Return
10%

EXHIBIT 1

Just as immediate fixed annuities are the primary
means of generating lifetime income in the riskless
model, in the model with risky assets, immediate variable annuities are the primary means of

=

the value of one unit of the risky portfolio
at time t
= the AIR in logarithmic form

The payout of the immediate variable annuity
for period t is:
= exp ( - t )
Without loss of generality, I assume that S₀ = 1,
as I do in in E X H IB IT 2 , which shows percentiles
of how the payout of an immediate variable
annuity could evolve over time, depending on the
performance of the underlying portfolio.
The payouts of an immediate variable annuity can
be replicated with an ordinary investment
account of risk assets, so long as the portfolio is
the same as the underlying portfolio of
the annuity. To do this, in each period t, sell
a fraction of the portfolio, wh where:
wh = 1 - exp ( - )

For example, with = 3.40%, wh = 3.34%. With
S₀ = 1, the immediate variable annuity pays \$1 at
time t = 0. To have a portfolio that pays \$1
at time 0, its initial value needs to be 1/wh. With
= 3.40%, this comes to \$29.91.
The key difference between the immediate variable
annuity and a strategy that replicates it using an
ordinary investment account is that while
withdrawals from the investment account can in
principle go on indefinitely, the annuity stops
making payments at the death of the investor. This
makes the annuity less costly than an ordinary
investment account, which is the reason, in
general, immediate annuities can be advantageous

Equity Allocation of Risky Assets
IND

100%
4.38%

75

REN
IFFE

E
8
URV
CE C
NTIER
O
R
F
T
N
EFFIC IE
Eﬀective Equity
Allocation %

90
80
70
60
50
40
30
20
1020

ER
Selected Portfolio

Certainty Equivalent Return

50Assumed Interest Rate
SD

An immediate variable annuity pays the investor
the ratio of the value of one unit of a portfolio
of risky assets (similar to a share of a mutual fund)
to the value of \$1 growing at a fixed rate,
called the assumed interest rate, or AIR. Let:

Effective Equity Allocation With Riskless and Risky Assets

25

6.97%

0%
0
Standard 0%
Deviation

5

10

20
40
Percentage of Portfolio in Risky Assets

15
60

80

6
4
2
0

100

Source: Morningstar.

EXHIBIT 2

Percentiles of Payouts of an Immediate Variable Annuity Over Time
Payout
\$3.50
2.50

Percentile

95th
90th
75th
50th
25th
10th
5th

Logarithmic Scale

1.50
1.00
0.50
0
10
Years in the Future

20

30

40

50

Source: Morningstar.

Equity Allocation of Risky Assets
over ordinary accounts when the investor seeks
100%
only to fund personal consumption. To see the cost
savings of the immediate variable annuity,
75
we need to price it. The fair value of an immediate
variable annuity at time 0 is:

25
where q ( t ) is the probability of surviving to
0at least time t. Using the same value for that I
40
0%and using the20Gompertz model
used earlier
Percentage
of
Portfolio
in
Risky
Assets
of mortality that I reviewed in the last issue of

When there is a variable bequest, the investor
Eﬀective Equity
accumulates a portfolio of risky assets Allocation
that
%
90
becomes the bequest upon death. By assuming
80
that the investor's level of risk tolerance
70
for the variable bequest is the same as for lifestyle
60
consumption, it follows that the investment
50
portfolio in the regular account is the same
40 as that
30
of the immediate variable annuities. Hence,
withdrawing from the regular account 20
at
10
the withdrawal rate wh and the payments from the
60
80
100 of funding
annuities
are complementary
ways
lifestyle consumption.

Quant U, assuming that the investor is 30 years old
at time 0, the price of the immediate variable
annuity is \$23.98. This is a discount of about 20%
from the initial investment needed to fund
the investment account.

So far, I have taken the investment portfolio and
the assumed interest rate as given. To see
where they come from, we need to consider 1) the

50
A₀ ( ) =

exp ( - t ) q ( t )

Portfolio Selection and the AIR

global.morningstar.com/Morningstarmagazine

45

```
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Contents
Morningstar Magazine - February/March 2018 - Cover1
Morningstar Magazine - February/March 2018 - Cover2
Morningstar Magazine - February/March 2018 - 1
Morningstar Magazine - February/March 2018 - 2
Morningstar Magazine - February/March 2018 - Contents
Morningstar Magazine - February/March 2018 - 4
Morningstar Magazine - February/March 2018 - 5
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