Morningstar - Q3 2022 - 48

Strategies
Immediate Variable Annuities:
Real and Imagined
Either way, they can be useful tools
for managing longevity risk.
QUANT U
Paul D. Kaplan
Many investors face the prospect of outliving
their money. Fortunately, this risk, called longevity
risk, can be mitigated with a class of financial
products known as annuities. However, there are
various types of annuities, and picking the
right one is key in dealing with longevity risk.
The most basic type of annuity is an immediate
fixed annuity, or IFA. Upon purchase (hence,
immediate), it pays a fixed amount (hence, fixed) to
its holder at regular intervals for the person's
remaining life. If the IFA is held by a couple, it can
have a provision that should one member of the
couple die first, the surviving member continues to
receive a payment until death, possibly in
an amount different than the original payments.
In previous issues of Quant U (Kaplan 2018a,
2022), I discuss how in the context of a
lifecycle mode, investors can use IFAs and life
insurance to guarantee lifetime income and
guarantee a bequest in a world where market
returns are constant.
Immediate Variable Annuities
Of course, market returns vary. In other previous
issues of Quant U, I discuss lifecycle models
with risky market returns (Kaplan 2018b, 2020).
When market returns are risky, the relevant
type of annuity is an immediate variable
annuity, or IVA.1
Rather than making fixed
payments, an IVA 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.
To see how an IVA works, let
.2
denote the
realized return from year t to year t + 1 on a
portfolio of risky assets formed for an investor with
risk tolerance
These returns can be linked
over time to form an evolving cumulative index,
from year t to year v as follows:
Vv = Vt (1+
Letting h denote the assumed interest rate, the
payoff in year v of an IVA bought in year t is:
Pv
t
Vv /Vt
=
Vv = Vt (1+ )
(1+h
T
L =tt =∑ /1Vt
(1+kc
Pv
Vv
v= t(1+h )
T
T
H =t
In principle, investors should not only choose
an IVA with an underlying risky portfolio
that is consistent with their risk tolerance, but also
with an assumed interest rate that is equal
to the portfolio's certainty equivalent return. A
portfolio's certainty equivalent return is
the constant rate of return for which the investor
is indifferent to the risky return. Hence, it is
L =t
∑
T
H =t
Wt+1
t =∑ (1+ky ) t yv
= (1+
W + -Ht
v= t
Ft
1
W + -Ht
ct = +
t = Ft
t
Wt+1
g =
Wt
t
= (1+
1 See section 6.12 in Milevsky (2006) for a more detailed mathematical discussion of IVAs. Also, " [n]ote that immediate variable annuities are distinct from and should not be confused with
deferred variable annuities [VAs], which are tax-deferred accumulation polices that allow the investor to allocate funds to risky or investment funds. " (Milevsky 2006, p. 131).
2 I place a ~ over a variable to indicate that its value is unknown as of the present year (t) and will not be known until a future year.
ct = +
t
T
t =
=
48
Morningstar Q3 2022
1+h
1+
1+h
1+
g ∑ (qv
v= t
Wt
t
- 1
t ) - 1
1+g
1+h
v- t
95th
90th
75th
50th
25th
Lt
(
∑ (1+kc
1
v= t
v= t
( 1+ky ) t yvv
t
1
)
t
) Wt
Lt
) Wt
(
- +ct
t )
0%
Risk Tolerance
25
- +ct
t )
Source: Morningstar.
Percentile
10th
5th
50
75
o
100
D
P
c
q
tt
)
v- t
t
t
v
v- t
) (1+
)... (1+ )
v
Expected Return
) (1+
)... (1+ )
v
EXHIBIT 1
Expected Return and Certainty
Equivalent Return for Portfolios
Formed for Different Levels
of Risk Tolerance
Expected Return,
Certainty Equivalent Return
5%
3.6
4.7
Certainty Equivalent Return
o
2
D
1
P
3
4
f
n
P
f
n
P
P
The Optimal Spending Rule When Annuities
Are Not Available
In Kaplan (2020), I presented the optimal spending
rule for a retiree based on their preferences,
needs, and circumstances when annuities are not
available. Here, I generalize that rule for
any investor and show how it is related to IVAs.
a function not only of the expected return and
volatility of return of the portfolio, but also of the
investor's risk tolerance.
EXHIBIT 1 shows the expected return and certainty
equivalent return of portfolios constructed for
investors with different levels of risk tolerance. As
this exhibit shows, the certainty equivalent
return is always less than the expected return,
except in the case of zero risk tolerance, when
both are equal to the risk-free rate.
EXHIBIT 2 shows various percentiles of the payouts
of an IVA over time.
P
c
q
=
Pc
=
Morningstar - Q3 2022

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