Morningstar - Q3 2022 - 49

yq
Certainty Equivalent Return
2
2
1
0%
0%
To formulate the spending rule, I distinguish
between nondiscretionary and discretionary
consumption. Nondiscretionary consumption
includes any spending on essentials such as food,
clothing, and shelter. Discretionary consumption
is any spending beyond that. For year t,
denote nondiscretionary consumption t
, total
consumption ct, so that discretionary consumption
is ct - t. Nondiscretionary consumption
can vary over time in ways that are correlated
to asset-class returns.
Related to nondiscretionary consumption
is the value of liabilities, which is the amount of
wealth that the investor must dedicate to
the funding of nondiscretionary spending. The
value of liabilities in year t is:
) (1+
(1+h )
T
L =t
v= t
T
where:
H =t
v= t
∑ (1+kc
1
)
∑ (1+ky ) t yv
1
W + -Ht
kc
t = Ft
Wt+1
= (1+
ct = +
t
g =
t
v- t
t
v
25
25
Age
Age
t
T = the last possible year for the investor
to be alive.
Lt
= the expected return of a portfolio
that represents the uncertainty
of nondiscretionary consumption.
) Wt
(
Wt
t
- +ct
t )
= The expected value of nondiscretionary
consumption in year v as of year t.
Income and human capital are also components
of the spending rule. I denote income in year
t by yt
wages. Starting at retirement, it can include
government-sponsored social insurance (such as
Social Security in the United States), payouts
of defined-benefit pension plans, and any
annuities that were previously purchased. Like
nondiscretionary consumption, income
can vary over time in ways that are correlated
to asset-class returns.
. Before retirement, this mainly consists of
T
1+h
1+
t =∑ (qv
v= t
Vvt
c =
Nv
Pv
t
=
t
+
t )
1+g
1+h
Vt (1+ Pv
t
Nv
t = qv
T
( Vv /Vt
(1+h )
t )
t
Lt = ∑ (1+kc ) t
T
L =t
∑
qv
1
v= t
v= t
T
1+kc
t
H =t
H =t
∑ ( 11+ky ) t yv
T
qv
v= t
Lt
Lt
W =
=
W (1+ qt
t+1
)
(
t+1 ) W c- +
(Wt
- +ct
)
t )
g =
)
t
t ) (1+
(1+g)
v- t
v-t
Wt
t
Human capital is the economic value of
present and future income. Human capital in
year t is: (
t
v
t
v
t
∑ (1+ky ) t yv
v= t
W + -Ht
W + -Ht
t = Ft
t = Ft
(1+
t
The return that we introduced earlier,
ct = +
t
Wt
t
1+h
1+
- 1
W + -Ht
t = Ft
= (1+
Lt
Wt+1where Ft is financial wealth.Wt
) (
- +ct
t )
o
, is
v- t
)... (1+ )
v
- 1
Vv = Vt (1+
25
25 =Pv
t
Age
Age
T
L =t
Vv /Vt
(1+h )
v- t
Source: Morningstar.
1
v= t
The economic value of wealth that is available for
discretionary consumption is a person's net worth.
Net worth in year t is:
T
H =t
v= t
∑ (1+ky ) t yv
1
t
) (1+
)... (1+ )
v
50
50
∑ (1+kc
t
)
t
v
H =t
75
75
50
50
Source: Morningstar.
75
75
)... (1+ )
v
Pv
t
Vv /Vt
=
Vv = Vt (1+
1.0
1.0
Risk Tolerance
25
Risk Tolerance
25
EXHIBIT 2
Percentile
95th
95th
50
50
75
75
100
100
1
Percentiles of the Payout of an IVA Over Time
Percentile
90th
90th
75th
75th
50th
50th
25th
25th
10th
10th
5th
5th
Payout
Payout$6
$6
5.1
5.1
3.8
3.8
f = n- qT
n = qT
100
n
P =
EXHIBIT 3
o
c
q
∑
j=1
n
Number of IVAs With and Without Actual Annuities
With Annuities Without Annuities
With Annuities Without Annuities
P(y; c, T) = ∑(1+
j=1
c
q
D(y; c, T) =
∑ n
j=1
j -f
q
y
q
)
f-j
+ 1+
(
( ) (1+
c
q
Vv = Vt (1+
c
q
Pv
t
=
(1+h )
T
L =t
=
v= t
∑ (1+kc
(y)
c
q
1
∑ (1+ky ) t yv
TcP =
1
v= t
D =
c (y)
t = Ft
1
y
1
y
actually the return on net worth. Hence, net worth
evolves as follows:
f = n- qT
n = qT
c
q
ct = +Dm (y; c, T) =
The decisions of the investor are constrained by
an intertemporal budget constraint that says
that net worth must be able to pay for present and
all future discretionary consumption. Imposing
t
g =
∆P
P
P(y; c, T) = ∑(1+
j=1
Wt
t
c
q
1+h
1+
≈ -Dm∆y
- 1
morningstar.com/products/magazinev- t
t =∑ (qv
D(y; c, T) =
TDV01 P=
t )
v= t
*
q
1+h
(
)(
)
f
∑ n
1+g1+ y = Dm
P
D
j=1
q
D(y; c, T)
n1+ y
y
q
j -f
q
)
f-j
+ 1+
(
( ) (1+
c
q
y
q
)
P(y; c, T)
49
y
q
f-j
)
- qT
+ 1+T
(
y
q
W + -Ht
Lt
Dm (y; c, T) =
n
P(y; c, T)
P (y; c, T)
Wt+1 = (1+P = ∑ () Wt
j=1
- +ct+ 1+)
-j
(
t
y
q
)
-n
(
o
1+
y
q
)
f
(
(
v- t
(
y
1+ q
)
t
P(y; c, v
)100
)100
-j
t
n
q
1+ y
t
q
y
q
)
f
120
120
f
P(y; c, T) = P(y; c,
) (1+
(
Vv /Vt
P(y; c, T)
1+
y
q
)
f-j
=
c
q
n
q
)(1+
(
y
q
P(y; c, T)
)
1+
P(y; c,
n
q
f
)... (1+ )
f-j
v
y
q
)
)(1+
y
q
)
f
0.0
0.0
Number of IVAs
100K
y
q
)
+ 1+T
(
y
q
Number of IVAs
100K
75
75
50
25
50
25
y
q
f-j
)
- qT
100 + 1+
-j
(
y
q
)
-n
(
120
1+
y
q
2.3
1.3
0.7
0.4
0.3
2.3
1.3
0.7
0.4
0.3
120 )
f
2
3
4
5
2
3
4
5
1
1
)
- qT
1+ )- q
)
- qT
http://www.morningstar.com/products/magazine
Morningstar - Q3 2022

Table of Contents for the Digital Edition of Morningstar - Q3 2022
