Morningstar - Q2 2020 - 66

v=

Strategies

T

∆t =

∑

v= t

Percentiles of Spending
10th

25th

50th

75th

90th

Annual Spending

95th

0
60
40
20
0
0
Year

10

20

30

40

Source: Morningstar.

Percentiles of Financial Wealth
10th

25th

50th

75th

90th

Financial Wealth

95th

00
600
400
200
0
0
Year

10

20

30

Percentiles of Equity Allocation
25th

50th

75th

90th

Equity Allocation

95th

60%

40

20
0
0
Year

10

Source: Morningstar.

66

Morningstar Q2 2020

∑
∑

∑

∑
∑
∑

( )
∑ ( )

EXHIBIT 5

10th

∑
∑
∑

( )

40

Source: Morningstar.

5th

1+ r
g = T ˉ -1
1
∆t = 1 +
-t
+ r ) vhave
( 1must
the retiree
enough wealth to pay out
v= t
t
t
nondiscretionary
spending
in present and all
RFv = ( qv )
Wt The present discounted value of
future
years.
ct =
v-t
presentT ∆and
future
spending is
t
1 + g nondiscretionary
∆t =retiree's
RFv liability. The liability in year t is:
the
1 + rˉ
v =T t
1
∆t =
T
( 1 +c rˉ ) v - t
v= t
ˉv
Lt =
+
(
1
r )v-t
v =Tt
1+ g v- t
∆t =
1F+-rˉ L
v= t
t
t I assume that no annuities
that
because
=
+
c
cNote
ˉ
t
t
∆
are available,
is no adjustment for survival
t
1 + rˉ there
g=
-1
probabilities.
Also,
because
we are treating
T1 +
yv
Ht =
nondiscretionary
as certain, it is
v -spending
t
(1+ r )
v= t
discounted
at the risk-free rate.
RFvt = ( qvt )
Ft + Ht - Lt
cLt is=the
cˉTt +minimum
v-t
of wealth that the
t
+ gamount
1∆
∆t = T must
RFv havet in order to fund present
retiree
1 1 + rˉ
v= t
and
nondiscretionary
spending. Any wealth
∆t =future
Ht - vL-t t
+
(
+
xt = v =1t L is1 forr )funding discretionary spending.
beyond
T
F
t
cˉ v t
Lt = discretionary
Taking
spending into account,
Wt( 1 + r ) v - t
v= t
~
=
cthe
spending
rule
becomes:
t
u'(ct+1 )
~
1 ∆
t
= Qt
1+
u'(cFt) - L t+1
ct = cˉTt + t1 t
∆t
∆t =
1- 1
( 1 + rˉO) v=- t1 c O -1 , O 1
u(c) =v = {t ln(c),
T
1- O1
yv
where
Ht = Ft
T
( 11++ rg) v -Fvt--t L . This is because L is that
v= t
replaced
t
t
t
-1
∆t = c~Wt with
O
~that
1
t
t+11
+
amount
of
wealth
needs to be invested
rˉ = Q
v= t
+
-
H
L
F
t+1
t
t
c1t += cˉ t +ct t
∆
1 + rˉ
t
gc~ =
-1
1 - ~t - O
+
t+1
1
=
H O LtQt+1
)
xtct= 1 +(1 + t spending.
discretionary
Ft
t
t
RFv = ( qv )
~t - O
Taking
Retirement
~ )Qt+1 Income Into Account
~
u'(c
1 =T t+1 ~ + ~1 -t Ov - t - 1
ROt+1
1t= gQretirement
Finally, I introduce
income into the
t+1
+
Et v ) Qt+1
∆1t = RF
u'(c
t denote the amount of income in year
model.
Let
y
+
v 1 rˉ
v= t
v. This includes
1 insurance payments
1 - Osocial
~t any
~
c1- O -1~, O 1
E
Q
c
T
t
t+1
=
{
ln(c),
O
=
1
u(c)
U.S.) that the retiree
(such
as Social
t+1
cˉ v Security 1in+1the
1- O ROt+1
Ltct= = To(1take
O
+ v -)these
t
receives.
payments
into account,
+
(1 r )
v= t
assuming that1their values are known with
-
O
c~t+1
~their
1
certainty,
we take
present discounted value,
t
- =Lt Qt+1
F
t
+
ct
1 = cˉ I +denote
for
human
capital. We have:
H
cwhich
t
t
∆t t
-O
c~t+1 T 1
y Q~t
=
Hct t = (1 + )vO v - t t+1
(1+ r )
v= t

∑
∑

EXHIBIT 4

5th

v-t

∑

EXHIBIT 3

5th

1+ g
1 + rˉ

20

30

40

∑

∑

( )

(

)

( )

~t - O
Ht - Lt
Ft Q+t+1
~
=
+
cRHuman
c
=
-1
ˉ
tOt+1 t capital
1 - Oaddition to total wealth.
~tis∆ an
Et Qt+1 t
Hence, the spending rule becomes:
H -L
x~t = 1 +E Qt~t t1 - O
ct+1
~
Ft+1
t
t
=
1 + ROt+1
ct
(1 + ) O
~
~t
1 u'(ct+1 )
= Qt+1
1+
u' c

( )
( )

( )

(

)



Morningstar - Q2 2020

Table of Contents for the Digital Edition of Morningstar - Q2 2020

Contents
Morningstar - Q2 2020 - CT1
Morningstar - Q2 2020 - CT2
Morningstar - Q2 2020 - Cover1
Morningstar - Q2 2020 - Cover2
Morningstar - Q2 2020 - 1
Morningstar - Q2 2020 - 2
Morningstar - Q2 2020 - Contents
Morningstar - Q2 2020 - 4
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Morningstar - Q2 2020 - Cover3
Morningstar - Q2 2020 - Cover4
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