Morningstar Magazine - August/September 2018 - 50

=

(

)

( 1+
)2 - 1
Strategies

f = n- qT

D

n

sD
s=
( 1+ ) 3

K = probability
The
density function of a smoothed
( 1+ ) 4
discrete return distribution function is given by: 2
n

∑ p f (R; R , )
j N

-j

∑

(

Dj

y

+ 1+ q

j =1

KD

f (R) =

c
q

P=

n

-n

) (1 + )

y f
q

f- j

- qT

∑ (1 + ) + (1 + )
Cumulative Density Function of

EXHIBIT 2

EXHIBIT 3

Probability Density Function of
Smoothed Discrete Distribution Based
on Monthly Returns on the S&P 500,
January 1926-May 2018

Smoothed Discrete
Distribution
f- j Based - qT
y
j -f c
y
n
∑ j =1 ( qon
q (1 + q ) + T (1 + q )
)
S&P
500,
on Monthly
Returns
the
D ( y; c, T ) =
P ( y; c, T )
January 1926-May 2018

c
q

P ( y; c, T ) =

y
q

y
q

j =1

j =1

DiscreteDiscrete
Values ofValues
Returnof Return
where: n
Probability
Probability
Density Density
FunctionFunction
f (R) = pj FN R; RDj ,
j =1
n
= the number of scenarios in the
underlying discrete distribution
= the probability of scenario j
pj
= the return under scenario j
RDj
f(.; RDj , ) = the probability density function of
the normal distribution with mean
RDj and standard deviation

∑ (

)

E X H I B I T 2 shows the probability density function
based
on the
-15
-150
0 15
1530
3045
45
= ( 1+
) Dmonthly returns on the S&P 500 -30% -30%
MonthlyMonthly
Return Return
from January 1926 through May 2018 with
Source: Morningstar Direct, Author's Calculations.
= 2%. The continuous function captures all of
the=points
the underlying discrete
( 1+in the
) 2 -tails
1 of
D
distribution, but without gaps. The lines
below the
in the tails of the underlying discrete distribution,
sD probability density function show the
s=
3
but without gaps. The lines below the cumulative
locations
of
the
discrete
values
of
return.
( 1+ )
Note that in the central part of the curve, the gaps
density function show the locations of the
discrete values of return. As with the probability
are so small
KD that the lines appear to form
= area. 4However, moving out into the tails,
aK solid
density function, in the central part of the
( 1+ )
curve the gaps are so small that the lines appear
the gaps became larger and more frequent.
to form a solid area. Moving out into the tails,
n
the gaps became larger and more frequent.
The cumulative density function of a smoothed
f (R) = pj fN R; RDj ,
Because the curve is everywhere upward sloping,
discretej =1return
distribution function is given by:
the gaps in the right tail are more pronounced
n
than they are in the left tail.
f (R) = pj FN R; RDj ,

(

)

∑ (

∑ (
j =1

)

)

Where:
F (.; RDj , ) is the cumulative density function of the
normal distribution with mean RDj and standard
deviation .
shows the cumulative density function
based on the monthly returns on the S&P 500
from January 1926 through May 2018 with = 2%.
As with the probability density function,
the cumulative function captures all of the points
EXHIBIT 3

Maximum Flexibility
A major goal of the Markowitz 2.0 portfolio optimization framework is to have maximum flexibility
with respect to return distributions. While a
scenario-based approach achieves this, it does so
at the expense of having discontinuous probability
distribution functions with gaps between
the possible values. These gaps can be quite large,
especially in the tails. The smoothed discrete
distribution approach fills these gaps by adding
a normally distributed disturbance term.
Smoothed discrete distributions maintain the
flexibility of the scenario approach while removing

DiscreteDiscrete
Values ofValues
Returnof Return
y f
= P y;Density
Cumulative
P ( y; Cumulative
c, T )Density
Probability
Probability
1 + q Cumulative
c, qn Cumulative
100% 100%

)(

(

c
q

=

f- j

(1 + )
y
q

c
q

=

P ( y; c, T )

)

f- j

(1 + )
P ( y; c, ) (1 + )
y
q

y f
q

n
q

-j

(1 + )
P ( y; c, )
y
q

c
q

n
q

(

y f

)

(

)

Pc ( y ) = y1 1 + q
-30% -30%
-15
0-15
015
30
15
MonthlyMonthly
Return1 Return
y
f
= Direct,
q - Calculations.
q
y 1+
nDc=( y ) qT
Source: Morningstar
Author's

50

50

25

25

0
45

0

f = n- qT
P ( y; c, T )
Dm (y; c, Tn ) =
-n
f
discontinuities.
Thus,Pthey
( y;- jc,are
c
y suited to
y form
T ) well
+ 1+ q
P= q
1+ q
the basis for scenario-based
optimization.
j =1

(

∑

) (

)

Dn( y; c, T )
c, T ) of
= cQuant
Dm (y;issue
- qT
f- j
yU, yI will
yhow to
In the next
P ( y; c, T ) = q 1 +1q+
+show
1+ q
q
use smoothed discrete
j =1distributions to measure
risk and∆expected
return
in the context f-ofj
P
y
y
≈ - D ∆y n j -f c
Markowitz
P 2.0. Km ∑ j =1 q q 1 + q + T 1 + q
D ( y; c, T ) =
P ( y; c, T )
Paul D. Kaplan, Ph.D., CFA
D , is director of research with
DV 01Canada.
= P He isy a=member
P Dm of the editorial board
Morningstar
1+ q

∑(

) (

( )(

)

)

(

*

of Morningstar magazine.

(

P ( y; c, T ) = P y; c,

n
q

) (1 + )

c
q

y f
q

P ( y; c, T )
f- j
y
P ( y;c c, T ) 1 + q
= q
y
P ( y; c, T )
P y; c, qn 1 + q

C (y; 1c,+T y) =f- j

(

q

)

(

)
)( )

(

C ( y; c, T1) += y - j
q
= qc
n
c
f - j c, f - j - 1
n P y;
q
q ∑
j =1

Pc ( y ) =

(

f

( )
(( ) ( ) ) (1 + )
q

y
q

q

P1( y; c, Ty)
y 1+ q

(

)

f- j-2

f

)( )
( )
y

1
T T+ q 1+ q
1
+Dc ( y ) = y 1 + qy -
P ( y; c, T )

- qT - 2
f
q

2
∆P
≈ - Dm ∆y P+ (12y;Cc,( ∆y
T))
DPm (y; c, T ) =
P ( y; c, T )

Dm (y; c, T ) =

Morningstar August/September 2018

75

30
45

2 This formula for the probability density function, and the one for the cumulative density function, shows how a smoothed discrete distribution is a special case of what in probability and
Wt = Ht + Ft D ( y; c, T )
statistics is called a "mixed normal distribution."

50

75

y

1+ q

Wt = (1 + r) (Wt -1 - ct-1 )
∆P
≈ - Dm ∆y
PT

- qT

)



Table of Contents for the Digital Edition of Morningstar Magazine - August/September 2018

Contents
Morningstar Magazine - August/September 2018 - Cover1
Morningstar Magazine - August/September 2018 - Cover2
Morningstar Magazine - August/September 2018 - 1
Morningstar Magazine - August/September 2018 - 2
Morningstar Magazine - August/September 2018 - Contents
Morningstar Magazine - August/September 2018 - 4
Morningstar Magazine - August/September 2018 - 5
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