Morningstar Magazine - August/September 2018 - 49

EXHIBIT 1

To illustrate the problem of gaps in the tail,
I sorted the monthly returns on the S&P 500 from
January 1926 through May 2018 to create
a scenario-based distribution from historical data.
E X H I B I T 1 shows the 10 largest gaps, all of which
occur in the two tails.
Solving the Gaps Problem
To fill the gaps, Savage and I created a two-stage
distribution model, which we call a smoothed
discrete distribution.1 In the first stage, a value
is selected from a discrete nonparametric
distribution. In the second stage, a value is drawn
from a normal distribution with the value
from the first stage as the mean parameter and
with the standard deviation parameter, ,
based on the standard deviation of the underlying
discrete distribution. Another interpretation
of the model is that the return is the sum
of a discrete term and a normally distributed
disturbance term with a mean of 0 and a standard
deviation of .
The statistical properties of the smoothed discrete
distribution are a bit different from those of
the discrete distribution from which it is derived.
We control these differences when we set
the standard deviation of the normally distributed
component. Let:

D

= the standard deviation of the smoothed
discrete distribution
= the standard deviation of the underlying
discrete distribution
= the parameter that controls the difference
between and D

The relationship between and
= ( 1+ )

is:
D

K

=

s=

Right End

Month

Return (%)

( 1+ ) 2 - 1
KD

)

13.12

May 1933

16.83

June 1938

25.03

8.20

Sept. 1931

-29.73

March 1938

-24.87

4.86

Aug. 1932

38.69

April 1933

42.56

3.88

April 1938

14.47

Sept. 1939

16.73

2.26

Oct. 1929

-19.73

Feb. 1933

-17.72

2.00

March 1938

-24.87

May 1940

-22.89

1.98

June 1930

-16.25

Aug. 1998

-14.46

1.79

Oct. 1987

-21.54

April 1932

-19.97

1.56

May 1940

-22.89

May 1932

-21.96

0.93

Source: Morningstar Direct, Author's Calculations.

For example, suppose that D = 20% and we set
= 2%. Then, we have ≈ 4%. The standard
deviation of the smoothed discrete distribution,
, is 20.4%.
= ( 1+ )
The skewnessD of the smoothed discrete
distribution is a bit less than that of the underlying
discrete
= (distribution.
1+ ) 2 - 1 It is given by:

(

s=

)

D

sD
( 1+ ) 3
KD

For
K = example,4 using the monthly returns of the
( 1+ )
S&P 500 from January 1926 through May
2018, the skewness of the discrete distribution
n
is about
= ( 1+0.34,
) and with = 2%, the skewness
f (R) = pj fND R; RDj ,
of the smoothed discrete is about 0.32.

)

)

= (n1+
) 2 - 1 kurtosis
Similarly,
the excess
of the smoothed
D
f (R) = distribution
pj FN R; RDj ,is a bit less than
discrete
sD underlying discrete distribution.
j =1
that of the
s=
( 1+ by:
)3
It is given

K

=

∑ (

)

KD

( 1+ ) 4
n

( 1+ ) 3

∑ (

)

1 See Appendix 26A in Frontiers of Modern Asset Allocation, edited byf (R)
Paul=D. Kaplan.
Hoboken,
R , N.J.: John Wiley and Sons, 2012.
p f R;

∑ (
n

Using the monthly returns of the S&P 500 from
January 1926 through May 2018, the excess
kurtosis of the discrete distribution is about 9.79,
and with = 2%, the excess kurtosis of the
smoothed discrete is about 9.04.
Thus, the smoothed discrete distribution is a bit
more like a normal distribution, with slightly
higher standard deviation and slightly
lower skewness and excess kurtosis than the
underlying discrete distribution. This is the
result of the addition of the normally distributed
disturbance term.
Distribution Functions
Probability distributions can be characterized by
several related functions. The probability
density function gives the shape of the distribution,
and the cumulative density function gives all of
n = qT of the distribution.
the percentiles

f = n- qT
A big advantage of smoothed discrete distributions
n
-n
-j
over their underlying
discrete
nonparametric
c
y
y f
+ 1+ q
P= q
1+ q
distributions
n = qTisj =1that their distribution functions are
continuous. This makes them much easier to use in
n
- qT
f- j 2.0 optimization.
f = n-such
qT asc the Markowitz
applications
y
y
P ( y; c, T ) = q
1+ q + 1+ q

(

∑

j N

Dj

j =1

)

n

∑ p F (R; R , )
j N

Dj

)

) ( )
+ (1 + ) (1 + )
P= ∑
∑ ( ) (1 + ) + T (1 + )
D ( y; c, T ) =
P ( y; c, T )
P ( y; c, T ) = ∑ (1 + ) + (1 + )
j =1 - j

c
q

n
j =1

j =1

c
q

j -f
q

n

c
q

f (R) =

) (

∑(

n

D

n

%

38.15

( 1+s ) 4

f (R) = K pj fN R; RDj ,
D
j =1
K =
( 1+ ) 4

Return (%)

July 1932

(

D

Month

25.03

j =1

)

Gap

June 1938

∑ (

(
(

Left End

D

For=this (relationship
1+ ) 2 - 1 toD hold, we set the standard
deviation of the normally distributed disturbance
= (as
1+follows:
term
sD ) D
s=
( 1+ ) 3
=

10 Largest Gaps in Monthly Returns of the S&P 500: January 1926-May 2018

y
q

y
q

-n

y
q

f- j

f- j

y f
q

y
q

- qT

y
q

j =1
global.morningstar.com/Morningstarmagazine
n

jn-f

c

49

f- j
y fy

+

- qT

+T

+

y

- qT


http://global.morningstar.com/Morningstarmagazine

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
Morningstar Magazine - August/September 2018 - 6
Morningstar Magazine - August/September 2018 - 7
Morningstar Magazine - August/September 2018 - 8
Morningstar Magazine - August/September 2018 - 9
Morningstar Magazine - August/September 2018 - 10
Morningstar Magazine - August/September 2018 - 11
Morningstar Magazine - August/September 2018 - 12
Morningstar Magazine - August/September 2018 - 13
Morningstar Magazine - August/September 2018 - 14
Morningstar Magazine - August/September 2018 - 15
Morningstar Magazine - August/September 2018 - 16
Morningstar Magazine - August/September 2018 - 17
Morningstar Magazine - August/September 2018 - 18
Morningstar Magazine - August/September 2018 - 19
Morningstar Magazine - August/September 2018 - 20
Morningstar Magazine - August/September 2018 - 21
Morningstar Magazine - August/September 2018 - 22
Morningstar Magazine - August/September 2018 - 23
Morningstar Magazine - August/September 2018 - 24
Morningstar Magazine - August/September 2018 - 25
Morningstar Magazine - August/September 2018 - 26
Morningstar Magazine - August/September 2018 - 27
Morningstar Magazine - August/September 2018 - 28
Morningstar Magazine - August/September 2018 - 29
Morningstar Magazine - August/September 2018 - 30
Morningstar Magazine - August/September 2018 - 31
Morningstar Magazine - August/September 2018 - 32
Morningstar Magazine - August/September 2018 - 33
Morningstar Magazine - August/September 2018 - 34
Morningstar Magazine - August/September 2018 - 35
Morningstar Magazine - August/September 2018 - 36
Morningstar Magazine - August/September 2018 - 37
Morningstar Magazine - August/September 2018 - 38
Morningstar Magazine - August/September 2018 - 39
Morningstar Magazine - August/September 2018 - 40
Morningstar Magazine - August/September 2018 - 41
Morningstar Magazine - August/September 2018 - 42
Morningstar Magazine - August/September 2018 - 43
Morningstar Magazine - August/September 2018 - 44
Morningstar Magazine - August/September 2018 - 45
Morningstar Magazine - August/September 2018 - 46
Morningstar Magazine - August/September 2018 - 47
Morningstar Magazine - August/September 2018 - 48
Morningstar Magazine - August/September 2018 - 49
Morningstar Magazine - August/September 2018 - 50
Morningstar Magazine - August/September 2018 - 51
Morningstar Magazine - August/September 2018 - 52
Morningstar Magazine - August/September 2018 - 53
Morningstar Magazine - August/September 2018 - 54
Morningstar Magazine - August/September 2018 - 55
Morningstar Magazine - August/September 2018 - 56
Morningstar Magazine - August/September 2018 - 57
Morningstar Magazine - August/September 2018 - 58
Morningstar Magazine - August/September 2018 - 59
Morningstar Magazine - August/September 2018 - 60
Morningstar Magazine - August/September 2018 - 61
Morningstar Magazine - August/September 2018 - 62
Morningstar Magazine - August/September 2018 - 63
Morningstar Magazine - August/September 2018 - 64
Morningstar Magazine - August/September 2018 - 65
Morningstar Magazine - August/September 2018 - 66
Morningstar Magazine - August/September 2018 - 67
Morningstar Magazine - August/September 2018 - 68
Morningstar Magazine - August/September 2018 - 69
Morningstar Magazine - August/September 2018 - 70
Morningstar Magazine - August/September 2018 - 71
Morningstar Magazine - August/September 2018 - 72
Morningstar Magazine - August/September 2018 - 73
Morningstar Magazine - August/September 2018 - 74
Morningstar Magazine - August/September 2018 - 75
Morningstar Magazine - August/September 2018 - 76
Morningstar Magazine - August/September 2018 - 77
Morningstar Magazine - August/September 2018 - 78
Morningstar Magazine - August/September 2018 - 79
Morningstar Magazine - August/September 2018 - 80
Morningstar Magazine - August/September 2018 - Cover3
Morningstar Magazine - August/September 2018 - Cover4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019winter
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019fall
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019summer
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019spring
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20191201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20181011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20181201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20171011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20171201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20161011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20161201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20151011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20151201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20141011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20141201
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20131011
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20130809
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20130607
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20130405
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20130203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20131201
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20121011
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20120809
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20120607
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20120405
https://www.nxtbook.com/nxtbooks/morningstar/investorconference2012
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20120203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20121201
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20111011
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20110809
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20110607
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20110405
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20110203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20111201
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20101011
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100809_lincoln
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100809
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100607_lincoln
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100607
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100405_lincoln
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100405
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20101201
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20091011
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20090809
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20090607
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20090405
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20090203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2008fall
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2008summer
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2007spring
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2007fall
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2007summer
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2008spring
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2008catalog
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2008winter
https://www.nxtbookmedia.com