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
