Transcript The Math and the Magic of Financial Derivatives

On The Mathematics of
Income Inequality
Klaus Volpert, PhD
Villanova University
February 11, 2011
Acknowledgements: This is joint work with Bob Jantzen.
Thanks for helpful conversations with Tim Feeman, Jesse Frey and Bruce
Pollack-Johnson.
Stunning inequality
In 2010, hedge fund manager
John Paulson took home
$5 Billion.
Stunning income differences
In 2010, hedge fund manager
John Paulson took home
$5 Billion.
That’s more than all the salaries
of all the full-time faculty of 100
Villanova-like universities
combined. (50,000 professors)
On the other hand, there is no doubt that one
could find 50,000 of the world's poor whose
combined wealth is less than each one of ours.
On the other hand, there is no doubt that one
could find 50,000 of the world's poor whose
combined wealth is less than each one of ours.
Does it have to be that way?
Even if we could distribute all wealth equally, inequality would
come back immediately. For one man would take his money to
the bank, and one man would take it to the bar.
Herr Procher (my 10th grade English teacher)
Inequality seems to be built into life itself. . .
As is our struggle with it. . .
So is there anything we can do?
The first thing we need to realize is that. . .
Inequality might be inevitable,
But it is not static!
Inequality varies a great deal from country to country.
Even within the same country it can change dramatically over time!
So what causes such differences?
To help figure this out, we can measure the degree of inequality,
so that we can
• understand trends
• see connections
• discern forces that increase or decrease inequality
• and possibly effect change
Q: What do you think: over the last 100 years, has
inequality in the US increased or decreased?
Q: What do you think: over the last 100 years, has
inequality in the US increased or decreased?
30.00
25.00
Share of total income (in percent) of top 1% Earners in
the US, 1913-2008
20.00
15.00
10.00
5.00
1913
1918
1923
1928
1933
1938
1943
1948
1953
1958
1963
1968
1973
1978
1983
1988
1993
1998
2003
2008
0.00
Discussion:
•What caused the decreases and increases in inequality?
• Partisan politics?
• Tax rates?
• Immigration?
• Globalization?
• Labor Movements?
• A fundamental change in American values from greed (1920’s)
to the contentment with the small house and white picket fence
(1950’s), back to greed (today)?
There is a Problem with the measure used on the previous slide:
the percentage share of the top 1% ignores the middle class as well as
the poor. It says nothing about poverty rates, for example, which follow
quite a different graph.
We need a more comprehensive measure.
The Lorenz Curve and the Gini-Index
L(x)= share of total
income earned by all
households combined
that are poorer than the
xth percentile, when
ranked by income.
0.1
e.g., L(.4)=.1 means
that the poorest 40% of
the population have a
share of 10% of the
total income.
The curve has two anchors: (0,0) and (1,1)
The Lorenz Curve and the Gini-Index
If income was perfectly
evenly distributed, then
L(x)=x, and the graph
would be a straight line
Perfect equality
The Lorenz Curve and the Gini-Index
Perfect
inequality
If one household
received all the income,
and no one else
received anything,
the curve would look like
this
The Lorenz Curve and the Gini-Index
Realistic curves are
somewhere in
between.
The closer to the
diagonal the more
equally distributed the
income.
The Lorenz Curve and the Gini-Index
The Gini-Index is the ratio
of the area between the
line of equality and L(x),
and the entire area under
the line of equality
(which is ½).
So, 1
G  2 x  L( x )dx
0
Perfect Equality → G=0
Perfect Inequality → G=1
Example: If L( x)  x2 , then
1
2
3
1
x
x
1
G  2  x  x dx  2(  ) 
2
3 0 3
0
2
Example: If L( x)  x2 , then
1
2
3
1
x
x
1
G  2  x  x dx  2(  ) 
2
3 0 3
0
2
So the Gini-index is a
summary measure,
giving weight to the
poor as to the wealthy
Example: If L( x)  x2 , then
1
2
3
1
x
x
1
G  2  x  x dx  2(  ) 
2
3 0 3
0
2
So the Gini-index is a
summary measure,
giving weight to the
poor and to the wealthy
This allows us to
make comparisons
between countries,
as well as look at
trends over time.
The Gini-Index for various Countries:
(according to data from the World Bank and CIA)
Country
Gini-Index
Sweden
.25
Germany
.28
Egypt
.33
China
.42
Russia
.44
Brazil
.55
South Africa
.65
Discussion: there are several reasons why one must take
these numbers with a grain of salt. . .
(what are the basic economic units? What constitutes income?
What about adjustments for cost of living? Etc.)
Still. Overall the comparative impression is correct.
What about the US?
Here are Some Raw Data from the Census
Bureau:
US-Income Data 1967:
Published by the Census Bureau
Quintile
How can we calculate the Giniindex from these Quintiles??
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
Connect the Dots??
US-Income Data 1967:
Published by the Census Bureau
Quintile
This would clearly underestimate the
area and the Gini-Index.
We would rather have a smooth
function through the data points. . .
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
A First Model for the Lorenz Curve:
Matched with real income data,
however, no power function fits
the whole population very well:
Best fit:
L( x)  x
2.287
L( x)  x
p
US-Income Data 1967:
Published by the Census Bureau
Quintile
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
L( x)  x
p
However!
If we only try to match the
bottom 60%, then it does fit well!
We’ll come back to that. . . .
Best fit:
L( x)  .849 x
1.90
US-Income Data 1967:
Published by the Census Bureau
Quintile
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
A Second Model for the Lorenz Curve: A Quintic Polynomial:
L( x)  ax  bx2  cx3  dx4  ex5
A quintic polynomial fits much
better:
US-Income Data 1967:
Published by the Census Bureau
Quintile
Lorenz
value
.2
.04
.4
.148
.6
.321
.8
.563
1
1
L( x)  .13x  .23x2  3.9 x3  5.8x4  2.99 x5
2 Problems: (a) We calculate G=.391, while the official number is G=.397
(b) The coefficients tell us nothing.
Q: Can we come up with a function that has some economic meaning?
The rich don’t feel so rich. . .
The self-similarity phenomenon:
In 2007: (according to data from Saez and Picketty)
•The top 10.0% received 48% of total US income
•The top 1.0% received 24%
•The top .1% received 12%
•The top .01% received 6%
The inequality repeats among the rich:
no matter how rich you are, there are always some who are far richer,
in a very peculiar and predictable pattern.
Q: what function for L(x) would describe such a pattern?
L( x)  1  (1  x) , 0  q  1
q
Here is a family of such
functions.
q=1 → total equality
q=0 → total inequality
1
q   log10 ( S )
2
where S is the share
received by the top 1%.
In 2007: the top 1% received 24% of total, so q=.31
(roughly the blue line)
The Principle of Self-Similarity `to the right`
These functions have this remarkable property:
The Gini-index among the top x percent is the
same as in the whole population, for any x.
=
=
This can be called a (one-sided) `scale-invariance’,
also known as the `Pareto Principle’.
This function works great for the Saez/Picketty data of the top 10%,
but, alas, even this model does not fit very well for the whole
population:
So what to do? We have a model that fits very well at the top, and,
remember, we have a model that fits very well at the bottom. . . Let’s
p
take another look at
L( x)  x
L( x)  x
p
exhibits the opposite phenomenon:
Self-similarity to the left:
The inequality repeats among the poor
The Gini-Index among the whole population is the same as the
Gini-Index restricted to just the poorest 50%, say.
Or the poorest 20%. Or the poorest 2%. Etc.
So one model works well at the top, one at the bottom, so for the whole population we
try a hybrid model, that asymptotically equals the two separate models at either end:
The Hybrid Model:
L( x)  x  (1  (1  x) )
p
q
Note: asymptotically we have
as x  0, L( x)  qx
p 1
as x  1, L( x)  (1  (1  x)q )
but the self-similarity is
`softened’ on either side.
Empirically, the model fits
amazingly well:
L( x)  x (1  (1  x) )
p
q
1967
With this model, G=.395.
Official value:
G=.397
L( x)  x.7438 (1  (1  x).6823 )
Sum of Squares of Residuals: 0.00001989
L( x)  x (1  (1  x) )
p
2008
2008
L( x)  x.7987 (1  (1  x).5608 )
Sum of Squares of Residuals: 0.00000396
q
With this model, G=.468
Official value:
G=.466
Quintic model: G=.457
We matched this model to
every year 1967-2007.
Each time the predicted
Gini-index, based on just 4
data points each year,
is within 0.25% of the
official Gini-Index computed
by the Census bureau from
a full set of raw data.
Softening the Self-Similarity:
The Gini-Index from the point
of view of the rich:
The Gini-Index from the point of
view of the poor:
2008
2008
1967
1967
This graph shows as a function of r, the
Gini-index restricted to the poorest r%
This graph shows the Gini-index
restricted to the richest (1-s)%.
The Gini-index, between 1967 and
2007, localized to the rich has
increased far more than the Giniindex localized to the poor.
1967-2008: The Trends:
In this time span, most of the increase of the nation’s wealth has been absorbed
by the rich, in such a way as to vastly increase the gradient among the rich, while
hardly causing a ripple among the middle class and poor.
←p
Inequality among the poor
has slightly increased
←q
Inequality among the rich has
increased significantly
←GiniIndex
What have we gained?
Differentiation and Understanding:
1.The Gini-Index has increased a great deal from 1967 to 2008, from
.396 to .466. But that does not tell the whole story.
By creating a well-fitting two-parameter model for the Lorenz curve, we
can now see what happened: The increase is mostly due to an accrual
of new wealth at the top, increasing the wealth-gradient among the rich.
2. This helps us understand the instability in the financial markets that
led to the crash of 2008: As the wealth accrues to a smaller and smaller
sliver of the population, so does the risk. The increased gradient
(actually convexity) among the rich creates a `gold-rush’ mentality. That
induces financial players to take ever-increasing risks with financial
instruments, such as credit default swaps and other derivatives. Instead
of dispersing risk, as most notably, Alan Greenspan conjectured, these
instruments served to concentrate risk, and thereby destabilize the
markets.
Future investigations:
1. Obtain raw data from other countries, and investigate whether
there are examples of countries with similar Gini-Index, but very
different p- and q- parameters, thereby pointing to a
differentiation in the structure of inequality
2. Obtain raw data from the US from earlier periods, in order to
learn more from the evolution of the p- and q- parameters.
Concluding Thought
It is easy to forget among all the models and data, that there are real
people affected by poverty and inequality.
So I would like to finish with a quote from a girl at a school in South Africa
that is under the care of the Augustinian Mission.
While this good work of the Augustinians and others might hardly register
on the Gini-Scale, they are actually doing something about poverty.
Instead of turning their attention to the rich and the race to the top, they
have turned around and seen the poor, and decided that this is not right.
Andile is one of the reasons for their work, and one of their triumphs:
“I am somebody. I may be poor, but I am somebody.
I maybe make mistakes, but I am somebody.”
Andile
Student, St. Leo’s School, Durban, South Africa