So welcome one and all to the
latest installment
of the Brooklyn
Quant Experience BQE
>> And we originally
scheduled to have
Professor Kim Western from
Rutgers University
present tonight,
but she came down to cool.
So we asked Professor Stern
to step in and
he kindly agreed
>> So professor Stern is
visiting NYU Tandon
and I saw him
give this talk
recently at Fordem
and I really liked it.
>> And I have to say I didn't  understand
everything I wanted
to hear again.
>> So so I'd like
to read Professor
Stearn's bio.
So he is Associate
Professor of
Mathematical
Sciences at
Werchester Polytechnic
Institute.
>> I think I said
that badly anyway,
WTI Massachusetts and
currently spending is
sabbatical at
the Chinese University
of Hong Kong
and here at NYU Tandon
So after obtaining  his PhD in
math from TU Berlin in Germany,
he became a Postdoc
research associate
and lecturer at
the Department
of Operations
Research and
financial engineering
at Princeton University.
Before joining the
APIs faculty member,
Professor Stearn's
research covers
many different areas and
financial mathematics,
but he's interested also in
stochastic modeling
in general,
such as applications to
climate science. In finance
his work is devoted in
particular to questions of
value adjustments for
jury securities
This is called with SPAs,
optimal portfolio
selection and
systemic risk in
financial markets.
Professor Stern. Thank you Peter
>> I have to say I
have a really great time.
in WIT Massachusetts
and  NYU. I am
more than happy to step in.
>> And of course
I cannot give
you what Professor western would have given you.
but I think it's, it's
an interesting talk
and I think it has
many different levels.
>> So I've things
as core message,
which is relatively
easy to understand
and send it to different
levels of detail.
>> So to set it up on a  value-based care.
>> So what we have
is as one of, say,
classical questions
of quantitative finance.
quantitative finance
basically is
ease of derivatives
pricing.
>> What portfolio
management?
>> And no, maybe
a mismatch.
>> But these are
the three big
topics but we will discuss
one of them,
portfolio management
>> And we take however
little bit different angle than
what most people are doing,
authentically in the
industry,
and I will tell you why.
>> But before I do this, I
should point
out this is not
only work by myself but with 
to collaborators.
>> One is Mauricio Enriqui
who is currently
a PhD student in
Madrid University 
Autonoma and
the other one is Carole Bernard
>> She is a professor in
finance at
business
school in France.
>> And we worked together
on different
not supposed to talk.
>> We are doing
things maybe
a little bit different than what
You might have
experienced somewhere else
>> Let's maybe first take
value-based back
and let's think in general
about what
is portfolio optimization
and how to do
what is portfolio
optimization
is I think an easy
question. Question is
usually you have
a certain amount of
money and you invest,
want to invest it optimally in
different assets in
way which is good for you
>> But we have
to discuss is
what means it
is good for you
and in general it's good for you can
be understood as many different ways
On a big level,
One could say that
we try to find
some general of the
amount of (inaudible)
which has nothing
to do with you and your proximal preferences
but just tries to
say, let's do best.
>> When I was the
best decision that
things like this for
businesses are called Kelly criterion
Kelly backing
maximum common as an approach
This is an approach to
twice takes it to
preferments of an investor into account
>> There many ways
to do this is
for instance,
macrophytes approach
after that is 
the CAPM which
basically relies
Mr. Bushman and
understanding of valency is
a trade off between
risk and variance.
>> And if their hands,
well, we want to have
return as high as possible
and this as small
as possible.
>> And you kind of a chief
bosses the same time.
>> And you have
to Chagas disease
more channels
more but tingle.
>> Or used in academia,
in particular
business schools
and finance departments,
is additional vouch
and one agenda,
which is you call
using so-called
utility functions,
which means it's a graph
of answers are forgiven,
invest, or encoded
a function
which is called
utility function.
>> And then this is used
for optimal investments
This is actual as I am saying
Well, is a very very long history
much long than
everything else we know
>> While we usually think
about financial mouth,
we think about
developments which have
had since the late
sixties, early seventies.
Our founding father of
math shows Fischer
Black scholes formula.
about an optimal
investment is the things
which were developed
50 to 60 years ago.
>> If you think, maybe
stoic way about financial mouth,
maybe somebody comes up
and tells you, well,
I heard about this guy
Ursula in France,
they don't.
who did already 1900
thesis on house (inaudible)
, but this is basic
resolved is what
people usually
>> Tell the facts
of the matter.
>> This Swiss guy,
Daniel Bernoulli, child of
very famous
family of mathematicians
in Switzerland,
introduced  this
utility function
already in the first half
of the 18th century.
>> So how does he
came to do swell?
>> Said big thing.
>> And it was at this time
I thought about investing,
but more about gambling
is this problem,
which is known as the
St. Petersburg Paradox.
>> I don't know
if you have heard it but
if you just think
about a single game,
you flip a coin,
you come back and
if you get heads,
you get what you have
back plus the same amount.
So you're doubling, or
if you are losing
sandwiches, rules
what you have.
>> And this St.Petersburg Paradox,
doubling strategy
tells you about,
a gateway to
always make the
gain into scheme.
>> Since you play for $1,
if you win, you are happy
you go off with dollar and extra
>> If you're losing,
what do you doing can play.
>> But now respecting
two dollars  well,
if you win you win $2
was the first game,
you lost one dollar
so in total
>> You're still $1 up
>> You go out. Well if
you are losing something not
good since first you lost one dollar
Now you lost two dollars
so you have negative three dollars
>> But you are now
starting to bet $4
If you are betting $4 and you win
you are in total plus one
if you're losing,
well you can just continue to
increase your bet
size. Well you bet for 8,
for 16 and well you know
a coin flip can
not forever show tails.
>> Sometimes heads has
to appear and hence
at some point you will
have one dollar
>> Well, this is
something which doesn't
seem plausible and say
reason why its not
very plausible
maybe you could say, well,
we have not an
infinite amount of
credit or blame
for eight for
$16 is maybe something
that I'm happy to do.
1024 dollars may be
I have already
to link up my bank account,
how things are doing.
And if idealists,
our powers
of two up my bank for.
No. You don't get this money
We don't give you this money to spend
>> So disadvantages,
however,
we thought in a different way
>> And actually this year
his original drawing of this
what he calls a
utility function.
>> And he says, well,
in general well one dollar is not one dollar
How you value $1 depends
very much on how much
money you already have.
>> If you're very rich,
one additional dollar is
only increases your
happiness own live event.
But of you are very poor
and you have zero dollars
getting $1 is
absolutely great.
>> And he takes explains this in
a very nice way
>> He's say well says that
a poor man walks on
the street and finds on the ground
>> A lottery ticket. This
lottery ticket is already
filled out and paid for
and it's
very simple lottery 
ticket.
the bids with
probability 50% and
then 10 million or 50%,
you lose all the money
just great sense or have
the  poor guy gets
in this case for free,
five-minute but now
a rich man comes along
>> He sees this poor guy
found
a lottery ticket and he is a nice rich man
he doesn't calls
the police or doesn't take it for himself
No. He makes the poor guy an offer.
he said that I would 
be willing to buy
this ticket for four million dollars
poor guy agrees and say yes yes
and gives it for four million dollars
>> Well, why isn't two
people making this deal?
Poor guy situation is well
maybe
this 50% probability
and value of H and have
$10 million on this
50% probability
I'm might have 
absolutely nothing.
>> And I have to sell to half
as all the other
days before.
>> So it's much better
to have  four million dollar
His prospective is not much different
>> A $10 million, it's
just a huge
amount of money.
And so this is a
good deal for him
>> And he
can eat very well for the rest of his life and
can have a nice housing
for the rich guy he doesn't care
He has already
a couple of
billion dollars
and having an another 10 or four is not a big of difference
>> But he counts like this.
>> Well, on an average
this ticket
gives 5 million.
I payed 4 million.
on an average I made 1
million gain
>> So it makes sense.
>> So this is a
story which makes
off for bosses,
the guy sense.
>> And to get through
this story to
a mathematical representation,
He said, well,
people have
usually a utility
function which
looks like this,
which tells you about how
much you value a certain
amount of money.
>> And this function
is on the one hand
side increasing.
>> But it's such a
function is increasing,
is a little bit clean you
know everybody prefers
more money than less.
Well, nobody will tell you I prefer 3 dollars
over 5
>> That's another
feature is
not completely
clear and this is
this function is
convex and concave, convex.
>> And we shall visit
really about now.
>> And this is exactly
what it represents,
say you would amounts
of money are you amount of
wealth which is on the x-axis
>> If this is
relatively small.
>> Say, gain of happiness.
>> Does utility
function (inaudible)
>> I'm so glad
you're happiness
with money you gain from
an additional dollar is much
higher said if your
analysis is very high,
Sen, going an
additional dollar
survive on the x axis.
>> Oops, see y, x is up.
>> And this is
basic concept
how economists
measure correct
answers (inaudible)
>> So use utility
functions to represent
preferences. Bernoulli
was the first one.
>> And this can
be done by phenomena
which you might
know as one of
the parents of
say, modern computer.
>> And also Morgenstern
which is a very well-known
economist say
came up with an axiom.
>> I think theory say crucial
as long as people
have somehow
consistent preferences
then that has to
be such a complicated
truth was antics.
>> If i, for each
structure concave
utility function
corresponds to obese and
that will set preferences.
>> And this has been
since the start
of modern portfolio theory,
which was eligible
evangelist
or say Black-Scholes-Merton
Formula.
>> He also came up
in the late sixties.
>> List is using
utility functions for
portfolio optimization.
academically, I
think if I say a couple
of thousand
papers are underestimates
I would say a couple
of 10 thousand papers
tweeting
detriment variations
on this problem.
>> Now this is all
nice and good.
I don't want to talk
too much about this,
why Wellson says
actually a lot of
criticism on this
kind of utility.
>> And I want to
discuss mainly 2 points
>> So first one is maybe
the most common one,
which was first developed
mostly by people who
copying from psychology,
but have now we found
a large foothold in
economics departments.
>> And this is
what is in general
called behavioral finance.
>> Let's study not
what people should do,
but what people
are really doing.
>> Ads of actually,
although Merton
won a Nobel advice,
but also for behavioral
finance,
a couple of people won Nobel prizes
for this Kahneman for instance was
one of the first one who did it.
There was also another one
for, for which your talent.
>> So this is probably
something on block.
>> You can be Nobel
Prizes resist,
but that's the
end all what
we are saying  is we are
trying to measure
what people are
really doing
and what people are really
doing doesn't look like
these nice concave
shaped function.
>> What people
are really doing
is many things for a
couple of packets.
>> Because the one
that and as well
for 3D modeling is
something live,
pretty reasonable,
non-cognitive, find mature.
But for losing money,
most people
behave like it would
be a convex utility function.
>> What does this mean?
well if we 
lose some money
instead of just cutting
>> So losses
>> So stop.
>> As is the St. Petersburg
paradox to gamble
>> This happens
in particular
financial endlessly,
that traders who make
big offers tried to
hide this losses and get
additional money to make
bigger and bigger
pads to get out
of the situation,
to not fight.
>> And if you go through
the list of big
trading scandals,
I think at least
half of sand
can be explained by
such a behavior.
>> Said, people
try to overcome
losses sets I
get from (inaudible)
at least, at
least not easily.
>> Seasons, at least
at least it's a
bearing bang.
>> So unless
you're home alone,
bail us very often.
>> This kind of service
is the one thing,
the other thing
is a reality.
>> So you need
to differentiate
both utility function,
but in reality also,
people don't see all
kinds of events equally.
>> And in particular,
think about banks
which are very severe,
very extreme, and very unlike
so it gives them
a higher probability of happening
And so you cant even do
classical probability
theory and you
need something,
what is culture
k integrals?
>> And yes, that is
also a big bunch.
I personally think this is
not for what we
want to do, portfolio
optimization.
>> I don't think this is
a very valid line of criticism
since I actually don't
care about what people,
which, which stupid things
people actually do.
>> People do stupid things,
but I want to
give some advice.
>> And they don't
want to give
some advice to do
stupid things
they are doing.
>> Always NO I want
to give some advice.
>> How they can do better
I want to give some
rationale advice.
>> So if my goal is not
just describing what
people are doing,
but giving some advice.
>> It's a good argument to
make that this is a nice line
of what
people are doing
>> In reality, but if you want
to give some advice.
>> This is actual
a sound advice.
However, there is another
problem which I actually
think is a little
bit bigger.
>> And this is
theoretically,
everybody of you should
have such a
utility fantasy.
>> But how to find
it out says no good
reliable way to find out.
utility functions or
even risk aversions of
people in a way that
allows many people to
try different things.
People giving you questionnaires
>> How does he react in a
certain situation
with money to get
some approximation
what I know
all of this is not very
reliable people the way,
again, in a slightly
different set up,
something completely
different is coming.
>> So stego is can't do
something about this.
>> Can we think a
little bit different
about investing?
>> And instead of having
this utility
function u, we can,
we do something else to get
around this problem of
finding out the people who
do what I say is I
liked the feeling.
And if you accomplish
your utility
factions and tell me,
I have found out this is
my utility factual
place it best for me.
>> I say, great,
I can do it.
>> But realistically,
most people
cannot come out.
>> So what can
we do instead
of getting utility function,
well, and this
is something,
we should, I buy that.
But another great mind,
another Nobel Prize
winner Sharpe
Sharpe might be somebody
you have heard of,
the most likely
someway or other you heard about Sharpe ratio
you might also
have heard about,
or if he was one
of the main guys
pushing kappa m and index
investing in channel
>> Well, to gather
this Jimbo.
>> And he came
up with this idea of
what he calls the
distribution builder
And he's coming,
he's coming together with
some co-authors
called Stein,
who is a psychologist,
live in journalism,
and see how exactly
it forms a point
>> Investors
are notoriously
bad at estimating
something.
>> Try to instead
to get away
which is more direct to find
out how people
react in specific,
way you  may choose a much
more direct value.
>> So say, okay,
you fix a time horizon
of your investment.
Let's say typically
for your retirement,
you have a certain
distribution of
wealth invested all
of his van don't,
you cannot guarantee
assaulted amount on what you can do
enums and say start
say this distribution
and sang
say, well, if costs go up,
gives me a distribution
>> I can find out what is
the minimum amount of
money I need  to get this distribution
>> And I'll tell
a little bit more
details a little
bit in a second.
>> But I will say
this is his approach.
So far
exists not much
literature about it.
>> So that other
Hague-Visby
himself has done
as far as I know,
in his own investment firm
he's doing this or
similar concepts
to find it out.
More academically, I
know one paper, Philip,
moment of this and
otherwise this is something
that is a little
bit on the sidelines
>> I tried a
little bit as we
select it and made
you or they have,
I think this is such
a good evaluators.
>> And to understand
a little bit better
what is going on,
let's look at what
Sharpe  himself has written
as he has very,
very nice book.
>> He gave a
series of lectures
on that person
and vote from
these that book
investors markets.
>> And it's very
understandable in a lot of
intuition built
simulation library.
>> If somebody
wants to look on it
and I liked it a lot.
They landed on a couple of
months ago, the
first lately.
>> And I like his thinking a lot
and a lot of insights.
I want just to focus on
one paragraph
and I complete (inaudible)
>> So useful a
useful phrase
exactly tell if I'm to choose
>> But it's the same
would be true for
Let's say CAPM
Markowitz model.
>> Well, what
does increase yet
well as one thing is
you have a bunch of,
you know, how much money
you have and
you can invest.
>> Often business
we know what is
asset prices?
>> How we model
stocks and bonds
and other assets
in which we want to invest.behave well.
>> A bit of a
model models and
we much, much different.
>> It can be a simple
geometric out and nodes by
Charles model can
be discrete time,
can be very complicated,
jumping stochastic
volatility models
>> But we have these things
and we have preferences.
>> Preferences needs,
usually easy utility,
funcyion or measures
of risk aversion.
>> And said,
classical theory
tells you always needed
 ingredients,
with these ingredients,
we can come up
with an optimal investment
strategy for you.
>> These optimal
investment strategy
tells you how to invest.
What is the result of
this optimal
investment strategy?
>> Well, depending on
the dynamics of surprises,
you will end up with
some distribution
of wealth at
your retirement.
>> And which
distribution depends
very much on how your
investment strategies,
and he's a as well as
the big problem is have
a very hard time to
find out references.
>> So let's try to
invert this
whole procedure.
And we don't worry
about the budgets
everybody knows
>> How much money they have.
at least they hope
And if they dont know
and they have enough money
and we don't have
to worry about
asset prices are valid and typically
>> We have typical
the model pricing
>> But we will
have this two.
>> We will try to get from the
people directly say
information
about the distributions
you would like to
have at the retirement.
>> And we said we
will from this
distribution,
figure out what is
the best way to
address and as
a side product,
find out about
some preferences
as a people and
a more classical
way for us to get 
findings of utility function
>> Yeah, so that
point last one,
let me enable the big distribution to
find so-called utility
function which is not concave.
>> No, no, I agree
>> In deeply not agree
as long as we have
a complete market
as long as we
have a complete market
we will always
be able to find
a concave utility function
which corresponds to
cities huge chosen.
>> I can't do it
in one minute.
>> So just a
little bit through
how this works.
>> Practically shop has
on his website, actually,
a pretty nice
model and this is
how people actually
use this discipline.
>> It looks like a
little bit like I
like a game of
gateways or so.
>> So you will have a
lot of the small box
>> 100 blocks.
>> But doesn't matter how
high are you can just these move blocks
around and you have
a level of a 100% of
the original bowls
And yes, just a scale
of how much you want.
>> You might have some
Lower level  if you
want to imply math
you might have some
reference point.
>> But the key point is
your can just move around
these blocks and what is
happening behind is that
this where you're
moving things in
a certain situation
behind this picture.
So as a calculation machine
will calculate exactly how
much your budget is
used to get to this
specific distribution.
>> And, well, you
want to be close to
100 thousands since if you are
below 100 thousand
well you are not using money for investments
and its
throwing money away.
>> And well, a
few 100 thousand
of your budget and
you cannot afford it.
>> So you can never be
above and you should
not be too much below.
>> So this is implicit.
you markets are complete
at this evaluation. I still have a question anyway
2n variables that are
distinct from each
other can have
the same probability
distribution? Yes. Okay.
>> So the bigger
distribution,
but how do we go from that
distribution to one of
the two
Although this question is
coming in two
slides from now.
>> Okay?
>> I recognize
this is exact.
>> They say,
Great question,
what people choose here?
>> Distributions of
random variables,
you're usually used to
deal with random variables
if you're doing,
I don't know, hedging
options was oh,
you know exactly about
this event of variable
>> Here we don't have
a random variable,
to head onto the distribution. So there are many
random variables
which are of same distribution
>> And not all
random variables
have the same cost of
hedges and has
the same price.
>> So the question is,
how can we find cheapest
random variable which has
exactly this distribution?
>> This is exact
as against.
>> And this is what
is happening behind.
>> And that's where
the advice, like,
let's say the
formally advice
is this is the cheapest way
to get what you're after.
>> She gives you out
of a hedging strategy.
>> As soon as you have
a random variable you
want to achieve in a 
complete market
at least formally,
you can write down
a hedging strategy.
>> Why?
>> Well, on the
risk-neutral measure you
want anyway to be a martingale. you have a martingale representation serum
you have explicit forms of
Some martingale  representation around which
can invite you down.
Basically, what is delta
with which you
should hedge to get
the tension between like
replication
because isn't it?
>> Yes, I know what you know
>> And distribution (inaudible)
can you just clarify what is the random variable you are talking about
you see the distribution
whats the random  variable you are talking about
delivers that distribution
yeah, okay, let's,
let's put it like this.
this this
is always a
distribution output there's
>> Something else which I
didn't tell you about
was we have to
some assumption
of the model.
>> We have to say, what is
our specific
complete mock-up?
>> To make things simple,
let's just say we're
living intellectuals
Now we have
stockprice which is
geometric value motion
and bonds which we,
which we can invest in
then these random
variables behind it,
all random variables
I can achieve by
hedging in this market
and having exactly this
distribution concept.
It doesn't matter
as long as
it's some continuous
market model.
>> We can also do
it with the screen
but you have just thoughts
and everything
is fine enough.
>> If it's continuums,
you can basically
always do it.
>> How I versus specific
price will depend.
>> Of course,
specific multiple.
>> So is it some function
of this thing yes, yes.
>> We're coming
in one moment to
this more
mathematical details.
I just want first to discuss general
understanding so we all on the same page
What we want to
do and
will space you talk
some off more. Wow.
>> Okay, so far,
is everybody
comfortable where we are
on a big, big picture,
all that much time to
do a little bit more.
>> So what is going
on for a moment,
we will just assume
we are in a complete
what does it means that
the market is complete?
Well, informally,
it means they
can also be as low as
possible once a market,
mathematically,
it means at least
for two hours,
the law does exist and I,
a unique one.
>> Sex is a unique measure.
>> Few democracies I
don't divide is
measured cubed.
>> I just five years.
>> Advice and counsel side,
this is a little
bit of glass to
form valuable coat coming.
But if you are
more coming from
the martingale
measure cloud,
something just about
sign is d u over d, f.
>> Visit, walk off the
knobs and its side.
>> Or at least this is
the case when you assume,
certainly those who
want to discount it
>> Discounting is
not a big issue.
>> So I just mutually
assume i is equal to c 0.
>> This incubation
period is okay.
>> What do I want to do
is we're going to do
is be assume says
this company market.
>> If you want
something specific,
just vague about
what Charles
and I think
people prefer to
comments the same moths
and models and like
shells as occupies
a Geometric Brownian
Motion saying,
well that's
thinking formula,
eventual, it's fine.
>> What do you want
to do is you want to
find land or variable
which has exactly
this distribution.
>> Your client gave us
the distributive law.
what does this mean you want to
find one random variable
and you take some minimum,
of all (inaudible)
>> This distribution
which is specified,
so F is the CDF
basically of the
random variable
comes from the
client and said,
we want to find
the cheapest way
to hedge row.
>> What is the
cheapest way to hedge?
in a complete market, well
just say expectation on
the, say,
risk-neutral measure.
>> What is an expectation
on the risk neutral
measure this is
just u of x,
which is nothing
else saying e of x,
i's dq over dp
by dz plus just
containing e of
x times psi.
>> Yeah, so this approach
sharpes actually
reminds me of
something in
the probability
literature called
an optimal stopping
problem for score.
Score starting from
anybody familiar with it.
>> But that's okay.
>> I know what it is,
but I am not an expert
So, it goes like someone has
to give a probability
distribution.
>> And it has 0 waste
or things he's
written on the
whole real line.
>> And then I have to go.
>> You have the right
to grant emotion
and you haven't
started timely supply.
And then the man stopped
brand-new motion have
this distribution
that we started from.
So I'm just thinking
it's pretty close to
being in that
sharp Marty model
>> Yeah.
>> That both
people start with
a distribution that's
given to them.
I create a flow
created mechanisms,
let's say for for
Sharpes its via
portfolio construction,
whereas for (inaudible) its
there's only one kind
of risky assets as peak as
well as tiny as
opposed to portfolio construction
So anyway, I'm just
saying there's similar.
>> Okay, so this is
a problem. What do
you want to do?
>> You can also
live differently.
You can just say this is
nothing else than
a cost function.
Now this is the cost of
a specific random
variable X
>> You want to
minimize your cost by
take the infimum of
all random variables
>> Which have the
specific distribution
>> Now, how can you
solve this problem?
>> Exists nice resolve,
which is calls Frechet Hoeffding bounds
in probability theory.
>> Saying that
something much simpler.
>> Peter gave me an
idea about this and is
tells you you can't
drive this optimal
random variables,
random variable,
it's minimizing this
functional as this for you.
>> Take (inaudible) function,
same CDF of what you
want and say you take one
minus the CDF of surprising kind
look like this advice,
What's going on?
>> What is going on is
a very simple idea,
which is called
Chebyshev's inequality.
>> Chebyshev's
inequality tells you,
if you have to
teach Chebyshevs inequality.
>> Because now
supposing Chebyshev
inequality,
it is for sounds
and novice.
>> Some Chebyshaves inequality
well,
you have two ordered
sequences of numbers,
a1 to an, and a second
one, b1, b2, bn.
>> What about number
that says the same amount,
the same amount of numbers
about to
make sum of a problem. They want to make
a sum over sum a i.
>> And then I take some other B
let's say sigma of i,
some permutation
of say xi
>> And they want
to sum of all i
meaning in ESA and take A1,
B2 plus b1 times
a3 plus a seven times
b eight and so on.
>> And sounds like (inaudible) is in which
way do I have to order us sequences?
>> Said I get e as a
maximum amounts was
a minimum amount.
>> This is exactly what
Chebyshev's
inequality tells you,
is tell that the sum 
is always the largest
if you sum up two 
in a monotonic way,
so you get summing up
out of a i times B i,
i equal to one to n.
>> And it is
always smallest.
If you sums
about inverters call
count the monatomic way
you meaning your pairing is the largest
Just a small p to second largest
Just aim is the second
smallest and so on.
>> So divide and form a,
i, b and minus i plus one.
>> So basic message is
just the same way at
maximum if I sang
in an opposite way.
>> Add this can easily
be channelized from sums
to general
two random variables
and I have
an expectation of sum
>> Said
>> It turns out it is
always say largest
>> If si and
x are basically
ordered the same way,
meaning the si is large,
L2 x should be large
>> And if x i is small
as x should be
small and it is
the smallest,
then x is large,
then si is small,
and delta different
size large axis is,
is what is called the
counter monotonic.
And I can finds hit count
a monotonic very
explicitly.
Why, what is going on?
>> First of all, I know
if I want to have a random
variable with a CFF,
what I can do is I can just
take general
uniform comparable
and I can apply this CDF,
let's say of x to
it and then know
the sampling
distribution of x. I
know this is something
what you very
often using if
you're implementing a
Monte-Carlo advance
to find Monte-Carlo distribution
tells you you're part C
has implemented well.
>> So this is a uniform distribution
>> But this uniform distribution well,
you want to have a
uniform distribution.
>> This is a uniform
distribution.
>> If you apply a CDF
for the random
variable itself,
you have a uniform
distribution.
>> But this is a bad
uniform distribution.
>> Since a cdf is
increasing if you
have a function,
but this is also
increasing function.
>> If you have not
this term here,
you will have an
increasing function
of an increasing
function of psi,
which means said, what
do you get out is 01?
>> And exactly the
same way as side.
>> This is bad.
>> We want to have to audit
the opposite way as
 cheap trick.
>> This is the meaning of
f inverse is o
times one minus
yes off see this
as a fact and
applied to a random number
>> Well, if I
have a uniform
standard uniform
random variable
on the interval
01, great news.
One minus
this uniform
random variable
is again a uniform value
and it changes exactly ypur oder this means.
>> Now if psi is large,
this random
variable is small.
this means my optimal random
the variable among
all the random variables
which have this
distribution
can be calculated.
>> But all what
I need to know
what is surprising column in the market
well like shelf
market, you know,
this is just coming
from you is enough
to transform
saying I apply the cdf of y
as a function of walked.
>> My client is giving me
this model and I have
some random variable.
>> What I need?
>> Well, if I have
a random variable,
and this I think hopefully
been very comfortable on
and just X,
the random variable is
the same way as
we hedge any kind
Was this in the Sharpes paper? This is yours
>> At the time
I would credit
mostly (inaudible)
for doing it exactly?
>> The same way down but
I have what I would
say this is v.
>> I launched this way
of thinking about sharpe
as i assume he's
aware of it
but I don't think he has
been mathematically.
>> Okay.
>> Message is
this problem on
which looked first very
esoteric, is
being transformed
into something
>> What you can do
relatively easily
by finding
this esoteric problem
optimizing visitor
given distribution by
a complete market.
>> I can tell you
exactly what this
among all the variables
which have this
distribution,
which is a cheapest edge.
>> And of course, this
is the one with which
you want to target
with your investment
strategy.
>> Okay?
>> So this is the solution for the complete market.
>> Yes, so too,
what happens?
Why is somebody with a
so-called risk lover
and somehow excluded here
is you're seeing
utility function.
Again, this has
to be concave.
So say someone actually is,
let's say the way Kahneman
and Tversky artery
was a part of the upside
and the downside of that.
So it seems that
they can't place,
they can participate in
a participate in this
to advice you get to know?
>> Or is it
only if you have a
utility function?
Wow, I was growth
assumption
which came into my
mind, which is,
are very nice mathematical
insight to groups
as man measured by the actions
>> So if somebody
is risk seeking
or not from this
perspective, we don't know.
>> All we know is
terminal distribution which  t
I can not get
into the mind of
people seeking an
awful lot of opioids.
>> And next slide is
I can how I am fine.
a very reasonable person
which will come up
exactly with the same.
>> Maybe this guy's
crazy, but to find 
you, a girl who is
valid rationale would
do exactly the same.
>> Okay?
>> Okay?
>> So the real statement
does there exist
a concave utility function
will behave exactly,
yes, right?
>> But let's
say, so that's,
I think you're
saying somebody
doesn't have
concave utility.
>> It's okay.
>> I mean they
supply out here,
let's say we use a
marking, partial sine psi
>> And we do the
calculation.
>> And now we know random
variable that they should
be trying to replicate
as a result.
>> Okay? So I completely,
completely agree,
but I want to make things
clear at this moment
what we assume.
>> We're assuming
actually two things.
So first thing I think
you said at the
beginning is radically,
everybody wants more money.
>> At last yeah, they don't
assumes that people are
so foolish that they prefer $3 to $5
 want more
money is better,
at least not worse than less money
>> But this is
something what
we have to assume
and I think I don't
have to discuss much.
So the second assumption
is implicitly clear here,
but we shouldn't
nevertheless
has covered a lot of
this is o, this
distribution.
>> That's to say something.
>> Of course, this says
only care about
the distribution,
but this means I don't care about the state of scenario
>> If you're an investor.
who says well i want 
evolved not only invest
in a vague way
distribution,
but I also want to be sure 
that if the market
is crashing
and doing particularly
well and
how the market is
doing very well.
>> I don't care much about
then this cannot be captured
Yes, it is all
what we are saying
a distribution that
has nothing else.
>> This, this,
and this is not,
this is the evolution
of a good thing,
since this is
what you're doing
when you're a
utility function,
is what you're doing
when you're cap,
and this is what you're
doing when you Markowitz,
this is even when
you're doing kathy,
All these things are
state independent.
So that I've just
stated is not anything.
>> What does it
have a isn't that
just the mainstream
assumption?
>> But things
should be clear
with such assumptions
as usual,
that people don't think
about this assumption 
>> But as soon as you
have just two assumptions,
you can find exact link
the delta function,
meaning irrational girl who
is doing exactly this.
>> This is actually
the next result,
what I have here.
>> And this is just a
combination of what we
have seen and
the classical result
from portfolio theory,
which is called
well-represented
age may not want to get in
and all the details
we have just seen,
we can find the
optimal 
random variable
in this way.
>> This is just what
I've written down as
a consequence of the
simple junction
>> On the other side,
classical theory tells you
you can ride this size also
as an inverse of utility function
of some constant times
basically
the constant is exactly
determined by your
budget constraints
that you can afford this.
>> So this is
what classical
theory tells you.
>> This is all sharpe
and benign for neutral.
>> And I tell them
You and hands.
>> Yes, this and
this. Here is this.
Well, how's event
as two people
who have no, yeah,
so the main point
here is use exact.
>> And if you have
these holes, move what you,
for any concave
utility function
if you have a concave
utility function,
you can bud size
die this way.
This is hence a generic
answer to say I have
any rational
investor because at
home have utility function,
then I can find
the optimal portfolio
in this way.
>> And I've just shown
you we can revise
optimal portfolios and we
can say red is
equal to blue.
>> And so
everything for you,
yeah, yeah, there's
some degree of
indeterminacy.
>> Then it might
be that C versus,
versus size is so you're
not saying that any
unique concave utility
function using as
a family concave utility
functions to scale.
>> It's a, or
something like that.
>> This is some, i agree,
but this is something that
is completely
true in channel
>> Utility functions
are always
unique only up to
affine transformations.
>> Functions increasing,
increasing affine
transformations.
>> To be, to be clear,
what does this mean,
we can shift
things up and down
and you can multiply
them is a positive number.
>> And it doesn't
change, say,
master behaviors.
>> You're basically
just changing states.
Yeah. So we're here and
this is exactly
encoded in DC.
>> Dc tells you
how much you
are shifting a utility
functional event
>> Yeah. Yeah.
>> So that's my question.
>> Suddenly with
the other that
the increasing part of
that is determined designs
like the smallest
automatic.
>> So you can check
then if you have
this function given as
an integral and we're
integrating over
the function.
>> Hence you will be
increasing for sure.
>> Yeah, you can
also check that
this function is
actually decreasing
function.
My well, since beside the
choosing these things
exactly this way,
and this function
is increasing,
this function
is decreasing.
>> We have line and
increasing found too good.
>> So this is still
a decreasing function
that if I integrate over
decreasing function,
I get exactly,
yeah, that's true.
>> So it is fine to
you is automatically
increasing.
>> Yeah, so it's just
automatically
concave
>> It's automatically
differentiable.
>> It satisfies
automatic way.
>> Some additional
condition, like I said,
always saying What
it does not satisfy
compared to
classical theory.
>> It's not
strictly concave.
>> It could be
>> This function here is
for some pot constant.
>> And if this
continuous constant,
this function here will be
concave but not 
>> But otherwise it
satisfies all of
its basically a
limiting case of
the classical theory
where you don't think,
wow, why is that
concavities strict?
>> So residential still
could turn out that it's
actually a
residential zone.
>> They function.
So as he says,
Master, they got so
good actually because,
you know, it can be
piecewise linear, yeah,
but it cannot be
linear overall
fall directly
from the graph
now of CDFs, basically.
>> Otherwise, otherwise,
your CDF will
someday degenerate.
>> But as long as
you assume you have,
you have classical CDFs,
anddegenerate CDFs,
and we don't have that
any issues. thank you
so basically
this is I would
say this here is some
main resolved in zed
beauty of Sharpe
>> And this here is
the main resolved how
you can connect it to
the classical theory
of utility function.
