Portfolio Optimization

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during this tutorial we will be
highlighting it
AP portfolio optimization model
as a vehicle for highlighting several
features
model risk in particular the
optimization
functionality built into model risk
professionals what we've done here as we
started with some stock
price prices for all of 2009
in so there's a closing value-free to
these stocks
then we take
the Delta or the change the day-to-day
change and convert that into law returns
the reason we do that is it easier its
it makes it easier later to
just and up the returns from every
future period
and convert them back catalogue space to
get the
total return so we take the
prices ending at each day and
I love 2009 we convert those into the
wall
in the law returns because we know
there's correlation
in the stock markets were gonna use a
copula
fitted to the log return data this case
refitted
a multivariate T copula this is Abe
feature unique to model risk yes
certainly in the
Excel environment we could in this case
for using a parametric
T copula we could also lemme have used
an empirical copula
another feature completely unique to
model wrist
so we have built our copula in then
what you've done is we fitted some
parametric distributions to each of our
stocks
log returns in this case so
stock A&B were fitted to a hyperbolic
see can't
stocks Eagle a plus stock d
to a logistic distribution again these
are parametric distributions which we
quite quickly fitted using model risks
fitting functions we also could have
used in ok I've
or an empirical distribution to
represent the historical return to these
now another feature very unique model
rest
what we've done then is
taking a sample for each of the days in
January of 2010 we've taken a sample
I love each stock from its
distribution and then it used the copula
to correlate them so that for each of
these days the returns
for all four stocks should
mimicked the correlation pattern
represented by the historical return to
the stock
because one large space for
each other future or the forecast values
we can just some nose up
for each value in a smile and then we
can
take the exponential a bad thing to
convert back out a blog space
in here is our actual total return and
so essentially if we were to multiply
each of these numbers
to the closing price at the end of 2009
that would give us the forecasted price
for the end of January
2010 if we go ahead and hit the f9 key
we can see that every time the
simulation runs
we take a new sample the copula in a new
sample
love the distributions add them up and
we get a new return
well we also done here is we
FN create a portfolio out of these for
stocks
based on these percentages talk about
how we determine those percentages in a
moment and then we've
just take proportionally added these
returns to get a total return e can see
the total return
changes with each trial the simulation
as well every time we recalculate
spreadsheet
okay for optimization there's a very
sophisticated very capable
optimization feature built in the model
risk it's called up quest
up quest is really a leading method
for doing simulation optimization
what you do first to run on optimization
is you
there's a number of things you need to
set up need a target
this is essentially a
an objective function in this case we
want to maximize the total return
have to identify some decision variables
in this case we've do
we've designated
a variable for each or four stocks as a
percentage
and we said that the lower bound is .1
or 10 percent upper bound
is point for forty percent so what were
essentially saying
is dead the allocation for each the
stocks
can be no more than forty percent and no
law less than 10 percent of the entire
portfolio
we've got a constraint that we've built
in to say that all four percent it does
have to add up to one
for one hundred percent and then we've
also said in order to try to minimize
the
the variation I'll the portfolio we said
we want the
coefficient of variation to be less than
.3
there's a number of options in ways you
can run eighty
optimization but to think about it
conceptually the
the what happens is we have designated
these
for values in the spreadsheet his
decision variables
so when we start and optimization what
we do is give over control
up these four values chopped quest
up question goes through a series about
a rhythms to
manipulate each of these four numbers so
they meet all our requirements
Afghan up to 100 percent and then once
its
once its chosen a set up these values 10
runs a very short simulation
in this case just 50 iterations looks at
the total return value based on that
and then tries to readjust and run a
simulation again so it's sort of a
nested loop kinda love
scenario you have the outside loop is
the optimizer
in picking a set of these and then
it run the simulation that's the inside
loop in then it resets the
parameters the decision variables runs
inside loop and so on
for time's sake I ran a optimization
earlier
miss I just ran it for a few minutes I
we can
look at the results
there's to basic windows in the
optimization result window or two basic
tabs
this one shows us a historical I'll
for each of the simulations is long x
axis and then the value of our objective
function it showed here see can see we
start actually pretty higher close to
one
and then very gradually over time at
Quest finds a better and better solution
over the course of the few minutes that
Iran this
then you can see here its maximize the
value the value is 1.04
or a the 'em about a four percent growth
a lease for the very few iterations that
we ran at the simulation
and then up percentage for each of the
decision variables
here's our requirement that the
coefficient variation be below
.3 and here's are constraint that
the percentage chest and up to one
hundred percent
we can also look at the results from
each other the simulations
and we here we've just look at the 10
best solutions and we can see that the
best solution was the
419 try
I love setting these particular values
for the
percentages if we were to let this run
longer
it may very well be that up quest finds
a better result
once we done at of quest automatically
put the best values found
back in our decision variables and then
I ran a
quick simulation a bad love are output
and what we can see is this is the
distribution a possible returns over
the one-month period and if we look at
these
a confidence interval we essentially had
an 80 percent probability of being
somewhere between
down about 5 .3
percent or up 9.7 percent a fairly broad
range
but given the example daily used there's
a higher probability of being
above one then below one
and we can actually see there the mean
value after five thousand trials
is a hundred in 2.1 so in other words
if we multiplied 1.02 one times the
up up beginning value of our portfolio
we could
therefore see the eigenvalue if you
found this
presentation interesting indeed maybe
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up the file or you'd like to learn more
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