So the seventh week of lectures, is
going to be split into two separate slide
decks.
So this one I've labeled 7.1, and it'll be
on Lagrange Method.
So I'll start off talking about the idea
of an
investment portfolio, and then how you
might want to optimize that.
So a portfolio, just, you know, if I buy
several
assets, put them all in one folder, that's
called a portfolio.
And then the decision I have is, well,
it's sort of two parts.
Is A, what am I going to buy?
And I'm going to sort of assume that
that's already been answered for us.
And then B is going to be, how much of
each thing should I buy?
And, so, if you have investment goals, you
can choose exactly
how much you want to buy of each thing, to
satisfy them.
So, that's what I'll be talking about, so
to setting up
the problem I want to solve in the first
group of slides.
Then I'll talk about finding.
So it's, labeled Relative Extrema of
Functions of Several Variables.
So we talked about first and second order
conditions, for maximizing a function of a
single variable.
And now that we've done some linear
algebra, I can tell
you what the conditions are for functions
of more than one variable.
And then we'll look at Lagrange Method,
which is essentially
trying to do number two, except it also
has some constraints.
So it says I want to maximize this
function,
subject to another function being
satisfied.
in the fourth bit, I'll given example of
that.
And then in the fifth bit, I'll go back to
one of the
concepts I'm going to introduce in the
first bit, called a Minimum Variance
Portfolio.
And just go through the calculation of
finding that, using the Lagrange Method.
We'll get started with Optimal Investment
Portfolios.
So like I, like I just said, a portfolio
is just a collection of n assets.
So n is going to be some positive integer.
portfolio just means a folder.
So in the old days I think you, you know,
if I went and bought some
stock, I actually got paper certificates
that represented
those assets, and I put them in a folder.
And so then my folder full of all of my
stocks became known as a portfolio,
and so the folders are gone but the, the
word has stuck around.
And I'm going make a vector w where, so i
is going to run from one up to n.
So n's the total number of assets.
And then w sub i is going to be the
proportion of the portfolio.
So value wise, that's invested in each of
the assets.
So if I made it equally weighted
portfolio, w i would
be equal to 1 over n for every asset in
the portfolio.
but there
are a lot of other ways you could do that.
You know, I could say well, I think I'm
going to
make more money on a certain, like in
technology stocks.
So I might buy technology stocks in a
higher proportion.
So those w's would be higher, than the w's
for the rest of the assets in the
portfolio.
And I'll end up the constraint that, so if
w i is proportions, all of the wi's have
to sum up to one, so I have one portfolio.
And so you can think about you know, if
you wanted to make this into dollars.
This one isn't a very useful number, but
you can
think about it as maybe a thousand or a
million.
So I have a million dollars to invest, or
I have a thousand dollars to invest.
And then the Wi would be, the amount of
money invested in each of the assets.
And just to normalize that, we divide by
the
total value of the portfolio.
So you always end up with one on this
side, and proportions on this side.
And we're also going to say for, to make
the, the problem
mathematically more easy, we can do
something called take long and short
positions.
So, a long position in an asset, is
something everybody is familiar with.
You go to a market, you buy something, and
now you have the something.
A short position is sort of a strange
invention, where I can go to a market,
I can sell something I don't have, I can
take the money, but I have to promise
that later on I'm going to give the, the
person one of these things back.
And the way that's going to manifest
itself mathematically, is
it just means there's no constraints on
each individual Wi.
So even though they have to sum up to 1,
I could have one Wi greater than 1, and
then I
would also have to have some negative Wi's
to balance
that out, so that this sum is still equal
to 1.
Then I'm going to have mu IB, something
called the expected rate of return on an
asset.
And so there will be one of these for each
asset.
So that's why it get's a little subscript
i.
So different assets in my portfolio.
Some are going to have higher expected
rates of return, than others.
and generally, those are going to be the
ones that are more risky, too.
[COUGH]
So you can think of sigma i squared then.
So ui is the expected rate of return.
Sigma i squared, is going to be the risk
of asset i.
And there's also a kind of related
quantity,
just sigma sub i, which I'll call the
volatility.
So you can think of this as sort of a
variance
like quantity, and then sigma is a
standard deviation like quantity.
And so again, I get one of these for each
asset.
Then, there'll also be row ij.
So this is going to be the correlation
between the returns on the different
assets,
and I'm going to end up with one row ij
for every pair of assets.
So that means that row ij is equal to row
ji.
[COUGH]
And then once I have all of these inputs,
I can make the expected return for the
entire portfolio.
It's just going to be a weigted, so a
linear combination with the,
the weights, from w of my expected returns
on each of the assets.
And the risk has this more slightly
complicated formula.
So you have the weight squared,
times the risk of each asset.
But then, you
also have to add in, sort of those extra
contributions to the risk that are
coming from, from the correlation between
all of the different assets.
So, well I guess if just looking at this
thing, it's not a very pretty formula.
So then, first thing I want to do is see,
is there a nicer way I can write this
down?
So I'll try doing the same thing using
matrix notation.
So now instead of having all the, you
know, w1 up
to wn, I'll just think about that as, as a
single vector.
And remember when I put a row like this
in, in parentheses, I mean column vector.
So I'll
do one vector with the weights in it.
A second vector with the expected returns
in it.
So I have a weight for each asset, and an
expected return for each asset.
Then I can write the expected return, for
the entire portfolio.
So that's what's on the left-hand side
here, expected return here for the entire
portfolio.
And the previous slide, that was just the
sum of Wi times mu i.
But in matrix vector notation, I can just
write this as W transpose mu.
The risk, it turns out, that that sum that
I had, that kind of ugly sum on the
previous slide, can be expressed quite
simply at just W transpose sigma W.
And here this sigma, so this is not
singular values,
this is the covariance matrix for the
returns of the assets.
And so that's going to look like this.
So on the diagonal, I have the, the risk
of each individual asset.
And then in the off diagonal entries, I
end up with the volatility for here, for
instance,
the volatility of asset 1 times the
volatility of
asset 2 times the correlation between
those two assets.
And that's what's going to give me my
covariance terms.
And if you look back at that sum, I have
this sort of strange notation of
1, less than or equal to i, strictly less
than j, less than or equal to n.
And essentially what that's doing, I'm
just
trying to sum up over the lower triangle.
So all of the values that are strictly
below the main diagonal.
And then I have a 2 in front of that,
and that's just because this matrix is
going to be symmetric.
So if I, I have sigma12 row 12, it's the
same as sigma 21 row 21.
And so I'm basically, I'm summing the main
diagonal, and then 2 times the lower
triangle.
And so that, that's what my formula is
giving me.
And so I, when I'd have this as a, a
vector of
weight times a variance, covariance matrix
it, it gets a lot more
compact notation.
Okay.
So if I want to use these two formulas
I've come up with now.
The, the mu and the Sigma, these are
going to
be properties of the assets that I'm
looking at.
There, there's something that generally,
the, you know,
generally the investor doesn't have any
control over.
You just have to look at historical data
and say, you know, Boeing
has been producing a return of so much for
the last two years.
You know, I think it'll probably
stay about that for the next six months.
but it's not something, I can't go to
Boeing, for instance,
and say be less risky to get this number
to go down.
you know, you're stuck with whatever you
can, you can find out from the market.
The thing you can select, are the weights
in your portfolio.
So the investor can select, you know, how
much of Boeing am I going to buy?
How much of Starbucks am I going to buy?
How much of Microsoft am I going to buy?
And, once they've decided on a particular,
vector of weights w, then you can
figure out what the portfolio return, and
the portfolio risk are going to be.
And so, how should you go about picking w.
So we have two notions of optimality that
I'm going to talk about, now, and later
I'll
show you, and so in the fifth session I'll
show you how to solve one of these.
Then hopefully at the end of week eight,
we'll get around to solving the other one.
So, I can try and design a portfolio to
have a specific expected
return.
And then, once I have, once I've decided
I want to have this specific expected
return, then I, I choose w.
So there going to be a whole bunch of
choices, that will give me that one
expected return.
And among all of those, I would want to
choose the portfolio that's going
to have the minimum amount of risk.
And so that'll, that'll give me an
optimization
problem, that I can use to find w.
So I want to choose w so that the risk of
the portfolio is minimum, given a certain
level of expected return.
And then I can solve sort of a related
problem of, I could instead of targeting
the expected return level, I could say, I
want to have this much risk.
So if, for some reason, I, you know,
said, I, I want a very conservative
portfolio, or I want a high-growth
portfolio.
I can communicate, you know, some number
of
some level of risk I'm willing to
tolerate.
And so if I can tolerate a certain level
of risk, then I want to choose w so that
my portfolio gives me the highest level of
expected
return, that I can get at that level of
risk.
And so both of these turn out
to be something called constrained
optimization problems.
So the constraint, is I'm not trying to
maximize the expected return.
So that would be an unconstrained
optimization problem.
But I want to maximize my expected return,
subject to
a constraint oops, sorry I want to
minimize my portfolio risk.
So in the first one I want to minimize my
risk, subject to the constraint that I'm
hitting my target expected return.
And in the second, the second task.
So I have a target level of risk.
So my constraint is, I want a certain
level of risk in my portfolio.
And now I want to maximize the expected
return,
that I can get at that level of risk.
So they're optimization problems.
I'm trying
to find a minimum or a maximum, but I'm
subject to a certain constraint.
And then there's actually one more
constraint, that's hanging over from a
previous slide, is that the w vector has
to sum up to 1.
So, there's two constraints for each of
these.
So, the first one has expected return, has
to be equal to
some value, and my portfolio weights have
to sum up to 1.
And, satisfy those two constraints.
And then find the one that has the minimum
risk.
And, and then sort of, related problem for
the second goal.
And see, generally the way I like to write
these things down.
So I have a, a minimum variance
optimization.
So, what I'm trying to do.
This, this turns out to be something
called the portfolio variance.
I'm calling it risk,
because that's a, it's a mathematical
concept, that can be used to
describe sort of, how spread out the
return from my portfolio will be.
So how, how far away from the expected
return, do I expect it to be?
So I want to minimize this quantity.
So this is minimizing the variance or
minimizing the risk.
And then I write below that, subject to,
and
the I just list the constraints that I
have.
So
I've been using e to be a vector of all
one's.
So e transposed w.
That just means that the sum of w, has to
be equal to 1.
So my first constraint is that the weights
sum up to 1.
So I, I spend exactly the amount of money
I have to invest.
And then the second constraint.
So this, I argued before was the expected
return on my portfolio.
And I'm trying to minimize the variance,
at at given level of expected return.
So I have to choose an expected level of,
of a level of expected
return, that I'm calling mu p, so that's
the expected return for the portfolio.
And I'm limiting myself to vectors w,
that satisfy this linear equation here.
So this is the n asset case.
So, you can write it all out just, using
matrices and vectors.
And we know this doesn't get any more
complicated, as n gets bigger.
It's just the matrices and the vectors are
getting bigger, but writing it down stays
pretty compact.
you can already see what this looks like
just for the two asset case.
And this is
actually trivial because, if I look at
the, the two constraints, I have
two equations and two unknowns.
So there's only one point that's going to
satisfy the, the constraints here.
So there's actually no minimization to do.
There's just one one portfolio that I
would have here.
So, I just wanted to show what it looks
like written out.
So this a quadratic form, and it's just
equal
to what w transpose sigma w would look
like, if you did the, did the matrix
multiplication.
And then my few constraints, are just the
sum of the w's has to be 1.
And the expected return of the portfolio,
has
to equal my pre-specified level of
expected return.
So then the other possibility, was to
maximize the expected return of the
portfolio.
So, this was my expression for the
expected return.
So now I'm going to say maximize, and put
the
quantity that I want to maximize on the
first line.
And my constraints are going to be, well,
I still
have to have my vector w summing up to 1.
But now I have to choose a target level of
risk.
So I have to, have to say who ever is
going to do this portfolio
optimization problem, you know, this is
the risk level that I would like to have.
And that's then going to give this
quadratic form, as the constraint now.
So I'm, I'm limited to choosing from
vectors, that
are going to give me this level of risk.
And so if I write that out pretty much all
that happens is the, the constraint here,
or the, the objective here, the thing I'm
trying to minimize, becomes a constraint.
I still have the constraint w1 plus w2
equals 1.
And now, it's just my expected return that
I'm trying to optimize.