Financial Engineering is a multidisciplinary field involving finance and economics, mathematics, statistics, engineering and computational methods. The emphasis of FE & RM Part II will be on the use of simple stochastic models to (i) solve portfolio optimization problems (ii) price derivative securities in various asset classes including equities and credit and (iii) consider some advanced applications of financial engineering including algorithmic trading and the pricing of real options. We will also consider the role that financial engineering played during the financial crisis. We hope that students who complete the course and the prerequisite course (FE & RM Part I) will have a good understanding of the "rocket science" behind financial engineering. But perhaps more importantly, we hope they will also understand the limitations of this theory in practice and why financial models should always be treated with a healthy degree of skepticism.
Mean Variance with a Risk-free Asset
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來自 Columbia University 的課程
金融工程与风险管理，第 2 部分
429 個評分
從本節課中
Mean-Variance Analysis and CAPM
Problem formulation and solution; the efficient frontier; including the risk-free asset; the Capital Asset Pricing Model (CAPM);implications of CAPM: α, β, security and capital market lines
與講師見面
Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
In this module, we are going to walk you through how to construct the efficient
frontier for a market with a risk free asset.
We're going to see that we're going to define a new portfolio called a Sharp
Optimal Portfolio. That's going to play a very important role
in this module as well as in the next module on capital asset pricing module.
So, we have a new asset. It pays a net return rf, with no risk.
It's completely deterministic. And we're going to label this asset as x0.
And again, I want to construct the efficient frontier.
But this time, I'm going to use a different formulation of the mean-variance
optimal portfolio selection problem to construct the efficient frontier.
If you go back two modules, we had set up three different ways of doing efficient
frontiers. One was by maximizing the return for a
given value of risk, another was to minimize the risk for a given value of
return. And a third one was to maximize a
risk-adjusted return. And that's the formulation that I'm going
to use. So here's my return, r f times x0 is the
return on the zero asset, which is the risk-free asset, and this is stuff that we
have seen before. Mu i times xi sum from i equal to 1
through d is the return on the risk-free asset.
This here, is the variance of my return. And the thing that I want you to focus on
the fact is that here the indexes go form i equals 1 to d.
So, i equals 0 is absent. And why is this absent?
Because i equal to 0 corresponds to the asset that gives me a deterministic
return, it doesn't have a variance. Down here is a risk aversion parameter.
And we also know that the entire frontier can be computed by just taking different
values of tau. As you go from tau, tau must be greater
than equal to zero. It cannot take negative values.
But as you crank up the tau, from tao equal to zero to tau equal to infinity,
the entire efficient frontier is going to be computed.
What's down here? Just a portfolio constraint.
Now I have d plus 1 assets. And therefore, the fractions invested in
this d plus 1 asset must be equal to 1. So x0 plus, so I can solve for one of
these using that. And it is what I'm going to do in the next
calculation. One thing to keep in mind is that this
entire mean variance calculation is meaningful only for target returns r that
are greater than equal to rf. Again, I would like you pause here and
convince yourself that you will never want to consider a return less than rf.
Alright. So now, we have this equation that relates
x0 to the other x's. So, I'm just solving for it.
I'm substituting x0 equal 1 minus the sum of the x's and plugging it back into this
equation. The nice thing is that this equation does
not involve x0 at all so it remains the same.
The other ones do involve x0. So, what I've done is that this one gets
multiplied by rf, so that's that one over there.
These minus xi's also get multiplied by rf so they are sitting over there.
So it's mu i minus rf times xi. So, one thing that immediately falls out
of this calculation without looking further into what happens to the efficient
frontier and so forth is that the relevant quantity.
When you have a risk-free asset in the market, the relevant return that you're
interested in is the excess return on the asset.
So mu hat i, which will play a role later on, is mu i minus rf.
It's the excess return on asset i. It's the returning excess of the risk-free
rate and that is what going to determine what the portfolio is going to be.
We have an expression, now let's just optimize it.
This time we are lucky because this expression that we have here has no
constraints, no constraints on x. Why?
Because there was only one constraint which was a portfolio constraint and we
used that constraint to set the value of x0.
So, we just take the derivative with respect to all the xi's and set it equal
to 0. If you take the derivative of the first
term, you'll get mu hat i. If you take the derivative from the second
term, you'll get minus 2 tau, sum of j going from 1 through d sigma ij times xj,
and that must be equal to 0. So, d equations, d unknowns, I can solve.
What I've done is rearrange this equation in a matrix form.
I said 2 tau times this matrix V, x, this is a vector of x must be equal to mu hat.
You invert it, and you end up getting that for a given value of risk aversion tau.
So, the only thing that I'm trying to emphasize here is that x is a function of
the risk aversion parameter. So, you give me the risk aversion
parameter, and I will compute out what is going to happen to the portfolio.
So, it's going to be 1 over 2 tau V inverse mu hat.
So, what is the family of frontier portfolios?
Remember, I said that all the frontier can be generated by changing the value of tau.
So, the family of portfolios that sit on the frontier are simply x'd out.
But this x'd out was only the risky part of the portfolio.
So, what is the amount that was invested in the risk-free part of the portfolio?
It's just 1 minus the sum of all the components.
So, this is essentially x0 tau, the amount that you invested the risk-free asset, x
tau is exactly that. And this is called the risky part of the
portfolio, the risky portfolio. And as you crank up your tau for all
values of tau going greater than or equal to zero, you'll get a family of
portfolios. All of these portfolios are going to sit
on the efficient frontier. That's great, but again, we're going to do
an exercise very similar to the two frontier.
I want to understand the structure of this frontier better.
So, let's focus just on the risky asset in the frontier portfolio.
X is 1 over 2 tau v inverse mu hat. This is just a risky position, therefore
it does not add up to one. So, the first thing that I'm going to try
to do is construct another portfolio, meaning that components add up to 1 from
this position which doesn't add up to 1. And the easiest way to do it is to take a
position x, because if they don't add up to 1, and divide it by the sum of the
components. And call a new position s star.
But now, if you look at the sum of s star i, i going from 1 through d, that is
nothing going to be the sum of i going from 1 through d of xi divided by sum of i
going from 1 to d of xi. They will cancel and you will end up
getting the sum of this components of this vector s star is equal to 1.
So, s star in fact, is a portfolio. So, what's special about this portfolio is
if you plug in the value of x for any value of the risk of urgent parameter tou.
So, x is 1 over 2 tau, we inverse mu hat. In the denominator, I'm adding it up by
the same, same position, so you get 1 over 2 tau here.
So, the 2 tau's cancel. Which means that if I look at the risky
positions that an investor is holding for any value of the risk aversion parameter,
tau, and construct a corresponding portfolio, meaning normalize it, so that
its components add up to one, I get the same portfolio as star.
It doesn't depend on tau. Which means that I can look at the
efficient portfolios are reparametrize them, think of them differently.
Essentially, what everybody is doing is taking a certain amount of their dollar x0
and putting it into a risk-free asset and the remaining about 1 minus x0 is the
amount that they are investing in s star. Everybody in, investing in the same
portfolio s star. So, the family of frontier portfolio now
is very simple. Put the money into the risk-free, take the
rest of the dollars and put it in, invest it into s star.
So, we end up getting this theorem which says that all efficient portfolios in the
market with a risk-free asset can be constructed by diversifying between the
risk less asset and the single mutual fund, s star.
And in the next few slides, we're going to try to give you more structure as to what
is this portfolio, s star. Before we get there, let's just construct
the risk, the efficient frontier in a market with the risk-free asset.
Everybody invest in s star. Let mu s star be the expected return on
this portfolio and sigma s star denote the volatility of the portfolio.
And the return on the generic portfolio is going to be x0 times the risk-free rate
plus one minus x0 times the return on this portfolio s star.
And the volatility will simply be 1 minus x0 times sigma s star.
Why? Because a risk-free asset does not have
any volatility associated with it. So, let's take two points.
The point down here, which has no risk, zero volatility, and will give you a
return rf. Zero volatility will correspond to x0
equal to 1. 1 minus x0 is equal to 0.
And therefore mu of x, that portfolio will have a return rf.
Take another point over here, which corresponds to the portfolio s star.
That corresponds to volatility s star, which means x0 is equal to 0.
And mu x therefore turns out to be just mu s star.
Now, if you look at this curve as I changed my x0, it will trace a straight
line. And therefore the efficient frontier will
simply be a straight line that goes from the point that corresponds to the
risk-free asset, goes to the point that corresponds to this special portfolio, s
star, and it's a straight line. And because if you can get a return rf
with no risk, you will always want to demand a higher return than rf if you are
asked to take a risky position. So, that's going to be the efficient
frontier. Now, what we want to ask is how does this
efficient frontier relate to the frontier with only risky assets?
And is there and economic interpretation for this very special portfolio s star?
So, here's what's going to happen. Here's the, the green line here is the
efficient frontier with the risk-free asset in place.
It's that straight line that we talked about.
I know that the point sigma s star mu s star belongs to that efficient frontier.
Now, rf0, this is the point that corresponds to a risk-free asset, also
belongs to it and that's a straight line. And this blue line denotes whatever is the
frontier that corresponds to just a risky asset.
So, my claim is that s star must be an efficient portfolio, efficient risky
portfolio. Meaning that it must lie on that.
Suppose it's not. This is the counter-factual.
That maybe s star is actually here. Here, we know that the efficient frontier
for a market with the risk-free asset must go to s star.
So, the efficient frontier will look something like that.
But that cannot be the efficient frontier. Why not?
Because all of these points lie within the efficient frontier, for just the risky
assets. So I can, if you give me a point over
here, which has a particular amount of risk, I can give you a better return.
So, holding this point here cannot be efficient.
And therefore, whatever that s star is, it cannot be at a point such that the
straight line that goes through s star. Enters the region which is feasible but
not efficient. And yet, s star is a portfolio of just a
risky asset, and therefore it must be somewhere in the region that is feasible.
It cannot be in sight. It cannot be on the boundary such that the
line goes through it. Therefore, the only thing it can do is
that the line must be tangent. And that's exactly the picture that's
drawn here. So, let me just clean up the story here.
For the moment, I'm going to clean up my drawings so that you can go back to seeing
what the picture is. So now, I know that this s star, whatever
it is, must lie, must be such that it's tangent to the efficient frontier, right?
Everything that's in the efficient frontier, everything that's down here, can
be computed by diversifying between risky assets.
So no, now I want to understand what is go, how this s star is going to be
computed. So, let's look at this angle, theta.
My claim is that s star is a portfolio that maximizes this angle or equivalently
maximizes the tangent of that angle. Let's write out what the expression for
the tangent of that angle is. Take any, any point, any feasible
portfolio here for the moment. So, just do fixed ideas.
Let's say here's a particular point that corresponds to a particular portfolio,
it's volatility is sigma x and it's return is mu x.
So, what is the corresponding angle there? Let's call this angle sum theta of x.
The tangent of that I take the y-value. Y-value is just mu x minus rf, which is
what is written over here. This expression is noting but mu x minus
rf. What is the x-value?
Which is nothing but sigma x. So, the angle that is cast over here by
any portfolio which has volatility sigma x and return mu x, the tangent of that angle
is simply the excess return of that portfolio divided by the variance or the
volatility of that portfolio. This particular s star portfolio is the
one that maximizes this angle. So, it maximizes the ratio of the expected
excess return to volatility. This has a name.
The Sharpe Ratio of a portfolio or an asset is the ratio of the expected excess
return to its volatility. The Sharpe optimal portfolio is a
portfolio that maximizes the Sharpe ratio. The portfolio s star is just argued
maximizes the tangent of that angle. The tangent of the angle is the Sharpe
ratio. Therefore, s star is a Sharpe optimal
portfolio. Everybody in a market which a risk-free
asset diversifies between the risk-free asset and the Sharpe optimal portfolio.
The name Sharpe comes from the fact that this was, that this portfolio was
identified by Bill Sharpe, who got a Nobel Prize together with Markowitz and Lintner.
Alright, the last bit. Now, the investment in the very risky
assets are in fixed proportions. All the investors in a mean-variance
market are diversifying between the riskless asset and this particular Sharpe
optimal portfolio. And therefore, the demand for the
different assets will be perfect synchrony to each other.
And which means that if the demands are proportional, then the price and the
returns should be correlated. This should be just a one-dimensional
quantity that should just tell me what the returns on all the asset in the market is
going to be. Why?
Because everybody holds the same portfolio.
This inside will lead to the capital asset pricing model in the next module.
