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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.