This module is the first of three modules, where we'll be walking through the main

theoretical ideas behind mean variance portfolio selection.

We'll first define what a portfolio is. We'll talk about what the return and risk

of a portfolio is going to be. We're going to define what an efficient

frontier is. We're going to define how an efficient

frontier changes when there is a risk-free asset and how all of that leads up to the

capital asset pricing market. The overview of what we're going to be

talking about here is that we're going to define assets.

We're going to define portfolios. We're going to define how to measure

random returns in assets and portfolios. And for the mean-variance optimization as

the name suggests, we're going to quantify the random asset and portfolio returns by

their mean and variance. We're going to define something called

mean-variance optimal portfolios or mean-variance efficient portfolios.

We're going to define something called the efficient frontier, and the portfolios

that lie on this efficient frontier and how does one compute them.

Along the way We're also going to talk about Sharpe ratio and Sharpe optimal

portfolios. We're going to define something called a

market portfolio, and after defining the market portfolio, we'll get to something

called the Capital Asset Pricing Model. These are the various modules that we're

going to be walking through in this bigger topic of mean variance optimization.

So what's the goal here? What I really want to do is I've got a

certain amount of money and I want to split it among various assets that are

available for investment. I'm going to caracterize and asset by its

price. More often than not I'll be interested in

returns on these assets I'm going to invest in them today, I'm going to sell

them tomorrow and Whatever difference is the return that I make on it.

And I would like to maximize this return in some appropriate sense.

I'm going to define the random gross return on a particular asset to be simply

the price one time step later. The time step could be a quarter, could be

a year, could be six months. So it's p t plus 1 divided by p t.

The net return is simply r t minus 1. It's going to be p t plus 1 minus p t

divided by p t. I want to point out that both r t, and

little r t are random quantities. These are random because the price at time

t plus 1 is random. Price at time t, which is right now, is

know, but the price at time t plus 1 is random, and therefore these returns are

going to be random. I've got d different assets, and I want to

split an amount, capital W, that I'm going to assume is strictly positive.

This is the capital that I have, and I want to split it over the d assets that I

have. W sub i will be the total dollar amount

that I've invested in asset i. If w i is greater than 0, I'm going to say

that that's a long investment. If w i is less than 0, it's going to be a

short investment. For the purposes of modeling, we allow w

to be both positive or negative. Our sub w will be The net rate of return

or net return on the position w. By position I mean, the wealth and the

various assets. So what is the definition?

It's simply the total value of this 1 times detonator.

Which is the gross returns of each of the assets times the amount of money which was

put into those assets. Minus the initial values, which is sum of

the w's divided by the initial value which is capital w.

If you do the math, this 1 could be subtracted from ri.

And you end up getting, this is going to be the net return.

Little rit, wi, divided by the sum of the wi's.

Rearrange them a little bit, and you end up getting that the net return on position

w is the net return on each of the assets rit, which is a random quantity times wi

divided by capital W. So what is important for the net return is

not the absolute amount of wealth that is invested, but the relative amount, or the

fraction of the total wealth that is invested in a particular asset.

So instead of worrying about positions, we just have to consider a portfolio vector.

So x is a portfolio vector. It could be positive or negative.

Xi's represent the fraction invested in a particular asset.

So some of the xi's must be equal to 1. One thing that I want to point out here is

I, I've been talking about time t. In reality the return that you get,

whether it's a gross return or the net return changes over time.

The random properties change over time. The actual values that are realized change

over time. But in this set of modules we are going to

be looking at a very myopic investment strategy.

I'm sitting at a particular time t. I only want to invest at time T plus one.

When we get into more advance topics, we're going to talk about how to extend

this idea into a multi-period optimization.

People are interested in multi-period optimization because at the end of the day

they want to save money for retirement, they want to save money for buying a house

and so one, which is not a one period problem but a multi-period problem.

A multi-period problem is not simply a concatination of one period problems

because the space over which you can optimize becomes much larger when you look

at multi-period problem. But in this set of modules we're going to

be concerned only with one period of optimization, one quarter, one era, and so

on. So how does one deal with randomness?

The return on the portfolio R's of X is going to be defined as R I, X I.

And notice over here I've dropped the R I, the time in the R I, because I'm focusing

mainly on myopic optimization. This is a random return.

Why is it random? Because the each of the returns, each of

the net returns, R I's are going to be random.

How does one quantify these returns? Should I simply look at the, maximize the

expected value? Is that the right thing to do?

Should one be worried about the spread around the mean?

So the expected value or the mean value tells you what happens when you repeatedly

invest. Most of us don't have the ability to

repeatedly invest. If we go bankrupt, our investment is over.

You don't have the ability to return from bankruptcy then you might have to worry

about what happened to the spread around the mean.

How does one quantify the spread around the mean is going to be a question that

we'll have to deal with. The way we are going to do this in these

set of modules is we're going to talk about the mean and we're going to quantify

the spread around the mean by the variance.

So the d, the values defining the asset returns are going to be the mean return

which is the expected value of the net return, the variance of the asset return

which is simply the variance of the random variables.

Notice the mu I's and the sigma I's are assumed to be independent of time, again.

Either the market itself is stationary or we are only interested in myopic

investment therefore we don't have to worry about time.

The covariance between the asset return is the covariance between a random return and

a particular asset i and a asset j. Correlation again it's between two assets

and there's a relationship between covariance and In correlation.

Covariance is nothing but the correlation times the volatility of asset I and the

volatility of asset d. All parameter are assumed to be constant

over time. Now, I'm going to characterized the random

return that I get on a portfolio by looking at the expected return on the

portfolio. And the variance of the return on the

portfolio. So the expected return on a portfolio x,

Mu sub x, is going to be expected return on the net return of that portfolio r sub

x, by using linearity of expectations. We end up getting that this is nothing but

the expected return on each of the assets times the fraction invested in that asset,

xi. So it's just mu i times xi, sum from I

going from Y to D. The variance of the return again, just by

the expression, it's the sum of R I time X I, the variance of this random variable.

And if you expand it out it becomes the covariance of R I and R J times X I, X J.

Here's just a simple example to try to work you through it.

So I have 2 assets with normally distributed returns with mean mew and

variance sigma squared. So R1 is one asset it has a mean 1, and it

has a variance 0.1. R 2 is another asset, it has a mean 2, it

has a variance 0.5. The correlation between these two assets

is -0.25. If it translates this statement into those

parameters, mu 1 is 1, mu 2 is 2, sigma 1 squared is 0.1, sigma 2 squared is 0.5.

Sigma 1 2 is which is the covariance between r 1 and r 2.

Is going to be the correlation whose number is right here, times Sigma 1, Sigma

2 you plug in the answers you end up getting it's .0559.

In the 2 asset market, portfolios are very easy to define.

Remember portfolios were the fractions invested.

So if I invest fraction x in asset 1. And since the fraction invested in asset

1, and the fraction invested in asset 2 must add up to 1.

I should invest 1 minus x in asset 2. Later on, we will see how we can use this

x1 minus x to try to do efficient portfolio selection between these 2

assets. Plugging it then into the formula, the

expected return on this portfolio is going to be sum of mu i xi 1 through d, 2

assets, the first asset has x, its expected return is 1 so it's 1 times x,

the second asset has 1 minus x, its expected return is 2, so the total

expected return that you get on this portfolio is x plus 2 times 1 minus x.

What is the variance associated with the return of this portfolio?

The formula is Sigma i j x i j, x i x j summed from i equals 1 through d, which

can be equally written as sum of i going from 1 through d Sigma i squared, x i

squared plus 2 times j greater than i Sigma i j x i x j, plugging it in, .1

which is a variance of the first asset times the investment in the first asset

squared. .5, which is the variance of the second

asset, times the investment in the second asset squared.

Two times the variance, covariance between the two assets, which is just computed

here, times the investment in the first asset and investment in the second asset.

This exact thing can be generalized to multiple assets, and that is what we'll do

in later modules. Okay, the first thing that we want to talk

about is that diversification, or thinking about the spread around the means, is

important because it reduces uncertainty. So let's consider a very contrived market.

I have D different assets, all of them have the same expected return, mu, all of

them have the same volatility, sigma. And the correlation between the assets is,

is equal to 0. So each of the assets is basically

identical, and now let's think about 2 different portfolios.

In one portfolio, x, I invest everything in asset 1.

In the other portfolio, I equally distribute my initial dollar over all the

assets, so in every asset, I invest 1 over d.

Both of these are portfolios. The expected return of the first, of the

first portfolio which is x, is simply the expected return of asset one which is mu.

The expected return on the other portfolio, y, which equally invests in all

the assets, is going to be the average of the returns of all the assets, since each

of them has the same return mu, the average is also going to be mu.

So if we were just looking at the expected value, the two of these, these two

portfolios cannot be differentiated. They both give me the same expected value.

We should be as happy investing in X as we should be investing in Y.

But if we are as we should be interested in reducing incertainty, perhaps these 2

portfolios are different. So if you look at the variance of the

returns of the portfolios, what do you get?

Sigma x squared Which is the radiance of portfolio x.

We know that, that just invests in asset 1.

So it's nothing but volatility squared, sigma squared.

What happens to the variance of portfolio y?

If you plug in the formula, every one of them has over d invested in it.

So it's 1 over d squared, sigma squared. There are b terms.

So ultimately you get that the variance associated portfolio wide sigma squared

over d. Think about d of the order of 100.

Okay, I'm interested in the s and p 500 index.

So in the one case, I get volatility sigma square.

In the other case my volatility has dropped down 100 fold.

So just by diversifying between assets, identical assets, now I have been able to

reduce my volatility a lot. The mean-variance portfolio selection

problem, in the end, generalizes this idea.

Here I'm taking a very simple. Problem All the assets have the same

return. All the assets have the same variability,

and I know that equal spreading is the best thing.

Now what I want to do it sprea, move this idea to, to a case where the mean returns

are not the same, variances are not the same, the covariance may not be equal to

zero. How does one think about this problem?

How does one compute efficient portfolios meaning portfolios that have A good mean

invariance properties is going to be the main focus of these sets of modules.

So, in 1954, Markowitz proposed a portfolio selection strategy.

In his model, he suggested that the return.

And the return has been put in quotes. I wanted to read this more as the benefit

coming from a portfolio, to be the expected return of that portfolio.

And risk associated with that portfolio to be the volatility of that portfolio.

And what he suggested was that these are the 2 quantities that are going to be

interesting to investors. They would want to increase their return

and decrease their risk. So what I'm plotting over here are the

returns on some random portfolios. In the, in the next module I'm going to

show you a spreadsheet which shows you how these random returns were generated.

I have, I have 8 different assets, the details of which will be in the

spreadsheet. I've randomly generated positions on these

8 assets, figured out what the return is going to be, figured out what their

volatility is going to be and then plotted a point.

So all of these blue dots, Dots, are actually verious portfolios randomly

generated. And the efficient frontier, this line over

here, is genearted by the following procedure.

We pick particular value of relativity, of risk, and try to compute the largest

return that you can get on a portfolio. Here that has risk no larger than

particular bound. So let's say sigma bar is the bound here.

Figure out a portfolio. Compute a portfolio.

And I'll show you the spreadsheet, how these portfolios are computed.

Which has the largest return, with risk not exceeding sigma bar.

And that point will be right here. And similarly, you take different values

of these sigmas, compute out what is the maximum return that you are going to get,

and this blue line is generated by computing the maximum return for a given

value of risk. That frontier Is called the efficient

frontier. Why is it a frontier?

Because all portfolios, all feasible portfolios must lie below.

This is all the part that is feasible. For any portfolio that you choose, its

risk and its return values must be below the line.

All of this space is unachievable. You cannot create a portfolio whose return

and risk point lies in that region. Why is that?

Just the way it's computed. I take the value of sigma, which is the

risk, I compute the maximum possible return that I can get, and that's how I

get this point. This return up here is not achievable.

So everything above the frontier is not achievable.

Everything below the frontier, below or equal to the frontier is achievable, but I

would never want to be below the frontier. If I have a point over here, its risk is

some quantity over here. Let's called it sigma 1.

My frontier tells me that I can create another portfolio, a different portfolio

from the one that generated that point, whose return is going to be right here.

It's going to be on the frontier. So I would never want this portfolio.

I only want portfolios that lie on the frontier.

Above the frontier, unachievable Below the frontier inefficient.

Right at the frontier is the place where I want to be.

So the question that we will answer in the next few modules, is, how does one

characterize this efficient frontier? How does one compute efficient.

Or sometimes I'm going to call it optimal portfolios.

Portfolios that lie on this efficient frontier.

There are three different ways in which you can compute.

The mean variance optimal frontier, that line that I showed you before, which tells

you the maximum return for a particular value of risk.

One way is to minimize risk for a target return.

You can set the target return that you want.

You want to make sure that the expected value on the portfolio has to be greater

than or equal to R. And among all portfolios that satisfy

that, you want to minimize the variance, or minimize the volatility, which is EQ1.

If you expand this optimization problem, you can write it as sum of XI is equal to

1. So this is a portfolio constraint.

And here you're saying that the expected return on that portfolio must be greater

than equal to r. And this expression here just expands out

whatever is written over there. And for those of you who are,

mathematically sophisticated, this expression is nothing but the vector x

transpose, a matrix of covariance times x. And what is this matrix?

It's sigma 1 squared, sigma 1 2, sigma 1 3 and so on.

Sigma 2 1, Sigma 2 squared and so on. So this is the variance-covariance matrix.

And this expression there is simply x transposed variance-covariance matrix

times x. An equivalent way to get the entire

frontier is going to be to maximize return for a given value of risk, maximize mu of

x such that sigma squared of x is below some target number sigma bar squared.

If you write this in terms of the x i's it becomes some of the x i's must be equal to

1, again the portfolio constraint. X transpose sigma x, ir sigma ijxij summed

must be less than or equal to sigma bar and you want to maximize the expected

return. There is yet a third way of trying to get

the entire frontier. And that is, to maximize a risk adjusted

return. So maximize over portfolios, x.

Mu of x, the expected return minus tau, which is for the risk aversion parameter,

sigma X squared. So the risk aversion parameter is always

greater than zero. You don't like risks, therefore you want

to subtract from the extracted return a certain quantity that depends upon risk.

If you again this expression, you get a sum of XI is equal, from XI must be equal

to one, which is the portfolio constraint. This is just mu of x and that is sigma x

squared and that's the tau there. What do I mean by saying all these three

will generate the same frontier? There are parameters and there are three

parameters in all of this, all of these formulations, r, sigma bar squared and

tau. And as you Crank up these parameters for

different values of these parameters you will write out the same curve.

Overview of Mean Variance

金融工程与风险管理，第 2 部分

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