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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All

right, let's start. So first of all, I hope you've

been enjoying the class so far. And thank you for

filling out the survey. So we got some very useful

and interesting feedbacks. One of the feedbacks–

this is my impression, I haven't gotten a chance

to talk to my co-lecturers or colleagues yet, but

I read some comments. You were saying that some of

the problem sets are quite hard. The math part may be a bit more

difficult than the lecture. So I'm thinking. So this is really the

application lecture. And from now, after three

more lectures by Choongbum, it will be essentially the

remainder is all applications.

The original point

of having this class is really to show you

how math is applied, to show you those cases

in different markets, different strategies,

and in the real industry. So I'm trying to think, how

do I give today's lecture with the right balance? This is, after

all, a math class. Should I give you more math,

or should I– you've had enough math. I mean, it sounded

like from the survey you probably had enough math. So I would probably

want to focus a bit more on the application side. And from the survey also

it seems like most of you enjoyed or wanted to listen to

more on the application side. So anyway, as you've already

learned from Peter's lecture, the so-called Modern

Portfolio Theory. And it's actually

not that modern anymore, but we still call

it Modern Portfolio Theory. So you probably wonder,

in the real world, how actually we use it. Do we follow those steps? Do we do those calculations? And so today, I'd like to share

with you my experience on that, both in the past,

a different area, and today probably more

focused on the buy side.

Oh, come on in. Yeah. Actually, these

are my colleagues from Harvard Management. So– [CHUCKLES] –they will be able to ask

me really tough questions. So anyway, so how I'm

going to start this class. You wondered why I handed

out to each of you a page. So does everyone have

a blank page by now? Yeah, actually. Yeah. Could also pass to–? Yeah. So I want every one of

you to use that blank page to construct a portfolio, OK? So you're saying, well, I

haven't done this before. That's fine. Do it totally from

your intuition, from your knowledge

base as of now. So what I want you to

do is to write down, to break down the

100% of what do you want to have in your portfolio. OK, you said, give me choices. No, I'm not going

to give you choices. You think about whatever

you like to put down. Wide open, OK? And don't even ask me

the goal or the criteria. Base it on what you want to do. And so totally free

thinking, but I want you to do it in five minutes.

So don't overthink it. And hand it back to me, OK? So that's really the first part. I want you to show

intuitively how you can construct a portfolio, OK? So what does a portfolio mean? That I have to explain to you. Let's say for

undergraduates here, so your parents give

you some allowance. You manage to save a

$1,000 on the side. You decided to put into

investments, buying stocks or whatever, or gambling,

buy lottery tickets, whatever you can do. Just break down your percentage. That could be $1,000, or you

could be a portfolio manager and have hundreds of billions

of dollars, or whatever. Or then and say if they raise

some money, start a hedge fund, they may have $10,000

just to start with. How do you want to use

those money on day one? Just think about it. And then so while you're

filling out those pages, please hand it back to me. It's your choice to put

your name down or not.

And then I will start

to assemble those ideas and put them on the blackboard. And sometimes I may come back

to ask you a question– you know, why did you put this? That's OK. Don't feel embarrassed. We're not going to

put you on the spot. But the idea is I want to use

those examples to show you how we actually connect

theory with practice. I remember when I was a

college student I learned a lot of different stuff. But I remember one

lecture so well, one teacher told me one thing. I still remember vividly well,

so I want to pass it on to you. So how do we learn

something useful, right? You always start

with observation.

So that's kind of

the physics side. You collect the data. You ask a lot of questions. You try to find the patterns. Then what you do,

you build models. You have a theory. You try to explain what is

working, what's repeatable, what's not repeatable. So that's where

the math comes in. You solve the equations. Sometimes in economics,

lot of times, unlike physics, the repeatable

patterns are not so obvious. So what you do after this, so

you come back to observations again.

You confirm your theory,

verify your predictions, and find your error. Then this feeds

back to this rule. And a lot of times, the

verification process is really about

understanding special cases. That's why today I really want

to illustrate the portfolio theory using a lot

of special cases. So can you start to hand back

your portfolio construction by now? OK, so just hand back

whatever you have. If you have one thing on

the paper, that's fine. Or many things on

the paper, or you think as a portfolio manager,

or you think as a trader, or you think simply as

a student, as yourself. All right, so I'm

getting these back. I will start to write

on the blackboard. And you can finish

what you started. By the way, that's the only

slide I'm going to use today. I'm not concerned– you realize

if I show you a lot of slides, you probably can't

keep up with me.

So I'm going to write down

everything, just take my time. And so hopefully you get a

chance to think about questions as well. OK, I think– is

anyone finished? Any more? OK. All right, OK. OK, great. You guys are awesome. OK, so let me just

have a quick look to see if I missed any, OK? Wow, very interesting. So I have to say, some

people have high conviction. 100% of you, one of those. I think I'm not going to read

your names, so don't worry, OK? OK I'm just going to read the

answers that people put down, OK? So small cap equities, bonds,

real estate, commodities. Those were there. Qualitative strategies,

selection strategies, deep value models. Food/drug sector models, energy,

consumer, S&P index, ETF fund, government bonds,

top hedge funds. So natural resources,

timber land, farmland, checking account,

stocks, cash, corporate bonds, rare coins, lotteries,

collectibles. That's very unique. And Apple's stock, Google

stock, gold, long term saving annuities. So Yahoo, Morgan Stanley stocks. I like that. [LAUGHTER] OK. Family trust. OK, I think that

pretty much covered it.

OK, so I would say that

list is more or less here. So after you've

done this, when you were doing this, what kind of

questions came to your mind? Anyone wants to– yeah, please. AUDIENCE: [INAUDIBLE] how do I

know what's the right balance to draw in my portfolio? Whether it would be cash,

bills, or stuff like that? PROFESSOR: How do

you do it, really? What's the criteria? And so before we

answer the question how you do, how do you group

assets or exposures or strategies or even people,

traders, together– before we ask all those questions,

we have to ask ourselves another question. What is the goal? What is the objective, right? So we understand what

portfolio management is. So here in this class,

we're not talking about how to come up with

a specific winning strategy in trading or investments. But we are talking about

how to put them together.

So this is what portfolio

management is about. So before we answer

how, let's see why. Why do we do it? Why do we want to have

a portfolio, right? That's a very, very good point. So let's understand the goals

of portfolio management. So before we understand goals

of portfolio management, let's understand

your situations, everyone's situation. So let's look at this chart. So I'm going to

plot your spending as a function of your age. So when you are

age 0 to age 100, so everyone's spending

pattern is different. So I'm not going to tell you

this is the spending pattern. So obviously when

kids are young, they probably don't have a

lot of hobbies or tuition, but they have some basic needs. So they spend. And then the spending

really goes up. Now your parents have

to pay your tuition, or you have to borrow–

loans, scholarships. And then you have college. Now you have– you're married.

You have kids. You need to buy a house, buy

a car, pay back student loans. You have a lot more spending. Then you go on vacation. You buy investments. You just have more

spending coming up. So but it goes to

a certain point. You will taper down, right? So you're not going

to keep going forever. So that's your spending curve. And with the other curve,

you think about it. It's what's your income,

what's your earnings curve. You don't earn anything

where you are just born. I use earning. So this is spending. So let's call this 50. Your earning probably

typically peaks around age 50, but it really depends. Then you probably

go down, back up.

Right, so that's your earning. And do they always match well? They don't. So how do you make

up the difference? You hope to have a fund,

an investment on the side, which can generate those cash

flows to balance your earning versus your spending. OK, so that's only one

simple way to put it. So you've got to ask

about your situation. What's your cash flow look like? So my objective is, I'm

going to retire at age of 50. Then after the age of

50, I will live free. I'll travel around the world. Now I'll calculate

how much money I need. So that's one situation. The other situation is, I want

to graduate and pay back all the student loans in one year. So that's another. And typically people

have to plan these out. And if I'm managing a

university endowment, so I have to think about what

the university's operating budget is like, how much money

they need every year drawing from this fund.

And then by

maintaining, protecting the total fund for basically

a perpetual purpose, right? Ongoing and keep growing it. You ask for more contributions,

but at the same time generating more return. If you have a pension

fund, you have to think about what time frame

lot of the people, the workers, will retire and will actually

draw from the pension. And so every situation

is very different. Let me even expand it. So you think, oh, this

is all about investment. No, no, this is not

just about investment. So I was a trader for a

long time at Morgan Stanley, and later on a trading manager. So when I had many

traders working for me, the question I was

facing is how much money I need to allocate to each

trader to let them trade. How much risk do

they take, right? So they said, oh, I have

this winning strategy. I can make lots of money. Why don't you give

me more limits? No, you're not going

to have all the limits.

You're not going to have all

the capital we can give to you. Right, so I'm going to explain. You have to diversify. At the same time, you have

to compare the strategies with parameters–

liquidity, volatility, and many other parameters. And even if you are

not managing people, let's say– I was going to

do this, so Dan, [INAUDIBLE], Martin and Andrew. So they start a

hedge fund together. So each of them had

a great strategy. Dan has five, Andrew has

four, so they altogether have 30 strategies. So they raise an

amount of money, or they just pool

together their savings. But how do you

decide which strategy to put more money on day one? So those questions

are very practical. So that's all. So you understand

your goals, that's then you're really clear on

how much risk you can take. So let's come back to that.

So what is risk? As Peter explained

in his lecture, risk is actually not

very well defined. So in the Modern

Portfolio Theory, we typically talk about

variance or standard deviation of return. So today I'm going to

start with that concept, but then try to

expand it beyond that. So stay with that

concept for now. Risk, we use standard

deviation for now. So what are we trying to do? So this, you are familiar

with this chart, right? So return versus

standard deviation. Standard deviation is

not going to go negative. So we stop at zero. But the return

can go below zero. And I'm going to review one

formula before I go into it. I think it's useful to review

what previously you learned. So you let's say you

have– I will also clarify the notation as well

so you don't get confused. So let's say– so Peter

mentioned the Harry Markowitz Modern Portfolio Theory

which won him the Nobel Prize in 1990, right? Along with Sharpe

and a few others.

So it's a very

elegant piece of work. But today, I will try to

give you some special cases to help you understand that. So let's review

one of the formulas here, which is really

the definition. So let's say you

have a portfolio. Let's call the expected

return of the portfolio is R of P, equal to the

sum, a weighted sum, of all the expected

returns of each asset. You'll basically

linearly allocate them. Then the variance– oh, let's

just look at the variance, sigma_P squared.

So these are vectors. This is a matrix. The sigma in the middle

is a covariance matrix. OK that's all you need to

know about math at this point. So I want us to go through an

exercise on that piece of paper I just collected back to put

your choice of the investment on this chart. OK, so let's start with one. So what is cash? Cash has no standard deviation. You hold cash– so it's

going to be on this axis. It's a positive return. So that's here. So let's call this cash. Where is– and let's me just

think about another example. Where's lottery? Say you buy Powerball, right? So where's lottery falling? Let's assume you put

everything in lottery. So you're going to lose. So your expected value is

very close to lose 100%. And your standard deviation

is probably very close to 0. So you will be here. So some of you say, oh, no, no. It's not exactly zero. So OK, fine.

So maybe it's

somewhere here, OK? So not 100%, but you still

have a pretty small deviation from losing all the money. What is coin flipping? So let's say you decide to

put all your money to gamble on a fair coin flip, fair coin. So expected return is zero. What is the standard

deviation of that? AUDIENCE: 100%? PROFESSOR: Good. So 100%. So we got the three

extreme cases covered.

OK, so where is US

government bond? So let's just call it five-year

note or ten-year bond. So the return is better than

cash with some volatility. Let's call it here. What is investing in a start

up venture capital fund like? Pretty up there, right? So you'll probably get

a very high return, by you can lose all your money. So probably somewhere

here, you see. Buying stocks, let's

call it somewhere here. Our last application

lecture, you heard about investing

in commodities, right? Trading gold, oil. So that has higher volatility,

so sometimes high returns. So let's call this commodity.

And the ETF is typically lower

than single stock volatility, because it's just

like index funds. So here. Are there any other choices

you'd like to put on this map? OK. So let me just look at

what you came up with. Real estate, OK. Real estate, I would say

probably somewhere around here. Private equity probably

somewhere here. Or investing in hedge

funds somewhere. So I think that's enough

examples to cover. So now let me turn the table

around and ask you a question.

Given this map, how would you

like to pick your investments? So you learned about

the portfolio theory. As a so-called

rational investor, you try to maximize your return. At the same time, minimize

your standard deviation, right? I hesitate to use

the term "risk," OK? Because as I said, we

need to better define it. But let's just say you

try to minimize this but maximize this,

the vertical axis. OK, so let's just say you

try to find the highest possible return

for that portfolio with the lowest possible

standard deviation.

So would you pick this one? Would you pick this one? OK, so eliminate those two. But for this, that's

actually all possible, right? So then that's where we learn

about the efficient frontier? So what is the

efficient frontier? It's really the

possible combinations of those investments you

can push out to the boundary that you can no longer find

another combination– given the same standard

deviation, you can no longer find a higher return. So you reached the boundary.

And the same is true

that for the same return, you can no longer minimize

your standard deviation by finding another combination. OK, so that's called

efficient frontier. How do you find the

efficient frontier? That's what essentially

those work were done and it got them the

Nobel Prize, obviously. It's more than that,

but you get the flavor from the previous lectures. So what I'm going to do

today is really reduce all of these to the

special case of two assets. Now we can really derive a

lot of intuition from that.

So we have sigma, R.

We're going to ignore what's below this now, right? We don't want to be there. And we want to stay

on the up-right. So let's consider

one special case. So again for that, let's

write out for the two assets. So what is R of P? It's w_1 R_1 plus 1

minus w_1 R_2, right? Very simple math. And what is sigma_P? So the standard deviation of

the portfolio– or the variance of that, which is a

square– we know that's for the two asset

class special case. So let me give you a further

restriction– which, let's consider if R_1 equal to R_2. Again, here meaning

expected return. I'm simplifying some

of the notations. And sigma_1 equal

to 0, and sigma_2 not equal to 0, so what is rho? What is the correlation? Zero, right? Because you have no

volatility on it. OK, so what is– what's that? AUDIENCE: It's really undefined. PROFESSOR: It's

really undefined, yes. Yeah. AUDIENCE: [INAUDIBLE]

no covariance.

PROFESSOR: There's no–

yeah, that's right. OK, so let's look at this. So you have sigma_2 here. Sigma_1 is 0. And you have R_1 equal to R_2. What is all R of P? It's R, right? Because the weighting

doesn't matter. So you know it's going

to fall along this line. So here is when

weight one equal to 0. So you weight everything

on the second asset. Here you weight the

first asset 100%. So you have a possible

combination along this line, along this flat line. Very simple, right? I like to start with a

really a simple case. So what if sigma_1 also is

not 0, but sigma_1 equal to sigma_2. And further, I impose– impose–

the correlation to be 0, OK? What is this line look like? So I have sigma_2

equal to sigma_1. And R_1 is still equal

to R_2, so R_P is still equal to R_1 or R_2, right? What does this line look like? So volatility is the same. Return is those are the same

of each of the asset class. You have two strategies

or two instruments. They are zero-ly correlated. How would you combine them? So you take the derivative

of this variance with regarding to

the weight, right? And then you minimize that.

So what you find is that

this point is R_1 equal to 0, or– I'm sorry, w_1,

or w_1 equal to 1. You're at this point, right? Agreed? So you choose either, you will

be ending up– the portfolio exposure in terms of return and

variance will be right here. But what if you

choose– so when you try to find the minimum

variance, you actually end up– I'm not going to do the math. You can do it afterwards. You check by yourself, OK? You will find at

this point, that's when they are equally

weighted, half and half. So you get square root of that. So you actually have a

significant reduction of the variance of the portfolio

by choosing half and half, zero-ly correlated portfolio.

So what's that called? What's that benefit? Diversification, right? When you have less than

perfectly correlated, positively correlated

assets, you can actually achieve the same

return but having a lower standard deviation. I'll say, OK, that's

fairly straightforward. So let's look at a few

more special cases. I want really to have you

establish this intuition. So let's think about what

if in the same example, what if rho equals to

1, perfectly correlated? Then you can't, right? So you end up at

just this one point. You agree? OK. What if it's totally

negatively correlated? Perfectly negatively correlated. What's this line look like? Right? So you if you weight

everything to one side, you're going to

still get this point. But if you weight

half and half, you're going to achieve

basically zero variance. I think we showed

that last time, you learned that last time.

OK, so let's look

beyond those cases. So what now? Let's look at– so R_1 does

not equal to R_2 anymore. Sigma_1 equal to 0. There's no volatility

of the first asset. So that's cash, OK? So that's a riskless

asset in the first one. So let's even call that

R_1 is less than R_2. So that's the– right? You have the cash asset, and

then you have a non-cash asset. Rho equal to 0,

zero correlation. So let's look at what

this line looks like. So R_1, R_2, sigma_2 here. When you weight asset

two 100%, you're going to get this point, right? When you weight asset

one 100%, you're going to get this point, right? So what's in the

middle of your return as a function of variance? Can someone guess? AUDIENCE: A parabola? Should it be a parabola? PROFESSOR: Try again.

AUDIENCE: A parabola. PROFESSOR: Yeah, I know, I know. Thank you. Are there any other answers? OK, this is actually

I– let me just derive very quickly for you. Sigma_1 equal to

0, rho equal to 0. What's sigma_P? Right? And sigma_P is essentially

proportional to sigma_2 with the weighting. OK, and what's R? R is a linear combination

of R_1 and R_2. So it's still– so it's linear. OK, so because in these

cases, you actually– you essentially– your return

is a linear function. And the slope, what

is the slope of this? Oh, let's wait on the slope.

So we can come back to this. This actually relates back to

the so-called capital market line or capital

allocation line, OK? Because last time we talked

about the efficient frontier. That's when we have no riskless

assets in the portfolio, right? When you add on cash, then

you actually can select. You can combine the

cash into the portfolio by having a higher boundary,

higher Efficient Frontier, and essentially a higher

return with the same exposure. So let's look at a

couple more cases, then I will tell you– so I

think let's look at– so R_1 is less than R_2.

And volatilities are not 0. Also, sigma_1 is

less than sigma_2, but it has a negative

correlation of 1. So you'll have asset

one, asset two. And as we know, where you pick

half and half, this goes to 0. So this is a quadratic function. You can verify and

prove it later. And what if when

rho is equal to 0– and actually, I want to– so

sigma_1 should be here, OK? So when rho is equal

to zero, this no longer goes to– the variance can

no longer be minimized to 0.

So this is your efficient

frontier, this part. I think that's enough

examples of two assets for the efficient frontier. So you get the idea. So then what if we

have three assets? So let me just touch

upon that very quickly. If you have one more

asset here, essentially you can solve the

same equations. And when the– special case:

you can verify afterwards, if all the

volatilities are equal, and zero correlation

among the assets.

You're going to be able to

minimize sigma_P equal to 1 over the square root

of three of sigma_1. OK. So it seems pretty neat, right? The math is not hard

and straightforward. But it gives you the idea

how to answer your question, how to select them when

you start with two. So why are two

assets so important? What's the implication

in practice? It's actually a very

popular combination. Lot of the asset

managers, they simply benchmark to bonds

versus equity. And then one famous

combination is really 60/40. They call it a

60/40 combination. 60% in equity, 40% in bonds. And even nowadays, any fund

manager, you have that. People will still ask you

to compare your performance with that combination. So the two-asset examples seem

to be quite easy and simple, but actually it's a very

important one to compare. And that will lead me to

get into the risk parity discussion. But before I get to

risk parity discussion, I want to review the concept

of beta and the Sharpe ratio.

So your portfolio return,

this is your benchmark return, R of m, expected return. R_f is the risk-free return,

so essentially a cash return. And alpha is what you can

generate additionally. So let's even not to worry

about these small other terms– or not necessarily small,

but for the simplicity, I'll just reveal that. So that's your beta. Now what is your Sharpe ratio? OK. And you can– so

sometimes Sharpe ratio is also called

risk-weighted return, or risk-adjusted return. And how many of you have

heard of Kelly's formula? So Kelly's formula

basically gives you that when you have– let's

say in the gambling example, you know your winning

probability is p. So this basically tells

you how much to size up, how much you want to bet on.

So it's a very simple formula. So you have a winning

probability of 50/50, how much you bet on? Nothing. So if you have p equal to 100%,

you bet 100% of your position. If you have a winning

probability of negative 100%, so what does it mean? That means you have a 100%

probability of losing it. What do you do? You bet the other

way around, right? You bet the other side, so that

when p is equal to negative– I'm sorry, actually

what I should say is when p equal to 0, your

losing probability becomes 100%, right? So you bet 100%

the other way, OK? So that I leave to

you to think about. That's when you have

discrete outcome case. But when you

construct a portfolio, this leads to the next question.

It's in addition to the

efficient frontier discussion, is that really all

about asset allocation? Is that how we calculate

our weights of each asset or strategy to choose from? The answer is no, right? So let's look at a

60/40 portfolio example. So again, two asset stock. Stock is, let's say,

60% percent, 40% bonds. So on this– so typically

your stock volatility is higher than the bonds, and

the return, expected return, is also higher.

So your 60/40 combinations

likely fall on the higher return and the higher

standard deviation part of the efficient frontier. So the question was–

so that's typically what people do before 2000. A real asset manager, the

easiest way or the passive way is just to allocate 60/40. But after 2000, what happened

was when after the equity market peaked and the bond had

a huge rally as first Greenspan cut interest rates before

the Y2K in the year 2000. You think it's kind of funny,

but at that time everybody worried about the year 2000. All the computers

are going to stop working because old software

were not prepared for crossing this millennium event. So they had to cut interest

rates for this event. But actually nothing happened,

so everything was OK. But that left the market

with plenty of cash, and also after the

tech bubble burst.

So that was a good

portfolio, but then obviously in 2008 when the

equity market crashed, the bond market, the

government bond hybrid market, had a huge rally. And so that made

people question that. Is this 60/40 allocation of

asset simply by the market value the optimal

way of doing it, even though you are falling

on the Efficient Frontier? But how do you compare

different points? Is that simple choice of your

objectives, your situation, or there's actually other

ways to optimize it.

So that's where the risk

parity concept was really– the concept has been

around, but the term was really coined in

2005, so quite recently, by a guy named Edward Qian. He basically said, OK,

instead of allocating 60/40 based on market value,

why shouldn't we consider allocating risk? Instead of targeting a return,

targeting asset amount– let's think about

a case where we can have equal weighting of

risk between the two assets. So risk parity really means

equal risk weighting rather than equal market exposure. And then the further step

he took was he said, OK. So this actually,

OK, is equal risk. So you have lower return

and a lower risk, a lower standard deviation. But sometimes you will really

want a higher return, right? How do you satisfy both? Higher return and lower risk. Is there a free lunch? So he was thinking, right? There is, actually.

It's not quite free, but

it's the closest thing. You've probably heard

this phrase many times. The closest thing in

investment to a free lunch is diversification. OK, and so he's using a

leverage here as well. let me talk about it a bit

more, about diversification, give you a couple

more examples, OK? That phrase about the free

lunch and diversification was actually from– was

that from Markowitz? Or people gave him that term. OK, but anyway. So let me give you another

simple example, OK? So let's consider two

assets, A and B. In year one, A goes up to– it

basically doubles. And in year two,

it goes down 50%. So where does it end up? So it started with 100%. It goes up to 200%. Then it goes down

50% on the new base, so it returns nothing, right? It comes back. So asset B in year one loses

50%, then doubles, up 100% in year two.

So asset B basically

goes down to 50% and it goes back up to 100%. So that's when you look

at them independently. But what if you had a 50/50

weight of the two assets? So if someone who is

quick on math can tell me, what does it change? So A goes up like that,

B goes down like that. Now you have a 50/50 A and

B. So let's look at magic.

So in year one, A,

you have only 50%. So it goes up 100%. So that's up 50%

on the total basis. B, you'll also weight

50%, but it goes down 50%. So you have lost 25%. So at the end of

year one, you're actually– so this is a combined

50/50 portfolio, year one and year two. So you started with 100. You're up to 1.25

at this point, OK? So at the end of year

one, you rebalance, right? So you have to

come back to 50/50. So what do you do? So this becomes 75, right? So you no longer have

the 50/50 weight equal. So you have to sell

A to come back to 50 and use the money to buy B. So you have a new 50/50

percent weight asset. Again, you can

figure out the math. But what happens in

the following year when you have this move,

this comes back 50%, this goes up 100%. You return another 25%

positively without volatility. So you have a straight line.

You can keep– so

this two year is a– so that's so-called

diversification benefit. And in the 60/40 bond market,

that's really the idea people think about

how to combine them. And so let me talk a little

bit about risk parity and how you actually

achieve them. I'll try to leave plenty

of time for questions. So that's the return, and

so let's forget about these. So let's leave cash here, OK? So the previous example I gave

you, when you have two assets, one is cash, R_1,

the other is not. The other has a

volatility of sigma_2. You have this point, right? So and I said,

what's in between? It's a straight line. That's your asset allocation,

different combination. Did it occur to you, why

can't we go beyond this point? So this point is when we weight

w_2 equal to 1, w_1 equal to 0.

That's when you weight

everything into the asset two. What if you go beyond that? What does that mean? OK. So let's say, can we have w_1

equal to minus 1, w_2 equal to plus 2? So they still add up to 100%. But what's negative 1 mean? Borrow, right? So you went short cash

100%, you borrow money. You borrow 100% of cash,

then put into to buy equity or whatever,

risky assets, here. So you have plus 2 minus 1. What does the return looks

like when you do this? So R_P equal to w_1

R_1 plus w_2 R_2. So minus R_1 plus 2R_2. That's your return. It's this point here. What's your variance look

like, or standard deviation look like? As we did before, right? So sigma_P simply

equal to w_2 sigma_2. So in this case, it's 2sigma_2. So you're two times more

risky, two times as risky as the asset two.

So this introduces the

concept of leverage. Whenever you go short,

you introduce leverage. You actually– on

your balance sheet, you have two times of asset two. You're also short one of

the other instrument, right? OK so that's your liability. So your net is still one. So what this risk

parity says is, OK, so we can target on the

equal risk weighting, which will give you somewhere

around– let's called it 25. 25% bonds, 75%– 25%

equity, 75% of fixed income. Or in other words, 25%

of stocks, 75% of bonds.

So you have lower return. But if you leverage

it up, you actually have higher return,

higher expected return, given the same amount

of standard deviation. You achieved by leveraging up. Obviously, you

leverage up, right? That's the other

implication of that. We haven't talked about

the liquidity risk, but that's a different topic. So what's your Sharpe ratio look

like for risk parity portfolio? So you essentially

maximized the Sharpe ratio, or risk-adjusted return, by

achieving the risk parity portfolio. So 60/40 is here. You actually maximize that, and

this is– does leverage matter? When you leverage up, does

Sharpe ratio change, or not? AUDIENCE: It splits in half.

So you've got twice the

[? variance ?] [INAUDIBLE]. PROFESSOR: So let's look at that

straight line, this example, OK? So we said Sharpe

ratio equal to– right? So R_P, what is sigma_P? It's 2sigma_2, right,

when you leverage up. So this equals to R_2 minus

R_1, divide by sigma_2. So that's the same

as at this point. So that's essentially the

slope of the whole line. It doesn't change. OK, so now you can

see the connection between the slope of this

curve and the Sharpe ratio and how that links back to beta.

So let me ask you

another question. When the portfolio has higher

standard derivation of sigma_P, will beta to a specific

asset increase or decrease? So what's the

relationship intuitively between beta– so let's take

a look at the 60/40 example. Your portfolio, you have

stocks, you have bonds in it. So I'm asking you, what is

really the beta of this 60/40 portfolio to the equity market? When equity market, it

becomes– when the portfolio becomes more volatile. Is your beta increasing

or decreasing? So you can derive that. I'm going to tell

you the result, but I'm not going

to do the math here. So beta equals to– [INAUDIBLE]

in this special case, is sigma_P over sigma_2. OK. All right, so so

much for all these. I mean, it sounds like

everything is nicely solved. And so coming back

to the real world, and let me bring you back, OK? So are we all set for

portfolio management? We can program, make

a robot to do this. Why do we need all

these guys working on portfolio management? Or why do we need anybody

to manage a hedge fund? You can just give money, right? So why do you need somebody,

anybody, to put it together? So before I answer

this question, let me show you a video.

[VIDEO PLAYBACK] [HORN BLARING] [END VIDEO PLAYBACK] OK. Anyone heard about the

London Millennium Bridge? So it was a bridge

built around that time and thought it had

the latest technology. And it would really

perfectly absorb– you heard about soldiers just

marching across a bridge, and they'll crush the bridge. When everybody's

walking in sync, your force gets synchronized. Then the bridge was

not designed to take that synchronized force, so the

bridge collapsed in the past. So when they designed this,

they took all that into account. But what they hadn't

taken into account was the support of

that is actually– so they allow the horizontal

move to take that tension away. But the problem is

when everybody's sees more people walking in

sync, then the whole bridge starts to swell, right? Then the only way

to keep a balance for you standing

on the bridge is to walk in sync

with other people.

So that's a survival instinct. And so I got this–

I mean, that's actually my friend at

Fidelity, Ren Cheng. Dr. Ren Cheng brought

this up to me. He said, oh, you're

doing– how do you think about the

portfolio risk, right? This is what happened in the

financial market in 2008. When you think you got

everything figured out, you have the optimal strategy. When everybody

starts to implement the same optimal strategy

for your own as individual, the whole system is

actually not optimized. It's actually in danger. Let me show you another one. [VIDEO PLAYBACK] [CLACKING] OK. These are metronomes, right? So can start anywhere you like. Are they in sync? Not yet. What is he doing? You only have to listen to it. You don't have to see it. So what's going on here? This is not– metronomes

don't have brains, right? They don't really

follow the herd. Why are they synchronizing? OK, if you're expecting they

are getting out of sync, it's not going to happen. OK, so I'm going

to stop right here.

OK. [END VIDEO PLAYBACK] You can try as many–

how do I get out of this? OK, so you can try it. You can look at– there's

actually a book written on this as well, so. But the phenomena

here is nothing new. But what when he did

this, what's that mean? When he actually raised

that thing on the plate and put it on the Coke cans? What happened? Why is that is so significant? AUDIENCE: Because now

they're connected. PROFESSOR: They're connected. Right. So they are interconnected. Before, they were individuals. Now they're connected. And why did I show you

the London Bridge and this at the same time? What's this to do with

portfolio management? What's this to do with

portfolio management? AUDIENCE: [INAUDIBLE]

people who are trading, if they have the same strategy,

[INAUDIBLE] affect each other, they become connected

in that way– PROFESSOR: Right. AUDIENCE: If as

an individual, you are doing a different

strategy, if everybody has been doing

something different, you can maximize

[? in the space. ?] PROFESSOR: Very well said. So if you're looking

for this stationary best way of optimizing

your portfolio, chances are everybody

else is going to figure out the same thing.

And eventually, you

end up in the situation and you actually get killed. OK, so that's the thing. What you learned today,

what you walk away was this. OK, today is not what I

want you to know that all the problems are solved. Right? So you say, oh, the

problem's solved. The Nobel Prize was given. So let's just program them. No, you actually– it's

a dynamic situation. You have to. So that makes the problem

interesting, right? As a younger generation,

you're coming to the field.

The excitement is

there are still a lot of interesting

problems out there unsolved. You can beat the others

already in the field. And so that's one takeaway. And what are the

takeaways you think by listening to all these? AUDIENCE: Diversification

is a free lunch. [CHUCKLES] PROFESSOR: Diversification

is a free lunch, yes. Not so free, right, in the end. It's free to a certain extent. But it's something–

you know, it's better than not diversified, right? It depends on how you do it. But there is a way

you can optimize. And so it's– I want

to leave with you, I actually want to finish a few

minutes earlier so that you can ask me questions. You can ask. It's probably better to

have this open discussion. And so I want you to

walk away, to really keep in mind is in the

field of finance, and particularly in the

quantitative finance, it's not mechanical.

It's not like solving

physics problems. It's not like you can get

everything figured so it becomes predictable, right? So the level of predictability

is actually very much linked to a lot of other things. Physics, you solve

Newton's equations. You have a controlled

environment and you know what you're

getting in the outcome. But here, when you

participate in the market, you are changing the market. You are adding on

other factors into it.

So think more from a

broader scope kind of view rather than just

solve the mathematics. That's why I come

back to the original– if you walk away

from this lecture, you'll remember what I

said at the very beginning. Solving problems

is about observe, collecting data,

building models, then verify and observe again. OK, so I'll end right

here, so questions. AUDIENCE: Yeah, just

[INAUDIBLE] question. Does this have anything to

do with– it kind of sounds like game theory, but

I'm not exactly too sure. Because you have

a huge population and no stable equilibrium. Does it have anything to do

with game theory, by any chance? PROFESSOR: It has a lot

to do with game theory, but not only to game theory. So game theory, you have

a pretty well-defined set of rules. Two people play chess

against each other. That's where a computer actually

can become smarter, right? So in this market situation,

you have so many people participating without

clearly defined rules.

There are some rules, but

not always clearly defined. And so it's much more

complex than game theory. But it's part of it, yeah. Dan, yeah? AUDIENCE: Can you talk a little

bit about why some of the risk parity portfolios that did

so poorly in May and June when rates started to rise

and what about their portfolio allowed them do that? PROFESSOR: Good question, right. So as you can see here, what

the risk parity approach does is essentially to weight more

on the lower volatility asset. In this case, the question

is, how do you know which asset has low volatility? So you look at

historical data, which you conclude bonds have

the lower volatility. So you overweight bonds. That's the essence

of them, right? So then when bonds

to start to sell off after Bernanke, Fed

chairman Bernanke, said he's going to taper

quantitative easing.

So bonds from a very low high

yield, a very low yield level, the yield went much higher,

the interest rate went higher. Bonds got sold off. So this portfolio did poorly. So now the question

is, does that prove the risk parity approach

wrong, or does it prove right? Does the financial

crisis of 2008 prove the risk parity

approach a superior approach, or does the June/May

experience prove this as the less-favored approach? What does it tell us? Think about it. So it really is inconclusive. So you observe, you extrapolate

from your historical data. But what you are

really doing is you're trying to forecast volatility,

forecast return, forecast correlation, all based

on historical data. It's like– a lot of

people use that example. It's like driving by looking

at the rear view mirror. That's the only thing

you look at, right? You don't know what's going

on, happening in front of you.

You have another question? AUDIENCE: Given all

this new information, do you find that

people are still playing similar [INAUDIBLE]

strategy with portfolio management? PROFESSOR: Very much true. Why? Right, so you said, people

should be smarter than that. It's very difficult to

discover new asset classes. It's also very

difficult to invent new strategies in which you have

a better winning probability. The other risk, the other

very interesting phenomenon, is most of the traders and

the portfolio managers, the investors, they

are career investors– meaning just like if

I'm a baseball coach, I'm hired to coach

a baseball team. My performance is

really measured against the other teams

when I win or lose, right? A portfolio manager

or investor is also measured against their peers. So the safest way for them to

do is to benchmark to an index, to the herd.

So there's very little

incentive for them to get out of the crowd, because

if they are wrong, they get killed first. They lose their jobs. So the tendency is to

stay with the crowd. It's for survival instinct. It's, again, the other example. It's actually the

optimal strategy for individual portfolio manager

is really to do the same thing as other people are

doing because you stay with the force. AUDIENCE: So you said given

that we have all these groups, in the end, it's not just

that we could leave it to the computers.

We need managers. So what different

are the managers doing, other than [INAUDIBLE]? PROFESSOR: Can you try to

answer that question yourself? What's the difference between

a human and a computer? That's really– what

can human add value to what a computer can do? AUDIENCE: Consider the factors,

the market factors and news and what's going on. PROFESSOR: So taking more

information, processing information, make a judgment

on a more holistic approach. So it's an interesting question. I have to say that

computers are beating humans in many different ways.

Can a computer ever get to

the point actually beating a human in investment? I can't confidently tell you

that it's not going to happen. It may happen. So I don't know. Any other questions? Yeah? AUDIENCE: Just to add to that. I think there is some more to

management than just investing. I think managers also have key

roles in their HR, key roles in just like managing people

and ensuring that they're maximizing their

talents, not just like, oh, how much money did you make? But I mean, are you moving

forward in your career while you're there? So I think management has a

role to play in that as well, not just investment. PROFESSOR: Yeah, I think

that's a good point. Yeah. All right, so– oh, sure. Jesse? AUDIENCE: What is your

portfolio breakdown? PROFESSOR: My

personal portfolio? Well, I am actually very

conservative at this point, because if you look at my curve

of those spending and earning curve, I'm basically trying

to protect principals rather than try to maximize

return at this point.

So I would be sliding down

more towards this part rather than try to go

to this corner, yeah. So I haven't really

talked much about risk. What is risk, right? So I talk about volatility

or standard deviation. But as we all know that, as

Peter mentioned last time as well, there are many other ways

to look at risk– value at risk or half distribution or

truncated distribution, or simply maximum loss you

can afford to take, right? But looking at standard

deviation or volatility is an elegant way. You can see. I can really show you in

very simple math about how the concept actually plays out.

But in the end,

actually volatility is really not the best

measure, in my view, of risk. Why? Let me give you another simple

example before we leave. So let's say this is over time. This is your cumulative

return or you dollar amount. So you start from here. If you go flat,

then– does anyone like to have this

kind of a performance? Right? Of course, right? This is very nice. You keep going up. You never go down. But what's the

volatility of that? The volatility is

probably not low, right? And then on the

other hand, you could have– what I'm

trying to say, when you look at expected

return matching expected return and the volatility,

you can still really not selecting the best combination.

Because what you really

should care about is not just your volatility. And again, bear in mind all

the discussion about the Modern Portfolio Theory is based

on one key assumption here. It's about Gaussian

distribution, OK? Normal distribution. The two parameters, mean

and standard deviation, categorize the distribution. But in reality, you have many

other sets of distributions. And so it's a concept

still up for a lot of discussion and debate.

But I want to leave

that with you as well. Yeah? AUDIENCE: Just going back to

the same question about what these guys were asking

about management and how do they add

value, I think the people who added value– there

were some people who added a tremendous amount of

value in the financial crisis. And they were doing

the same mathematics. But a difference was in

their expected return of various assets was

different from the entire– the broad market. So if you can just know what

expected return is that, probably that is the only

answer to the whole portfolio management debate. PROFESSOR: Yes. If you can forecast expected

return, then that's– yeah, now you know the game.

You solved it. You solved the big

part of the puzzle. Yeah? AUDIENCE: What

management does is how good it can do [INAUDIBLE]

expected return, full stop. Nothing more. PROFESSOR: I disagree on that. That's the only thing. Because given two managers, they

have the same expected return, but you can still further

differentiate them, right? So that's– yeah. And that's what all this

discussion is about. But yes, expected return will

drive lot of these decisions. If you know one manager's good

expected return, three years later, he's going to make 150%. You don't really care

what's in between, right? You're just going

to ride it through. But the problem is you

don't know for sure. You will never be sure. AUDIENCE: I'd like

to comment on that. PROFESSOR: Sure. AUDIENCE: What

[INAUDIBLE] looked at in simplified

settings, estimating returns and volatilities.

And the problem, the

conclusion for the problem, was basically cannot

estimate returns very well, even with more data,

over a historical period. But you can estimate volatility

much better with more data. So there's really an

issue of perhaps luck in getting the return estimates

right with different managers, which are hard to prove

that there was really an expertise behind that. Although with volatility, you

can have improved estimates.

And I think possibly with

a risk parity portfolio, those portfolios are focusing

not on return expectations, but saying if we're going to

consider different choices based on just how

much risk they have and equalize that risk, then

the expected return should be comparable across

those, perhaps. PROFESSOR: Yeah. So that highlights

the difficulty of forecasting return,

forecasting volatility, forecasting correlation. So risk parity appears

to be another elegant way of proposing the

optimal strategy but it has the same problems. Yeah? AUDIENCE: Actually, I

also wanted to highlight.

You mentioned the

Kelly criterion, which we haven't covered the

theory for that previously. But I encourage people

to look into that. It deals with issues of

multi-period investments as opposed to

single-period investments. And most– all this classical

theory we've been discussing, or that I discuss, covers

just a single period analysis, which is an oversimplification

of an investment. And when you are investing

over multiple periods, the Kelly criterion tells you

how to optimally basically bet with your bank roll. And actually there's an

excellent book, at least I like it, called

Fortune's Formula that talks about–

[? we already ?] said the origins of

options theory in finance. But it does get into

the Kelly criterion.

And there was a rather major

discussion between Shannon, a mathematician at MIT, who

advocated applying the Kelly criterion, and Paul Samuelson,

one of the major economists. PROFESSOR: Also from MIT. AUDIENCE: Also from MIT. And there was a great

dispute about how you should do portfolio optimization. PROFESSOR: That's a great book. And a lot of

characters in that book actually are from MIT–

and Ed Thorp, for example. And it's really about people

trying to find the Holy Grail magic formula– not

really to that extent, but finding something other

people haven't figured out. But it's very

interesting history. Big names like Shannon, very

successful in other fields.

In his later part of his

career and life really devoted most of his time to

studying this problem. You know Shannon, right? Claude Shannon? He's the father of

information theory and has a lot to do with

the later information age invention of computers

and very successful, yeah. So anyway, so we'll end

the class right here. No homework for today, OK? So you just need to– yeah, OK. All right, thank you.