16. Portfolio Management

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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.

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