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16. Portfolio Management

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Jan 06, 2015

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16. Portfolio Management
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  • 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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MIT 18.S096 Topics in Mathematics with Applications in Finance, Fall 2013
View the complete course: http://ocw.mit.edu/18-S096F13
Instructor: Jake Xia

This lecture focuses on portfolio management, including portfolio construction, portfolio theory, risk parity portfolios, and their limitations.

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