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visit MIT OpenCourseWare at ocw.mit.edu. ANDREW LO: Well, let me pick
up where we left off last time and give you just a very quick
overview of where we're at now, because we're on the brink of
a very important set of results that I think will change
your perspective permanently on risk and expected return. Last time, remember, we
looked at this trade-off between expected
return and volatility. And we made the
argument that when you combined a bunch
of different securities that are not all
perfectly correlated, what you get is this
bullet-shaped curve in terms of the possible trade-offs
between that expected return and riskiness of various
different portfolios. So every single dot on
this bullet-shaped curve corresponds to a specific
portfolio, or weighting, or vector of portfolio
weights, omega. So now what I want to ask you to
do for the next lecture or two is to exhibit a little
bit of a split personality kind of a perspective. I'm going to ask you to
look at the geometry of risk and expected return,
but at the same time, in the back of
your brain, I want you to keep in
mind the analytics of that set of geometries. In other words, I want
you to keep in mind how we got this bullet-shaped curve. The way we got it
was from taking different weighted
averages of the securities that we have access
to as investments. So every one of these
points on the bullet corresponds to a
specific weighting. As you change those
weightings, you change the risk and
return characteristics of your portfolio. So the example that I gave
after showing you this curve where I argued that the
upper branch of this bullet is where any rational
person would want to be. And by rational, I've
defined that as somebody who prefers more expected
return to less, and somebody who prefers less risk to
more, other things equal. So if you've got those
kind of preferences, then you want to be
in the Northeast. You want to be as north,
sorry, Northwest as possible. And you would never want to be
down in this lower branch when you could be in the upper
branch because you'd have a higher expected return
for the same level of risk. So after we developed
this basic idea, I gave you this
numerical example where you've got three
stocks in your universe. General Motors,
IBM, and Motorola. And these are the parameters
that we've estimated using historical data. Now there's going
to be a question, and we've already raised
that question, of how stable are these parameters. Are they really parameters,
or do they change over time. And I told you, in reality of
course, they change over time. But for now, let's play
the game and assume that they are
constant over time, and see what we can do
with those parameters. So with the means, the
standard deviations, and most importantly,
the covariance matrix-- So this is the matrix of
variances and covariances-- With these data as
inputs, we can now construct that
bullet-shaped curve. The way we do it is of
course, to recognize that the expected
return of the portfolio is just a weighted average
of the expected returns of the component securities,
where the weights are our choice variables. That's what we are
getting to pick, is how we allocate
the 100% of our wealth to these three
different securities. And the variance,
of course, is going to be given by a somewhat more
complicated expression where you have the individual
security variances entering here from the diagonals. But you also have the
off diagonal terms entering in that same
equation for that variance of the portfolio. And when we put these
two equations together, the mean and the variance,
and we take the square root the variance to get
the standard deviation, and we plot it on a
graph, we get this. This is the curve, the
bullet-shaped curve, that we generate just
from three securities, and from their covariances. And where we left
off last time is that I pointed out a
couple of things that was interesting about this curve. One is that unlike the two asset
example, where when you start with two assets, the
curve, the bullet goes through the two assets. In this case, with
three or more assets, it's going to turn out
that the bullet is actually going to include these
assets as special cases, but they won't be on the curve. In other words, what
this curve suggests is that any rational
person is going to want to be on this upper branch. What that means is that
it never makes sense to put all your money
in one single security. You see that? In other words, if we agree that
any rational investor is going to want to be on that efficient
frontier, that upper branch, why would you ever want
to be off of that branch? You'd like to be Northwest
of that, but you can't. You'd never want to
be below that branch, or to the right of that branch
because you could do better by being on that branch. So what this suggests
is that we never are going to want
to hold 100% of IBM, or 100% of General Motors,
or 100% of Motorola. If we did, we'd
be on those dots, and those dots would lie
on that efficient frontier. But in fact, they don't. So right away, we
have now departed from Warren Buffett's world
of, I want to pick a few stocks and watch them very,
very carefully. Yeah, Brian? AUDIENCE: Would you expand
that to say that you'd never want to invest in less than
three stocks at a given time? ANDREW LO: That's
not necessarily true. There are points on
this line where-- and they may be pathological,
so in other words, they may be very rare-- but there may be
points on the line where you are holding two
stocks, but not the third. So you've got to be
careful about that. But those are exceptions. As a generic statement,
you're absolutely right. The typical portfolio is going
to have some of all three of them. And if you had four stocks,
the typical portfolio would have some of all four. Yeah, [INAUDIBLE]. AUDIENCE: You
answered my question, which is if you
take one more stock, you'll always have your package
[INAUDIBLE] n stocks include all the [INAUDIBLE], all the
n stocks so, at the limit you should have an infinite
number of stocks [INAUDIBLE] ANDREW LO: Well, let me
put it another way that may be a little bit more intuitive. What this diagram suggests--
you guys are already groping towards-- is the insight that
the more, the merrier. As you add more stocks, you
cannot make this investor worse off. So in other words, I've now
shown you an example with three stocks, we used to do two. Is it possible that by
giving you an extra stock to invest in, I've
made you worse off? Yeah? AUDIENCE: No ANDREW LO: Why AUDIENCE: Because you can
just not invest in that stock. ANDREW LO: Exactly. I can never make you
worse off in a world where you're free
to choose, that is. Because you always have
the option of getting rid of the stock
that you don't like. You can always put 0 on it. So to your point, [INAUDIBLE],
as I add more stocks, first of all my
risk-reward trade-off curve will get better. What does it mean to get better? What does it mean for
the risk-reward trade-off to be better? Yes? AUDIENCE: It means you
get a higher return for the same level of risk. ANDREW LO: That's right. A higher return for
the same level of risk, or a lower risk for the
same level of return. In other words, your
upper branch actually moves to the Northwest. That's what it
means to get better. As I add more stocks, this
will move to the Northwest. And therefore,
you have available all of the opportunities to
the south and to the east, but you would never take
those because you're rational in the
sense that you always prefer less risk to more,
and more return to less. Yeah? AUDIENCE: If we put all of
the stocks on the index, on [INAUDIBLE]. And if we looked at all the
possible combinations that-- we can look at them
all at the same time, but then all the subsets
that you can think of, then you must come up with
some most efficient frontier in that market. ANDREW LO: Hold onto that
thought for 10 minutes. We're going to
come back to that. Let's do three first, and
then we could do all of them. Yeah, Chris? AUDIENCE: Trying to better
understand Buffett's strategy relative to this one. Is the correlation of
these stocks due primarily to just psychological
factors of the market, or is it due to
intrinsic correlation? And then the follow on is when
Buffet says invest in one stock and just watch it carefully,
isn't that sort of assuming that the market will
determine at some point that the stock is
undervalued, and what was he-- ANDREW LO: So those
are two good questions. Let me take each
of them separately. Let's first talk
about the correlation. Why is there correlation? We haven't really
talked much about it, but it turns out that there
are many different arguments for why there is correlation. Probably the most compelling
is that a rising tide lifts all boats, and vice versa. In other words, when
business conditions are good, then that helps all companies. Just like when business
conditions are bad, it hurts all companies. So there's some
macroeconomic type of commonality among businesses
that create correlation. That's one reason. But the second reason is
something you pointed out, which is quite apt, particularly
over the last few weeks, which is the psychological factor. When the entire economy
is under stress, and people are scared
to death about what's going to happen to
the market, what they will do is withdraw money
in mass from equities and put them into safer
assets like cash, or treasury bills, or money market
funds, or whatever they can do to get to safety. So I would say that
the answer is both. There are good economic reasons
where correlations should exist among different companies,
but there are also psychological or behavior
reasons that exacerbate those kinds of commonalities. Now your second question about
Buffett versus this approach. There's one
fundamental difference between what Buffett
would say about a company that he decides to buy versus
how we're approaching it. The fundamental
difference is that Buffett would say that he's been able
to identify a severe mispricing. In other words, he would
argue that markets are not in equilibrium. He would argue that Goldman
Sachs is dramatically undervalued where it is today. And seven years from
now, he may be right. And that's the kind of
time frame he has in mind, if not longer. So far, I've made no
such argument at all about deriving these analyses. I've not made any argument about
whether prices are good or bad. In fact, I'm arguing, in
a way, that these prices I'm taking as given. And the question
is, what can I do to construct a good
portfolio irrespective of whether markets are crazy
or markets are rational. In a few minutes,
I'm going to argue that when markets are rational
and in equilibrium, then there is something
that we can say about the relationship
between risk and reward that's extraordinarily sharp and
meaningful from the perspective of financial decision making. And then at the
end of the course, I'm going to try to
explain to you what the limitations of that
set of assumptions are. Dennis? AUDIENCE: Just as we
wouldn't put anything in bond's half of this
frontier, does this graph imply that we strictly
prefer IBM over GM? That we pretty much never
weigh anything for GM? ANDREW LO: Well, from a
risk-reward perspective, let's take a look. IBM has a higher
expected rate of return, and it's got a
higher level of risk. So you really can't say that you
would never prefer GM over IBM, because GM has lower risk
and lower expected return. If on the other hand,
GM were over here, then you would be right. Because any point to
the direct Northwest of a particular point on this
curve is strictly preferred. And GM and IBM don't
have that relationship. In other words, the
way you can identify securities that are dominated
in both dimensions is-- So this is your risk dimension,
this is your expected return dimension. Pick a point in this space,
and ask the question, what are the other
portfolios that are strictly preferred to that point. Well the answer
is pretty simple. Any portfolio that has higher
expected rate of return for the same level of
risk, so the vertical line. Any portfolio that has less
risk for the same level of expected returns, So
the western direction. And anything in this segment,
that orthant, or quadrant, is strictly preferred. So in the case of
IBM, if you draw the vertical and the horizontal
and ask the question, does GM lie in that area? No. If you do GM, and you
draw the vertical and then the horizontal
and asked does IBM lie in that strictly
preferred quadrant? The answer is no. So the answer to your
question about IBM versus GM, no, there isn't any
strict relationship that would say one would
always dominate the other. But if GM were here,
then IBM is clearly contained in that
preferred quadrant. So then the answer to your
question would be yes. Yeah, Justin? AUDIENCE: Theoretically
then, wouldn't everyone just buy IBM, sell GM,
then wouldn't there be some sort of
equilibrium where then GM-- ANDREW LO: So the
answer is, it depends on other things going on. Everyone would not do that. Everyone would do
something else, and I'm about to tell you. So I'm about to
give you the tools to make that exact
conclusion, and the reason is that when I show you
what people will do, that's going to
far dominate what you think people want to do. Just with pairs. So instead of doing
it with pairs, let's do it with
all the securities, as Zeke wanted to do. We're going to do
that in just a minute. But I want to make sure
everybody understands this basic framework
first, because we're going to now start making this
a little bit more complex. Where we left off at the very
last moment of Wednesday's lecture was I showed
you this diagram with the tangency portfolio,
but we hadn't really gotten to talking about it. Remember the case where we had
only one risky asset and one riskless asset, treasury bills? And in that case, when you
are combining a portfolio with one risky asset
and one riskless, you've got a straight line. It turns out that that
is much more general. You get a straight line anytime
you combine a riskless asset with any number of risky assets. So let me give you an example. Suppose we picked an
arbitrary portfolio which is this red dot, p. And I wanted you
to tell me what is the risk-reward
possibilities that you could achieve by mixing
p with treasury bills. Well, you get that
straight line, right? We derived that last time. So any point along the straight
line is what you could achieve, right? Anybody tell me
where the portfolio would be that invests 100%
of your assets in T-Bills? Where is that on this graph? AUDIENCE: [INAUDIBLE] ANDREW LO: Right
this dot right here. How about 100% in portfolio p? Right, the red dot over there. How about 25% in
T-Bills, 75% in p? Where would that lie? AUDIENCE: [INAUDIBLE]
along the line. ANDREW LO: It would be
along the line, but where? Here? 25% T-Bills, 75%-- AUDIENCE: [INAUDIBLE] ANDREW LO: Right, exactly. It would be 3/4 of the
way up towards this dot, because it's 75% of the
risky, 25% of the riskless, so you're going to get
closer to the risky asset. OK, great. So we've now
demonstrated that what I can achieve as an
investor, just mixing portfolio p with
the risk-free rate, is anywhere along that line. Now this analysis applies
to any portfolio p. So for example, suppose
I wanted to ask you, what risk-reward
trade-offs could I generate by mixing the risk-free
rate with General Motors? What would that look like? Yeah, Ken? AUDIENCE: The line from
T-Bills through GM. ANDREW LO: Exactly,
that's right. If I wanted to mix T-Bills
with General Motors, I get that straight line
right through that dot. If I wanted to mix T-Bills with
IBM, I'd go through that dot, with IBM. If I wanted to mix
T-Bills with Motorola, I'd go through Motorola. And if I wanted to mix
T-Bills with any portfolio on that frontier, on
that upper branch, it would just be a
line between T-Bills and that point on
the upper branch. Right? So question. If I were to give you the
choice of mixing T-Bills with only one portfolio,
just one, which would it be? Which would you prefer? [INAUDIBLE]? AUDIENCE: The one where the
line is tangent to the curve. ANDREW LO: The one where
the line is tangent-- so you're talking about
right around here, right? Somewhere here. That's where the line is
just tangent to that curve. Now why is that? How'd you come up with that? Yeah? AUDIENCE: If you took
anything below that then it'd be, I would say,
preferable to stay back. [INAUDIBLE] ANDREW LO: Exactly. If you picked any
other portfolio besides the tangency portfolio,
let's pick one and see. If you picked, let's
say this one right here. If you drew a line between
this point and that portfolio, it's going to turn
out that there are other points over
here that are strictly in the Northwest of that line,
that you could do better. There exists only one
portfolio that you can mix with T-Bills, such that
you can never, ever do better in terms of generating
risk-reward trade-offs for everybody that
likes expected return, and doesn't like risk. And it turns out that
that portfolio happens to be the tangency portfolio. That's the portfolio that
all of you in this room would love to have. I don't know anything about you,
I don't know your backgrounds, I don't know your
risk aversions, but I don't have to know. As long as I know that
you like expected return and you don't like risk,
those are the only assumptions that I need. Then I know, all of
you in this room, are going to want
that portfolio. You may not be at
that portfolio. For example, some of
you who don't like risk, you're going to be down here. Those of you who are
budding hedge fund managers, you're going to be up here. But the point is, you're
going to be on this line. You're not going to be
on this line down here. Why? Because why be on that line
when you could get higher return for a given level of
risk, or a lower risk for a given level of return. You're giving up something
for no good reason. So this is a remarkable insight
of modern portfolio theory. This basically tells
us that regardless of our differences
in preferences, as long as we satisfy
the hypothesis that we like expected return
and we don't like risk, that means that everybody
in this room will agree that the only line
that they would ever want to be on is that tangency line. Questions? Ingrid? AUDIENCE: Is there a
particular level of risk that makes you accepting
to the tangency fund? ANDREW LO: Yes. In fact, this tangency
portfolio is one very particular and
special portfolio. So in other words, it's a
particular weighting of IBM, General Motors, and
Motorola, that gives you this particular portfolio. AUDIENCE: Which one? [INAUDIBLE] something intuitive? ANDREW LO: It's something
you can solve analytically. It has a solution, and if we
were using matrix algebra, I can actually solve it for you. But it's a little
bit complicated, so I'm not requiring that
people know how to do that. Only that you know
that it exists. Yeah? AUDIENCE: You go
above the red dot, and into leverage [INAUDIBLE]? ANDREW LO: Exactly AUDIENCE: So it means that
I have costs for my debt. ANDREW LO: Yes AUDIENCE: So maybe
some debt, it would be more efficient to buy a
portfolio with other weights-- ANDREW LO: Yes. If you assume that there
are borrowing and lending differences, then obviously
these analyses don't apply. So in particular, if you're
here, you're actually lending. If you're here, you're fully
invested in the stock market. If you're here,
you're borrowing. If you're borrowing and
lending rates are different, then it turns out that the curve
that you want to be on actually has a kink in it. And that means that there
is a potential for being on this curve, and then there's
another tangency line that goes out at a different slope. That's possible. But that's more complicated
than what we want to talk about at this point. So here I am assuming borrowing
and lending rates are the same. Zeke, and then Rami. AUDIENCE: If I had
a choice of, if I had control over the
volatility of the market, then if the yield goes
down, of the deals, then I would want to have
a more volatile market so that I can intersect the curve
at the higher return point. Tangent to the curve-- ANDREW LO: OK, hold on. You're changing the
assumptions here. Why are you controlling the
volatility of the market? The volatility of the
market is a data point that you're basically
using as an input. OK? AUDIENCE: I know, I know. I'm just trying to figure out
what I am-- because you're basically connecting, I
see this as a connection between the yield
curve and the market. Because it's not only
retrospective in the sense that if the yield
goes down, there's cash flowing from the market-- ANDREW LO: Let's not
worry about the dynamics. This is not meant to
be a dynamic story. I didn't say anything about
this happening over time, and there's lots of
different changes going on. This is a static snapshot,
today versus next period. These returns and
covariances and all that apply to the returns from
this period to the next, whether it's monthly
or annual, that's a static snapshot as of today. So we're not talking about any
term structure effects yet. Yeah, Rami? AUDIENCE: This is
assuming three stocks. So if you had four
or five, are you going to actually
move the bullet left? And then you're gonna change-- ANDREW LO: Yes, absolutely. If you start adding more
stocks to this cocktail, what's going to happen is the bullet
is going to shift to the left, and it's going to shift up. And so the tangency
point will change. But the curve, that straight
line, the tangent line, what you're going to
see is that tangent line is going to go like that. The slope, that's right. You're going to get more
expected return per unit risk. And that is
something we're going to take as a measure of how good
this particular trade-off is. We're going to look at
that slope of this line. And the slope of this
line will give us a measure of the expected
rate of return per unit risk. That's exactly what
it's going to do for us. AUDIENCE: So if you're
in the upper right beyond the tangency
[INAUDIBLE], you know that line? Then you're borrowing
[INAUDIBLE] portfolio? ANDREW LO: That's correct. AUDIENCE: If you were to
extend the line leftwards, down to the left, would you be
then shorting the market to invest in T-Bills? ANDREW LO: Yes. And if that happens, you
know what you would do? It would not go this
way, because of course standard deviation
can't be negative. It would go like this. It would go this way. Because standard deviation
is always non-negative. It's the square root
of the variance, which is always positive. So if you decided to short
the tangency portfolio and put it in T-Bills, well,
you'd be a knucklehead. But you would be on
this line right here. You would have higher and
higher risk, because you're taking a short
position on equities, and you'd have a
lower and lower return because you're shorting
the high yield asset and buying the low yield asset. Any other questions about
the geometry of this point. It's very important,
this is a major insight. Yeah? AUDIENCE: Earlier we
discussed about how we bring market [INAUDIBLE]
portfolios and expected return changes as a function of
n and standard deviation from certain changes [INAUDIBLE] ANDREW LO: Yes. AUDIENCE: Is that the reason why
the shift is more towards left? Because as we add more
and more portfolios, the n dominates the [INAUDIBLE]? ANDREW LO: That's
right, that's right. As we add more securities,
you get more and more impact of diversification. So that increases your expected
rate of return per unit risk because you can't make
somebody worse off by giving them choices. They can always put a
0 for the new stocks that you give them if
they don't like it. So the only thing you can do is
to make you better off, meaning the only thing you
can do is to give you a higher level of expected
return per unit risk, or a lower level of risk
per unit of expected return. So by adding more
securities, you're basically increasing
the slope of this line. So let's talk about
the slope of the line. The slope of that line is
equal to the expected return of that tangency portfolio,
minus the T-Bill rate, divided by the volatility of
that tangency portfolio. If you just calculate
rise over run, that's what you
get as the slope. There's a name for this. The name for this is
called the Sharpe ratio. You may have heard
of this, particularly those of you who have interest
in hedge fund investments. Hedge fund managers will often
quote their Sharpe ratio very proudly. The Sharpe ratio
is simply a measure of that risk-reward trade-off. The higher the Sharpe ratio,
the better you're doing. If you're a mean variance
optimizer, meaning you prefer more expected
return and less risk. So the idea behind
the tangency portfolio is that it is the one
that will give you the highest Sharpe ratio. Let's look at it again. If you pick a portfolio
like here, take a look. Look at the slope. The slope is going to be lower. Take a point over here,
in the inefficient branch of the bullet. Then the slope is
going to be even lower than the upper branch. The biggest slope occurs when
you invest between T-Bills and that tangency portfolio. That's what you're optimizing. Yeah? AUDIENCE: I have a
hard time understanding why the bullet would go left
when I add additional stocks. I understand it analytically,
standard deviation goes down. But then on the other
hand, wouldn't it happen that the
likelihood of correlation between these
additional stocks would decrease and therefore as we-- ANDREW LO: How would
the correlation increase by adding another stock? AUDIENCE: I mean the likelihood
that I have 20 stocks, the overall correlation is
higher rather than [INAUDIBLE] ANDREW LO: How could that be? You've got 20
stocks, and they've got a correlation
among those 20 stocks. Now, I want you to think
about adding a 21st stock. When you add that
21st stock, you don't affect the existing
correlations, right? I mean, it is whatever it is. Those are parameters. At least for now, we're going
to call them parameters. When I add my 21st stock,
I'm giving the investor an extra degree of freedom. Now instead of investing
among 20 securities, I'm going to let
you invest among 21. You don't have to
invest in the 21st. Or another way of
thinking about it is that when you
only had 20 stocks, you really had 21
portfolio weights. But the 21st weight,
I've arbitrarily constrained to be 0. Now, I'm going to
loosen the constraint and I'm going to
say, OK, now you can invest in the 21st stock. You won't affect the
existing correlations, but the new stock
that you add in can benefit in providing
additional diversification benefits. AUDIENCE: But as a
general rule, we've always been trying to have
negative correlation so that the bullet was left. ANDREW LO: Oh,
well actually, you don't need a
negative correlation to make this go to
the left, you just need to have
something less than 1. Remember? From the last lecture? This is a case where you
had perfect correlation. Anything less than
perfect correlation, brings you to the left. So as long as my
21st stock is not perfectly correlated
with the existing stocks in that
portfolio of 20, I'm going to move
things to the left. AUDIENCE: In general,
[INAUDIBLE] any stocks? ANDREW LO: In
general, that is true. AUDIENCE: I mean I would try to
have the negative correlation. ANDREW LO: You would,
but what this suggests is that negative correlation
is a very rare thing. AUDIENCE: It's difficult. ANDREW LO: It's very difficult,
it's extremely difficult. Now, from the
analytical perspective, we can conclude
it's very difficult. Let me ask you from an
economic perspective, why is it difficult to find
a stock that's negatively correlated with
all other stocks? Anybody give me a business
rationale for that? Yeah, Ingrid? AUDIENCE: There's
what you said before that when the economy goes
down, everything goes down and we get vice versa. Just that we mention
[INAUDIBLE] different countries, in different economic regions,
in different industry, and they should not be-- ANDREW LO: Let's actually
spend a little bit more time thinking about this. I want you guys to tell
me right now, give me a stock that you would put
your money in right now, today. S&P has gone down by 45% since
the high several months ago. The stock market's
doing terribly, and it doesn't look like
it's getting any better. So you tell me, what stock
would you put your money in right now, today? Yeah, Terry. AUDIENCE: Campbell's Soup. ANDREW LO: Campbell's Soup. Why is that? AUDIENCE: It's a food stock. It's a foodstuff people need,
will purchase, inexpensive, pretty much just for-- ANDREW LO: OK. But on the other hand, if
people are poorer all around, might not they start consuming
even less of canned soup and try to make their own soup
from little packages of ketchup and hot water? I saw that on an I Love
Lucy episode years ago. It's pretty cool. So are you sure? Are you sure that
Campbell's Soup is going to go up over
the next few months, in response to the
current crisis? AUDIENCE: It'll
stay pretty stable. ANDREW LO: It'll stay stable. Ah, but that's not
negative correlation. That's 0 correlation. I want something
that's going to go the opposite direction of
where the economy is heading. Tell me where that is? Yeah? AUDIENCE: [INAUDIBLE]
short financials. ANDREW LO: OK, fine. So you're going to
short the market. That's a cheap answer. Sorry, you don't get
any credit for that. I want to answer
the question that was raised by David
which is, show me a stock that can get me
even more to that left. I want a negatively
correlated stock. Yeah, [INAUDIBLE]. AUDIENCE: Wal-Mart. ANDREW LO: Wal-Mart? AUDIENCE: It's been going up. ANDREW LO: Well, that's
not the same thing as saying that it
is going to go up over the next several
months in response to this economic crisis. You don't think
that there's going to be a decline in
consumer spending that will affect retail as well? AUDIENCE: So far, everybody's
going to Wal-Mart. If they don't go to Wal-Mart,
where would they go? ANDREW LO: Well, that's
what I'm asking you. Where are you going to go? So you're telling me
now that you believe that Wal-Mart is the answer? You think it will be
negatively correlated? Historically, just
to let you know, retail has not been
negatively correlated with the business cycle. Yeah, Zeke? AUDIENCE: What
about Freddie Mac? ANDREW LO: Freddie Mac? AUDIENCE: Yeah. [LAUGHTER] ANDREW LO: If you
like that investment, I have something else for you-- [INTERPOSING VOICES] --afterwards. AUDIENCE: We could
do it this time. ANDREW LO: I don't
know if you want to argue that Freddie Mac
is negatively correlated with market downturns. I mean, the reason that Freddie
Mac got into the trouble that it did was because
of the economic downturn. All right, one more. [INAUDIBLE]? AUDIENCE: Philip Morris. ANDREW LO: Philip Morris. That's an interesting one. Obviously, people
are very nervous now. When you're nervous,
you're going to be smoking. On the other hand,
again, one could argue that it's not
negatively correlated. It might be either slightly
positively correlated, but even there,
people have argued that cigarettes are a
consumption good that can get hit with a downturn in markets. The bottom line is
that it's really hard to come up with negative
correlated stocks. Let me tell you, if you
found one that was negatively correlated, if you found
one that was really, really, negatively correlated,
what would all of you do? AUDIENCE: Buy it. ANDREW LO: Exactly,
you'd buy it. The effect of that would
be to increase the price and depress the expected return. Remember what the
expected return is. It's the expected future price,
divided by the current price. If now all of you go
out and buy Wal-Mart, or whatever stock you think
is negatively correlated, that would have the impact of
increasing the current price and therefore decreasing
the expected return. Now if you have a
stock that's got a negative covariance
and a negative return, that doesn't help. Because in fact,
that was a suggestion that was put forward here. Let's just take the
S&P and short it, and then you get a
negatively correlated stock. The problem is that it's also
got a negative expected return and then you're
not helping things. The key is to find
negative correlation with a positive return. If you can find
that, then you've really found
something worthwhile. But my guess is it won't
last, for exactly this reason. Other questions? OK, so now, let's go back
and ask the question, what does this mean if we
agree that all of us want to be on that
tangency portfolio. What does that tell us? Well, that allows us to
then make an argument that managers that are
trying to provide value added services for us, they
need to be doing something above and beyond what we
can do ourselves. Now, here's where Warren Buffett
meets modern finance theory, in a way. If I want to see
whether or not Warren Buffett or any other
managers are adding value, one simple criterion that
I can put forward is this. This Is what I can do on my own. I can get that line pretty much
by just using my basic finance skills that I've
learned here at MIT. If you're going to manage my
money and charge me 2 and 20, show me what you can do
above and beyond this. I want you to tell me where
you can get me on this graph. Can you get me up here? Can you get me over here? Can you get me anywhere
either to the left or above that curve? We can use that as a
measure of performance, and there's a name for that. It's called Alpha. Typically, when people
talk about Alpha, they're talking about deviations
from a line like this. We're going to get to
that more formally, you don't have to write it down
or make note of it just yet. It's on the next slide. But we're going to show you
how to measure that explicitly so now, not only is this a
good idea for you as a baseline to manage your own
portfolio, but you can then use it as a metric to gauge
whether other people are adding value to you. So Warren Buffett
would say, no problem. I think I've got Alpha. So I'm not going to
bother with this, I think I can get you up here. That is, if you want
to invest with me. And in fact, if you looked at
Warren Buffett's performance over the last 25 years that
he's been doing it, or 30 years, his Sharpe ratio is a lot better
than the tangency portfolios. So he actually has added value
if you use this as a criterion. But the problem is, you
have to identify the Warren Buffetts before they
become Warren Buffetts. Because after they
become Warren Buffetts, it's not clear
that they're adding the same amount of value. It's already, the
cat's out of the bag. Yeah, [INAUDIBLE]? AUDIENCE: Question
about the Sharpe ration. So is this a stag
ratio, or is it dynamic? Because in my mind, as you
gain more stock options, it's going to become sharper. ANDREW LO: Yes AUDIENCE: Opportunity cost
for switching to T-Bills is going to be
greater, so you're going to shift preferences
away from T-Bills. Then isn't that point going
to increase and flatten out? ANDREW LO: Yes, so the dynamics
of this are very complex. This is, right now,
a static theory. Static meaning today
versus next period. We're not looking at
the dynamics over time. In order to do that, there's
lots of different effects that are much, much
more complicated. For that, you've really
got to take 15 433 and even 433 won't cover
those kinds of questions in complete detail because they
rely on some very complex kinds of analysis. But I'm going to get
to that at the end. So if I don't, please
bring it up again. I want to make a
comment about that, and how you can take this
relatively simple static theory and make it dynamic
in an informal way even though the analytics
become very hard when you try to do it formally. So the key points of this
lecture are, oh, sorry, question? AUDIENCE: [INAUDIBLE]
every point on the line, it's indifferent? Or-- ANDREW LO: No, no no. Not at all, not at
all indifferent. Any point on this line is
a different risk-reward combination. So in other words, it
depends on your preferences. AUDIENCE: You take a
function for the industry? ANDREW LO: Yes, yes. Now, we haven't talked
about utility functions yet, but we're going to
in a little while. Let me preview that,
since you asked. You all remember what
indifference curves are? From basic economics? An indifference curve, when
I first came across that, I was rather offended
because I don't view myself as an indifferent individual. I have lots of passions. And so, why should we be
indifferent about two choices? In fact, that's
an economic term. It simply means that you are
just as well off between two combinations and
therefore, these two combinations you're indifferent
to, you're indifferent between. So if I had to ask you to draw
on this graph, an indifference curve of risk-reward
trade-offs for you, the typical individual,
what would it look like? Can anybody give me a sense
of what different kinds of risk-reward trade-offs you
would be indifferent among? And to make the question
a little bit simpler, let's start off with
a particular point. So let's suppose that
this point right here is the point that I want
you to draw the indifferent curve from. Which is a standard deviation
on a monthly basis of about 6%, and an expected return of
about, say 1.4% or something. So you've got a monthly return
of 1.4% and a risk of about 6%. Give me another point that you
would be indifferent between, versus that one? Anybody? Any volunteers? Yeah? AUDIENCE: There could be a
point above the tangent line, but to your right. Somewhere where you're pointing. ANDREW LO: OK, so do you have
a particular number in mind? In other words, let
me ask you this. If I cranked up your
volatility from 6% to 8%, how much extra
return would I have to give you in order for
you to be just as well off as you were at 6% and 1.4%? AUDIENCE: Something slightly
higher than the 1.8% [INAUDIBLE]. ANDREW LO: OK,
higher than 1.8% OK. AUDIENCE: Or what about
the corresponding value. 1.41%. 1.41% ANDREW LO: OK, this is 1.41%. And if I said, now
I want to be at 8%, how much risk do I
have to give you, how much expected return do I
have to give you to make you just as well off as this point? AUDIENCE: Higher
then 1.2%, right? Or 1.4%, higher than 1.4%. ANDREW LO: Right,
but how much higher? That's the question. It's a personal question. Rami? AUDIENCE: 1.3 times your
initial expected return. So 33% on top. ANDREW LO: You
would have to have an increase in 33% of
your expected return, even though I'm only giving
you a 25% increase in the risk. AUDIENCE: Well, no. You'd have to at least
do 25% of the-- sorry. Put 6%-8% is-- ANDREW LO: That's a
third, you're right. So I'm increasing
the risk by 1/3, you want me to increase
the expected return by 1/3. So your trade-off is
linear, is that right? You're looking at it linearly? Anybody else? You know, you may
want to translate this into annual numbers. Because I'm sensing that
you may not have a good feel for what your
own preferences are. And by the way,
this is a challenge. Not everybody understands what
their own personal preferences are for these numbers. This is not a natural
act of human nature, that we automatically have
preferences on these numbers. But the bottom line is that
if I make you take more risk, I'm going to have to
compensate you and give you more expected return. There's got to be a reason why
you want to take that risk. For some people, it's linear. For other people, it's
much more than linear. They don't want to
take any more risk. In fact, right
now most investors don't even want to
answer that question. Because they don't
want to take more risk. And you say, well
what if you did? Well I don't want to. Well, but just what if? I don't want to what if. I just don't want
to take the risk. So they can't even
answer that question. But if they could, my guess is
that it would be way up here. So you'd have to give them
a lot of expected return to make people take
more risk today. Alternatively, if you want
to give people less risk, my guess is that
you can actually subtract a lot of return
in order to take away a little bit of risk. How do I know that? Take a look at the yield on
the three month treasury. So an indifference curve. It's going to look like this. It's going to look like
it'll be increasing, but it'll actually
be bowed this way. And the theory behind why
it's going to be convex, holds water as opposed
to spills water, the reason it's
going to be convex is because there is a
decreasing, or diminishing, marginal utility between risk
and expected rate of return. Like anything else,
economists have this notion of diminishing marginal utility
between any two commodities. If you've got ice cream
sundaes and basketballs, there's only so many
basketballs that you can enjoy before the next
incremental basketball provides relatively little
pleasure for you. The same thing with
ice cream sundaes. You can only consume so
many ice cream sundaes before the next
incremental sundae provides somewhat less benefit to you. That kind of diminishing
marginal utility gives you this kind
of a bowed curve. So where you are on
this straight line depends upon how
bowed your curve is. Somebody that's really
risk-averse has a curve that looks-- let me draw this because
it's a little bit easier to see rather than trying
to follow my laser pointer. So here's the trade-off,
this is the line. Somebody that's
extremely risk-averse is going to have
curves like this. Those are indifference curves. And as you go to the Northwest,
you're happier and happier. So the optimal point is
where this indifference curve hits this particular line. On the other hand, if
you're very risk-seeking, if you don't need a
lot of compensation of expected return
per unit risk, then your difference curve's
not going to look like that. It's going to look like this. In which case, you're
tangency point will be farther to the Northeast. You'll be taking
more risk and getting more expected rates of return. But the bottom line for
this graph and this lecture is that everybody, no matter
what your risk preferences are, everybody's going to
want to be on that line, that tangency line. And it turns out
that that insight is going to translate into
a remarkable, remarkable conclusion about
risk-reward trade-offs. So the key points
for this lecture are diversification
reduces risk. In diversified
portfolios, covariances are the most important
characteristics of that portfolio. It's not the variances,
but the covariances. Investors should try
to hold portfolios on the efficient frontier,
that upper branch. And with the riskless
asset, everybody is going to want to be
on the tangency line. Those are the major
conclusions from this analysis. And you can work all of
this out analytically using the mathematics
of optimization theory, but in fact, all of this
can be done graphically as we have geometrically. Question, [INAUDIBLE]? AUDIENCE: Sorry, this is sort
of a simple-minded question, but I'm having trouble thinking
of the expected return. I know it's absolute, but
so much of the portfolio metrics, I guess, are relative
to the benchmark of the market rather than [INAUDIBLE]. So I don't know. ANDREW LO: OK, so we're
going to get to that. We're going to talk about
benchmarks because you're right, that most investments
today are all benchmarked against something, right? And you're probably
wondering how that got to be. That whole direction of analysis
and performance attribution, that came out of this. In other words,
it was because of this particular
academic framework that was developed by
Harry Markowitz, and Bill Sharpe, and others,
that indexation and benchmarking came to be. So I'm going to get to that. Let me put that off for
another lecture or so. After we derive the implications
of everybody wanting to hold the tangency
portfolio, it's going to turn out that that
tangency portfolio happens to be the benchmark. So we'll get to that. Yeah? AUDIENCE: Is there
any assumption behind all this
analysis that would say, if you have these
preferences, then you should choose this portfolio. If everyone did that, does
that change [INAUDIBLE] ANDREW LO: So you would think
that it would, but in fact, I'm going to show you that there
is exactly one case where it doesn't. And that's the case
of the equilibrium that I'm about describe. So let me turn to
that right now. There any other questions? AUDIENCE: We've been using
risk and standard deviation kind of interchangeably,
whereas I think of risk as the risk
of not making anything. Is there a way to
mathematically translate from a standard deviation
in your portfolio to the risk of not
making [INAUDIBLE]? ANDREW LO: Well, there is. Although, I would have
to say that if you have a preference
about the downside, so not making anything
as you point out. Then that changes this analysis. So this analysis really requires
that you use standard deviation as the sum total
of your perception of the risk of a portfolio. If you have other
kinds of sensitivities, then you need to bring
them into the analysis, and that will change
these outcomes. AUDIENCE: The way
we do this, if we measure the risk of a company,
if they're historically-- Nevermind. I guess if they
were historically, they varied higher, if that
wasn't strictly normal, and they end up being higher
in market [INAUDIBLE] lower, they would still have
a larger deviation so you're correlating that
with companies that are also [INAUDIBLE]. Does that make sens? ANDREW LO: That's
true, but again, you've made an assumption there
that I'm not making. Which is you're assuming
companies are outperforming or underperforming. I'm assuming that
the data are given, and I'm not making a bet on
whether any companies are likely to succeed or fail. I'm merely looking at companies
as investment opportunities that provide certain
expected returns, volatilities, and covariances. You want to go down the
path of Warren Buffett, and I'm resisting that
because I don't have the skills of a Warren Buffett. So I don't know
what's a good value and what's not a good value. And the case in point is a
discussion we just had today. You tell me what is
a good value today? Do you really believe that
Campbell's Soup or Wal-Mart should be the companies
you invest in today? I don't know. I mean, another argument
is entertainment. Why don't you invest
in movie theaters? Lots of people now are going
to see the James Bond movie, and they want to
escape from reality. Wouldn't that be
a growth industry, given market conditions? Well, that's true,
but how many people have $12 to spend on a movie? Plus, you've got to get the
popcorn and the bonbons, and all those. And by the time you're done,
it's like a $60 evening. I mean, I don't know. So the point is
that unless you are willing to make
predictions, this is the only alternative
that provides a disciplined approach to investing in
so-called good portfolios. So it's a different approach. So now, let me turn
to the next lectures. Lectures 15 through 17,
where we're now going to talk about equilibrium. We've already identified
that all of us in this room, assuming we have mean
variance preferences, that's an important
assumption, I grant you, but it's not an
unreasonable one. It's just, it is an
important assumption. We've all agreed
that we're going to take on portfolios that lie
on that line, and therefore, the portfolio that is
the tangency portfolio, I'm going to give
it a special name. I'm going to call it M,
portfolio M. What we now know is that, given a
choice between holding n securities and
T-Bills, versus holding T-Bills and a single
portfolio, all of you would be indifferent
between those two choices, if that single portfolio were
M. The tangency portfolio. Do we agree on that? So therefore, I
could, in principle, construct a mutual fund called
M. This mutual fund holds stocks in exact proportion
to the weights given by that tangency portfolio. In other words, it is
the tangency portfolio. So what that suggests is
that all of you in this room would be absolutely indifferent
between investing among the n stocks and T-Bills
on the one hand, versus investing in two
securities on the other. One security is T-Bills,
and the other security is shares of mutual fund
M. Do we agree on that? Any controversy there? I know I've made a number of
assumptions to get us here, but given mean variance
preferences, which is not an unreasonable
assumption, and given that we've assumed
these parameters are stable over time,
that's where we are. Rami? AUDIENCE: Somebody
might have said this, but you assume all
fees are trading fees? ANDREW LO: Forget about fees. There are fees no
matter what you do. So for now, I'm going
to forget about fees. I'll put fees back in
later, and if I do that, then it's going to look
even more compelling for you to want to invest in
mutual fund M, versus n stocks. I don't know how many of you
have traded individual stocks, but if you ever try to manage
a portfolio of 1,000 stocks, it's actually fairly
time consuming, right? And by the way, there are
more than 1,000 securities. I mean the S&P 500 you
can think of as being M, but that's an
approximation, right? There's probably 7,000 or 8,000
securities that trade today. Probably only 2,000 or
3,000 that you would really take seriously, and
probably only 1,500 that you really need from a
diversification perspective. 1,500 stocks. Would you want to trade
in that, or would you want to trade in
one mutual fund? Yeah? AUDIENCE: Can I ask, I mean
knowing that with M you're trying to get on that
tangent portfolio? And you said, for
example, Warren Buffet beats it the whole time. Why don't you just buy one
share of Berkshire Hathaway, and you'd have a
higher Sharpe ratio? ANDREW LO: Because Warren
Buffett beat it in the past, do you think he's going
to beat it in the future? AUDIENCE: I would
[INAUDIBLE] it. ANDREW LO: I don't know. That's right. Good question, good question. I mean, if you're thinking about
Warren Buffett as a 10 year investment, I think
I might short that. I mean, you know,
he seems healthy, but you know those Cherry
Cokes have to have an impact. I'm sorry. You know, you eat enough steaks
at that Omaha restaurant, I don't know what it is,
and those Cherry Cokes, I don't know. OK, so fine. Let's not do Warren Buffett,
let's do somebody else. Fine. You tell me who that is? Tell me who the next
Warren Buffett is? Can anybody tell me? I'll be happy to do that, I'll
be happy to invest in them. Who is it? AUDIENCE: Andrew Lo. ANDREW LO: Thank you, but
those who can't do teach, those who can't
teach, teach gym. And at least I don't teach gym. The point is that we don't
know who the next Warren Buffett is going to
be, and I don't want to have to figure that out. I mean, that's a
pretty tall order to tell an investor that
they've got to figure out who the next investment genius is. If they knew, they wouldn't
have to ask them to invest. They'd invest themselves, right? So what I'm showing you is
a simple way of investing that may not be as
good as Warren Buffett, but it's certainly
better than trying to pick the next
Warren Buffett if you don't know what you're doing. Jen? AUDIENCE: Is it easier
to kind of figure out the future covariances
of the different than it is to pick the
next Warren Buffet? ANDREW LO: Thank you, that's
another way of looking at it. If you ask the question,
is it easier to try-- is the historical
covariances and variances and expected returns more
predictive of the future than your ability to find
the next Warren Buffett, then yes, that's
another good argument. That in other words,
this framework relies on less
ability to forecast. It doesn't completely rule
it out because, as I said, these parameters,
they change over time. And you have to think
about that impact. So it's not totally
trivial, but from the theoretical
perspective, it seems like it's a very internally
consistent approach. Now, let me go on
for a little while longer because if it
were just this, then this would be an
interesting rule of thumb. But this is not a theory of
financial markets just yet. I haven't really done
anything truly astounding because you're still left
with the question of, what's the appropriate
risk-reward trade-off? What should I use
for my discount rate? A lot of financial
decision making is not just picking stocks
and making good investments. But it's whether or not should
I invest in nanotechnology as a corporate officer of
a particular tech company, or should I invest in
green technologies? What discount rate should I use? How should I engage in capital
budgeting or project financing? All of these questions
seem like they have nothing to do with investments. So I don't want to make this
course into an investments course. There's a lot about corporate
financial management that relies on being able
to understand these markets. So let me show you
where we go next, because we're very close
now to the big payoff. We've already identified
the tangency portfolio as being special. I'm going to call
that portfolio M, and I'm going to argue that
everybody in their right minds are going to be indifferent
between picking among these two investment opportunities,
T-Bills and M, versus the n plus 1 investment
opportunities of all stocks, plus T-Bills. It turns out that
portfolio M, therefore, has to be a very
specific portfolio. And it turns out
that that portfolio is the portfolio of all
assets in the entire economy, in proportion to their
market capitalizations. Now what I just said is
an incredibly deep result, so I don't expect
you to just get it. Let me say it again. First of all, I want
you to understand it, and then I'm going to try to
give you the intuition for it. If it's true that everybody,
not only in this room, but in the world, if
everybody in the world is indifferent between investing
in those n plus 1 securities, and in two, then we can argue
that those two securities play a very special role. In particular, think about what
that mutual fund M has to be. Everybody in the
world wants to hold M. So, let's make the leap of faith
that everybody does hold M. So in other words, now we're
in a world where everybody is already mean
variance optimizers, and they already hold two
assets in their portfolio. The treasury bill asset, and the
mutual fund M. So you hold M, you hold M, you hold M,
you hold M, you hold M, everybody holds M. We hold
different amounts of it, so as a hedge fund
manager, you're holding a large amount
of M. In fact, you're holding twice as much M
as your wealth allows, and you're borrowing
T-Bills to do so. Somebody who's very
conservative is holding a very tiny
little bit of M. Mostly, that person is
invested in T-Bills. But the point is that every
single person's portfolio you look at, when you
look at their portfolio, it's M. If that's true, if
what I just said is true, what portfolio
does M have to be? There's only one that
it can possibly be. And that is the
portfolio of all equities in the marketplace, held in
proportion to their market value. Do you see the beauty of that? Now, let me try to explain it. I hope you understand
it, let me explain it. Why does that have to be? This has to do with
supply equaling demand. Now, I'm going to make an
argument about equilibrium. I haven't done so up until now. Up until now, I
haven't said anything about supply equaling demand,
but I'm about to do so. If everybody is holding
this portfolio M, that's the demand side, right? Everybody is demanding
M. On the supply side, I'm assuming that all stocks
that are being supplied are held. If all stocks that are being
supplied are held by somebody, but if everybody in the world
is holding the same portfolio M, when you aggregate
all of the demands. So I'm going to add up your
demand, and your demand, and your demand, and you're. We're going to go
through the class, and go through the
world, we're going to add up everybody's demand. In every single case, your
weights are identical. You're holding the
same portfolio M. So when I aggregate
the entire world, and I get the portfolio M,
what does it have to equal? It can only equal the sum total
of all assets in the world, right? Supply equals demand. And therefore, when I aggregate
all of your holdings of M into one big fat
M, that big fat M can only be equal to
one thing, which is all the equities in the world. And the weightings are just
simply their market caps, right? There's only so much
of General Motors. Take the entire
sum total of that, that's the global investment
in General Motors. And then you do that
for every single stock, and you divide that by the total
market capital of all stocks, you get the market portfolio, M. So this shockingly, simple,
but extraordinarily powerful result is due to Bill Sharpe. Harry Markowitz came up
with portfolio optimization. He applied mean
variance analysis to portfolio optimization
and argued that everybody has to be on the line. Bill Sharpe looked at
this and said, aha. If everybody's on
that line, that means that everybody's
going to be either holding M or T-Bills, or
both, and therefore, the only thing that
M could possibly be is the market portfolio. And now we have a proxy for the
market portfolio, the Russell 2000. Or the S&P 500. Both of those are very well
diversified stock that have lot-- they don't have
everything in it-- but they have a lot of things in
it, that proxy for everything. The Russell 2000 has 2,000
stocks weighted by market cap. That's as close as you're going
to get to everything that you care about. So now, you'll see we're
benchmarking is coming from, but I'm going to get back
to that in more detail. So this equilibrium result
that says supply equals demand, identifies this portfolio
M. And what it says is that if everybody does this,
if everybody takes finance here and learns how
to do this, it's not going to kill the idea. It's going to lead to a
very well-defined portfolio M. Now, let me take
it one step farther, then I want to ask
you to ask questions. If I know what that
portfolio M is, then I've got an
equation for this line. I can write down a relationship
between the expected return and risk of a
portfolio on this line. And this is it. The expected rate of return
of an efficient portfolio, by efficient I mean a
portfolio that's on that line. Anything that's not on that
line, if it's below that line, it's inefficient, right? You're not getting as much
expected return per unit risk, and you're not reducing your
risk as much as you can, per unit of expected return. The expected return of
an efficient portfolio is equal to the risk-free
rate, plus the ratio of the standard deviation
of that portfolio, divided by the standard deviation
of the tangency portfolio, or the market, multiplied by
the excess return of the market portfolio. This result is a risk-reward
trade-off between risk and expected return. You see, what it says is really
something quite astounding. It's telling you that,
here's the risk-free rate. That's the base return
for your portfolio. And what this is
telling you is that what you should expect
for your portfolio is that base return,
plus something extra. And the extra is the
market's excess return, multiplied by a factor. And the factor is simply
how risky your portfolio is relative to the market. Let's do a simple example. Suppose that your portfolio
is the exact same risk as the market. Well, if that's
the case, then what is your expected rate of return? AUDIENCE: The market. ANDREW LO: It's the market. So it's the risk-free rate,
plus the market excess return, which, when you add it
together, is just the market. Suppose you're holding
a portfolio that's more risky than the market. Is your rate of
return greater or less than the rate of
return of the market? AUDIENCE: Greater. ANDREW LO: Greater. Suppose that your
portfolio has no risk. Suppose that sigma p is 0, then
what's your rate of return? AUDIENCE: [INAUDIBLE] ANDREW LO: Exactly. Makes sense, right? This is very intuitive. What this tells us, now,
is that we can figure out what the fair rate of return
is for an efficient portfolio. For any portfolio
on this line, I can tell you what my
fair rate of return is, and it's an objective measure. It's not just theory now. Now, I can go into
the marketplace, I can measure the expected
return of the market. You know what that
is, historically? Not including the
last few months. AUDIENCE: 7%. ANDREW LO: It's about
7%, historically. Over the last 100
years, 7%, the expected rate of return of the market. Sorry, the expected
risk premium, the excess rate of return. About 7%. What about the
volatility of the market? It's been about
15% historically. So according to
this relationship, I've already figured
out what this number is. It's like 7%. I've already figured
out what this number is. It's like 15%. So now, you should be able
to get a benchmark for what to expect when you've got
a particular level of risk in an efficient portfolio. You've got all the ingredients. What about risk-free rate? Well, it depends on
what risk-free rate, but let's talk about
over a one year period. Right now we're
looking at somewhere between, I don't
know, 1%, 2%, 3%, depending on what day of
the week you're looking at. So one year T-Bill rate
is about 1% or so, yeah? AUDIENCE: I think it
was the last class we talked about unsystematic risk. ANDREW LO: Yes. AUDIENCE: Is that defined
by [INAUDIBLE] in this case? ANDREW LO: No, the
unsystematic risk is risk that is not
measured by sigma p, so we're going to
come back to that. Let me hold off on
that for now, because I want to come back to it after
I finish developing this. There's going to be
a connection between systematic and
unsystematic risk that's going to come right out
of this relationship. Yeah, Brian? AUDIENCE: So if you take
the S&P 500 as M here, the market portfolio, and the
capitalization is the weight, so you've got non-zero weights
for all the different stocks there. Does that imply that there's
no stocks in the S&P 500 that are Southeast
of any others? ANDREW LO: No, no,
there could be. AUDIENCE: Why would
you have them, because we said those are
strictly non-preferred? ANDREW LO: Well, that's
if you're looking at a pairwise comparison. If, now, you're trying to
create an entire collection of these portfolios
of securities, that's a different story. That's why I answered in
response to Justin's question. Justin said, why not
just trade off those two? Why not? It's because you can do far
better by using all of them in this way. You see, by looking at pairwise,
you can no doubt do better. But if I use all of them,
I get this entire line. And you can't get
that entire line just from looking at
two of these stocks, you need all of them. AUDIENCE: So in this
portfolio of three, if you kick GM over to
the right a little bit, and made it strictly
non-preferred to IBM, then you still might
have a positive portfolio weight on GM? ANDREW LO: You might,
but more likely, it'll be a negative
portfolio weight. It'll be negative, and
you'll be shorting it somewhere along the line here. However, the tangency
portfolio, by assumption, if it's the market portfolio,
cannot have negative weights. And so there, what will happen,
is that all of the stocks will change in
their relationship based upon various different
kinds of equilibrium, so that you won't get into
a lot of those situations where you're going to be
shorting these negative stocks. Yeah? AUDIENCE: So basically,
according to the Sharpe theory, every stock that the
market, the capital is not 0 is worth holding
in some portfolio? ANDREW LO: That's right. AUDIENCE: Diversifying
your portfolio. ANDREW LO: That's right. Every stock has some
benefit in adding to this particular
risk-reward trade-off, and the sum total benefit
is summarized by this line. That's the ultimate objective. AUDIENCE: If I don't hold a
specific stock in the market and I gain a
diversification [INAUDIBLE]? ANDREW LO: Sorry? If you hold a specific stock? AUDIENCE: If I don't
hold it, because it has market [INAUDIBLE]. ANDREW LO: Oh, if
you put 0 weight. Yes. What Sharpe would argue,
based upon this theory, is that you want to
hold as many stocks as you can to get the
most diversification. Now, that's the theory. In practice, it may well
be that the benefits do not outweigh the costs, because
when you hold multiple stocks, you have to manage them,
and so it may cost more. So a mutual fund
that has 3,000 stocks may have a higher expense ratio
than a mutual fund with 500. It may not. Nowadays, actually, the
technology is so good that probably it doesn't. But 15 years ago,
that was not true. But apart from the
transactions cost, the theory suggests
more is better. Because it will always give
you more opportunities, and it can never hurt you because you
could always put a 0 weight on them if you don't like them. Now, it turns out that this is
a trade-off between the expected return of an
efficient portfolio, and the risk of that portfolio. In other words,
this applies only to portfolios on
that tangency line. What if you want to know what
the expected rate of return is for Wal-Mart? We just said that
no individual stock is going to be likely to be
on that efficient frontier. And therefore, no
individual stock is likely to be on this line. So this is great if what
you're talking about is investing in
efficient portfolios, but how does that help the
corporate financial officer that's trying to
figure out how to do capital budgeting for a
particular pharmaceutical project? It turns out, it doesn't. It doesn't help. This doesn't answer
that question. It turns out, you need to have
an additional piece of theory that allows you to derive
the same results, not just for the efficient
portfolios here, but for any portfolio. And this is another
innovation of Bill Sharpe. This is actually why Bill
Sharpe won the Nobel Prize. It was not for this
little picture here, but it was for this
equation right here. What Bill Sharpe discovered is
after computing the equilibrium relationships among various
different securities, he's demonstrated that there
has to be a linear relationship between any stock's expected
return and the market risk premium. Just like here, where you've
got the risk-free rate, plus some extra premium. So this is the premium,
the second term. But what Bill Sharpe showed was
that if this portfolio is not an efficient portfolio,
if it's not on that line, the linear relationship
still holds. But it turns out that this
particular multiplier is no longer the right one to use. It turns out that the right
parameter to plug in there, is something called beta. Now, you've heard all
about beta, I'm sure. But now, I'm telling you
exactly what beta is. Beta is the multiplier
that is defined by the covariance between
the return on the market and the return on the
individual asset, divided by the variance of
that market return. If the portfolio happens to
be on that efficient frontier, then this beta reduces
to this previous measure. So this is a special case of
the more general relationship where beta is used
as the multiplier. So let me repeat what beta is. Beta is the ratio of the
covariance between the return on the particular asset or
portfolio, that may or may not be efficient, it's any
asset, with the return on the market portfolio. So this numerator is a
measure of the covariability between the particular
asset that you're trying to measure the
expected return of, and that tangency
portfolio, divided by the variance of that
tangency portfolio. Beta, it turns out, is
the right measure of risk, in the sense that it is the
beta that determines what the multiplier is going to be on
the market risk premium, which is to be added to your asset's
expected rate of return, above and beyond
the risk-free rate. That's how the cost of capital
is determined for your asset. So I think you all saw
how I derived this, but I didn't derive this. I'm just telling
you this is really where Sharpe's ideas became
extraordinarily compelling. And in order to
understand how to derive that, I'm going to refer
you to 433, because in that investment's
course, we really delve into the underpinnings
of that kind of calculation. It's a little bit
more involved, it involves some matrix algebra. But it's not terribly
difficult or challenging, and certainly be happy to give
you references if any of you are interested. I believe it's in
Brealey, Myers, and Allen. But the bottom line
is that this gives you an extraordinarily
important conclusion now to the several weeks that we've
been working towards this goal. Which is now, finally,
after eight or nine weeks, I can tell you how to come up
with the appropriate discount rate for various
NPV calculations. The answer is the
expected rate of return, the appropriate
fair rate of return, or the market equilibrium
rate of return, is simply given by the beta
of that security, multiplied by the expected excess return
on the market portfolio. So now, this has a lot
of assumptions, granted. We're going to talk
about those assumptions over the next
couple of lectures. But what we've done today
is move the theory forward by quite a bit, because we've
identified a particular method for coming up with the
appropriate cost of capital as a function of the risk. Where the risk is measured,
not by volatility anymore, but by the covariance between an
asset and the market portfolio. And next time, I'm
going to try to give you some intuition for
why this should be, why this makes sense, and why,
in a mean variance efficient set of portfolios, why it
reduces to something that we know and love. Any questions? OK. I'll stop here, and I'll
see you on Wednesday.
