0:41

It's a heavily described in that book that we've been using, Grinold and Khan.

You can also find other papers about it online of course.

Anyways, the reason that this fundamental law is important is it provides a nice

framework for looking at helping us make quantitative decisions like,

should we expend effort to make our forecast better?

Or should we expend effort to enable us to make more trading opportunities?

Both of those are important to overall performance, and

this framework proposed by Grinold helps address that.

1:37

And all that means is we're just gonna take a look at something a little

bit different than investing and look at how the numbers come out,

then consider how that relates to investing.

Now a quote that I think is really relevant

here is this one by Warren Buffet.

2:02

Wide diversification is only required when investors do not

understand what they are doing.

Now, of course, Buffet understands deeply each one of his investments,

and so, indeed, it turns out that when we work out this math,

we'll see that in some sense he's right.

But it's still a valid approach to approach investing where you make

lots of little bets, where for each bet you don't have deep information.

On the other hand, Buffett makes a few large bets

where he has deep information about each bet.

So this framework is gonna allow us to consider both of those approaches

together.

2:49

Okay, bunch of things to cover in this module, the first three information ratio,

information coefficient, and breadth are components of the fundamental law.

And we'll touch upon all those.

Then of course, we'll present the fundamental law.

And a side effect of this module is to understand why

lots of little bets are better than a few large bets.

And then we'll apply the fundamental law to a couple examples.

Okay, getting back to our thought experiment, okay,

we're gonna think, I'm gonna suggest, that we think about

3:33

betting on the outcome of a coin flip, as being like making a trade.

And when I say trade, I mean we enter a position and

then exit it a few days later.

That's a bet.

Let's face it, the kind of approach

to the market that this course is about is about making bets.

And in order to make those bets effectively and scientifically,

we need to think about, what's my expected outcome statistically?

And how many of those bets can I make?

Okay, so we're gonna assume that the information we have about the probability

of the outcome of a trade is encapsulated in a coin flip,

and it's expressed as a biased coin.

We somehow know that the coin is gonna come up heads, or got a little bit of

information that tells us that it's more likely to come out heads than tails, okay.

So, our uncertainty about how it's gonna come up,

that's a lot like what we don't know about the market as a whole.

So the uncertainty component of which way the coin is gonna go is like beta.

The bias in the coin, that small sliver of probability that it's gonna come up heads,

that beyond just randomness, that's like alpha.

So we have essentially a 2% advantage in each coin flip because the coin is biased.

Okay, so when we make a bet,

we put a chip on the table, a token.

It might be a $1000 chip.

It might be a $1 chip.

6:15

An analogy is you may have information about 1,000 stocks and

when you go in in the morning you place 1,000 small bets instead of 1 large bet.

So we're gonna consider the case, we're gonna compare two cases.

One case where we make just one bet, a single $1,000 bet,

and another case where we make simultaneously 1,000 $1 bets.

We flip 1,000 coins at once.

6:48

And we're going to compare the outcome and

the risk reward ratio of those two approaches.

Okay, so the approach in this class is that we

view things from a reward risk point of view.

That's essentially the Sharpe ratio.

7:11

We get a certain return, but we want to divide that by the risk so

we know what this ratio is.

In other words, if we get a rate of return but

it's extremely risky, that might not be such a good strategy.

Whereas a smaller reward with very, very low risk could be much better.

Okay, what is our expected return from these two cases?

We bet $1000 at once versus $1 1,000 times.

In the single bet case where we make just one bet,

there's a 51% chance it's gonna come up heads, and we make $1,000.

49% chance it'll come up tails, and we'll lose $1,000.

The expected outcome then is 0.51 *

1000 + 0.49 * -1000.

That's our expected, if we did this many times, that's our expected result, $20.

Okay, let's compare that to the multi bet case,

where we're making 1,000 simultaneous bets.

Each individual bet is subject to the same mathematics,

and it turns out that each individual bet, our expected return is $0.02.

There's a 51% chance we'll make $1, 49% chance we'll lose $1.

Work that out, our expected return is two pennies.

Well, we're doing that 1,000 times in parallel, so

9:23

Now let's look at the multi-bet case, where we're making, again, 1,000 $1 bets.

In order to lose all that $1,000 in that one parallel bet,

we'd have to lose the first one.

We've got a 49% chance of that, and the second one, and the third one, and

the, and the.

We'd have to lose all 1,000.

The probability of that occurring is the probability of each one occurring

all multplied together.

So it's 0.49 multiplied by 0.49 1,000 times, or

0.49 raised to the 1000th power, and that's a very, very, very small number.

In fact, I tried to calculate it once on my calculator, and

it gave me an error because the number was so small.

You can, however, calculate it.

Okay, another measure of risk and the one that we use over and over and

over in this class is standard deviation of returns.

Okay, in the multi bet case where we're making 1,000 bets simultaneously,

we can do it once and see what heads tails, plus $1 minus $1,

plus $1, minus $1, plus, plus, minus, minus, and so on.

We can take the standard deviation of all those 1s and -1s.

And if we did that a few times, on average we would discover

that the standard deviation of that set of bets is $1.

The single bet case is a little bit tougher to measure because

standard deviation is not really defined for a single event.

So in order to compare apples to apples,

what I've suggested we do is in the single bet case,

let's treat that as if we made one bet for

$1,000 and then 999 bets of $0.

So there's 999 bets that we risked no money, so

the outcome in all those cases is 0.

11:33

That first bet it'll either be negative 1,000, or positive 1,000.

You take the standard deviation.

Compute the reward risk ratio for the single bet case, it's 0.63.

For the multi bet case, it's 20.

So our reward risk ratio in the multi bet case is more than 20 times better.

12:00

And this measure of course is very, very similar to the Sharpe ratio.

Okay, now I'm not gonna go through the math to derive this, but you could do it.

And if you look at the result if you analyze this a little more deeply,

you'll see that the Sharpe Ratio or the reward risk ratio for

the multi bet case is equal to the reward risk ratio of the single bet case.

Times the square root of 1,000,

or times the square root of how many bets you're able to make.

I worked the numbers out there, and you can see it works out to be 20.

What this is telling us, and we can refine it a little bit more,

let's think about the reward risk ratio of the single bet case,

as being some coefficient C times alpha.

And that alpha in this case, is what we know about the biased coin,

that 2% knowledge that we have, that the coin is likely to come up heads.

13:46

Okay, so in both cases we get the same return,

same expected return, but the multi bet case is better in a lot of different ways.

Lower risk that we're going to lose everything, much lower standard deviation,

and much higher reward risk ratio.

14:08

Take home lessons.

You can make this ratio better by improving your alpha,

make that coin a little bit more biased.

Or in the case of investing, what that boils down to is refine your

methods of making forecasts about individual stocks.

You can improve your Sharpe Ratio by making more bets.

14:32

In other words, if you can find a strategies that scales

where you're able to find many more investment opportunities,

you can improve your reward risk ratio substantially.

The issue is that sometimes to find more of those

opportunities you have to use a strategy that has lower alpha.

So this relationship shows us though that we can overcome

lower alpha if we can have more bets.

And finally that ratio grows as the square root of the breadth.

In other words in order to double

15:39

That means, he can afford to make fewer bets.

However, if your alpha is lower, but

you can make more bets, you have to make a lot more bets.

But if you can make more bets, you can perform as well as Warren Buffett.

And that's the take home message.

I'll see you in the next video.

Thanks a lot.

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