>> And what we realize is that it takes a little bit of work.

So let me run you through a couple of equations.

So first of all, suppose there is a portfolio manager who observes a signal

about the accessory terms in the market next period.

And this is important.

So, essentially you have some ability to forecast the market next period,

call it MT or REMT, that is excess return on the market in time t.

But you observe the signal at time t minus 1.

That is the period before.

Except you don't observe the signal with zero error.

In other words, there is some uncertainty still

even after you observe the signal as to whether your forecast is correct or not.

And that uncertainty is represented by the delta t minus one.

That's okay.

Once you have a precise enough signal, what you can do as a market timer

is of course, you will adjust the beta of your portfolio in response to the signal.

So if your signal tells you that the market is going to outperform the risk

free rate next period, obviously you want to be in a high beta position.

If your signal tells you that the market is going to underperform the risk

free rate next period, you would want to be in a low beta position.

In other words, you want your beta to be a linear function of your signal.

And that's what I've shown you here in the slide.

And once you plug that into your standard market model, what you observe

is that there is now an excess market return square term on the right hand side.

In other words, not only do you have a beta term on the right hand side,

which is your traditional data of your fund to the market.

You also have a beta p 2 here, which is going to represent timing ability.

So if you observe in this regression a high beta 2,

let's say positive, then you know that you have a positive timing ability.

In other words, in response to an anticipated performance of the market over

and above the risk-free rate, you're able to adjust your beta in a positive fashion.

So that's one way to test for timing ability.

The other way to test for timing ability is a closed variant where you take not

the square of the excess return in the market on the right-hand side.

But you take this market return minus the risk-free rate times a dummy variable,

which represents whether the market outperformed the rf or not.

So the easiest way to read this equation is consider what happened in months when

Rm was greater than Rf,

that is the stock market return amount which is greater than the risk free rate.

Then the D will take a value of one,

then the quotient on the right hand side can be clubbed into beta 1 plus beta 2.

In months when the market underperformed the risk free rate,

what you would find is the D would turn off, that is go off to a value of 0,

leaving you with a quotient of beta 1 on the right-hand side.

In other words, the beta p 2 can be interpreted as a quotient

of how much you increase your beta in

those months where the market outperformed the risk free rate.

f that beta p2 shows up to be a positive number, then there is some evidence

that this particular manager has managed to somehow time the market.

That is to say,

adjust the beta of his portfolio in response to anticipated market movements.

Of course, having run these tests across, the evidence is almost near unanimous

that most fund managers out there fail to have any sort of timing ability.