Active Portfolio Management

[ckyeh]

Dear David:

On your webnair 「2010-8-a-Investment」, page 10:
ψAlpha forecast = Std{r}*Corr{r,g}*(g-E{g})/Std{g}.
I couldn’t get it!
I tried to simplified it ,since Corr{r,g}=(COV(r,g))/(Std{r}*Std{g}):
ψAlpha forecast = Std{r}*(COV(r,g))/(Std{r}*Std{g})*(g-E{g})/Std{g}
=(COV(r,g))/(Std{g}^2)* (g-E{g})=β*(g-E{g})
How come?
Does g mean market return? WhyψAlpha forecast becomesβ*(g-E{g})?
Where could I fine the related description on Grinold’s book Active Portfolio Management?

And could you give more description about IR = IC*√BR ?
An example would be even better.

Finally, about residual return and active return on「2010-8-a-Investment」, page 11,
I read Grinold’s book Active Portfolio Management page 100 and conclude as following:
Returns could be split into three parts:
1.fB (Intrinsic)
2.βPA*ΔfB (Timing)
3.αp(residual)

Risks could be also split into three parts:
1.σB^2
2.βPA^2*σB^2
3.Wp^2

So we can get:
θp (residual return) = rp -βp*rB=αp(residual)
rpa (active return) = rp –rB = θp +βp*rB = αp(residual) +βPA*ΔfB (Timing)
Is it correct? An example for calculation would be deeply appreciated!

Have a nice day!

 

[David Harper CFA FRM]

Hi ckyeh,

In regard to: ψAlpha forecast = Std{r}*Corr{r,g}*(g-E{g})/Std{g}.
This is from Grinold Chapter 10 and the technical derivation is in the appendix to Chapter 10.
(While Chapter 10 is not assigned, I included it because I felt it might be helpful prelude to the assigned chapters…)

FWIW, Table 10.1 is illustrated on final sheet of:
http://www.bionicturtle.com/how-to/spreadsheet/8.a.1_grinold_setup/

In regard to: IR = IC*√BR
Grinold establishes this in Chapter 6, alas.

Grinold: "A simple and surprisingly general formula called the fundamental law of active management gives an approximation to the information ratio. We derive the result in the technical appendix. The law is based on two attributes of a strategy, breadth and skill. The breadth of a strategy is the number of independent investment decisions that are made each year, and the skill, represented by the information coefficient, measures the quality of those investment decisions. The formal definitions are as follows:

BR is the strategy's breadth. Breadth is defined as the number of independent forecasts of
exceptional return we make per year.

IC is the manager's information coefficient. This measure of skill is the correlation of each forecast with the actual outcomes. We have assumed for convenience that IC is the same for all forecasts.

The law connects breadth and skill to the information ratio through the (approximately true)
formula:

IR = IC* SQRT(BR)" -- Grinold page 148

David here. Grinold goes on to evaluate three strategies, each were an IR of 0.5 is desired. The stock selector who follows 100 companies requires an IC (skill) of 0.25 in order to achieve IR = .025*SQRT(100) = 0.25; e.g., to double IR requires double skill or 4x the breadth.

In regard to the your final re page 100, that conclusions agrees with mine, so that looks okay.
Apologies I don't have time to illustrate with calculations. Rather I focused on the assignment in Chapter 17 with: http://www.bionicturtle.com/how-to/spreadsheet/8.a.3_grinold_perf_ch_17/

Thanks, David

 

[ckyeh]

Dear David:

So on your webnair 「2010-8-a-Investment」, page 10:
Z score is *√BR, right?

BR is the strategy’s breadth. Breadth is defined as the number of independent forecasts of
exceptional return we make per year.

Then why √BR could be negative ( -0.25, -0.48)?
And could you explain the implication of Z score is *√BR,?

Thanks

 

[David Harper CFA FRM]

Hi ckyeh,

Re: Z score is *√BR: did I say that somewhere, I did not mean to?

Grinold's phi is defined as:
E(r|g) - E(r); i.e., "we will define the refined forecast as the change in expected return due to observing g"
Where (r) is excess return and (g) is forecast
… such that he defines "skill" as correlation(r,g); e.g., perfect skill = 1.0 as in, our forecast tracks perfectly with return.

"z score" refers to [g - E(g)] /StdDev(g). This is just a transform of (g) into a standard normal variable: Z = (g - mean)/StdDev(g)
… by definition, it has mean of 0 and StdDev = 1, such that negatives are likely.

David