[MUSIC]

So let's now look at how we go to the implementation and the estimation issue.

So as I said before, the strategic asset allocation undertaken by most

practitioners use a constrained mean-variance optimization technique,

and for that purpose, it will diversify in the simple case that we saw before,

between three asset classes, cash, stocks and bonds, fixed income.

But in a larger institution like, think about UBS, Credit Suisse,

JP Morgan, Goldman Sachs, we can have up to 20 asset classes.

20 asset classes means we have to estimate about 20 mean returns for these classes,

and we have to estimate 490 parameters for the variance-covariance matrix,

so you see quite a large number of parameters to estimate.

I'll come to that point in a minute.

The portfolio weights are generally constrained to be equal to one,

no leverage, and to be positive, but that's not necessary,

some hedge funds may actually allow for short selling.

The mean variance and covariance estimates are to be for your horizon.

For instance, if I'm looking at my horizon up to my retirement,

this is going to be a relatively not very short horizon but

moderate horizon that is of about 10, 15 years.

If you're a young investor, you may have a 40-year ahead of you over

which you will do these optimizations, so the horizon is agent-specific.

Now, the question is, typically, remember we are still looking at

horizons of three to ten years, and over these horizons,

how do you estimate means, how do you estimate variances and covariances?

Now, if you look at variances and

covariances, in fact, they're much more easy to estimate.

If you have long sample data, a long history of data, and

you sample very frequently, let's say weekly or even daily,

you can get quite good estimates of the variance-covariance matrix for

these 20 asset classes, for instance.

Where the problems comes is when you have to estimate the mean, so

the mean will not be more precise.

So for instance, suppose I gave you 240 monthly observation

over a 20-year period, well, the mean that you would estimate for

each asset class, the mean return, would not be much more precise than the one that

you would get by simply taking the mean over the 20-year holding period.

And this has been acknowledged by many, many studies, in particular,

one of Goyal and Welch, who said it is so difficult to estimate the equity premium.

So the equity premium would be the expected return minus the risk-free rate,

and most models that are used would be unstable and spurious.

And how do banks, financial institutions here can bend this problem?

Well, they would do a mixture between using historical data and

estimates from the chief investment office practitioners, or somebody would sat

estimate domain return on the US stock market to be 10% next year.

Another guy would say 8%, some would say 7, and then you take an average between

all these forecasters to estimate your mean return on the SMP and

from the risk premium.

But, but, but this is not an easy task, and for many people,

the mean-variance optimization is actually called an error maximizer.

And let me explain why this is the case.

There was an interesting article in the journal of portfolio management, and

I'll just show you one graph where you have,

on the horizontal axis, the size of the error that you make.

It could be 0.05, 0.10, up to 0.20, and

the loss in term of the cash equivalent is on the y-axis above.

If you estimate means and covariances, you'll see that this loss is hardly

ever reaching 0.5%, never reaching 1%.

You'll see that as the error increases for

the mean, the cash equivalent loss can reach ten times more than

the one that you had when you were estimating the means and the covariances.

So in other words, it is much harder to estimate means and

risk premium, and it's much more costly to do a mistake at this level.

So let me try to give you some final words of caution.