[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. The result of mean-variance estimation is an efficient frontier. You will know that. You would position yourself on this efficient frontier by looking at your utility or preference function, and the point that you would choose on the efficient frontier corresponds to your risk appetite and also gives you the weights that you would allocate into the different asset classes. Well, while the theory is appealing, we have seen that implementing the SAA is not easy, and primarily, it's not easy because of estimating problems, and these estimation problems are even more acute with the mean returns than with estimating the parameters of the variance-covariance matrix. So then, you may ask yourself, well, if we're so much subject to estimation risk, how much should we care about SAA? So typically, in a very well-known article in the Financial Analysts Journal, a big debate over big confusion was settled. And the confusion started with an article published in 1986 by Brinson, where this author was perceived incorrectly by practitioners as saying that 90% of the performance of a given strategy originates from the strategic asset allocation component. In fact, he never said that. He said 90% of the variations in the time series of returns is due to the SAA. And then in 2010, Ibbotson and his colleagues, in a Financial Analysts Journal, settled the issue. And they showed something which I think we should all know and be aware of, which is that the time series of returns variation, if you look at the two bars on the right, are mostly due to market movements, which are uncontrollable. In fact, 75% of the time series return variation is not manageable, neither by the SAA nor by active management. And the residual 25% of the time variation can be explained in half, so 12.5% by the SAA and 12.5% by Active Portfolio Management, so be aware of this word of caution. And to summarize, SAA plays a key role for passive investors, but on top of that, many active investors will implement a so-called tactical asset allocation, and that's something that you will see in the next lessons. Thank you very much. [SOUND]