Let's talk about ratios again. Here I have beautiful visualization of performance of Japanese funds. It's called upside downside capture ratio. At the center, we have the benchmark which is topics in our case. Points on the plane are the draft of Japanese funds, then on the vertical axis we have upside captures. Basically this is just part Beta or the fund when markets moves upward. The downside capture is beer Beta, or average capture of downside movements of the market. Funds which are below dashed line are performing worse than the market. They have more than one unit of downside for the unit of upside taken. Funds which are to the left of dashed line higher than that are having more upside than downside capture. The bigger is the distance to the dashed line, for funds which are to the left part of it, then the bigger is the difference between upside and downside capture. If fund is managed passively, it is very close to the unit, to the center of the plane. Still we have several outliers here and all funds are interesting; J. P. Morgan Japan technology fund, Diam Domestic, Bailie Gifford Japan Trust, and Legg Mason. But probably these funds are managed using different strategies. Some of them use more active strategies, some of them use less active strategies. Maybe some of these funds are just tracking index. Can we use the same ratio to judge all of them? If we would use Sharpe ratio, we will have the problem of high volatility of the benchmark. Here we have a plot for Sharpe ratio, funds that are higher than third dashed line have Sharpe ratio bigger than one, then nobody is having Sharpe ratio bigger than two or three. You can see that the horizontal line is for annualized risk, vertical is for annualized return. Can we actually propose better measure which would allow us to compare both active and passive funds? The answer is yes. We have just slightly change previous visualization. If we would plot excess risk on horizontal line and excess return on the vertical line, then we will have it. Here we have Diam Domestic, which have this ratio over two, we have JPM Japan Technology fund, Japan, which have that ratio almost equal to one. Then all others have Sharpe ratio worse than that. Now what is information ratio again? This is excess return or return of the fund minus return of benchmark divided by tracking error. What is the essence of that ratio? The main idea is that now we can compare funds which is using different strategies. It doesn't depend whether this fund is using active or passive strategy or semi-active strategy, we will discuss this approach in details later. We can judge all that funds using this same measure. Here we have Legg Mason Japanese Equity Fund, this is absolute killer. They have 20 percent excess return, according to Bloomberg. Then we have another fund, which is call it Bailie Gifford Japan Trust. This fund have a little bit low excess return but their information ratio is better than in the case of Legg Mason. Finally, we have another fund which is called Diam Domestic. This fund have even more lower mean excess return but their information ratio is best for the whole crowd of funds managing equities in Japan. Let's look at the plot. You see that for every downside movement they have less than the movement of the market. For every upside they capture more than market moves upside. They probably producing something like enhanced indexing strategy. Legg Mason have slightly different approach. They have bigger volatility, they performed very good in 2016, but probably due to that volatility and higher tracking error, their skill of management is still lower. We have another important thing connected with information ratio, it's called Fundamental Law of Investments. It could be proved actually that information ratio is proportional to information coefficient times square root of breadth. Information coefficient is the correlation between independent predictions, and realizations of predicted variables for these independent predictions, breadth is the number of predictions. Basically, manager have two strategies to improve his information ratio. He can either make more precise predictions or he can leave precision of predictions intact so he could leave information coefficient intact. Instead, he can increase the number of independent bits. This is very important implication, for example, we will speak about core satellite approach. It is based on fundamental law, we'll speak about enhancing and mixing, the idea of that is also based on fundamental laws. Basically, fundamental law is at the core of modern portfolio management. As a home exercise, you can think based on fundamental law, what is better? To time the market or to select stocks? That's it. Let's think about that.