One very interesting fact that our metrics help reveal, is many different rates of return that a money manager might claim are actually equivalent to each other in terms of that manager's skill. And the only difference between them may be whether they use leverage or investing in part with borrowed money to achieve their results. Let me explain what I mean. In the following discussion, I'll assume that a money management firm can borrow at 1% rate of return. >> Assume a manager has used his skill to find the stock that has an annual return of 9% with an expected volatility of returns of 15%. So in this standard diagram we have volatility on the x-axis and portfolio returns on the y-axis. So we're looking at a point at X equals 15, Y equals 9 which would be just about right about here. So we would say volatility of returns is 15% and expected return is 9%. So, let's assume that a manager is investing $20 million. And they've generated this return by investing $20 million and getting back 21.8 million. So they have 20 million (1.09) = 21.8 million, and our discrete return is calculated as (21.8 / 20)- 1, 9%, okay. Okay. Now let's suppose that this same manager borrows money at 1% interest. Let's say that they borrow $10 million. Now they have 20 + 10, or 30, to invest at the beginning of the year. And at the same return, they would have, 32.7 million, okay. But before we can calculate the return, they would need to give back the $10 million dollars, excuse me, that they had borrowed and pay back the interest of 1%. So the actual amount that they have would be $2.6 million,right, that's (32.7- 10.1) / 20, minus one, which would be equal to 13%. So, simply by borrowing $10 million, they were able to boost returns on the same stock to 13%. However, you don’t get something for nothing in investing. The cost, or the price, that you would pay here is that you would also increase your volatility by 50% so you would have volatility of 22.5%. You have a point here that is at 22.5% and 13%. Okay. Now I'm running out of room, but we could try one more. We could say, well what if I borrowed, my investor borrowed $20 million. So, now that's an initial equity position of 20 million and 20 million of borrowing, or 100% leverage. So it is (20+20)(1.09) which would give us 43.6 million minus 20 minus .2, which would give us 23.4 million, I'm running out of space but we'll put it right here. (23.4/20)- 1 would be a return of 17%, okay. However, volatility of this return would be 30%, so it's like I said, I'm running out of space, I'm all the way over here, at 30%, 17%. But what you'll notice, if I've drawn this picture correctly, is that all of these points essentially lie on a single line. And what's happening is, we're borrowing money at 1%, and we can actually achieve a return that is anywhere on this line. What that means is that in theory, a money manager could generate any return they wanted if the took enough leverage. But, their volatility of return would increase just as quickly. >> This idea is captured in the Sharpe ratio. The excess return of the asset investment over the risk free rate divided by the volatility of return of the investment asset. The Sharpe ratio is very interesting, it's a revenue metric divided by a risk metric. So it represents how many units of revenue can be achieved for how many units of risk. >> Sharpe ratio of the original investment, with return of 9% and volatility of return of 15%, is calculated as follows. 9%- 1%, so this is the portfolio return minus the risk free rate. Divided by the volatility of the portfolio return, so divided by 15% = 0.533. If I pick a return of 13%, now I'm going to have a volatility of, going to have a return of 13%- 1%. So this is excess return over the risk free rate divided by the volatility. It would be 22.5%, and again I get the same Sharpe ratio .533. And finally for my 17% return, I'm going to have (17%- 1%) / 30% = .533. Logically, that all makes sense because we’re talking about the slope of this line, and we’re talking about points that fall on this line with constant slope. So, any point on this line reflects the same skill level. And this is why you cannot compare two managers performance without knowing the volatility of returns, because otherwise you could always have one manager increase their returns at the expense of much greater volatility and much greater risk. >> The investment paying 9%, and the one paying 17% are identical in terms of manager skill. And this equivalence is captured by the Sharpe ratio. If I were hiring this manager, I would say focus on stock picking skill. And if I want to increase both my return and my risk, I will borrow money and use it to invest in your fund. Let me decide where I want to be on the revenue-risk continuum. You simply pursue the highest possible Sharpe ratio.