Okay, so let me just recap what we just did. So we just derived the capital allocation line right. Which tells us all the risk and return combinations, right, that we can achieve given one risky asset that's called the optimal risky portfolio and a risk free asset. Right so the opportunity set of risky and inventory combinations that we can achieve by combining these two axis is represented by this line right. And analytically we also derive the expression for it which says that the expected return that we can achieve by combining them together, on this portfolio, is going to be the risk-free rate plus, right Whatever risk we accept, right times reward to risk ratio, right. So the excess return we're going to get for investing in the risky assets per unit of risk okay. And I said in the last lecture that this is called the sharp ratio okay. So now the question is where would you like to be on this capital allocation line. Well, of course that's going to depend on your attitude toward risk, right. And how do we represent our attitudes towards risk or preferences. Remember by indifferent scripts. So let me draw again for you, right, how do we choose the optimal allocation. Given this capital occasion right line, right. Okay so here are my again [INAUDIBLE] return on volatility space. Here is the risk free rate, the risky portfolio. Right, so let's suppose this is the capital occasion line. The question is, where along this capitol allocation line we're going to be. Well, remember we can represent our preferences by indifference curves. Let me change the color. Here we are, right. Those are our indifference curves. And remember that the higher the indifference curves, the higher results we get. Well where is it going to be. Well it's going to be the indifference curve, the highest possible indifference curve that just touches the capital occasion line, all right. So it's going to be the highest indifference curve right. Maximizing utility in this case is going to mean we want to get on the highest indifference curve possible. That is, just tangent to the Capital Allocation Line. Right, there's another indifference curve here but of course we can't get to that one, because Capital Allocation Line gives us the opportunities that we can create right. So, for somebody with this type of preferences. The optimal way to obey. The optimal common allocation will be somewhere in there, right. Now, I illustrated the solution to you, graphically. What would it look like, analytically. Okay, well let me find my notes here. Well that's going to given by. Okay, so what is the problem we're solving. All right, we're trying to maximize Our expected utility. Choosing the weight on the risky assets, right. And what is the utility function? Remember it's mean variance utility so we're maximizing, This particular Utility function. Right. Where A is the risk aversion coefficient. Okay. And I had expressions for these, right. So let me plug them in. What are we trying to do. We're trying to choose an optimal weight, right, on the risky asset. All right, that's coming from plugging in for the expect return, plus the variance. Again. Represented in terms of weights, right. So, remember, if you want to maximize this, you need to set the first order condition. The optimal solution to this problem solves the first order condition. So we're going to take the derivative with respect to W, set it equal to zero. Solve for W. What does that give us. Well, the optimal date weight in the risky asset is given by this expression. Okay, so what is this weight tell us, well it's the analytical solution to their optimal allocation problem right. So, it is the It is how much you want to invest in the risky asset, okay. And look what it's saying. It actually is very intuitive. It's saying that you want to invest more into risky assets the more excess return you get from the risky asset, right. The higher the excess return, over and above the risky rate, the more you want to invest in the risky asset. What else does it say. Well it says the more risk adverse you are, the higher the risk aversion coefficient, right, the less you want to invest in the risky asset. That also makes since, right. And finally, right, the riskier Is the rescue portfolio. The volatility of the rescue portfolio, unless you want to invest in the risky asset. So this gives us an analytical solution, while the picture gives you the graphical solution, to the optimal allocation decision. All right so. In these two lectures, we talked about the capital allocation line which describes the set of risk-and-return combinations, or portfolios, that we can construct by putting together a risky portfolio. Well it's going to be the optimal risky portfolio, and a risk-free asset right. The slope of that Capital allocation line gives us the sharpe ratio, right? It is the excess return that we can get from the risky asset per unit of risk, right. That's the Sharpe ratio. And finally, right. The optimal assets allocation Is found by maximizing utility, right. Being utility which means that we want to get on the highest indifference curve, right, the tangency point with the capital allocation line.