In this set of modules, set of three modules, we're going to talk about liquidity, how liquidity of assets affect trading costs and how these trading costs have implications for portfolio selection. In the first module, we are going to be mainly focusing on measures of liquidity, how do you set up a portfolio selection problem that takes liquidity into account? What is liquidity, and what is a liquid security is very hard to define in practice. What we do know is, a liquid security is one that can be traded very quickly, has very little price impact, meaning, when I try to put in an order to buy or sell it moves prices very little. It can be bought and sold in large quantities. That means, it has a very deep order book. There are different measures of liquidity. Some of these measures are based on volume, and these include the trading volume and the turnover, which is defined as the trading volume divided by the shares outstanding. Both of these measures try to get at the idea, that liquidity refers to the fact that one can execute your trades quickly. So if something has a very high trading volume or high turnover, then my order will get executed very quickly. There are possibilities measures, which look at the impact of liquidity onto the cost of creating a particular set of securities, particular amount of securities. One of the measures that gets at this is the percentage bid-ask spread. Ask price is the price at which people are willing to sell a particular security and bid, the bid price is the price at which people are willing to buy a security. As always, the ask price is going to be greater than the bid price, and the percentage bid-ask spread is defined as A minus B, the difference divided by A plus B, divided by two. So the mid-price times 100. A volume weighted average price is another quantity that gets us this notion of liquidity. So what happens is that you want to sell a total amount of shares V. This total amount gets split up into smaller trades, V_1 through V_m. Each of these traits gets executed at a different price, P_1 through P_m. So the volume weighted price that you got for a particular order would be simply P_i, V_i, summed from i going from one to m, divided by the total volume V. So this quantity is known as the volume weighted average price. It tells you the price that you ended up getting. If you're trying to sell a particular commodity, what happens is that the price start to slip downwards, so you instead of, P_1 could be the opening price, P_m could be the price at which your last bid got sold. Typically, P_1 is greater than P_2, is greater than P_3 and so on. So the average price turns out to be less than the price that was existing in the market just before you put in your trade. Other authors have tried to get at the price impact function directly. The first of these functions was introduced by Loeb in 1983 called the Loeb price impact function. More recently, Kissel and Glantz have introduced a price impact function, which measures how expensive or what is the cost of putting in a particular trade. In the next slide, I'm going to talk about the Kissel-Glantz price impact function and in that context, I'll tell you what the Loeb function is as well. In the rest of this module, I am going to assume, that we are going to be looking at basically a cost based measures of liquidity, and we are going to incorporate those cost-based measures into my portfolio execution or portfolio selection procedure to get at what is the optimal choice for my portfolios. The trading cost function, the cost of creating a particular block of shares is typically assumed to be separable across assets, which means that it ignores the cross asset price impact. This is an assumption which is not valid anymore. There is cross-price impact and in the last module of the series, we're going to start discussing some ideas of why this cross-price impact occurs and what can be done to sort of take it into account in your asset allocation decisions. So the Kissel-Glantz function says, that if I want to trade Q shares of a particular asset, then the cost of trading these shares meaning, the slippage, the extra price that I have to pay if I'm buying or the loss in price that I'm going to see if I'm selling, is given by this function c of Q. It has three components. The first component is just a constant a_3. So the cost is dependent on the total dollar amount that I'm going to sell, it's dependent on the volatility. So there is no index i there, if a particular stock is very volatile, you expect to pay more, prices move around and so when you've put in your order to the time that it actually got executed, you will end up getting a higher volatility. Then the third component that depends on basically the percentage of your trade Q to the daily volume V. So Q over V is the percentage of the daily volume that you are trying to take, times 100, that gives you the percentage raised to the power beta and these are some constants a_1. How does one estimate a function like that? What one does is, postulates that there are three factors in the regression function. This is factor number one. Volatility is factor number two, and this is basically an interceptor. Then one runs a regression, one records over a history of traits, how much extra cost that you ended up paying for that particular trait? So Q_t is a particular trade that was executed at time t, or trade t. P_t, Q_t should have been the price that you should have gotten. CQ_t is the extra price that you had to pay, or the extra revenue that you lost. You divide that, that gives you one observation of this regression function, and then you regress it to compute out what a_1 is going to be, what a-2 is going to be, what a_3 is going to be. This is what was proposed by Kissel and Glantz in mid 2000s and this has sort of become the standard function that people use for trading costs. There was another function that was introduced by Loeb before, and which was slightly different. In his model, the cost versus volume initially grew linearly, and then it grew with a power law. So the cost was some Alpha_1 times q up to some q_max, and then it was some Alpha_2 times q_1 plus Beta, after q_max. Beta was estimated to be approximately 0.65. So this is the cost under q and that's q and this was the Loeb function. So Loeb function is, was suggested in 1983, and it was relatively simple, it did not take into consideration the volatility, it did not take into consideration the average daily volume, but it was the inspiration that led to other liquidity functions later on, in particular, the Kissel-Glantz function. We will focus with the Kissel-Glantz function in this module. Alright. So once we have price impact function, we can include that into our portfolio selection problem.