So there are two approaches to introducing liquidity on portfolio selection. One approach is to do the usual portfolio selection and then account for liquidity in executing traits. In the second module of the series, we're going to talk about how to include liquidity in executing traits. The other approach is to incorporate liquidity concerns directly into the portfolio selection problem. That way you're choosing position. So you're choosing portfolios that will have low cost of execution. The best practice is to do both. Account for it in the portfolio selection, and then when you do trades, you account for it via trade execution as well. So the generic problem that one solves for the second approach, which is to incorporate liquidity concerns directly into the portfolio selection is as follows. You take your usual mean-variance optimization problem. So I have a current position y, one transpose y tells me the total wealth that I have. X is a new set of positions. So in this particular problem, the x's do not add up to one they are just dollar amounts or any other units which add up to the initial amount of money that I had. Mu transpose x minus lambda x transpose x, this quantity, is our usual mean-variance objective. Mu is the mean return and V is the covariance matrix, lambda is the risk tolerance or the risk aversion parameter. Now, instead of just stopping there, what we're going to do is subtract from it at the trading costs. This is the extra cost that I have to pay and that actually reduces my mean return. I'm going to add an eta to try to incorporate the effect that I can control the amount of liquidity cause that I'm going to incorporate into my portfolio selection problem. So some part of it I might include here, some part I might handle while execution or I might just want to use eta as a way to trade off between mean-variance returns and the trading costs. What is C [ x, y]? C [x y] is the cost of moving from the current position y to the new position x, and we can write it as using the [inaudible] function as this expression down here. x_i minus y_i is dollar amounts and that's why I've now divided by P_i instead of just V_i to include the fact that it's not the presented dollar transacted to the power beta. This sigma i remains the same, a_3 remains the same and x_i minus y_i is actually the dollar amount transacted. Now, if you expand it, you end up getting, you can take the constant over here, it just becomes x_i minus y_i absolute value. You can take this one and take out the extra part so it's a_1, 100 over P_i i V_i to the power beta, and x_i minus y_i to the power one minus one plus beta. Now I have a function. I can incorporate this function into my portfolio selection problem and then I can solve that portfolio selection problem to compute what my new positions x are going to be. In the next module, which is going to be an Excel module, I'm going to show you how to solve set up and solve this optimization problem and we're going to play around a little bit with what happens when Eta changes values and so on. In the rest of this module, I'm going to talk about a very simple model that has become popular. That was introduced by Andy Lowe in one of his papers, and it's an easy model that incorporates some aspects of liquidity. So this approach is taken from a paper by Lo, Petrov, and Wierzbicki, and the title of the paper is very interesting, It's 11:00 pm - Do you know where your liquidity is? The mean-variance liquidity frontier. What they do is that they ascribe to each security a certain normalized liquidity measure. So let l tilda it denote the measure of liquidity where high values implies more liquidity. So if you're talking about turnover, high turnover is a good measure. If you're talking about volume then high volume is a good measure. When you're talking about trading costs or bid-ask spreads, you take the reciprocal of those numbers. So high percentage bid-ask spread is bad. Low percentage [inaudible] spread is good, and therefore, when you define this measure of liquidity in the model introduced by Andy Lowe, you take one over the percentage bid-ask spread to define your l_it. Then you normalize this over a certain period. So what you do is you look at l_it per a particular amount of time, you take the minimum value that this particular measure could take over all assets. So it's i prime ranging over all assets and all times and divide it by the maximum value that can be achieved over all assets and all times minus the minimum. So this number, whatever it is now, becomes a number between zero and one. So this lies between zero and one. They assumed in their model that all the wealth is in cash and formulated three different optimization problems that get at this notion of how do you incorporate liquidity into portfolio selection. The first method they call Liquidity Filtered Portfolio Selection. So you do the usual mean-variance portfolio selection. So one transpose x equals one, mu transpose x minus lambda over two x transpose V_x. But now we insist that x_i is equal to zero for all i's that do not meet a particular liquidity threshold. So l bar is your liquidity threshold, and if it doesn't meet the liquidity threshold, you cannot hold that particular asset. Another one is to use a mean-variance liquidity objective. Again, this time I'm just adding to it l_ix_ i and then saying this is different from the formulation in the previous page because this is no longer cost, but this is a quality measure. So l_i high is good. So I want to add that liquidity measure instead of subtracting as if it was a cost. A third version that they suggested was liquidity constrained portfolios. So you do not put a cost, but you say that l_i x_i over the average value of x_i is greater than equal to l bar. So difference between the first model and the last one is that the only thing this is measuring liquidity on a portfolio level. So this is a portfolio level measure instead of asset. In the first formulation over here, this is asset by asset. If a particular asset does not meet the liquidity threshold, you throw it away. In this particular case, you just want to make sure that on overall portfolio level this happens in that way. The expression for overall liquidity in the portfolio is slightly different here. This is to prevent short positions in illiquid stocks canceling long position in other liquid stocks because of that instead of taking just x_i, we take the absolute value of x_i. All of these portfolio selection problems can also be solved in Excel in much the same way that the first formulation is going to be solved.
