John Dodson:Hello there Greg:Hi John Sharon Linden:Quick question regarding the project John Dodson:sure, Sharon Greg:yes Fanda:yes john Sharon Linden:would you like us to attempt to quantify our confidence or simply which view we have more confidence in? John Dodson:Sharon, for the outline you do not need to quantify views confidence. We will talk about how to do that on Wednesday. Peter:I haven't had a lot of time to look at the homework due this week (HW4), but can you talk about how to get started or how are we suppose to incorporate covariance? John Dodson:Peter you can use the results for allocation-implied parameters on the last slide from week three. John Dodson:The basis of this is that the cap-weighted portfolio should be optimal to some representative agent. John Dodson:To make this argument, we assume that the covariance is readily observable be that agent, but the drift is not. John Dodson:So the cap weightings should represent the optimal portfolio given the represenative agent's preferences ($\zeta_0$). Fanda:Hey John, is the exponential utility function our satisfaction index or the certainty equivalent of it? John Dodson:Fanda, the satisfaction index would be the CE version, but the optimal portfolio would be the same if you were to use just the utility. John Dodson:And I do not think you need to use the Arrow-Pratt approximation. John Dodson:The opportunity cost is in terms of certainty-equivalent satisfaction. Fanda:I am not sure what you mean by saying "but the optimal portfolio would be the same if you were to use just the utility" Fanda:you mean use the utility as satisfaction? John Dodson:Fanda, sorry, I am confusing things. You need to work with the CE satisfaction, because I asked you to measure opportunity cost. John Dodson:My point before was simply that the portfolio that maximizes the exponential utility is also the portfolio that maximizes the CE satisfaction. Fanda:i see John Dodson:sorry, _expected_ exponential utility Fanda:What do we know about the distribution of \psi as the independent variable of the utility? i am wondering either taking the expected value of the utility or going through the Arrow-Pratt approximation John Dodson:Fanda, this is worked out in 7.4.1 in the book. The expectation is easy, because $\Psi$ is normal. John Dodson:That's why we use exponential utility, because there are analytic results for the expected utility. John Dodson:with normal markets, at least Fanda:i see. thanks john Fanda:Hey John, this might seem dumb. but i don't understand the business behind the love triangle (kidding, Feb 14th symptom) investor - investment manager - representative agent John Dodson:Fanda, we are acting as investment manager, so we need to measure both risk in the market and investor objective and preferences. Fanda:yep John Dodson:There are effectively two investors here: the investor paying us and the investor representing the overall market John Dodson:the latter is a composite, of course. But Markowitz says we can invent a fictitous "representative agent" with ordinary attributes John Dodson:this is where the $\zeta_0$ comes from. John Dodson:But our client has different views. Hence an opportunity cost. Fanda:i failed to incorporate this basic background to the fact that opportunity cost is the difference from optimal satisfaction Fanda:it seems that there are two satisfactions here with two investors John Dodson:yes, although we really only care about the satisfaction of our client. Fanda:so am i right that we use \zeta_0 to get this \nu pretending \alpha_0 is unknown, and then subsitute the \mu into the satisfaction formula to get the satisfaction of the investor? Fanda:i am talking about the things on the last page in the 3rd slides John Dodson:We use $\zeta_0$ to get $\mu$ based on the observed $\alpha_0$. Then we use this $\mu$ to measure our client's satisfaction for both $\alpha_0$ and $\alpha^\star$ Fanda:sorry i meant to say \mu not \nu Fanda:i am still confused about \alpha_0 and \alpha^*. so we have the \mu from \zeta_0. when you say "measure our client's satisfaction", do you mean we maximize the satisfaction to solve \alpha_0 and \alpha^*? Fanda:or \alpha_0 is known? John Dodson:$\alpha_0$ is given. But you do need to solve for $\alpha^\star$. Fanda:ok. i see Scott:so for solving for alpha star you mean taking the argmax of S(alpha)? Fanda:Hey John, can I just argue that the opportunity cost is the cost away from maximized (optimal) satisfaction so that the OC is non-neg? John Dodson:Scott, yes. This is the optimal portfolio specific to your client. Fanda:Hey John, for the second part, can I just argue that the opportunity cost is the cost away from maximized (optimal) satisfaction so that the OC is non-neg? John Dodson:Sorry Fanda. We know it is true in general; but it is good practice to check, especially if it is not obvious. John Dodson:I am imposing my own defensive math style on you. Fanda:no it is ok. i just want to know whant you want us to do Fanda:there is no need to solve \alpha^* for the second part, right? cuz constraints are unknown John Dodson:Use the global optimum. Fanda:you mean... taking the matrix derivative and ...... John Dodson:yes. It's not too bad. I think there is a derivation in the book. In any case, you know the answer from 7.4.1. Fanda:it's getting interesting now John Dodson:Any more questions? Peter:Is this the right equation for our S(alpha^star)? S(alpha^star) = (alpha^star)' (1/zeta^star)*E*alpha_0 - (1/2*zeta^star)*(alpha^star)' *E*(alpha^star) Peter:That's based on the last slide from lecture 3...but I thought there was only a zeta and zeta_0. John Dodson:Peter, I don't recognize your use of E here. Do you mean $\mu$, the mean of $M$? Peter:oh sorry...i mean E is the covariance. John Dodson:oh, ok John Dodson:Yes, I think somthething like this is the max value of the client satisfaction. John Dodson:wait... What is $\zeta^\star$? John Dodson:do you mean just $\zeta$? Peter:yeah. so there should be a $\zeta$ in the S(alpha^star) eqn and a $\zeta_0$ in the S(alpha_0) eqn. Correct? John Dodson:actually both zeta should appear in both satisfactions. The one $\zeta$ because it is the satisfaction of your client. The other $\zeta_0$ because that is how you calibrated the market vector parameters. Peter:ok, thanks. Peter:Thanks for your help John. I'm signing off. Have a good night. John Dodson:good night yiran:Hi, John. I am just back home from tutoring. John Dodson:OK, I am going to sign off soon. Any last questions?