 Hi everyone, it's MJ the fellow actuary and in this video we're going to be looking at a question regarding the efficient frontier which is part of portfolio theory which is part of subject CM2. Now for those of you who don't know I do offer tuition to the South African students but because we're living in the time of COVID we have turned all of our tuition to make it more virtual and because it's virtual I kind of feel like we can open it up to international students. So if you are an international student and you want some actuarial tuition check out the form in the description below fill that in and you might be able to join for the next semester but what we're going to be doing in this video is this efficient frontier question and what we're going to be doing is jumping straight into it. So it says consider two independent assets asset A and asset B with expected return of 6% per annum and 11% per annum and standard deviations of return of 5% and 10% respectively. Let xi denote the proportion of the portfolio invested in asset i so what we want to do is basically just get some key points from the question and essentially we have this asset A which has got an expected return of 6% standard deviation of 5% so this is our low risk low return asset then we have asset B which has the expected return of 11% and the standard deviation of 10% this is your high risk high return. Now you want to make sure and have a little check here sometimes writing these things down someone might have oh asset A is 6% expected return and 11% standard deviation and then asset B and they get the little things confused. Essentially what you want in these type of questions is an asset that has low risk low return and the other one that has high risk high return. If it's a bit of a blend you want to maybe just check if you've read the question correctly that is a common mistake some students make. Essentially we then want a bit of a risky portfolio where we're going to have xi in asset A and 1 minus xi in asset B because essentially what we're doing is we're looking for the optimal combination of these two assets we're answering that question you know how should I invest my money and you could visually represent this on the graph this is our mean variant space or our mean standard deviation space where we have A you can see low risk low return and then B high risk high return. Now part I of the question says if only asset A and B are available determine the equation of the efficient frontier in the expected return standard deviation space. So essentially what we need is the equation of this efficient frontier. Now our efficient frontier like I say we need in our expected return which is going to be given by E and our risk which is given by our standard deviation or sigma. Now we can express the proportion or the portfolio proportions as functions of the portfolio's expected returns. So the expectation of the portfolio is going to be you know coming from A and it's coming from B and it's given by the following equation E is equal to xi times the expected return of asset A plus one minus xi times the expected return of asset B. Now what we can do is we can substitute those amounts that we looked at earlier which was six percent for asset A and eleven percent for asset B and now what we want to do is just kind of clean it up we can now have our expected return of the portfolio is going to be equal to eleven minus five xi where xi is the proportion invested in asset A. Now what we want to do with these efficient frontier equations is we want to get this xi on its own so when we look at the variance side we can then plug it in and have an equation that has got the variables of the expected value and of sigma. So what I'm going to do here is I'm going to get our xi on its own and we also want one minus xi on its own because that's going to be the proportion invested in B and that's going to be the other part of our equation that we want to substitute out. So one minus xi we simply take five divided by five minus our xi and then we can clean it up and we have the following and these are two important things we want to keep in the back of our mind and we're going to see we're going to be substituting them in on the next slide. So sticking with the efficient frontier because the two assets are independent that's great for us that makes the question a lot easier the portfolio variance is given by the following formula where the variance of the portfolio is going to be equal to the proportion squared invested in asset A times the variance of asset A plus one minus the proportion invested in asset A squared times the variance of asset B. Now like I said from the previous slide we had determined what that proportion was equal to as well as what one minus that proportion was equal to. We can now substitute those in to the portfolio proportions and we're going to get the following equation. Okay we now can substitute the risk of asset A which was equal to five squared and of asset B which is equal to 10 squared. Now the reason why we're squaring them is because we were given them as standard deviations and in this equation we're using them as variances because the variance is equal to the squared standard deviation that's why it's five squared and 10 squared. Okay so just if you're wondering why we're forgetting those squares there. So what we then do is we have the following equation we want to simplify tidy it up and we'll have our variance of our portfolio is equal to five times the expectation of the portfolio squared minus 70 times the expected return of that portfolio plus 265. Now we want our efficient frontier to have expected return with E and we want our risk to be the standard deviation but we currently have it with the mean and the variance we like I said we want instead of the variance we want the standard deviation and we know that the variance is equal to the standard deviation squared so in order to now get to the standard deviation we're going to need to take the square root so you can think of that equation instead of v as sigma squared and then to get sigma on its own we're going to take the square root which means the efficient frontier is the part of the curve above the point at which the variance is minimized. Now in order to find the minimum what you want to do is you want to take you know the derivative so to find this point we're going to differentiate so we're going to take the derivative of v divided by the derivative of e and it's going to be equal to 10 e minus 70 which means once we set it to 0 e is going to be equal to 7 so the efficient frontier is the part of the opportunity set where our expected value or our expected mean is going to be greater than 7 percent so our return in other words is going to be greater than 7 percent and so we have the following as our answer sigma is equal to the square root and we've got that equation over there where e is greater than 7 percent and we can represent this visually to see what it would look like so remember we had our two assets the whole idea this is showing the benefits of diversification like like I said you should probably have watched first all my videos on the theory of the stuff before jumping into this question because that'll make it a little bit clearer but here we have that 7 percent so all that highlighted area of the curve that is now going to be our new efficient frontier on the opportunity set cool now we can move on to the second part of the question which says a third asset asset c is a risk-free and has an expected return of 4 per cent per annum now i'm going to probably mispronounce this name but a lagarian function is used to calculate the equation of the new efficient frontier write down but do not solve the five simultaneous equations that result from this procedure so this is a very theoretical question but essentially what we're doing is we're introducing a risk-free asset we're calling it c and this is how it would be represented on our graph this is our mean standard deviation space so simultaneous equations this is just going to be a lot of theory because we've got the lagarian function which is given by the following form and just to note that lambda mu or your lagarian multipliers and so for our question this function is going to be and also just note that because we're now introducing asset c i'm going to be using xb as a proportion invested in asset b instead of that 1 minus xi because now we have three assets that we could potentially be investing in and essentially we use that function above we just make it you know for our values over here i'm not going to read that all out because it's a lot but essentially what we now need to do is we now only need to differentiate the function with respect to its five parameters and set it to zero in order to complete the question so we're going to be taking partial derivatives with regards to each of these parameters setting them to zero and that essentially is this question done so when the parameter is xa it's given by this equation xb by that equation xc that equation by lambda by that and mu by that and essentially we're not going to be using this in the next two questions that are coming up so part three says use your simultaneous equations to derive the relationship between xa and xb on the new efficient frontier and please note that this is making the question quite easy the fact that they're breaking it down and they're saying you know how to do each step sometimes the questions just are like you know find the new efficient frontier once you've introduced the risk-free asset so yeah please appreciate that the question is holding our hand whereas in the exam it might not be um you know that step forward they might not be giving us the recipe they might require us to know this whole procedure off by heart but anyway let's jump into the relationship between xa and xb so we're going to use that third simultaneous equation from the previous question which is the partial derivative of the variance with regards to partial derivative of xc which is the proportion that we've invested in asset c and we see it's equal to negative four lambda minus mu is equal to zero or in other words mu is equal to negative four lambda now what we want to do is substitute this into the first two equations and essentially this was the first equation over here we substitute mu and we get the following answer that the proportion invested in asset a is going to be equal to eight lambda divided by 200 we're going to do the same with the proportion invested in asset b and we're going to see that it's actually quite similar that the proportion invested in asset b is seven lambda divided by 200 now that's great because now we can find the relationship between xa and xb and we see that the difference is seven eighths so the relationship between xb is equal to seven eighths of xa i mean this is great because now we can find the final part of the question which says hence derive the equation of the new efficient frontier in expected return standard deviation space so like i said our asset c is a risk-free asset so the new efficient frontier is going to become a straight line because we're now going to have a combination of our risk-free asset c and our risky portfolio or the market portfolio then in this instance contains asset a and asset b so visually it would look like this we now have introduced c the new efficient frontier is the straight line our task is now to find well what is the formula or the equation of this straight line and we're using the information from the previous question so we know that the proportion invested in asset b is equal to seven eighths of the proportion invested in asset a and please note your seven plus eight is going to be equal to two fifteen which means that the efficient portfolio consisting entirely of risky assets so this is when our amount invested in asset c is zero we're going to have this point where xa is equal to eight divided by fifteen and xb is equal to seven divided by fifteen now the expected return of this portfolio is equal to the following so our expected return of our portfolio is equal to the amount invested in a times the you know expected return of asset a plus the amount invested in asset b times the expected return of asset b so now what we're doing is we're plugging in the values the proportion invested in a which is eight divided by fifteen times that six percent which was the expected return of asset a plus seven divided by fifteen the proportion in asset b times the expected return of asset b which is eleven and we solve that and we get the expected of our portfolio when we're entirely in our risky asset is going to be eight point three three percent and now we're going to do the same with regards to the standard deviation of this portfolio so again we're taking the standard deviation of the portfolio formula we're substituting in the values we're solving for it and we're getting five point three seven five percent and what's great about this is we can now kind of wrap up this whole question so therefore the efficient frontier is a straight line in our expected or our mean standard deviation space and it's going to join the following points okay we've got the first point out with zero risk and a return of four this is you know when we've gone a hundred percent on our risk free asset we got this because we were told that the return of asset c is four percent it's risk free which means the sigma is equal to zero and then we have this other point which we've just determined in the previous slide and this is when we go a hundred percent in the risky portfolio when we put zero percent in asset c and now what that means is because we've got two points and we know it's a straight line we can determine what that efficient frontier should be so using you know a formula to find the straight line we can simplify that and we have our answer that e is equal to four plus zero point eight oh six sigma and that concludes the question thank you so much for watching and i'll see you guys soon for another one cheers