[Music] well let's put this to work and let's see if it can help us do anything over here you see that I have my risk-free rate on the intercept and my risk-free rate I'm saying we're confronted with a five percent risk-free rate here I have my minimum variance frontier I draw my capital market line just tangent to that and where it is tangent creates a point that will be our market portfolio here we have a expected return of fifteen percent with a standard deviation of 20 percent and from this what we want to do is we want to look at what happens when we change our weightings on these so what I want to do is I want to show you how this this works right along this long in here and to do that let's let's first calculate the market price of risk which is just the slope of the line so the expected return on the market minus the risk-free rate divided by the standard deviation of the market gives us point one five minus point zero five or point two or 0.5 so what does this mean how do we interpret this let's be clear about this first that means that if we increase the portfolio risk by one unit by one unit the return would increase by point five units now I want to be very clear here because there's a trap of of falling into these deterministic relationships thinking that this is a this is a solid relationship this is an expected return based on the level of risk and everything that got us to this point was built on a foundation of probability not certainty nowhere in here is there certainty this is all still probabilistic so probably if we increase if we assume more risk probably we would squeeze out 0.5% more return probabilistically speaking so let's let's be clear that we're still even at this level we're on the top floor of the building but we built our foundation on a foundation of probability this whole structure were we're building is all based on probability so that doesn't mean that's what we're gonna get it means that's what we think we're gonna get based on everything we've done so far okay so every unit of standard deviation delivers 0.5 units all that expected return on the capital market line so I'm gonna show you how this works and and we can do it in a very quick way and then we'll do it with the formula just to show what's going on let's say we're 100% in the risk-free asset over here and zero percent in this particular asset what is our expected return and standard deviation well that's just the risk-free asset we know that's 5% and the standard deviation of that is zero nice and simple right well what if we were at the bottom here and we are not in the risk-free asset and we're 100% in the risky asset well then it is just the characteristics of the risky asset that apply which would be 15% and 20% so if we know the market price of risk at 0.5 if we were 50/50 if we're going from 0 to 10 if we were 50/50 we could just say well that's 10 and we're going from 5 to 15 the distance is 10 we're gonna cover half it should be 10% and we're going from 0 to 50 if 25 is in the middle it should be half that distance at 5% which would make this at 15% because that's halfway through halfway between here should be 7.5% and halfway between here should be 12.5% so if we know the slope of the capital market line it's easy to find any return and risk combination it's easy we don't really have to use the formulas but if we did let's use the 7525 if we waited asset 1 at 75 at 0.75 multiplied by point 0 5 plus 0.25 multiplied at 0.15 you would get 7.5 percent if we try to determine the standard deviation of the portfolio using this particular function we would get 5 percent and just a brief note why is the standard deviation of the portfolio just this we've been through this but just as a refresher all I'll explain remember the first term was w1 squared the standard deviation of the first asset squared well the first one is the risk-free asset it has a standard deviation of 0 so the first term just cancels right out altogether the second term was 1 minus w1 squared standard deviation of the second asset squared that stays the third term was 2 times the waiting on the first waiting on the second times the covariance of the two returns well because this has a standard deviation of 0 by definition its covariance will be 0 that term cancels out so the variance of the portfolio was equal to 1 minus w1 squared and the variance of the second asset squared so to cancel the square you just cancel the square what are you left with there we go just a quick one to show you how we got there in case you forgot so if you take each of these each of the standard deviations and use the waiting's on w1 in here you will get exactly this so you can rely on the formula but once we know the slope of the capital market line we can just use that directly and solve for anything now everything we solve for lies in between 0% on the risky asset to 100% all along this line well let's take that that same scenario this time we're going to use leverage we're going to use leverage at 25% 50% and a hundred percent and we're going to create a borrowing portfolio so a 25% let's solve our our problem here at 25% where you use 25% leverage which means we're gonna borrow at the at the risk-free rate so RW 1 which is how much we have in the risk-free rate is now going to have a negative weighting negative 25% RW 2 will now be 125 percent so let's let's see if we can't solve this we just figured out our expected return my expected return on the portfolio will equal negative 0.25 times point zero five plus one point two five times point one five equals seventeen point five percent asset the market portfolio returns 15% we've used leverage we're getting up to seventeen point five percent what happens to our standard deviation standard deviation of the portfolio well we're up here we should expect it to be higher than the 20% we're currently getting right now right so it's just 1 minus w-1 times the standard deviation of 2 which is what do we have 1 minus and w1 is negative 25 negative 0.25 1 minus negative 0.25 times 0.2 well that's one point two five times point two which is 25% so yes we can leverage and we can get higher than the market portfolio on the capital market line but it will come with proportionately more more risk what about 50 percent can we calculate one for 50 percent well we can our w1 in this case is going to be negative point five our w-2 in this case will be one point five as long as they add to one so the expected return on the portfolio is going to be negative 0.5 times point zero five plus 1.5 times 0.15 which will give us 20% standard deviation the portfolio is 1 minus negative 0.5 times point two which will be 30% fairly straightforward the last one if it were 100% let's just get our weightings down or waiting on number one would be negative one are waiting on asset two would be two so the expected return on the portfolio is going to be negative one times point zero five plus two times point one five or 25% plugging the same numbers in for our standard deviation of the portfolio we're gonna end up with 40% so the capital market line can stretch on and on and on we can use as much leverage as we want to push ourselves further up this curve depending on our risk profile so to recap what do we have here we have one risky portfolio m-mom and a risk-free rate and we can choose how risky we want to be by either lending or borrowing so we don't have to go around looking for a bunch of assets to create a certain amount of return we don't even have to do that all we have to say is look this is gonna be really easy I got a well diversified portfolio here the market index why should I even try why the work is already done now if I'm trying to target a specific return all I have to do is say well where's the return on trying to target up here well let's come across here there we go there's there's where I should be on the capital market line how much leverage do I need to get there I'll borrow here at the risk-free rate invest in the portfolio and leverage it up to that amount knowing that I am of course taking on what a terrible line there right knowing of course that I am taking on more risk at that point in time but if that's what I was looking for I don't have to go looking around anywhere else I can do it all with two assets the market portfolio and the risk-free rate I can do it all with two assets so I can find my position anywhere on here depending on how risk tolerant or risk-averse I am I can that's all I need job done right well let's look at lending and borrowing here we can lend we can in our lending portfolio we can lend up the risk-free rate so that creates this type of slope as we show over here but now we have to borrow what 7% we can't lend at 5% we have to borrow what 7% well that's easy enough to do all we have to do is let's just pretend that the risk-free rate was 7% and let's just do the cat the the the line again so if we if this is 5% and this is 7% over here actually that's five let's say that that's 7% right there all week and all we need to do is draw the line so let's anchor our point let's go over here and there we go we don't draw the first part because we can land at 5% we'll just draw the second part which is what we have to borrow at there we go and to determine the slope of that line we'll just assume that the line extends all the way down to two to the line and how do we do it rise over run right so what is the slope of that line it is the expected return if we take this point over here and we extended this down to here it is just the rise over the run the expected return on the market minus our B naught our F naught the risk-free rate but the borrowing rate we're just going to substitute the borrowing rate in divided by the standard deviation of the market and since we already know what those are we can figure this out we know that the return on the market was 0.15 - we're not going to subtract point zero five which is the risk-free rate we're going to subtract point zero seven at this point we're still dividing by the same standard deviation point two which means point four zero so now we're only getting paid point four percent in units of return for every unit of variance we take on our payoff just got lower as it should be because our cost of funds just got hired so we can continue this on what is our expected return on the portfolio keep in mind now it's going to be different right it will be the waiting on the expected we turn ber be why because we're borrowing remember now we're constructing a borrowing portfolio so up here our expected return cannot be a function of the risk-free rate we're gonna be waiting how much borrowing we're doing and of course w2 and will be the expected return on the market and of course our variance is still going to be one minus w-1 variance on the market it doesn't change just because we borrow at 7% doesn't change the volatility on the market portfolio the market as far as volatility is concerned doesn't care what we can borrow money at doesn't know what we can borrow money and it doesn't change anything it just really doesn't change anything so it stays the same so let's say that we're gonna borrow 75% so in other words are waiting on one RB will be negative 0.75 and w2 will now be 1.75 so what is the expected return on the portfolio we know we're creating a borrowing portfolio not a lending portfolio let's write that in so that we know that this is of course the borrowing portfolio we know we're creating a borrowing portfolio so let's expect a return on the portfolio will equal negative 0.75 times we're going to use our B here point zero seven if we were lending this would be positive would have a positive weight and that would be point zero five right plus the weights have to add up to one so that's negative 0.75 this must be one point seven five and of course the same point one five twenty one percent twenty one percent and the variance on this portfolio will still be one minus negative 0.75 times the standard deviation you'll still get the same standard deviation 35 percent versus versus if you could if you could borrow at the risk-free rate if you could borrow at the risk-free rate your expected return on the portfolio would be twenty two point seven five percent and your standard deviation of the portfolio would be 35% so notice that the standard deviation is the same 35% it's the same you're creating the same portfolio in terms of risk it's just that your return is going to be lower because your cost of funds is higher [Music]