 Hello and welcome to you with a risk-free asset. In the prior session, we looked at the portfolio that's composed of only two risky assets. So it's very important that you understand how we form a portfolio of two risky assets because in that portfolio I explained and we illustrated the covariance, the correlation coefficient. In this session, in addition to the two risky assets, we're gonna be adding a risk-free asset, which is gonna add little bit more information about what we know, but it's very important that you understand the prior session. Just look in here for the prior recording or in the description in the video below. This topic or these topics are covered on the CFA exam as well as essential or principles of investments. As always, I would like to remind you to connect with me only then if you haven't done so. YouTube is where you would need to subscribe. I have 1,800 plus accounting, auditing, finance, as well as Excel tutorial. If you like my lectures, if you like my lessons, please like them and share them with others. If they benefit you, it means they benefit them as well. Connect with me on Instagram. On my website, farhat-lectures.com, you will find additional resources to complement and supplement your accounting as well as your finance and this scores specifically education. So I strongly suggest you check out my website. So on the prior session, we looked at two risky portfolios, I'm sorry, at two risky assets in a portfolio and we try to make sense of this information. And this is the information that we are using. Basically, we were looking at a stock fund with an expected return of 10%. We are looking at a bond fund with an expected return of 5%. The standard deviation for the stock is 19, the standard deviation for the bond is eight and the coefficient or the coefficient correlation between the two is 0.2, which is a positive. It's less than one. It means diversification would help us. This is what we look at in the prior session. And what we did is we composed a portfolio with different weight, which I'm gonna show you how to do so or how we did so earlier. This way it's very important that you understand how we did so because we're gonna be adding to that portfolio. So let's take a look at this. So first what we did is we have on one side, we have risk and return. So we have risk and the technical word is not risk, the technical word is standard deviation. And we have the reward on the y-axis, the reward we'll call it expected return. And what we're gonna do, we're gonna plot some points to show you what it looks like, to show you what the graph would looks like of the investment opportunity set. What does that mean? It means we're gonna be looking at different sets. I'm not gonna plot every one of them, but I'm gonna plot enough where we have a graph. It will remind you of what we did earlier because we're gonna carry this graph and add to it a risk-free asset. That's why I'm doing this. So the first point I'm gonna look at is if we invested, first let's kind of basically, this is let's assume 8% standard deviation, 10%, 12, 14. Don't worry, I'm gonna show you this graph on the actual graph. So I'm just showing you how to build it because if you don't know how to build it, it's a little bit difficult for you. So these are the standard deviation and the returns are four, five, six, seven, eight, nine, 10, 11, 12. And this is the return starting with three, four, five, six, seven, eight, nine, 10, and 11. Okay, now my scale may not be perfect. It won't be perfect at all, but it's gonna get you the point because it's very important that you understand how you do this, how you build the graph. I'm gonna show it to you at the end. Okay, so let's start by investing 9.2% of our money in stocks and 90.8 in, 90.8 in bonds. If we do so, our expected return will be 5.46, which is we learn how to compute this in the prior session. I will not do that in this session. So 5.46 someplace here, this is the return. Let me use a different color. So you see what I'm doing, 5.46 here, and the standard deviation is 7.8. So the standard deviation is someplace, 7.8 is someplace here. So here's the first point, okay? And what is, if you remember, what is this first, what does this first point represent? This first point represent, if you don't know what this is, this is the minimum variance portfolio. So I'm gonna explain real quick what minimum variance portfolio is because we need to use this. What does that mean? It means at this level, if we invest 9.2 in stocks, 9.8, 90.8 in bonds, we expect to return 5.46, and the standard deviation is 7.8. Let's see what happened if we invest 10% in stocks and exactly 90% in bonds. Our return will go up 5.5, but our risk will go up as well. The standard deviation will go up. If we invest 20%, again, return will go up, but risk will go up. Let's see what happened if we invest zero in stocks, zero in stocks, and we invest 100% in bonds. So we didn't invest anything in stocks. We invest 100% in bond. The bond has an expected return of 5. Therefore, I'm gonna use a different color. This is the 5. The 5 is right here, and the standard deviation for the 5 will be 8. And some place here. Okay, and let's do one more, and let's graph if we invest, let's make it 20%. Let's invest 20% in stocks and 80% in bonds. The expected return will be 6. Let's use a different color for the third point, the third and last, I'm not gonna do any more points, 6%, and the standard deviation will be 8.7. Let's just make it someplace here, so they meet here. Now, here's what's gonna happen. We're gonna go ahead, I'm gonna draw the line if we graph all of those. It's gonna look, again, don't worry, I'm gonna show you the complete line. It's gonna look something like, whoops. Let me go back one more time. Okay, it's gonna look something like, this, okay? And to be more specific, to show you what it looks like, it's gonna look something like this. So let's work with a complete now picture. So this is the picture I was trying to draw for you. This is the minimum variance portfolio. Minimum variance means what? You're taking the minimum amount of risk, minimum amount of risk, given all these combination in the portfolio. So, and the bonds here is only the bond portfolio. So notice, if you have to choose, and hopefully you know this from the prior session, if you have to choose something here, or something here, like which is at 6%, which is 6% is what? 6% is 2080. So let's assume this is portfolio A. If you have to choose between portfolio A and the bond portfolio, no questions about it. You would choose portfolio A. Why? Because portfolio A, your return equal to 6%, your risk or your standard deviation equal to something around 8%. It's gonna make it eight. The bond, your return is five. Those are not 100% accurate, the standard deviation, but you guys get the point and the standard deviation equal to eight. Hold on a second. If I'm taking the same risk, standard deviation is the same. I will definitely would invest in portfolio A because I'm making more return. Now, this is an important concept that we're gonna be using. This is called the sharpie ratio. Now, how can we compute the sharpie ratio? You will see that later. We can compute this and we can find out that the sharpie ratio for A is more than the standard deviation. Actually, we can do it right now. If we take the risk, we don't have a risk-free return here. If we take the risk for this portfolio here and we take 0.06, we're gonna take, I'm sorry, yes, the risk premium, which is 0.06, because we don't have a risk-free divided by the standard deviation of A. This is for portfolio A and we'll do the same thing for portfolio B. 0.05 divided by 0.08. We could compute the sharpie ratio and let's do so here before we add anything. So if we do so, 0.06 divided by 0.08, your sharpie ratio is 0.75, 0.05, 0.05 divided by 0.08. It's 0.625. So you will definitely choose this portfolio Y because the sharpie ratio, you're getting more return for your risk. That's why you're getting more return for your risk. So this is actually everything that I did so far is something that we learned about. We saw in a prior session, but it's very important to understand how we do this. So please make sure you know how to draw this before we go to the next slide. On the next slide, we're gonna be adding what we want to do for this session is adding a risk-free asset to the picture. So when choosing the capital allocation between risky and risk-free portfolio, investors naturally would want to work with the risky portfolio that offered the greatest reward for assuming risk. And how do we measure the greatest return for assuming risk? It's this sharpie ratio, this sharpie ratio, okay? So the higher the sharpie ratio, the greater is the expected return corresponding to any level of volatility, which is volatility is risk or standard deviation. And this is how we compute the sharpie ratio, the expected return of the portfolio minus the risk-free, which is the risk-premium, the risk-premium divided by the standard deviation. Suppose then we are still confined to the risky bonds and fund, but now we can add to our portfolio a T-bill yielding 3%. T-bill means risk-free asset. Now we're gonna be introducing this picture, introducing risky-free asset. And let's see what's gonna happen when we do so. The resulting opportunity set is the set, the resulting opportunity set is the straight line that we called the capital allocation line. And we looked at the capital allocation line in a prior session. If you don't know how to do the capital allocation line, please look in the description. I should have the prior session for that. Now on the next slide, we will consider the various capital allocation line constructed with risk-free bills and the variety of possible risky portfolio, each form by combining the stocks and the bonds funds in an alternative proportion. So simply put, what I'm gonna do, I'm gonna change the date a little bit and now we're gonna be introducing risk-free asset. So here's what we have. Here's what we are weighted in stocks and whatever we are weighted in stocks, we are weighted in bonds. So if we invest 0% in stocks, it means we're investing the remaining in bonds, okay? So here's the weighted in stocks. This is the expected return of the portfolio and we're gonna be using a correlation coefficient is 0.2. So we're gonna be, they already computed the standard deviation for us as well. And our minimum variance portfolio is when we invest 0.09 in stocks. Our expected return is 5.46 and our standard deviation is 7.8. Simply put, we are looking at this point here about the minimum variance portfolio. This is still our minimum variance portfolio. So what are we going to do now? What are we going to do now? I'm gonna use the same graph. I'm gonna use the same graph, but I'm gonna use a different scaling. So I'm gonna draw another graph here with the same information with different scaling. So let me go ahead and do so. Expected return and the standard deviation. And basically we have a standard deviation of, I'm gonna use five, 10, 15, 20, 25, and 30. So those are the, I'm just using different scale because we're gonna be adding a little bit to this graph. So I prefer to draw it, then show you the final product, show you what it looks like at the end. And for the return, we're gonna do three, we're gonna do four, we're gonna do five, return six, seven, eight, nine. I'm not sure how much the scaling is going to work, but let's go ahead and do so. So remember the minimum variance portfolio is the return is 5.46. So 5.46 and 7.8 someplace here. So this is the minimum variance. So let's go ahead and draw like an approximate line. Okay, now, so I'm gonna go like this, something like this. So all what I'm doing now, all what I did now is draw the same, the same picture as here with different scaling. And you're gonna see why in a moment. Now we're gonna be introducing, so this is the minimum variance. This is the minimum variance point, minimum variance. Now remember, we're introducing a new asset and with this asset, that risky free asset has a 3% return. So anytime we start now, we're gonna be starting at 3%. If we invest everything in risky free asset, we'll earn 3%, so we'll be right here. Now let's graph some combinations. Let's graph some combination. So 0%, if we invested 0% in, I'm sorry, if we invested nothing in stocks and we invested everything in bonds, what's gonna happen? Bonds, it's gonna give us 5%. So we're gonna have 3% from the risk free rate and 5%. So simply put, this is the first Cal, the capital allocation line that goes through, that goes through the minimum variance portfolio, the minimum variance portfolio. So there we go, this is the first, this is the, I'm gonna call it CAL minimum. Now notice what happened. The line is no longer curved. So this is the curve line. The line is no longer curved. The line is straight because we are starting from the y-axis and we're drawing a line. Therefore, the curve will be straight. Let's draw another combination. So rather than zero and one, let's put 20% in stocks and what would be the expected return? The expected return will be around, notice the expected return will be 6% now. The expected return is six. So the expected return will be someplace like this. So we're gonna have, the line will go from, let me use a different color for this portfolio, 3% and will go through six. And this is, we're gonna call this portfolio CALA, portfolio A. Now we can draw, we can keep on drawing these portfolios. We can go and draw these portfolios. But the question is, what is the best portfolio? What's the optimal portfolio? What's the composition of the optimal risky portfolio? So here's what we can do to find this because we can also do, we can grab this. We can grab this. So how do we find out that we got to our optimal risky portfolio? So to find the composition of the optimal risky portfolio, we search the weight and the stock and bounce fund that maximize the portfolio sharpie ratio. Now let me, before we proceed, let's compute the sharpie ratio for these two portfolios. Let's compute the minimum variance. The minimum variance has a return of 5.46, 5.46 minus 3% the risk-free rate. This is the risk-free premium and the standard deviation is 7.8. So the sharpie ratio is 0.32. The sharpie ratio for A, the return is 6 minus 3% the risk-free and the standard deviation, we're gonna say 8% or around 8%. It's specifically 8.07, 0.37. So notice here, we would definitely prefer this portfolio over this portfolio. Yes, we'll prefer the portfolio A over the minimum variance because look, our sharpie ratio is higher, okay? Relative to this portfolio. Now again, we're gonna have to keep on finding out. Where do we find the maximum sharpie ratio? Because at some point it's gonna go down again. So we use this formula and you don't have to worry about this formula for now. We use this formula to find out the optimal weight. So to find the optimal weight, we find out that the optimal weight is 0.568 for bonds and 0.432 for stocks. So some place, so fives, we have the stocks here. The weight for stocks is 0.432. So at some place in this area, a little bit then 4.42 and the remaining is unbounds. And when we do the computation, basically when we compute the expected return of this portfolio, we find out that the expected return is 7.16 and the standard deviation equal to 10.15. And the sharpie ratio for this portfolio will be 0.41. So the expected return is 7.16 minus the risk free rate divided by the standard deviation. Again, how did we come up with this? We just assume and we kept on plotting all of these until we find out and computing the sharpie ratio. Again, the formula, don't worry about the formula. Don't worry about how we find out the formula. There's a formula to find out this optimal risky portfolio. Now let me show you the graph itself rather than my crappy graph. And this is the graph that I was telling you about. So if we keep on, once we introduce the risk free rate, first of all, the CAL is basically straight line not curved, okay? Because you're starting from the Y. So what's gonna happen is this, the optimal risky portfolio is at this point. This is an important point. So this is the efficient frontier. Let me do this in red. This is the efficient frontier, okay? And where the tangent point, where this line, where the CAL hit the efficient frontier at that point, this is when it hit it, like tangent. It's the tangent point. We call it the optimal risky portfolio. This is the optimal risky portfolio. This is the target portfolio, okay? And this is where you have the highest sharpie ratio or sharp ratio. I just, I always call him sharpie, I don't know why. The highest sharp ratio, which is 0.41, okay? Now we find out what's the combination. What's the best combination of stocks and bonds? It doesn't mean you, that's if you invested everything, you're not gonna invest 100% in here, but this is the optimal weight. This is the optimal weight, okay? So the best is to have 56 in bonds and 43 in stocks. So the preferred complete portfolio from a risky portfolio risk-free asset depends obviously on your risk aversion. More risk aversion person will prefer low-risk portfolio despite the expected return. And if you are more risk-tolerance, you choose the higher risk, which is we already learned about this in the prior session. Now, both type of investors will choose portfolio O. Why? Because portfolio O gives them the highest return per risk. It provides the highest return per risk that tangent point when you find this tangent point, this is your highest return per risk. So that's why whether you are risk averse or risk tolerant, you will prefer your best option is this combination. So now you don't, you know, investor will defer only in their allocation. Now they may want to put only 80% in this and remaining in the risk-free asset or they may wanna put 90% or they wanna put 80% here and 20% here. It just depends on how they split it. But the point is we know how to split it. So let's take a look at one possible choice, one possible choice of the preferred portfolio C. Let's assume the investor places 55% of their wealth in portfolio O and 45% in treasury, in treasury bill. Simply put, to find the expected return, we'll take 3% times 45. This is what they invested in the treasury bill and the remainder is invested in the risky portfolio that earned 7.16. They're not putting everything in 7.16. If they put all their money, in this portfolio, 100%, then the expected return will be 7.16, they're only putting, that's what they want. They're a little bit more than half, they want to go risky and a little bit less than half, they want to be risk-free. So the expected return of this complete portfolio is 5.29 and the standard deviation of this portfolio is only the standard deviation of the risky assets, which is the 10.15 from the previous slide. Therefore, the standard deviation of the portfolio is 5.18. Therefore, it would look something like this. So this portfolio, what they did is said, look, we're gonna invest 45% of our fund at 3% and 55% at 7.16. And here we have, when we compute the expected return, where we combine those two, compute the expected return, overall, we're gonna be expected to earn 5.29 with a standard deviation of 5.58. So simply put, here's what's gonna happen to be more specific, because portfolio is a mix of bond fund and stock fund with a weight of 56 and 43.2. Here's how we're gonna allocate it. For the risk-free asset, we're gonna give them 45% and the remaining 55% will be allocated using 56.8 to the bond fund and 43.2 to the risk fund. Simply put, another picture of this is something like this. So I would say this is a conservative portfolio because only little bit than half is invested in something other than risk-free and half of that, well, little bit more than half is even invested in bonds. So notice, I would say this is a conservative portfolio. What else can you do? For example, you could have done is 20%, keep 20% here and take the 80% and split the 80% 56 to bonds 43.2 to stocks and you'll be little bit more riskier. And if you want to, you could only do 5% and T-bills and risk-free and the remaining 95% split at 56 and 43. It all depends on your risk, risk tolerance. In the next session, we would look at an example that deals with the optimal risky portfolio with a risk-free asset. As always, I'm gonna remind you to like and share my recording if you like them and don't forget to visit my website, farhatlectures.com. If you want to complement and supplement this course as well as other courses, Accounting Finance. If you're studying for your professional certification, invest well in your career because that certification will pay dividend for years. Study well and stay safe.