So far we have looked at how to calculate the expected return through the mean and understand how risk is interpreted through the variance of an asset. Sometimes this theory is called the mean-variance investment theory. But the question now becomes, how do we incorporate this notion of the mean and the variance or this notion of expected return and risk? And how do we put them together into our model? There is a trade-off. The underlying principle is that the riskier the investment the greater the required potential return. So if I'm going to invest in something risky, loan you money or put some money in a stock, then I want a big return, because I want to be compensated for that risk. However, if the risk is not that great, then I'm not going to be expecting a higher return. If investors are willing to bear that risk, they expect to earn a risk premium, versus something that has a risk-free rate, something that's almost has zero-risk. This trade-off is referring to the possibility of higher returns on investment, but it does not guarantee that you'll actually get those high returns. So let's take a look at this graph, and on the x-axis, we see the variance here, and that is interpreted to be risk. On the y-axis, we have returns, expected returns through the means. Here we have plotted four stocks: Apple, Intel, Cisco, Nvidia. Let's try to understand a little bit about these stocks in terms of the risk return trade-off. If we look at this green spot here, which is Cisco, it has a low variance, but it also has a low return. If we have a look at Texas Instrument, it also has a low variance and a high return, it has the highest returns of all the stocks. Whereas, if we looked at Apple, it has a larger variance, so there's more risk to Apple. So the question I want you to think about is, where do we want to be on this graph? Do we want to be in the bottom right, the top right, the bottom left, or the top-left? What this theory assumes is that, for any investor, we want to minimize our risk and maximize our returns. So we want to be in this direction on risk and in this direction on returns. So if you were to pick between say Intel here, and purple is Texas Instruments, which of these two stocks would you pick? Well, if I draw a line here like this and a vertical line like this, we can see that Texas Instrument is preferable to Intel because it gives you a higher return for less risk. Whereas, if we look at Intel versus Apple, then we don't really know because Apple does have a higher return, but its also has a higher risk. If Cisco is almost here, and this also is not clear which of the two you choose, the risk is about the same, but Intel might be more. This really depends on the individual investor's ability to accept risk or liking risk, etc. One thing that we can look at in order to think about the risk reward trade-off is something known as the Sharpe ratio, which is defined as the expected return minus the risk-free rate over the standard deviation. So in the previous slide, it's the returns over the variance. In terms of this risk-free rate, if you think of it as 0 for now, we have the expected return over the risk. This little adjustment here for the risk-free rate allows you to compare it to something like a T-bill. A T-bill is considered a risk-free rate. If you invest in T-Bills, you will get money back almost guaranteed. So we really are not interested in just how much you get back but how much you get back above the risk-free rate, and that's why we take the difference. The ratio is the average return in excess of the risk-free rate per unit of volatility or risk. So what does that mean per unit of volatility or total risk? That's really standardizing this number by standard deviations. Then finally, the greater a portfolio Sharpe ratio, the bigger the ratio, the better its risk-adjusted performance. So let's do a little calculation, a practice calculation on the Sharpe ratio. Let's consider the example used to calculate our previous portfolio parameters. So we've calculated the expected return to be 13.5 with a standard deviation of 0.05416. Let's assume for now the risk-free rate to be 0, and the Sharpe ratios calculated simply taking the difference between the expected return minus the risk-free rate and then dividing by that standard deviation, and so we have a Sharpe ratio of 0.58.