Portfolio expected return and standard deviation.

We can calculate expected returns and standard deviations for single assets, but we can also do it for broad portfolios with multiple assets.

Now, the expected return calculation for portfolio is very, very straightforward. It's just a simple weighted average of all the expected returns of the individual securities. And here's your typical formula where we weight each security by the weighting, in the portfolio multiply it by its expected return, and sum them all up. Very, very straightforward. Now, the standard deviation of a portfolio or also known as volatility, is a lot more complex. And unfortunately, that complexity increases as we add more and more investments or assets to the portfolio. It's not just a simple weighted average because we also need to take into account how the assets within the portfolio move together. And here's your typical standard deviation formula. As you can see, it's a little bit of a mouthful. Let's dig in further into that standard deviation formula. Now, this is a standard deviation formula for a portfolio of just two assets. You can see the first term we're looking at the variance or the variability of each individual asset and multiplying it by their waiting squared. So, so far it's pretty similar to an expected return calculation where we're taking the values for each individual asset and weighting them by how big they are in the portfolio. But this third term here is where the big difference comes in. It's it measures, it incorporates correlations, it measures how the assets move together. So the risk of a portfolio isn't just made up of the risk of the individual assets. It also needs to take a look into account the interaction of the assets together which we measure with a statistic called correlation. Now this is just a formula for portfolio of two assets. As I mentioned, the complexity of this calculation gets larger and larger as we add assets to the portfolio. So a portfolio of four assets, for example would not only have four individual weights of securities and variances but it will also have six correlation terms. A portfolio of five assets will have 10 correlation terms. So as you can see, it gets more and more complex. Now you may also see a similar formula but called variance. Now variance is not a particularly easy statistics to interpret on its own. So most analysts calculate the standard deviation which is simply the square root of the variance. It's a lot easier to interpret and a lot easier to compare across assets. Now let's dig into how correlations affect portfolio variability in standard deviation. Again, here's that typical standard deviation formula for two asset portfolio. The first scenario that we'll look at is when both assets are perfectly correlated they have a perfect positive correlation or a correlation of one. In this scenario, both assets move perfectly together. They both move up or down at the same time by the same proportion. So their variability or their variance is equal. As a result, if both assets move perfectly together the portfolio standard deviation is equal to the individual security standard deviation. And in this scenario, there's no offsetting impact since both assets are moving together perfectly there are no benefits from diversification. Now the more likely scenario, and this is what you'll see, I'd say 99% of the time, is when the correlation value is between negative one and positive one. In this scenario, the standard deviation of the portfolio is actually gonna be less than the weighted average of the individual security standard deviations. And that's because of diversification. And that diversification benefit increases as correlation continues to get closer to that negative one value. And lastly, we'll look at a perfect negative correlation. So a correlation of negative one. In this scenario, both assets move in different directions but at the same proportion. So it's essentially a perfect hedge. Asset A moves up by X, asset B will move down by minus X.

So the variability of the portfolio is actually zero and you have a perfect hedge. And because you have a perfect hedge and the variability of the portfolio is zero, it's essentially a risk-free portfolio. Some other key points to keep in mind, generally as you add more and more assets to a portfolio you see an incremental diversification benefit. Assuming of course that the correlation is not a perfect positive one, however, you do get to a point where you get diminishing effects. Obviously, owning five stocks or asset classes is better than owning one single position, but there comes a point where adding more investments to a portfolio stops making a meaningful difference. At that point notwithstanding, investors tend to broaden their universe when looking for assets to increase diversification. So they look across industries, across other asset classes, styles, and other parts of the world. Now, as we look to add more assets or different asset classes to our portfolio to increase diversification, we must be careful not to add highly correlated assets. This would be very redundant. It would add a little benefit on the diversification front and just add costs. And lastly, while we've talked about how adding assets and asset classes and securities with different characteristics increases diversification we must also take into account what it does to our expected return. And in fact, when we're creating optimal portfolios, our goal is to reduce risk without sacrificing our expected return.