If you can find another portfolio that has a higher expected return but doesn’t have a higher risk or doesn’t have higher volatility that will be an inefficient portfolio. So for example, let’s say we’ve got portfolio Y and portfolio Z and they each have the volatility of 20%. They each have the same total risk, but we find that portfolio Y has an expected return of 32% and then portfolio Z has an expected return of this 27%. Now they have the same risk, but portfolio Y has a higher expected return so we would then say that portfolio Z is inefficient. This is an inefficient portfolio because why would you ever pick portfolio Z when you could go and get portfolio Y and have the same amount of risk and yet have a higher expected return?
So I’ll grab this out and show you a more in-depth example. So, let’s say that we have two stock portfolios and we can invest in stock A and stock B, and we can invest in different weights. So we could have 10% of the portfolio stock A and 90% of Stock B, we could do 50/50, we can assign different weights to stock A and stock B. And so, we have 80/20, we have 70/30, we do different weights and we see for each weighting what is the volatility in the expected return. So what’s volatility and expected return for 80% of the portfolio and stock A and 20% in stock B. What’s the volatility expected return for the 70/30 split? So we look at these different splits and then we plot it out.
So we’ve got volatility on the x-axis, which is the standard deviation of the expected returns. So that’s our measure of total risk. Then on the y-axis, we have the expected return of the portfolio. So let’s just say we graph it out and then when we graph these volatility's and expected returns for the different combinations we'll end up with a curve that looks like this. Now I’ve put the curve in different colors and I’m going to show you why. So, we’ve got yellow at the bottom and then we’ve got purple up there.
Let’s take two points just say that is 10/90 and when I say 10/90 I mean that It’s a portfolio where 10% would be invested in stock A and 90% would be in stock B. So when we have 10/90 we have a volatility of 15%. We could just see here the volatility is 15% and then our expected return is 5%. Now, notice if we were to take another combination up in the purple line. Let’s just say that it’s 80/20. So is 80% of the portfolio is invested in stock A and 20% is in stock B. So instead of 10/90 now we’re going to look at 80/20. It looks like about the same amount of risk we’ve got volatility of 15% for this portfolio weighting, but look, when we trace over to see, what is the expected return for this portfolio we see that it’s 20%.
So, think about it just use common sense and kind of forget about all the finance and so forth, if I offer you two opportunities, and they each have identical risks, but one of them has an expected return of 20%, and the other one has an expected return of 5%. Which one are you going to pick? You’re going to pick the 20%. We would say therefore that this portfolio here with the 10/90 split, 10% in stock A and 90% in stock B this is an inefficient portfolio. It’s inefficient because we could go and find a different portfolio weighting that has a higher expected return but does not have a higher risk. So there’s no reason to accept additional risk when you’re not being compensated for that risk.
So we would assume that all the investors would go and say "If I’ve got two portfolios and they have identical risk. I’m going to pick the one with a higher return." So, all these ones that are on the purple line here, all of those are different portfolio combinations of stock A and stock B that are efficient. Because for any given point like let’s say this (80/20) portfolio weighting, we cannot find
another portfolio that has a higher expected return without having a higher risk. Therefore we would see this as efficient.
