Hello. Welcome back. In previous lectures, you learned how to describe the return distribution of a single asset. You learned how to find the expected return for a single asset given a probable distribution of the possible outcomes or given a time series of returns. You also learn how to measure the risk of a single asset. Now, we rarely hold a single asset however. Investors never usually arrive, just say usually, hold a single asset. Typically, we hold a collection of different assets, we hold the portfolio. So what we need to do is we need to develop similar measures for the expected return and risk for a combination of securities for a portfolio assets, rather than individual securities. So in this lecture, you will learn how to measure the expected return for portfolio securities. So let's first recall how we measure the expected return for a single asset. Remember that, given the distribution of returns, right, we use the central tendency, or the average, as the measure of the expected return. So for example, if we had a probability distribution of possible outcomes, the expected return would be simply, the probability weighted average of the possible outcomes. So it would be, the expected return would be, The probability weighted of the possible outcomes. So let's do an example. So here suppose we have shares of stock of Toyota, Walmart and Pfizer and we have four possible states, we have expansion, we have normal times. We have recession and we have a depression. So these are the states of the economy with, let's say, these given probabilities. So you learned before how to find the expected return given these probability distribution. So what is the expected return for Toyota, for example. Remember it's the probability way that average of the possible returns in it's state. So it's going to be time for some probability expression, times return in that state plus 40% probability of a normal times, times the return in that state. 30% probability of recession times the return in that state. Finally, plus 20% productivity in depression minus times the return in that state. And if you computed all that you would find that the expected return for Toyota, given these probability distribution these probable outcomes, is going to be 3.6%. You can find the expected return for Walmart and Pfizer similarly. What if you have a portfolio of assets? What is a portfolio? Well, a portfolio is a collection of assets that we can treat as if it's a single asset. So how would you find a portfolio expected return? So for example, suppose you have a portfolio that consists of Toyota and Pfizer. Suppose you invested 50% of your wealth in Toyota shares and 50% of your wealth in Pfizer shares. How would you compute the expected return on that portfolio? Well, let's first find the return to that portfolio in each state, and then find the expected return. So since Toyota and Pfizer is equally weighted in this example, we can just take the average of these two possible returns in each state. In an expansion, the portfolio return in this case would be 4.25%. In normal times it would be 3.5%. In a recession it would be 1.5% and in depression it would be 5%. And now, we can find the expected return, again, just we did before, as the probability weighted average of these possible outcomes. So the portfolio return will be 0.1 x 4.25% + 0.4 x 3.5%, 1.5%, 0.2 x 5%, which would give you 3.275%, which is the expected return to this portfolio. Now, alternatively, notice that we could've also find the expected return, by Using the, taking the average of these, the expected returns of the individual assets. We could've found the expected term on this portfolio, as the weighted average of the expected return on each assets. In other words, we could have written the weight in Toyota, times its expected return plus the weight in Pfizer, times its expected return. And again, it would have given us 3.275%. In general, the expected return of a portfolio is simply equal to the weighted average of the expected returns in each asset. So if you have two assets, asset 1, with weight W1, and asset 2, with weight W2. What is the portfolio return? The expected expenditure on the portfolio? Well, it will be that the weight of the first one times the expected return on the first one plus the weight of the second one times the expected return on the second one. Now, if you have N asset, if you have more than two assets, if you had N assets, each with weight WI in expected return, RI, what is the portfolio expected return on that portfolio, it would be the weighted average of the expected returns on the individual assets. So we already looked at two asset portfolios. What if we make an equally weighted portfolio of all three assets? Well, it's simply going to be that one third times, they expect a return on Toyota one third times the expected the return on Walmart, one-third times the expected return on Pfizer which will give you the expected return on a portfolio that is equally weighted of the all three assets. So in this lecture, we talked about the fact that investors rarely hold only a single asset. Instead, they typically hold a collection of assets, or a portfolio of assets. So you learn how to compute the portfolio expected return. So how do you compute the portfolio expected return? Well, given the portfolio weights for individual assets, the portfolio expected return is simply the weighted average of the individual assets' expected returns.