[BLANK AUDIO] Learning outcomes. After watching this video, you will be able to calculate the expected return and variance of portfolio, of two risky assets. Plot the investment opportunity set for various weights in the risky assets. Understand how the investment opportunity set changes with the correlation between the two assets' returns. Risky portfolios. In this video, we will see what our set of feasible investment choices are. When we have a number of risky assets. Let's expand the available investment choices to end risky assets and one risk-less or risk-free asset. How should you invest your money across these N plus one assets? Before we answer that question. Let's look at the portfolio math for larger portfolios. For a portfolio with two risky assets, the expected return Sub P is is W1 times Er1 plus W2 times Er2. Where W1 is the weight in the first risky asset, and Er1 is its expected return. Similarly, W2 and Er2 are the weights and expected return of the second risky asset. Remember, weights must still add to one. That is, W1 + W2 equals 1. The portfolio variance, sigma sub B squared is W1 squared, times sigma 1 squared, plus W2 squared, times sigma 2 squared, plus two times W1, times W2, times sigma 1, 2. Where the sigmas are the standard deviations of the two risky asset, and sigma one, two is the covariance between the returns of the two assets. This covariance further equals rho one, two times sigma one times sigma two. Where rho one, two is the correlation between the two assets returns. Taking the square root of the expression for portfolio variance, gives us a standard deviation of portfolio returns sigma sub P. Remember that the correlation coefficient can take values only between minus one and plus one. The math can be extended to a portfolio of N risky assets. The expected return of a portfolio Sub P equals W1 time Er1 plus W2 times Er2, so on, plus the last term WN times Ern, where, again, all the weights must add to one. The varies of this portfolio, sigma sub P squared equals W1 squared, times sigma 1 squared, plus W2 squared, times sigma 2 squared, plus so on until the last term, Wn squared times sigma n squared, plus 2 times W1 times W2, times sigma 1, 2 plus 2, times W1, times W3, times sigma 1, 3 plus so on. The last term being 2 times Wn1, times Wn, times sigma N minus 1N. Let's consider a portfolio of two risky assets now. Asset X has an expected return of 10% and a standard deviation of returns of 7%. And asset Y has an expected return of 20% and a standard deviation of returns of 10%. The expected return of this portfolio Sub P is equal to W times 0.1, plus 1 minus W, times 0.2. And its variance sigma sub P squared is W squared times 0.07 squared, plus 1 minus W squared, times 0.1 squared, plus 2 times W, times 1 minus W, times rho, times 0.07 times 0.1. Note that W is the rate asset X, and 1 minus W is the rate and asset Y. Plotting expected returns and standard deviation will depend on the values for W and rho. A given pair of risky assets will always have only one rho, so let's start off by setting rho equal to plus 1. Then for different values of W between zero and one, we can compute a portfolio expected return, it's variance, and it's standard deviation. And then plot them on a graph that has expected return on the vertical axis and standard deviation on the horizontal axis. We use values of zero, 0.2, 0.4, 0.6, 0.8 and one for W, while generating the plots. In the figure that you see, this is represented by the straight black line which is called the investment opportunity set. This represents all possible combinations of expected returns and risk given the two risky assets X and Y, and the correlation of plus one between the returns. If the correlation coefficient is 0.5, we get the dark blue color. If rho is zero, we get the light blue curve. At the value of negative 0.5 for rho, we get the green curve, and finally at a value of negative one for rho, we get the pink curve which you can see bends all the way back to the vertical axis. There are a couple of interesting things to note about these graphs. One, when the correlation changes, only the portfolio variance changes. Portfolio expected returns for a given W does not change. This is expected, as the formula for portfolio expected return, does not include the correlation coefficient rule. Secondly, as the correlation decreases from plus one to negative one, the plot goes further to the left, that is we have portfolios with lower risk for the same expected return as the correlation decreases. In fact, some of the plots goes so far back to the left that the standard deviations of a number of portfolios formed with X and Y are far lower than the individual standard deviations of assets X and Y. This is the idea of diversification which we will discuss in greater detail next time.