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 assets of visible investments are And we have a number of risky assets. Let's expand the available investment choices to N risky assets and one riskless 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 2 risky assets the expected return E(rp) = w1E(r1) + w2E(r2) where w1 is the weight in the first risky asset. And E(r1) is its expected return. Similarly w2 and E(r2) are the rates and expected return of the second risky asset. Remember, weights must still add to one, that is w1 + w2 = 1. The portfolio variance sigma sub b squared is w1 squared times sigma 1 squared + w2 squared times sigma 2 squared + 2(w1)(w2) times sigma1,2. Where the sigmas of the standard deviations of the two risky assets. And sigma 12 is the covariance between the returns of the two assets. This covariance further equals rho 12 times sigma 1 times sigma 2, where rho 12 is the correlation between the two assets returns. Taking the square root of the expression for portfolio variance, gives us a standard duration of portfolio returns, sigma sub b. 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 off of portfolio E(rp) = w1E(r1) + w2E(r2), so on plus the last term wn times E(rn), where again all the rates must add to 1. The variance of this portfolio, sigma sub p squared equals w1 squared times sigma1 squared + w2 squared times sigma 2 squared, plus so on until the last term, wn squared times sigma plus 2 times w1 times w2 times sigma 12, plus 2 times w1 times w3 times sigma 13 Plus so on, the last term being 2 times wn1 times wn times sigma, n minus 1 and n. 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 E(rp) = w0.1 + (1- w)0.2. And its variance sigmap 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 0.7 times 0.1. Note that w is the weight an asset x and 1- w is the weight in asset Y. Plotting expected returns and standard deviations will depend on the values for w and rho. A given pair of risky assets will always have a only one rho. So let's start off by setting rho = +1. Then, for different values of w, between 0 and 1, we can compute a portfolio expected return, it's variance, and its standard deviation. And then, plot them on a graph that has expected returns on the vertical axis, and standard deviation on the horizontal axis. We use a value of 0, 0.2, 0.4, 0.6, 0.8 and one for w while generating the plots. In the figure that you see, this is representative 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 a correlation of plus one, between the returns. If the correlation coefficient is 0.5, we get the dark blue curve. If rho is 0, we get the light blue curve. At the value of -0.5 for row, we get the green curve. And finally at the value of -1 for row, 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 the portfolio expected return, does not include correlation coefficient rho. Secondly as the correlation decreases from plus 1 to a negative 1, 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 go so far back to the left, that the standard deviations of a number of portfolios fall with x and y are far lower than the individuals standard deviations of assets x and y. This is the idea of diversification. Which we will discuss in greater detail, next time.