in this video we will cover a very important topic in the field of financial economics called portfolio theory and also cover some of the concepts that have advanced the field of portfolio theory and asset management over the number of years to begin with the most logical question is what is portfolio theory about portfolio theory is essentially about risk return preferences and opportunities suppose we have three assets a B and C and let's say we know the expected return and volatility of each of the three assets comparing asset B to asset a investors would prefer asset B over asset a and the reason is that even if a and B have the same volatility the expected return of B is greater than the expected return of a similarly when comparing asset C to asset a investors would prefer C over a and that's because even if a and C have the same expected return asset C has much lower volatility than asset a so portfolio theory is essentially about maximizing a portfolios return and minimizing its risk consider a portfolio with two assets a and B where expected return of asset a is greater than the expected return of asset B and the volatility of asset a is greater than the volatility of asset B then the volatility of the portfolio with assets a and B equals proportion of asset a the portfolio square times the variance of a plus proportion of acid being the portfolio square times the variance of B plus 2 times a B volatility of a volatility of B times a correlation between a and B the right hand side of this equation so far follows from the standard definition of variance of a portfolio with two assets and we take the square root to compute the vol 3 an important unknown variable here is the correlation between assets a and B let's suppose that the two assets are perfectly positively correlated in that case the volatility of the portfolio reduces down to the weighted sum of individual volatilities in risk-return space the feasible set becomes this straight red line that connects a and B and as we vary the weights of a and B the portfolio we move along the straight line now what is the correlation between the two assets is negative one so they are perfectly negatively correlated in that case the volatility of the portfolio reduces down to plus minus because we're taking the square root to compute the volatility a time's the volatility of a minus B times the volatility of B which is greater than equal to zero let the volatility of the portfolio be equal to zero in that case the ratio of A to B just equals the ratio of volatility of B to the volatility of a in risk-return space the point on the y-axis where volatility of the portfolio is 0 corresponds to correlation of negative 1 at this point a over B equals the volatility of B over the volatility of a finally if the correlation between assets a and B in the portfolio equals zero in other words there's no correlation between the two assets then the volatility of the portfolio equals proportion of acid aid the portfolio square times the variance of aid plus the proportion of acid being the portfolio square times the variance of B and we take the square root of this which is less than the weighted sum of two volatilities question is why is the portfolio volatility less than the weighted sum of qualities and the reason is that in any portfolio the volatility comprises of two parts the volatility of individual assets plus the covariance between assets which comes in through correlation when correlation itself is zero there's no covariance between the assets and hence portfolio volatility will be less that if assets were perfectly positively correlated in that case the feasible set becomes this curved mean various boundary that lies between the straight lines that connect and be when assets are perfectly positively correlated and this point on the y-axis when assets are perfectly negatively correlated so our feasible set is represented by this green curve boundary when correlation between assets is zero so far we've been dealing with portfolios with two assets question is what happens when the number of assets in a portfolio is greater than two let n represent the number of assets in a portfolio that if n is greater than equal to three our feasible set becomes this area bounded by this mean various boundary this curve is called the mean variance frontier or boundary and the point on this mean variance boundary that has the lowest volatility is called the minimum variance portfolio the part of the mean various frontier that lies above the minimum variance portfolio represented by this red curve line is called the efficient frontier any point on this efficient frontier has greater expected return for a given level of risk than any other point inside the feasible set represented by this blue shaded area based on to fajn separation the mean values frontier can be created from any two portfolios on the frontier and the entire efficient frontier can be created by using any to mean various efficient portfolios finally let's say we have n assets that have identical variance and then there's covariance between these assets then the variance of the portfolio that has all these n assets equals one over N times the variance of each individual asset plus and minus one divided by n times the covariance between the assets if the number of assets in the portfolio grows really large let's take the limit as n approaches infinity then the variance of the portfolio reduces down to covariance between assets let me explain what's going on here as we increase the number of assets the variance of the portfolio comes down and that's because of the Coriolis effect each asset has a unique or diversifiable risk and it has market or systematic risk so as we increase the number of assets in the portfolio the unique risks tend to diversify away and the only risk we're left with is the market or systematic risk and in our example that non diversifiable risk comes in from the covariance between the asset note that if all asset returns are independently distributed then the covariance will be 0 and the volatility of the portfolio will also be 0 so this tutorial has given us an overview of what portfolio theory is about and how investors aim to maximize return and minimize risk and they make use of correlations to construct portfolios that suit their needs we have also seen the efficient frontier includes portfolios with greater expected return for a given level of risk than any other portfolio inside the feasible set and finally as we increase the number of assets in a portfolio the unique risks of those assets tend to diversify away and what you are left with is not the rest viable if there are any questions or comments please feel free to post thank you