This movie/ video uses Excel to demonstrate how diversification helps reduce overall risk, continued by the concept of efficient frontiers. The following is a part of the transcript.

Assume there are only 2 equities in the market, A and B. A has expected return E_A =4% and B has expected return E_B = 8%. Their risks (standard deviation) are s_A = 9% and s_B = 15%. Therefore, the maximum return is 8% with pretty high risk (15%) or the minimum risk is 9% with pretty low return (4%). Is there anyway, we can either reduce the risk while keeping the maximum return or increases the minimum return with minimized risk?

The answer is no to the first question and yes to the second question.

For the yes part, we can achieve by diversification, i.e. investing both assets at certain proportion. First, let’s assume that the correlation of A and B is 0.2. So, we can compute the expected return of the portfolio according to the following equation

E(portfolio) = W_A*E_A + W_B*E_B

where W_A and W_B are the weightings of A and B in the portfolio.

Then we may compute the risk of the portfolio (as s) using:

s (portfolio) = sqrt((W_A* s_A)² + (W_B* s_B)²+2*W_A*W_B* s_A*s_B*r)

where r is the correlation between A and B.

Then we can plot the E(portfolio) vs. s (portfolio). You see that by diversification (adding B into the A), we are able to increase the expect return but with lower risk than investing A alone! Why is that? This is because when A is reduced and B is added, the decrease in the risk contributed by A is faster than the increase in the risk contributed by B. And the increase in the return contributed by B is faster than the decrease in the return contributed by A. Therefore, we are able to achieve lower risk with higher return.

How much can this be reduced depends on the correlation between A and B also. You can see that the diversification is the most significant when r = -1.

In any market, there are finite number of equities (though a lot!), so by figuring out the return and risk of each equity and correlation between each pair of equity, we can plot a tradeoff curve like this!

Now, we can easily form the efficient frontier. The efficient frontier is just the points on this trade-off curve that give the maximum return for a given risk.