Mean and Standard Deviation of a Portfolio

Now, let's combine the two securities we introduced above, A and B, to form a portfolio. You may think of this portfolio as a simple mutual fund. Suppose 10% of your money is invested in Security A and the rest (=90%) is in Security B. To calculate the expected rate of return and the standard deviation of the portfolio, we need to first calculate the rate of return on this portfolio in the three states of the economy. The calculation is shown in the next table.

State of Economy	Probability of this state occurring	rA (Return on Security A)	rB (Return on Security B)	rP (Return on Portfolio P) = 0.1 x rA + 0.9 x rB
Boom	0.2	70%	-15%	0.1 x 70 + 0.9 x (-15) = -6.5
Normal	0.5	15%	11%	0.1 x 15 + 0.9 x 11 = 11.4
Recession	0.3	-30%	15%	0.1 x (-30) + 0.9 x 15 = 10.5

Based on the portfolio's returns calculated above, we now calculate the expected return as E(r_P) = 7.55%, ²_P = 49.50, _P = 7.04%.

See the same Excel file posted in this lecture for the calculation of portfolio standard deviation using Excel. See different worksheets (taps) for different settings.

The expected return of 7.55% is very close to 7% (= E(r_B)) because 90% of your money is invested in B. How about the standard deviation of the portfolio? Is it between _A = 34.73% and_B = 11.14%? The answer is NO. Why is the standard deviation of the portfolio lower than the standard deviation of A or B? Please go to the Discussion Board and participate in the Discussion for this Session.

Discussion Questions #1 In the above example, the standard deviation of the portfolio's return is lower than those of the two securities, A and B. What is the intuition behind this? Is it always the case? Examine how the returns on these two securities behave. Do they move together or in opposite directions?