Question

There are 2 stocks in your portfolio. You own 65% of Stock A and 35% of stock B. The expected returns on each stock with the probability of those returns are:

State of the Economy Probability Stock A Stock B

Recession 30% -15 -6

Normal 45% +12 + 8

Boom 25% +30 +19

What is the variance of each stock?

What is the standard deviation of each stock?

What is the expected return of each stock and the expected return of your portfolio?

How much does the expected return change if instead of the existing ownership, you own 50% of each stock?

Answer 1

The formulas for Expected return and Expected Standard Deviation

  

where pi represents the individual probabilities in different scenarios

Ri represents the corresponding returns in different scenarios and

represents the expected return calculated as above.

So, Expected Return of Stock A =30%*(-15%)+45%*12%+25%*30% =8.4%

Expected Return of Stock B =30%*(-6%)+45%*8%+25%*19% =6.55%

Standard Deviation of Stock A = (0.3*(-0.15-0.084)^2+0.45*(0.12-0.084)^2+0.25*(0.30-0.084)^2)^0.5

=0.169334 = 16.93%

Variance of Stock A = 0.169334^2 = 0.028674

Standard Deviation of Stock B = (0.3*(-0.06-0.0655)^2+0.45*(0.08-0.0655)^2+0.25*(0.19-0.0655)^2)^0.5

=0.0932456 = 9.32%

Variance of Stock B = 0.0932456^2 = 0.0086948

Expected Return of portfolio is the weighted average return of constituent stocks

So.

Expected Return of portfolio = 65% * Stock A's return + 35% * Stock B's return

=0.65* 8.4%+ 0.35*6.55%

=7.7525%

If the ownership is 50% in each stock, then

Expected Return of portfolio =0.50* 8.4%+ 0.50*6.55% = 7.475%

So, expected portfolio return will change (decrease) by 7.7525%-7.475% =0.2775%