Question

The Lubin’s Investment Team is considering investing in two securities, A and B, and the relevant information is given below:

State of the economy	Probability	Return on A(%)	Return on B(%)
Trough	0.05	-20%	2%
Recession	0.4	-5%	2%
Expansion	0.5	15%	2%
Peak	0.05	20%	2%

What should the team do if they wish to earn 10% expected return on their portfolio?

What is the standard deviation of the team’s portfolio?

Answer 1

1. Find probability weighted expected return of both the assets:

   • State of the economy	Probability	Return on A(%)	Probability weighted	Return on B(%)
     Trough	0.05	-20%	-1.00%	2%
     Recession	0.4	-5%	-2.00%	2%
     Expansion	0.5	15%	7.50%	2%
     Peak	0.05	20%	1.00%	2%
     		Expected return = sum of probability weighted returns	5.50%	2%

   • As asset B has the same return in all state of economy, it has the same expected return of 2% and 0 standard deviation
   • To earn 10% we need to solve the following equation:
   • 
     • 
     • 
     • 
     • 
     • 
     • 
     • So basically the 10% return can only be achieved if we short 1.3x times B and long 2.30x times the money in A,i.e the proceeds from shorting + the entire amount we have, we invest in A
2. As B has no standard deviation, the portfolio standard deviation is completely dependent on A

   • State of the economy	Probability	Return on A(%)	Probability weighted	Return on B(%)	P x (return - Expected return)^2
     Trough	0.05	-20%	-1.00%	2%	0.325%
     Recession	0.4	-5%	-2.00%	2%	0.441%
     Expansion	0.5	15%	7.50%	2%	0.451%
     Peak	0.05	20%	1.00%	2%	0.105%
     		Expected return	5.50%	2%
     			3.500%	Variance sum of P x (return - Expected return)^2	1.32%
     				Standard deviation = square root of variance	11.50%

   • 
   • So the portfolio standard deviation is 26.45%