Question

A financial institution has the following options for investing funds to a basket of asset consisting of fixed income security, bond and highly risky asset. The distributions given below are the two investment options in terms of their return and its associated probability. We define that the profit= 0.9*return – expense.  

We also assume that the expense for Option A is $1 million and the expense for Option B is $1.8 million.

Option A			Option B
Return ($ millions)	Probability		Return ($ millions)	Probability
-2	0.2		-4	0.1
1	0.4		-1	0.2
5	0.2		5	0.5
10	0.2		8	0.2

What is the expected value of the profit for Option A? (in $ million)

What is the expected value of the profit for Choice B? (in million)

What is the standard deviation of the profit for Option A? (in $ million)

Answer 1

The probability distribution for profits from A and B are obtained here as:

Probability	P_A	P_B
0.2	0.9*(-2) - 1 = -2.8	0.9*(-4) - 1.8 = -5.4
0.4	0.9*1 - 1 = -0.1	0.9*(-1) - 1.8 = -2.7
0.2	0.9*5 - 1 = 3.5	0.9*5 - 1.8 = 2.7
0.2	0.9*10 - 1 = 8	0.9*8 - 1.8 = 5.4

a) The expected value of profit from option A is obtained here as:

= 0.2*(-2.8) + 0.4*(-0.1) + 0.2*(3.5) + 0.2*8 = 1.7

Therefore the expected value of the profit for option A here is given as: $1.7 million

b) the expected value of profit from option B is obtained here as:

= 0.2*(-5.4) + 0.4*(-2.7) + 0.2*2.7 + 0.2*5.4 = -0.54

Therefore the expected value of the profit for option B here is given as: -$0.54 million

c) The second moment of profit from option A is first computed here as:

= 0.2*(-2.8)² + 0.4*(-0.1)² + 0.2*(3.5)² + 0.2*8² = 16.822

Therefore the standard deviation here is computed as:

Therfore $3.73 million is the required standard deviation here.