Suppose that $4 million is available for investment in three projects. The probability distribution of the net present value earned from each project depends on how much is invested in each project. Let It be the random variable denoting the net present value earned by project t. The distribution of It depends on the amount of money invested in project t, as shown in Table (a zero investment in a project always earns a zero NPV). Use dynamic programming to determine an investment allocation that maximises the expected NPV obtained from the three investments.
Table
|
|
Investment (millions)
|
Probability
|
|
Project 1
|
$1
|
P(I1 = 2) = 0.6
|
P(I1 = 4) = 0.3
|
P(I1 = 5) = 0.1
|
|
$2
|
P(I1 = 4) = 0.5
|
P(I1 = 6) = 0.3
|
P(I1 = 8) = 0.2
|
|
$3
|
P(I1 = 6) = 0.4
|
P(I1 = 7) = 0.5
|
P(I1= 10) = 0.1
|
|
$4
|
P(I1 = 7) = 0.2
|
P(I1 = 9) = 0.4
|
P(I1= 10) = 0.4
|
|
Project 2
|
$1
|
P(I2 = 1) = 0.5
|
P(I2 = 2) = 0.4
|
P(I2 = 4) = 0.1
|
|
$2
|
P(I2 = 3) = 0.4
|
P(I2 = 5) = 0.4
|
P(I2 = 6) = 0.2
|
|
$3
|
P(I2 = 4) = 0.3
|
P(I2 = 6) = 0.3
|
P(I2 = 8) = 0.4
|
|
$4
|
P(I2 = 3) = 0.4
|
P(I2 = 8) = 0.3
|
P(I2 = 9) = 0.3
|
|
Project 3
|
$1
|
P(I3 = 0) = 0.2
|
P(I3 = 4) = 0.6
|
P(I3 = 5) = 0.2
|
|
$2
|
P(I3 = 4) = 0.4
|
P(I3 = 6) = 0.4
|
P(I3 = 7) = 0.2
|
|
$3
|
P(I3 = 5) = 0.3
|
P(I3 = 7) = 0.4
|
P(I3 = 8) = 0.3
|
|
$4
|
P(I3 = 6) = 0.1
|
P(I3 = 8) = 0.5
|
P(I3 = 9) = 0.4
|