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 I_{t} be the random variable denoting the net present value earned by project t. The distribution of I_{t} 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(I_{1} = 2) = 0.6
P(I_{1} = 4) = 0.3
P(I_{1} = 5) = 0.1
$2
P(I_{1} = 4) = 0.5
P(I_{1} = 6) = 0.3
P(I_{1} = 8) = 0.2
$3
P(I_{1} = 6) = 0.4
P(I_{1} = 7) = 0.5
P(I_{1}= 10) = 0.1
$4
P(I_{1} = 7) = 0.2
P(I_{1} = 9) = 0.4
P(I_{1}= 10) = 0.4
Project 2
P(I_{2} = 1) = 0.5
P(I_{2} = 2) = 0.4
P(I_{2} = 4) = 0.1
P(I_{2} = 3) = 0.4
P(I_{2} = 5) = 0.4
P(I_{2} = 6) = 0.2
P(I_{2} = 4) = 0.3
P(I_{2} = 6) = 0.3
P(I_{2} = 8) = 0.4
P(I_{2} = 8) = 0.3
P(I_{2} = 9) = 0.3
Project 3
P(I_{3} = 0) = 0.2
P(I_{3} = 4) = 0.6
P(I_{3} = 5) = 0.2
P(I_{3} = 4) = 0.4
P(I_{3} = 6) = 0.4
P(I_{3} = 7) = 0.2
P(I_{3} = 5) = 0.3
P(I_{3} = 7) = 0.4
P(I_{3} = 8) = 0.3
P(I_{3} = 6) = 0.1
P(I_{3} = 8) = 0.5
P(I_{3} = 9) = 0.4