Stochastic Programming - Index of

DYNAMIC SYSTEMS 125
Stage 0
A
B
8%
12%
7%
Stage 1
A B A B A B
5% 9% 5% 5% 9% 5%
5%
9%
5%
Stage 2
Figure 7
Stochastic decision tree for the simple investment problem.
into account A or into B. IfwechooseA, we shall experience an interest rate
of 8% or 12% for the first year. After that we shall have to make a new decision
for the second year. That decision will be allowed to depend on what interest
rate we experienced in the first period. If we choose A, we shall again face
an uncertain interest rate. Whenever we choose B, we shall know the interest
rate with certainty.
Having entered a world of randomness, we need to specify what our decisions
will be based on. In the deterministic setting we maximized the final amount
in account B. That does not make sense in a stochastic setting. A given series
of decisions does not produce a certain amount in account B, but rather
an uncertain amount. In other words, we have to compare distributions. For
example, keeping the money in account A for both periods will result in one
out of four sequences of interest rates, namely (8,5), (8,9), (12,5) or (12,9).
Hence, if we start out with, say 1000, we can end up with (remember the fees)
1083, 1125, 1125 or 1169 (rounded numbers).
An obvious possibility is to look for the decision that produces the highest
expected amount in account B after two periods. However—and this is a very
important point—this does not mean that we are looking for the sequence
of decisions that has the highest expected value. We are only looking for the
best possible first decision. If we decide to put the money in account A in the
first period, we can wait and observe the actual interest rate on the account
before we decide what to do in the next period. (Of course, if we decide to use
B in the first period, we can as well decide what to do in the second period
immediately, since no new information is made available during the first year!)