3. Suppose you have $5,000 to invest. You will have an opportunity to invest in
ID: 3361117 • Letter: 3
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3. Suppose you have $5,000 to invest. You will have an opportunity to invest in either of two investments (A or B) at the beginning of each of the next 3 years. Both investments have uncertain returns. For investment A you will either lose your money entirely or (with higher probability) get back $10,000 (i.e., a profit of S5, 000) at the end of the year. For investment B you will get back either just your S5,000 or (with low probability) $10, 000 at the end of the year. The probabilities for these events are as follows: Amount Investment Returned (S) Probability 10,000 5,000 10,000 0.3 0.7 0.9 0.1 You are allowed to make only (at most) one investment each year, and you can invest only S5, 000 each time. (Any additional money accumulated is left idle.) (a) (points: 5+6+1) Use dynamic programming to find the investment policy that maxi- mizes the expected amount of money you will have after 3 years. Specify stages, states, decision variables, the value function, its recursive relationship and show the interme- diate steps of the solution procedure. Identify the optimal investment policy and the optimal expected amount of money that you will have (b) (points: 5+6+1) Use dynamic programming to find the investment policy that max- imizes the probability that you wil have at least $10,000 after 3 years. Specify stages, states, decision variables, the value function, its recursive relationship and show the in- termediate steps of the solution procedure. Identify the optimal investment policy and the optimal probability of having at least $10,000 after 3 yearsExplanation / Answer
My approach was to start in the last period (n=3). In this period, I will either: (1) have at least$5000 so that I can make an investment, or (2) have no remaining capital and not be able to make an investment. In the second scenario I obviously don't have any decision to make. In the first scenario, the optimal choice would be AA, as
E(A)=0.30+0.710000=7000>5500=0.95000+0.110000=E(B)
where E() is expected value. Now,this conclusion is that E(A)>E(B) - means I should choose A in every period.
It ignores the expected returns from future periods. Investment A may have higher expected returns in a single given period, but it also has the risk of zero return and hence the risk of not being able to invest in future periods.
In the second period, for example, suppose you have $5000 to invest. The expected return of A is:
E(A)=0+0.710000=7000,
where the second term captures the expected return of being able to invest in A (the optimal choice) in period 3. The expected return of B, however, is
E(B)=5500+7000=12500
because there will be enough money to invest in A in period 3 with a probability of 1 (B returns at least $5000).
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