Self Study Problem Set
1.
Suppose that the average salary offer for graduates at a prestigious business school is $100,000
with a standard deviation of $10,000. Suppose a student receives an offer once a month, and
must decide either to accept it or to turn it down in the hope of receiving a better offer later. She
feels that each month that goes by without a job costs $2000 in terms of search costs, including
any psychic costs of being jobless.
The student would like to determine an optimal strategy for accepting a job offer. In particular,
she would like to set a “reservation level”, which is the minimum job offer that she will accept. If
she sets a very high reservation level, then she will eventually end up with a high paying job, but
it may take a long time to receive such an offer. She wishes to set an optimal reservation level
that maximizes her payoff net of the search costs.
Set up a spreadsheet model to solve the optimal stopping problem using simulation. Set the
random number seed to 1234 and the number of trials to 1000.
2.
The House Store buys Christmas trees from the wholesalers in the end of November and sells
them at their retail outlet till December 25, the Christmas day. A tree is purchased by House
Store at a price $70 and it is sold at $100 per unit. The store is somewhat far from the residential
area. This adds to the uncertainty in the number of trees sold during the season: This uncertainty
depends on factors such as the pricing by the competitors, the amount of snowfall in that region
etc. Any trees leftover are returned to the wholesalers at $25 a piece.
The wholesaler delivers the trees at one go every year (last week of November). Each year
House Store faces the problem of determining how many trees to purchase from the wholesaler.
If it orders too many, it loses $45 a tree on leftovers. If it orders too few, it foregoes the
opportunity to make profit of $30 a unit, not to mention the customer dissatisfaction.
From past experience, the estimated frequency distribution of demand for trees during the season
is as follows.
Demand 100 150 200 250 300
Frequency 0.2 0.2 0.35 0.2 0.05
1. Suppose the amount ordered is 180 trees.
a. Simulate the tree demand during a season and House Store’s total profit for this order
size.
b. In all the experiments use the initial seed 1.
c. The store manager considers the above demand distribution as a very rough
approximation to the actual demand. (Due to lack of specific data, this was a
reasonable guess available to her). She believes that the demand is actually Normally
distributed. The mean and the standard deviation may be estimated by taking them to
equal the mean and standard deviation of the above distribution. Under this
assumption of Normality,
i. Again compute the total profit for order size 180.
ii. Construct a 95% confidence interval for the total profit estimate.
iii. What is the probability that the demand exceeds the order quantity?
2. Determine the optimal order quantity House Store should purchase in order to maximize the
expected profit under the above two different set of distributional assumptions on the
demand.
3.
Consider a simple case of the classical mean variance portfolio theory pioneered by Markowitz
in early 1950’s. Consider three securities associated with Reliance, Infosys and Tata’s. We want
to divide our wealth in these three securities in the most risk effective manner (short selling of
any of these securities is not allowed).
