Business Modeling

Applied business modeling

Problem 1 Two players A and B throw a fair coin N times. If head, then A gets 1 point; otherwise B gets 1 point. What happens to the absolute difference in points as N increases?

Solution:

The problem is like a random walk, and the dynamics of the system can be described as follows: , ,where is the indicator variable of whether the ith flipping is head（with a value of 1）or tail （with a value of 0）, is the stochastic process recording the differences of points A and B get and is the absolute difference in points. We assume when A gets 1 point plus 1.We use simulation to see how the random walk goes with N=10000. Figure 1 and 2 shows sample paths of and respectively.

Figure 1

Figure 2

Now we want to see how the absolute points difference goes with the playing time N, so changing N and observing the last value of absolute points difference seems enough. But Figure 2 is only one sample path of the stochastic process so a simulation with a considerable number of repeating times is required and the results from statistical average is more convincing. So we let N change from 500 to 10000 with a step length of 500 and in each loop with fixed N, the simulation is run by 1000 times. Since the total program has 3 loops it takes quite a long time to get the result so more accurate result is omitted here, Figure 3 shows the simulation result. It offers a rough description of the relationship between N and average absolute points difference. While it is hard to say whether average absolute points difference is convex/concave /linear in N, it looks like a increasing concave function.

Figure 3

Problem 2 Suppose there are 2 stocks denoted as A and B whose prices are normally distributed random variables with respective expected return and standard deviations as follows: ， ， ， . The returns are independent random variables. Please find the optimal portfolio with smallest coefficient of variation composed of the two stocks.

Solution:

It is easy to formulate the problem as a portfolio selection procedure, here our object is to minimize the CV of portfolio. We assume short sales is not allowed（investors cannot borrow securities from stockbrokers）.So when the ratio of Stock A on the investor's total assets is , the ratio of Stock B on the investor's total assets is . Since the returns of 2 stocks are independent RVs, the expected return and standard deviation of the portfolio is ， . A mathematical programming model can be presented below：

The objective function may be a little complicated