1. Theory.

The most obvious way to represent a sequence of objects is simply to list them, one after the other, like this.

Note that the same objects often appear many times in a row. This is called a run of those objects. In the example sequence, there is a run of 2 a’s, a run of 3 b’s, a run of 1 c, a run of 2 a’s, and a run of 4 d’s. You can represent a sequence with runs by listing its objects, along with the number of times each object appears. For example, you can represent the sequence shown above like this.

Representing a sequence in this way is called run-length encoding. If a sequence has long runs, or many runs, then run-length encoding will represent it more efficiently than simply listing its objects. However, if a sequence has short runs, or few runs, then run-length encoding may represent it less efficiently, because extra space is needed to store the lengths of the runs.
      Since a stack is just a simple kind of sequence, you can use run-length encoding to implement it. In this assignment, you will write a Java class called RunnyStack that implements a stack which uses run-length encoding. Here are some examples of how it works. Suppose you push an object a on an empty RunnyStack. Then the stack will look like this, with a run of 1 a.

Now suppose you push b. The stack now looks like this, with a run of 1 b, and a run of 1 a.

If you push another b on the RunnyStack, then the length of the run on top of the stack is incremented, so the stack looks like this.

If you push yet another b, then the length of the run at the top of the stack would increase to 3. However, suppose that you pop the RunnyStack instead. Then the length of the run at the top is decremented, so that the stack looks like this.

If you pop the RunnyStack one more time, then the length of the run on top of the stack is decremented to 0. However, a run of 0 objects is like no run at all, so it vanishes, and the stack looks as it did after the first push.

Stacks with run-length encoding are used internally by some compilers and interpreters, because they often push the same objects over and over again.

2. Implementation.

You must write a class called RunnyStack that represents a stack. Your class must implement run-length encoding, as described previously. It must also hold objects of type Base, so it will look like this.

class RunnyStack<Base>
{
  ⋮
}

Your class must define at least the following methods, as described below. To simplify grading, your methods must have the same names as the ones shown here.

• public RunnyStack()

  Constructor. Make a new, empty instance of RunnyStack.

• public int depth()

  Return the depth of the stack: the sum of the lengths of all the runs it holds. This is not necessarily the same as the number of runs it holds, which is returned by the method runs.

• public boolean isEmpty()

  Test if the stack is empty.

• public Base peek()

  If the stack is empty, then throw an IllegalStateException. Otherwise, return the Base at the top of the stack.

• public void pop()

  If the stack is empty, then throw an IllegalStateException. Otherwise, decrement the length of the run on top of the stack. If this leaves a run of zero Base’s on top of the stack, then remove that run.

• public void push(Base base)

  If the stack is empty, then add a new run of one Base at the top of the stack. If the stack is not empty, then test if base is equal to the object in the run at the top of the stack. If it is, then increment the length of that run. If it isn’t, then add a new run of one base at the top of the stack. Note that base may be null.

• public int runs()

  Return the number of runs in the stack. This is not necessarily the same as its depth, which is returned by the method depth.

Here are some hints, requirements, and warnings. First, all these methods must work using O(1) operations, so they are not allowed to use loops or recursions. You will receive no points for this assignment if you use loops or recursions in any way!
      Second, your RunnyStack class must have a private nested class called Run. You must use instances of Run to implement your stack. Each instance of Run represents a run of Base’s. You will receive no points for this assignment if you use arrays in any way! The class Run must have three private slots that have the following names and types. The slot base points to the Base that appears in the run. The slot length is an int that is the length of the run. The slot next points to the instance of Run that is immediately below this one on the stack, or to null. It must also have a private constructor that initializes these slots.
      Third, your push method must test non-null Base’s for equality using their equals methods. It must use the Java ‘==’ operator only for testing null Base’s. It is helpful to define an extra private method called isEqual that takes two Base’s as arguments, and tests if they are equal. If either Base is null, then isEqual uses ‘==’. If neither Base is null, then isEqual uses equals.
      Fourth, RunnyStack’s methods are not allowed to print things. If you were writing RunnyStack in the Real World, then it might be part of some larger program. You don’t know if that larger program should print things.

1.Briefly explain the Pure Expectations Hypothesis (PEH). Explain why a risk premium related to maturity is not consistent with the PEH. Is the historical empirical evidence (historical data) consistent with the PEH?

2.Define and explain Key Rate Durations. Why do we look at Key Rate Durations for portfolios only? How could two portfolios have the same weighted Modified Duration, but very different Key Rate Durations?

3.Use the following information to calculate the no arbitrage price for a Treasury note futures contract. P = 103.65, Coupon Rate = 4.50%, borrowing and lending rate = 3.00%, t = 0.35. The current quoted futures prices for the same contract is 106.20. State whether you would execute a Cash and Carry or Reverse Cash and Carry trade to create arbitrage profit. Calculate the profit per contract from the trade.

4.Explain the difference between Option Adjusted Spread (OAS) and Z-Spread (Static Spread). Separately discuss how a call and a put option impact the relationship between the two spread measures.

5.One of the key functions of a credit analyst is deriving pro forma projections for future financial data. Explain why an analyst runs a pro forma and then runs a downside or "worst case scenario".

6.Briefly explain the major differences between reduced form and structural models of default risk. Include the inputs, outputs, and uses.

In a tennis match, the first serve percentage is a key statistic. Having an accurate (and fast) first serve is seen as an important advantage to winning a match. Let theta be the probability of getting the first serve in on any point in a tennis game. For each of the four situations below, give a prior distribution for theta, explaining each in a sentence. You may assume the winner is determined from the best of three sets, (so the first player to win two sets will win).

(a) The point is the first point of the match.

(b) It is the beginning of the second set and the server has won the first set.

(c) It is the beginning of the second set and the server has lost the first set.

(d) If the server wins the next point, they will win the match.

Please Explain!

An urn contains seven chips labeled 1,2,...,7. Three of the chips are black, two are red, and two are green. The chips are drawn randomly one at a time without replacement until the urn is empty. Answer both questions for i = 1,...,7.

a. What is the probability that ith draw is chip 5?

b. What is the probability that ith draw is black?

Each unit of A is composed of one unit of B, two units of C, and one unit of D. C is composed of two units of D and three units of E. Items A, C, D, and E have on-hand inventories of 20, 10, 20, and 10 units, respectively. Item B has a scheduled receipt of 10 units in Period 1, and C has a scheduled receipt of 50 units in Period 1. Lot-for-lot (L4L) lot sizing is used for Items A and B. Item C requires a minimum lot size of 50 units. D and E are required to be purchased in multiples of 100 and 50, respectively. Lead times are one period for Items A, B, and C, and two periods for Items D and E. The gross requirements for A are 30 in Period 2, 30 in Period 5, and 40 in Period 8. Find the planned-order releases for all items.

Lower-of-Cost-or-Market Inventory

Data on the physical inventory of Katus Products Co. as of December 31 follows:

Quantity and cost data from the last purchases invoice of the year and the next-to-the-last purchases invoice are summarized as follows:

Required:

Determine the inventory at cost as well as at the lower of cost or market, using the first-in, first-out method. Record the appropriate unit costs on the inventory sheet, and complete the pricing of the inventory. When there are two different unit costs applicable to an item:

Insert the quantity and unit cost of the last purchase.

On the following line, insert the quantity and unit cost of the next-to-the-last purchase.

Total the cost and market columns and insert the lower of the two totals in the LCM column.

The first item on the inventory sheet has been completed as an example.

Given below are seven observations collected in a regression study on two variables, X (independent variable) and Y (dependent variable).

a. Develop the least squares estimated regression equation.

b. State the hypotheses to test for the significance of the regression line.

c. Perform an F test to determine whether or not the model is significant. Let α = 0.05.

Calculate calorific value of E15 fuel. A car run by gasoline has highway mpg 33, if it runs by E15, calculate the new mpg. Ignore all other effects. If it runs only by ethanol what would be new mpg. (Points 10+10=20)

A rocket is fired straight upward, starting from rest with an acceleration of

25.0 m/s2. It runs out of fuel at the end of 4.00 s and continues to coast

upward, reaching a maximum height before falling back to Earth. (a) Find the

rocket’s height when it runs out of fuel; (b) find the rocket’s velocity when it

runs out of fuel; (c) find the maximum height the rocket reaches; (d) find the

rocket’s velocity the instant before the rocket crashes into the ground; and

(e) find the total elapsed time from launch to ground impact.

SPK Industries uses the LIFO inventory costing system, use the following information to calculate the COGS if they sold 50 units in July.