1. When proving If p then q.”

DIRECT PROOF you need to:

CONTRAPOSITION you need to:

CONTRADICTION you need to:

2. Prove by direct proof that if m and n are integers, with m odd and n is even, then 5n + m2 is odd.

3. Prove by contraposition that if x 6= 5 and is irrational, then 4x x − 5 is irrational.

4. Prove the following existential statements by providing a value for x. In both cases, the universe is the set of all real numbers.

a) ∃x x 2 + 5x − 7 = 0

b) ∃x x < 10 → (x − 2) 2 < 0

5. Prove that for any integer n, there exists an even integer k so that n < k + 1 < n + 3.

6. Prove or disprove: If x is rational and y is irrational, then xy is irrational.

7. Prove that there is no positive integer n so that 49 < n 2 < 64.

8. Prove or disprove: ∀x∃y ((x − 3)y = 4x), where the universe of discourse is R for both variables.

9. Prove, by contraposition, that if the product of two real numbers is irrational, then at least one of the two numbers is irrational. (In other words: If x · y is irrational, then x is irrational OR y is irrational.”)

10. Prove, by contradiction, that √ 3 is irrational. You may use the Little Theorem: If m2 is a multiple of 3, then m itself is a multiple of 3

A bag of 30 tulip bulbs contains 13 red tulip​ bulbs, 10 yellow tulip​ bulbs, and 7 purple tulip bulbs. Suppose two tulip bulbs are randomly selected without replacement from the bag.

(a) What is the probability that the two randomly selected tulip bulbs are both​ red?

​(b) What is the probability that the first bulb selected is red and the second​ yellow?

​(c) What is the probability that the first bulb selected is yellow and the second​ red?

​(d) What is the probability that one bulb is red and the other​ yellow?

Consider an MNC has two choices on investment. One of them is to invest the project in Taiwan only. The other is to invest 75% funds in Taiwan and 25% in Singapore. Both investments give the same return. The standard deviation of exchange movements is 7% for Taiwan dollars and 5% for Singapore dollars. The correlation coefficient between movements in the value of the Taiwan dollar and the Singapore dollar is .7. Based on this information, which investment should the MNC chose? Show the calculations

QUESTION 2 [20] The number of aces served by Novak Djokovic in the last 20 tournaments that he has participated in, is shown below. 12 17 13 7 8 14 11 14 10 12 15 9 11 13 6 15 18 5 19 24 2.1. Using the raw data, determine the range. (2) 2.2. Group the data into a frequency distribution with a lowest class lower limit of 1 ace and a class width of 5 aces. (7) 2.3. Determine the mean number of aces served, using the raw data. (3) 2.4. Draw a less than OGIVE curve corresponding to the data and use it to estimate the median. (8).

1. A stock records the following returns over the last 6 months.

1. What are the arithmetic returns for the stocks?
2. What are the geometric returns for the stocks?
3. What are the variances for the stocks?
4. What are the standard deviations for the stocks?
5. Compute the coefficient of variation of stocks.
6. Which stock would you choose? Why?

1. A stock records the following returns over the last 6 months.

1. What are the arithmetic returns for the stocks?
2. What are the geometric returns for the stocks?
3. What are the variances for the stocks?
4. What are the standard deviations for the stocks?
5. Compute the coefficient of variation of stocks.
6. Which stock would you choose? Why?

1. A stock records the following returns over the last 6 months.

1. What are the arithmetic returns for the stocks?
2. What are the geometric returns for the stocks?
3. What are the variances for the stocks?
4. What are the standard deviations for the stocks?
5. Compute the coefficient of variation of stocks.
6. Which stock would you choose? Why?

Imagine the following scenario at a company where you are the computer specialist: Your company recently installed high-speed Internet access at the office where you work. There are 50 workstations connected to the network and the Internet. Within a week, half the computers in the office were down because of a virus that was contracted by a screen saver. In addition, network personnel from a university in England contacted the company, claiming that your computer systems were being used as a part of a DDoS attack on their Web site. By the time the damage was repaired, your company had lost about 500 work hours when employees could not use their computers. The company also lost approximately $10,000 in potential revenue by not being able to respond to purchase requests, and narrowly averted a lawsuit from the university. Take steps to prevent this from happening again: • Research the price of at least three antivirus and firewall packages on the Internet and determine the most cost-effective package for the company to implement on 50 workstations. • Draft a memo to the CEO with your recommendations. Include the following in your memo: o A list of the antivirus and firewall packages you researched. You should research and list at least three different antivirus and firewall packages. o The antivirus and firewall package you recommend and your reasons for recommending it over the other two packages. o An explanation of how the antivirus and firewall products that you recommend would prevent future problems with viruses and with DDoS attacks from your computers on web sites belonging to other organizations. o If you need help with creating a memo, please click How to Create a Memo to download the instructions. • Compile a list strategies for avoiding virus infections and incorporate this list into a document to be sent to all employees. • Use the Networks and Security thread in the Experts Exchange class forum if you need some help, or if you are knowledgeable about computer security and can help others identify antivirus and firewall software.

Introduction to Mortgage Mathematics and Mortgage-Backed Securities

(I ONLY NEED THE SOLUTIONS FOR 5,6,7 and 8, the previous exercises are given as a reference) 4 SOLUTIONS IN TOTAL

1.(Level-Payment Mortgages) Compute the monthly payment on a 30-year level payment mortgage assuming an annual mortgage rate of 5% and an initial mortgage principal of $400,000. Submission Guideline: Give your answer rounded to two decimal places. For example, if you compute the answer to be $73.2367, submit 73.24.

2 (Mortgage Pass-Throughs) Consider a $400 million pass-through MBS that has just been created (so the 'seasoning' of the pass-through is equal to 0). The underlying pool of mortgages each has a maturity of 20 years and an annual mortgage coupon rate of 6%. The pass-through rate of the mortgage pool is 5%. Assuming a prepayment multiplier of 100 PSA what is the total amount of interest paid to the pass-through investors? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46.

3 (Mortgage-Pass Throughs) Referring to the same mortgage pass-through of the previous question, what is the total amount of the prepayments? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46.

4 (Mortgage-Pass Throughs) Referring to the same mortgage pass-through of the previous question, what is the total amount of the prepayments if the rate of prepayments increases to 200 PSA? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46.

5 (Principal-Only MBS and Interest-Only MBS) Suppose we construct principal-only (PO) and interest-only (IO) mortgage-backed securities (MBS) using the mortgage pass-through of the previous questions. Assume a prepayment multiplier of 100 PSA. What is the present value of the PO MBS if we use an annual risk-free rate of 4.5% to value the cash-flows? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46.

6. (Principal-Only MBS and Interest-Only MBS) Referring to the previous question, what is the value of the IO MBS? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46.

7. (Principal-Only MBS and Interest-Only MBS) Referring to the previous question, what is the average life of the IO MBS? Submission Guideline: Give your answer in years rounded to two decimal places. For example, if you compute the answer to be 12.1234 years, submit 12.12.

8. (Principal-Only MBS and Interest-Only MBS) Suppose now that you purchased the IO MBS of the previous question and that the price you paid was the same price that you calculated in the previous question. The risk-free interest rate suddenly changes from 4.5% to 3.5%. Everything else stays the same. How much money have you made or lost on your investment? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46.

WestGas​ Conveyance, Inc., is a large U.S. natural gas pipeline company that wants to raise​ $120 million to finance expansion. WestGas wants a capital structure that is 50​% debt and 50​% equity. Its corporate combined federal and state income tax rate is 30​%. WestGas finds that it can finance in the domestic U.S. capital market at the rates listed:

Both debt and equity would have to be sold in multiples of​ $20 million, and these cost figures show the component​ costs, each, of debt and equity if raised 50​% by debt and 50​% by equity. A London bank advises WestGas that U.S. dollars could be raised in Europe at the following​ costs, also in multiples of​ $20 million, while maintaining the 50/50 capital structure.

Each increment of cost would be influenced by the total amount of capital raised. That​ is, if WestGas first borrowed​ $20 million in the European market at 8​% and matched this with an additional​ $20 million of​ equity, additional debt beyond this amount would cost 11​% in the United States and 10​% in Europe. The same relationship holds for equity financing.

a. Calculate the lowest average cost of capital for each increment of​ $40 million of new​ capital, where WestGas raises​ $20 million in the equity market and an additional​ $20 in the debt market at the same time.

b. If WestGas plans an expansion of only​ $60 million, how should that expansion be​ financed? What will be the weighted average cost of capital for the​expansion?