Assignment (Replacement for Midterm Exam)
Please read the instruction carefully and solve the following questions. This assignment should be done individually and each student should upload his/her own work to the
course blackboard by the above time and date. Your answers should be hand- written.
Once you finish solve the questions, you can take a photo of your work (or scan it) and upload it to the course blackboard. Since it’s an individual assignment, you are NOT allowed to discuss your works with others. If you work together and turns in, substantially,the same work, this violates the rules of the course and the rules on academic integrity, and
penalties will be assessed. Make sure to write down your student ID number and name on the front page of your answer. Please show all the intermediate steps and calculations when solving the problems and state your assumptions (if any). Please write your answers by hand.
1. Today is Jane’s 23rd birthday. Starting today, Jane plans to begin saving for her retirement. Her plan is to contribute $1,000 to a brokerage account each year on her birthday. Her first contribution will take place today. Her 42nd and final contribution will take place on her 64th birthday. Her aunt has decided to help Janet with her savings, which is why she gave Jane $10,000 today as a birthday present to help get her account started. Assume that the account has an expected annual return of 10 percent. How much will Jane expect to have in her account on her 65th birthday?
2. At an inflation rate of 9 percent, the purchasing power of $1 would be cut in half in 8.04 years. How long to the nearest year would it take the purchasing power of $1 to be cut in half if the inflation rate were only 4 percent?
3. Today, Kim and Lee each have $150,000 in an investment account. No other contributions will be made to their investment accounts. Both have the same goal: They each want their account to reach $1 million, at which time each will retire. Kim has his money invested in risk-free securities with an expected annual return of 5 percent. Lee has her money invested in a stock fund with an expected annual return of 10 percent.
How many years after Lee retires will Kim retire?
4. Jane is 30 years old and is saving for her retirement. She is planning on making 36 contributions to her retirement account at the beginning of each of the next 36 years.
The first contribution will be made today (t=0) and the final contribution will be made 35 years from today (t=35). The retirement account will earn a return of 10 percent a year. If each contribution she makes is $3,000, how much will be in the retirement account 35 years from now (t=35)?
5. Today is your 21st birthday, and you are opening up an investment account. Your plan is to contribute $2,000 per year on your birthday and the first contribution will be made today. Your 45th, and final, contribution will be made on your 65th birthday. If you earn 10 percent a year on your investments, how much money will you have in the account on your 65th birthday, immediately after making your final contribution?
6. Jack and John (2 brothers) are each trying to save enough money to buy their own cars. Jack is planning to save $100 from every paycheck. (He is paid every 2 weeks.) John plans to put aside $150 each month but has already saved $1,500. Interest rates are currently quoted at 10 percent. Jack’s bank compounds interest monthly. At the end of 2 years they will each spend all their savings on a car. (Each brother will buy a car.)
What is the price of the most expensive car purchased?
7. Kate wants to open a savings account, and she has obtained account information from two banks. Bank A has a nominal annual rate of 9 percent, with interest compounded quarterly. Bank B offers the same effective annual rate, but it compounds interest monthly. What is the nominal annual rate of return for a savings account from Bank B?
8. A baseball player is offered a 5-year contract that pays him the following amounts:
Year 1: $1.2 mil
Year 2: $1.6 mil
Year 3: $2.0 mil
Year 4: $2.4 mil
Year 5: $2.8 mil
Under the terms of the agreement, all payments are made at the end of each year.
Instead of accepting the contract, the baseball player asks his agent to negotiate a contract that has a present value of $1 million more than that which has been offered.
Moreover, the player wants to receive his payments in the form of a 5-year annuity due. All cash flows are discounted at 10 percent. If the team were to agree to the player’s terms, what would be the player’s annual salary?
9. You are saving for the college education of your two children. One child will enter college in 5 years, while the other child will enter college in 7 years. College costs are currently $10,000 per year and are expected to grow at a rate of 5 percent per year. All college costs are paid at the beginning of the year. You assume that each child will be in college for four years.
You currently have $50,000 in your educational fund. Your plan is to contribute a fixed amount to the fund over each of the next 5 years. Your first contribution will come at the end of this year, and your final contribution will come at the date when you make the first tuition payment for your oldest child. You expect to invest your contributions into various investments, which are expected to earn 8 percent per year. How much should you contribute each year in order to meet the expected cost of your children’s education?
10. Your father, who is 60, plans to retire in 2 years, and he expects to live independently for 3 years. He wants a retirement income that has, in the first year, the same purchasing power as $40,000 has today. However, his retirement income will be a fixed amount, so his real income will decline over time. His retirement income will start the day he retires, 2 years from today, and he will receive a total of 3 retirement payments.
Inflation is expected to be constant at 5 percent. Your father has $100,000 in savings now, and he can earn 8 percent on savings now and in the future. How much must he save each year, starting today, to meet his retirement goals?
11. Tom and Jane Roberts are starting to save for their daughter’s college education.
• Assume that today’s date is September 1, 2002.
• College costs are currently $10,000 a year and are expected to increase at a rate equal to 6 percent per year for the foreseeable future. All college payments are due at the beginning of the year. (So for example, college will cost $10,600 for the year beginning September 1, 2003).
• Their daughter will enter college 15 years from now (September 1, 2007). She will be enrolled for four years. Therefore, the Roberts will need to make four tuition payments.
The first payment will be made on September 1, 2017, the final payment will be made on September 1, 2020. Notice that because of rising tuition costs, the tuition payments will increase each year.
• The Roberts would also like to give their daughter a lump-sum payment of $50,000 on September 1, 2021, in order to help with a down payment on a home, or to assist with graduate school tuition.
• The Roberts currently have $10,000 in their college account. They anticipate making 15 equal contributions to the account at the end of each of the next 15 years. (The first contribution would be made on September 1, 2003, the final contribution will be made on September 1, 2017).
• All current and future investments are assumed to earn an 8 percent return. (Ignore taxes.)
How much should the Roberts contribute each year in order to reach their goal?
12. You have some money on deposit in a bank account that pays a nominal (or quoted) rate of 8.0944 percent, but with interest compounded daily (using a 365-day year). Your friend owns a security that calls for the payment of $10,000 after 27 months. The security is just as safe as your bank deposit, and your friend offers to sell it to you for $8,000. If you buy the security, by how much will the effective annual rate of return on your investment change?
13. A financial planner has offered you three possible options for receiving cash flows. You much choose the option that has the highest present value.
(1) $1,000 now and another $1,000 at the beginning of each of the 11 subsequent months during the remainder of the year, to be deposited in an account paying a 12 percent nominal annual rate, but compounded monthly (to be left on deposit for the year).
(2) $12,750 at the end of the year (assume a 12 percent nominal interest rate with semiannual compounding).
(3) A payment scheme of 8 quarterly payments made over the next two years. The first payment of $800 is to be made at the end of the current quarter. Payments will increase by 20 percent each quarter. The money is to be deposited in an account paying a 12 percent nominal annual rate, but compounded quarterly (to be left on deposit for the entire 2-year period).
Which one would you choose?
14. Jane needs a new car and she is deciding whether it makes sense to buy or lease the car.
She estimates that if she buys the car it will cost her $17,000 today (t=0) and that she would sell the car four years from now for $7,000 (at t=4). If she were to lease the car,she would make a fixed lease payment at the end of each of the next 48 months (4 years). Assume that the operating costs are the same regardless of whether she buys or leases the car. Assume that if she leases, there are no up-front costs and that there is no option to buy the car after four years. Linda estimates that she should use a 6 percent nominal interest rate to discount the cash flows. What is the breakeven lease payment?
(That is, at what monthly lease payment would she indifferent between buying and leasing the car?)
15. Your ski vacation last year was great, but it unfortunately ran a bit over budget. All is not lost: you just received an offer in the mail to transfer your $15,000 balance from your current credit card, which charges an annual rate of 17.5 percent, to a new credit card charging a rate of 8.9 percent. How much faster could you pay the loan off by making your planned monthly payments of $250 with the new card? What if there was a fee of 2 percent charged on any balances transferred?
16. You are given the following data:
• k* = real risk-free rate = 4%
• Constant inflation premium = 7%
• Maturity risk premium = 1%
• Default risk premium for AAA bonds = 3%
• Liquidity premium for long-term T-bonds = 2%
Assume that a highly liquid market does not exist for long-term T-bonds, and the expected rate of inflation is a constant. Given these conditions, the nominal risk-free rate for T-bills is , and the rate on long-term Treasury bonds is .
17. ABC Corporation’s 4-year bonds currently yield 7.4 percent. The real risk-free rate of interest, k*, is 2.7 percent and is assumed to be constant. The maturity risk premium (MRP) is estimated to be 0.1%(t-1), where t is equal to the time to maturity. The default risk and liquidity premiums for this company’s bonds total 0.9 percent and are believed to be the same for all bonds issued by this company. If the average inflation rate is expected to be 5 percent for years 5, 6, and 7, what is the yield on a 7-year bond for ABC Corporation?
18. The real risk-free rate is expected to remain constant at 3 percent. Inflation is expected to be 2 percent a year for the next 3 years, and then 4 percent a year thereafter. The maturity risk premium is 0.1%(t-1), where t equals the maturity of the bond. (The maturity risk premium on a 5-year bond is 0.4 percent.) A 5-year corporation bond has a yield of 8.4 percent. What is the yield on a 7-year corporate bond that has the same default risk and liquidity premiums as the 5-year corporate bond?
19. Three-year Treasury securities currently yield 6 percent, while 4-year Treasury securities currently yield 6.5 percent. Assume that the expectations theory holds. What does the market believe the rate will be on 1-year Treasury securities three years from now?
20. You observe the following yields on Treasury securities of various maturities:
Maturity Yield (%)
1 year 6.0
3 years 6.4
6 years 6.5
9 years 6.8
12 years 7.0
15 years 7.2
Using the expectations theory, forecast the interest rate on 9-year Treasuries, six years from now. (That is, what will be the yield on 9-year Treasuries, issued in 6 years’ time?)
21. Ten-year bonds have an interest rate of 6.5 percent, while 15-year bonds have an interest rate of 6.0 percent. If the expectations theory is correct, what does the market believe will be the interest rate on 5-year bonds, 10 years from now?
22. The Wall Street Journal quotes the yield on 5-year Treasury bonds as 5.4 percent. Also,the current 1-year Treasury bond has a yield of 5 percent. If the real risk-free rate is 3
percent and is expected to remain constant, and the expectations theory is correct, whatis the average annual expected inflation for the 4-year period during Years 2 through 5?
23. The real risk-free rate, k*, is 3 percent. Two-year Treasury securities yield 6.5 percent,while 3-year Treasury securities yield 7 percent. The Treasury securities have a maturity risk premium = 0.1%(t-1), where t = the maturity of the security. Assume that the default risk premium and liquidity premium on all Treasury securities equals zero.
The expected inflation rate for this next year (Year 1) is 3.25 percent. What does the market anticipate will be the inflation rate three years from now?
24. Assume that today is January 1, 2003. A 5-year corporate bond maturing on January 1, 2008, has a yield to maturity of 7.5 percent. A 10-year corporate bond maturing on January 1, 2013, with the same liquidity and default risk premiums as the 5-year corporate bond has a yield to maturity of 8.2 percent. The annual real risk-free rate of interest, k*, is expected to remain constant at 2 percent. The maturity risk premium equals 0.1%(t-1), where t = the bond’s maturity in years. (For example, the maturity risk premium on a 5 year bond is 0.4 percent or 0.004.) Inflation is expected to average 2 percent per year for the next five years. What is the average annual expected inflation between January 2008 and January 2013?
25. An investor in Treasury securities expects inflation to be 3 percent in Year 1, 4 percent in Year 2, and 5 percent each year thereafter. Assume that the real risk-free rate is 3 percent, and that this rate will remain constant over time. Two-year Treasury securities yield 6.8 percent, while 4-year Treasury securities yield 7.6 percent. What is the difference in the maturity risk premiums (MRPs) on the two securities, that is, what is ???4 − ???2?
26. Consider the following information for three stocks, Stock A, Stock B, and Stock C.
The returns on each of the three stocks are positively correlated, but they are not perfectly correlated. (That is, all of the correlation coefficients are between 0 and 1.) Stock Expected return (%) Standard deviation (%) Beta
A 10 20 1.0
B 10 20 1.0
C 12 20 1.4
Portfolio P has half of its funds invested in Stock A and half invested in Stock B.
Portfolio Q has one third of its funds invested in each of the three stocks. The risk-free rate is 5 percent, and the market is in equilibrium. (That is, required returns equal expected returns.) What is the market risk premium (?? − ???)?
27. You hold a diversified portfolio consisting of a $10,000 investment in each of 20 different common stocks (that is, your total investment is $200,000). The portfolio beta is equal to 1.2. You have decided to sell one of your stocks that has a beta equal to 0.7 for $10,000. You plan to use the proceeds to purchase another stock that has a beta equal to 1.4. What will be the beta of the new portfolio?
28. You are an investor in common stocks, and you currently hold a well-diversified portfolio that has an expected return of 12 percent, a beta of 1.2, and a total value of $9,000. You plan to increase your portfolio by buying 100 shares of AT&E at $10 a share. AT&E has an expected return of 20 percent with a beta of 2.0. What will be the expected return and the beta of your portfolio after you purchase the new stock?
29. Assume a new law is passed that restricts investors to holding only one asset. A riskaverse investor is considering two possible assets as the asset to be held in isolation.
The assets’ possible returns and related probabilities (that is, the probability distributions) are as follows:
Asset X Asset Y
P k (%) P k (%)
0.10 -3 0.05 -3
0.10 2 0.10 2
0.25 5 0.30 5
0.25 8 0.30 8
0.30 10 0.25 10
Which asset should be preferred?
30. ABC Inc. has a beta coefficient of 0.7 and a required rate of return of 15 percent. The market risk premium is currently 5 percent. If the inflation premium increases by 2 percentage points, and Oakdale acquires new assets that increase its beta by 50 percent, what will be ABC’s new required rate of return?
31. The realized returns for the market and Stock J for the last four years are given below:
Year Market (%) Stock J (%)
1 10 5
2 15 0
3 -5 14
4 0 10
An average stock has an expected return of 12 percent and the market risk premium is 4 percent. If Stock J’s expected rate of return as viewed by a marginal investor is 8 percent, what is the difference between J’s expected and required rates of return?
32. You have been scouring The Wall Street Journal Looking for stocks that are “good values” and have calculated expected returns for five stocks. Assume the risk-free rate (???) is 7 percent and the market risk premium (?? − ???) is 2 percent. Which security would be the best investment? (Assume you must choose just one.)
33. Company X has a beta of 1.6, while Company Y’s beta is 0.7. The risk-free rate is 7 percent, and the required rate of return on an average stock is 12 percent. Now the expected rate of inflation built into ??? rises by 1 percentage point, the real risk-free rate remains constant, the required return on the market rises to 14 percent, and betas remain constant. After all of these changes have been reflected in the data, by how much will the required return on Stock X exceed that on Stock Y?
34. The current risk-free rate is 6 percent and the market risk premium is 5 percent. Jane is preparing to invest $30,000 in the market and she wants her portfolio to have an expected return of 12.5 percent. Jane is concerned about bearing too much stand-alone risk; therefore, she will diversify her portfolio by investing in three different assets (two mutual funds and a risk-free security). The three assets she will be investing in are an aggressive growth mutual fund that has a beta of 1.6, an S&P 500 index fund with a beta of 1, and a risk-free security that has a beta of 0. She has already decided that she will invest 10 percent of her money in the risk-free asset. In order to achieve the desired expected return of 12.5 percent, what proportion of Jane’s portfolio must be invested in the S&P 500 index fund?
35. A fund manager is holding the following stocks:
Stock Amount Invested ($mil) Beta
1 300 1.2
2 560 1.4
3 320 0.7
4 230 1.8
The risk-free rate is 5 percent and the market risk premium is also 5 percent. If the manager sells half of her investment in Stock 2 ($280 million) and puts the money in
Stock 4, by how many percentage points will her portfolio’s required return increase?
36. A portfolio manager is managing a $10 million portfolio. Currently, the portfolio is invested in the following manner:
Investment Dollar Amount Invested ($mil) Beta
Stock 1 2 0.6
Stock 2 3 0.8
Stock 3 3 1.2
Stock 4 2 1.4
Currently, the risk-free rate is 5 percent and the portfolio has an expected return of 10 percent. Assume that the market is in equilibrium so that expected returns equal required returns. The manager is willing to take on additional risk and wants to instead earn an expected return of 12 percent on the portfolio. Her plan is to sell Stock 1 and use the proceeds to buy another stock. In order to reach her goal, what should be the beta of the stock that the manager selects to replace Stock 1?
37. Here are the expected returns on two stocks:
Probability Returns (%)
0.1 -20 10
0.8 20 15
0.1 40 20
If you form a 50-50 portfolio of the two stocks, what is the portfolio’s standard deviation?
38. An analyst has estimated how a particular stock’s return will vary depending on what will happen to the economy:
State of the Economy Probability of State Occurring Stock’s Expected Return if
this State Occurs (%)
Recession 0.10 -60
Below Average 0.20 -10
Average 0.40 15
Above Average 0.20 40
Boom 0.10 90
What is the coefficient of variation on the company’s stock?
39. A money manager is holding the following portfolio:
Stock Amount Invested ($) Beta
1 300,000 0.6
2 300,000 1.0
3 500,000 1.4
4 500,000 1.8
The risk-free rate is 6 percent and the portfolio’s required rate of return is 12.5 percent.
The manager would like to sell all of her holdings of Stock 1 and use the proceeds to purchase more shares of Stock 4. What would be the portfolio’s required rate of return following this change?
40. A portfolio manager has a $10 million portfolio, which consists of $1 million invested in 10 separate stocks. The portfolio beta is 1.2. The risk-free rate is 5 percent and the
market risk premium is 6 percent.
The manager sells one of the stocks in her portfolio for $1 million. The stock she sold has a beta of 0.9. She takes the $1 million and uses the money to purchase a new stock that has a beta of 1.6. What is the required return of her portfolio after purchasing this new stock?
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