Problem Sets and Solutions For FNCE 604 Alex Edmans Wharton School, University of Pennsylvania
[email protected] Summer 2012
2318 Steinberg Hall - Dietrich Hall, 3620 Locust Walk, Philadelphia, PA 19104-6367. A substantial amount of this material is taken from the problem sets of Professor Simon Gervais, who taught this course for many years years at Wharton. I am extremely extremely grateful grateful to Simon for allowing allowing me to use his notes. notes. I also thank Indraneel Indraneel Chakraborty, James Park and Laila Shabir for their help in typesetting this packet.
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Contents 1 Problem Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Compounding and Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Bond Pricing - Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The NPV Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Capital Budgeting under Certainty . . . . . . . . . . . . . . . . . . . . . . . . . 2 Solutions to Problem Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Compou pounding and Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bond Pricing – Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The NPV Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Capital Budgeting under Certainty . . . . . . . . . . . . . . . . . . . . . . . . . 3 Practice Placement Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Sample Placement Exam A: Questions . . . . . . . . . . . . . . . . . . . . . . . 3.2 Sample Placement Exam A: Solutions . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Sample Placement Exam B: Questions . . . . . . . . . . . . . . . . . . . . . . . 3.4 Sample Placement Exam B: Solutions . . . . . . . . . . . . . . . . . . . . . . . . 4 Additional Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Solution to DiMaggio’s Vow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 4.2 Addi Additi tion on to Ho How w Tre Treas asur ury y Sec Secur urit itie iess A Are re Quot Quoted ed:: Examp Example le . . . . . . . . . . . . 4.3 Solution to BICC’s Toad Ranch . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Capit Capital al Budget Budgeting ing Under Under Resour Resource ce Cons Constra train ints: ts: A More More Elabora Elaborate te Tec Techni hnique que . 4.5 Short Sales: An An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Mathematics and Statistics: A Rem emiinder . . . . . . . . . . . . . . . . . . . . . . 4.7 Linear Algebra: Si Simple Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 4 8 10 12 16 17 24 29 31 38 39 44 49 53 58 59 61 62 64 65 71 74 80
Contents 1 Problem Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Compounding and Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Bond Pricing - Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The NPV Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Capital Budgeting under Certainty . . . . . . . . . . . . . . . . . . . . . . . . . 2 Solutions to Problem Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Compou pounding and Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bond Pricing – Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The NPV Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Capital Budgeting under Certainty . . . . . . . . . . . . . . . . . . . . . . . . . 3 Practice Placement Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Sample Placement Exam A: Questions . . . . . . . . . . . . . . . . . . . . . . . 3.2 Sample Placement Exam A: Solutions . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Sample Placement Exam B: Questions . . . . . . . . . . . . . . . . . . . . . . . 3.4 Sample Placement Exam B: Solutions . . . . . . . . . . . . . . . . . . . . . . . . 4 Additional Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Solution to DiMaggio’s Vow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 4.2 Addi Additi tion on to Ho How w Tre Treas asur ury y Sec Secur urit itie iess A Are re Quot Quoted ed:: Examp Example le . . . . . . . . . . . . 4.3 Solution to BICC’s Toad Ranch . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Capit Capital al Budget Budgeting ing Under Under Resour Resource ce Cons Constra train ints: ts: A More More Elabora Elaborate te Tec Techni hnique que . 4.5 Short Sales: An An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Mathematics and Statistics: A Rem emiinder . . . . . . . . . . . . . . . . . . . . . . 4.7 Linear Algebra: Si Simple Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 4 8 10 12 16 17 24 29 31 38 39 44 49 53 58 59 61 62 64 65 71 74 80
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Problem Sets
In this section, you will …nd problem sets related to the material covered in the bulk pack and in the textbook by BMA. These problems sets will not be collected and graded; they are meant meant to facilitate facilitate your your preparatio preparation n for the exam. exam. The “checkpoint “checkpoints” s” contain contained ed in the lecture notes will guide you as to when you should start working on the di¤erent problem sets. These problem sets are more di¢cult than the problems in the textbook. If you …nd them too hard, please start with the textbook problems listed at the end of each section of the notes, to “work your way up” to these problems. I have have indicat indicated ed the proble problems ms that should should be viewe viewed d as optional optional.. These These proble problems ms are meant to complement the material covered in the other problems, but should not be considered required. required. I have also indicated indicated the problems problems that are more di¢cult. Do not panic if you need to spend a little more time on these problems.
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1.1
Compo Compoun undin ding g and and Disco Discoun untin ting g
1. Find the values of the 6-month, 6-month, 1-year and 5-year discount discount factors (DF ( DF 0:5 , DF 1 , DF 5 ) when (a) the annual annual percentage percentage rate is 7% compounded compounded annually; annually; (b) the annual percentage percentage rate is 11% compounded compounded semiannually semiannually;; (c) the monthly monthly interest interest rate is 0.8% 0.8%;; (d) the continuousl continuously y compounded compounded annual percentage percentage rate is 9%; (e) the annual annual percentage percentage rate is 14.6 14.6% % compounded daily (assuming (assuming 365 days a year). year). Show all your calculations. 2. Suppose that a savings savings account o¤ers o¤ers you an annual annual percentage percentage rate of 10% compounded compounded quarterly. Calculate the equivalent annual rate compounded (a) every every year; year; (b) semiannually; semiannually; (c) monthly; monthly; (d) daily; daily; (e) continuously continuously.. Your answers should all show 2 decimals (e.g. 9.12%). Show all your calculations. 3. You have have won the Florid Florida a state state lotter lottery y. Lo Lotte ttery ry o¢ o¢cia cials ls o¤er o¤er you you the choice choice of the following alternative payments:
Alternative 1: $10,000 one year from now. Alternative 2: $20,000 …ve years from now. Which alternative should you choose if the annual discount rate is (a) 0% 0%?? (b) 10 10%? %? (c) 20 20%? %? (d) What discount rate makes the two alternatives alternatives equally attractive to you? 4. Suppose you you deposit $1,000 $1,000 in an account account at the end of each of the next four years. If the account earns 12% annually, how much will be in the account at the end of seven years? 4
5. Find an expression for the exact e¤ective annual rate of interest at which payments of $300 at the present, $200 at the end of one year, and $100 at the end of two years will accumulate to $700 at the end of two years. 6. Suppose that you invest a certain amount of money in a savings account which is earning an annually compounded interest rate of r. In exactly how many years (as a function of r) will that amount of money be doubled? 7. Suppose that you invest $100 in a savings account which is earning an annual percentage rate of 8% (compounded annually). What is the …rst year in which the interest credited to your account for that year exceeds your original investment in the account? 8. The present value of two payments of $100 each to be made at the end of n years and 2n years is $100. Find a general expression for n if the annual percentage rate (compounded every year) is r. 9. Your parents make you the following o¤er: they will give you $500 at the end of every month for the next …ve years if you agree to pay them back $500 at the end of every month for the following ten years. Should you accept this o¤er if your opportunity cost is 12% a year (compounded annually)? 10. A man aged 40 wishes to accumulate a fund for retirement by depositing $1,000 at the beginning of each year for 25 years. Starting at age 65, he will make 15 annual withdrawals at the beginning of each year. Assuming that all payments are certain to be made, …nd the amount of each withdrawal starting at age 65 if the annual percentage rate (compounded annually) is 4% during the …rst 25 years but only 3.5% thereafter. 11. Vernal Pool wants to put aside a …xed fraction of her annual income as savings for retirement. Ms. Pool is now 40 years old and makes $40,000 a year. She expects her income to increase 2% annually in real terms. She wants to accumulate $500,000 in real terms to retire at age 70. What fraction of her income does she need to set aside? Assume that her retirement fund are invested at an expected real rate of return of 5% a year. Ignore taxes. 12. A pro-football team o¤ers one of its players the choice between the following contracts: (a) a yearly salary (payable at the end of every year) of $400,000 for 5 years; (b) a quarterly salary (payable at the end of every quarter) of $95,000 for 5 years; (c) a monthly salary (payable at the beginning of every month) of $31,000 for 5 years; (d) a …ve year contract with a lump sum salary of $1.5 million payable now.
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Assuming an annual percentage rate of 8% compounded semi-annually, which contract should this player accept? Show all your calculations. 13. Suppose that the annual percentage rate (compounded annually) is 10%. Calculate the present value of the following perpetuities: (a) a perpetuity that pays $500 every year with the …rst payment occurring in 2 years from now; (b) a perpetuity that pays $500 every year with the …rst payment occurring in 3 years from now; (c) a perpetuity that pays $500 every six months with the …rst payment occurring in 4 years from now; (d) a perpetuity payable annually with the …rst payment of $100 now and each additional payment growing at 2% annually. 14. A savings and loan association pays 4% annual interest on deposits at the end of each year. At the end of every three years, a 2% bonus is paid on the balance (after interest) at that time. Find the e¤ective annual rate of interest earned by an investor if he leaves his money on deposit (a) two years; (b) three years; (c) four years. 15. A credit card company charges its customers an interest rate of 1.583% on their monthly balance. The company claims that this is equivalent to an annualized rate of 18.996%. What’s wrong with their claim? What does the rate of 18.996% represent? 16. An oil production platform in Alaska’s Cook Inlet will operate for 15 more years. At the end of that time, environmental regulations require dismantlement and removal. The current cost of this would be $10 million, but the cost is expected to increase by 5% per year. (a) What is the present value of the future cost? Assume an annual discount rate of 11% (compounded annually). (b) Suppose new regulations require the oil platform’s owner to contribute an equal annual nominal amount to build up a trust fund su¢cient to cover dismantlement and removal at the end of year 15. The trust fund has to be invested in U.S. Treasury bonds yielding 6.5% per year. How much will the oil company have to put in at the end of each year? 6
17. (Optional): The present value of an annuity which pays $200 at the beginning of every 6-month period during the next 10 years and $100 at the beginning of every 6-month period during the following 10 years is $4,000. The present value of a 10-year deferred annuity which pays $250 at the beginning of every 6-month period for 10 years is $2,500. Find the present value of an annuity paying $200 at the beginning of every 6-month period during the next 10 years and $300 at the beginning of every 6-month period during the following 10 years. 18. (Di¢cult): Suppose that the annual percentage rate (compounded annually) is r. Calculate the present value of the following perpetuities: (a) a perpetuity paying $C every n years, with the …rst payment occurring at the end of n years; (b) a linearly growing perpetuity which pays $1 at the end of the …rst year, $2 at the end of the second year, $3 at the end of the third year, etc. 19. (Optional, Di¢cult): The one-year compounding factor when the annual percentage rate r is compounded m times a year is given by
r CF 1 = 1 + m
m
:
Show that this compounding factor tends to er as m tends to in…nity, i.e. show that lim
m
!1
1+
7
r m
m
= er :
1.2
Bond Pricing - Term Structure
1. A Treasury bond has the following characteristics:
Principal: $1,000. Term to maturity: 20 years. Coupon rate: 8 percent. Semiannual payments. Calculate the price of the bond if the stated annual percentage rate (compounded annually) is (a) 8 percent; (b) 10 percent; (c) 6 percent. 2. A riskless bond with a par value of $1,000 is sold at $923.14. The bond has 15 years to maturity and investors require a 5% semiannual yield on the bond. What is the coupon rate for the bond if the coupon is paid semiannually? 3. A two-year bond with a face value of $1,000 pays annual coupons at a rate of 10% (in other words, the bond pays interest of $100 at the end of each year, and its principal of $1,000 is paid o¤ in year 2). If the bond sells for $960, what is its yield to maturity? 4. You are given the following bond prices: Bond Type Face Value Coupon Maturity Price A Zero Coupon Bond $100 – 1 year $93.46 B Coupon Bond $100 4% 2 years $94.92 C Coupon Bond $100 8% 3 years $103.64 (a) Assuming that all the bonds make only annual payments, what spot rates are imbedded in these prices? (b) What forward rates are embedded in these prices? (c) What should the price of a 3-year bond with a face value of $100 and a 6% annual coupon be? (d) (Optional, Di¢cult): A 3-year bond with a face value of $100 and a 4% annual coupon is trading at $95.00. Show that this bond is mispriced by showing how you would take advantage of its price. In doing so, make sure that your arbitrage pro…ts are realized today. 8
5. You are given the following information on three traded bonds making annual coupon payments. Bond Face Value Rate A $1,000 0.00% B $1,000 5.00% C $1,000 10.00%
Maturity Yield to Maturity 1 year 5.00% 2 years 5.85% 2 years 6.00%
(a) What are the prices of the above three bonds? (b) (Optional, Di¢cult): Is it possible to make arbitrage pro…ts using only these three bonds? If so, give an arbitrage strategy (making sure that the pro…ts are realized today). Hint: Try to solve for the discount factors. 6. You are given the following spot rates and forward rates: f 1 = 6:0%, r2 = 6:25%, f 3 = 7:0%, and r4 = 6:75%. (a) Calculate r1 , r3 , f 2 , f 4 . (b) Describe the shape of the term structure of (spot) interest rates (increasing, decreasing, or ...). (c) Calculate the price of a 4-year bond with a 9% coupon rate and annual coupons. Assume that the face value of this bond is $1,000. (d) Is the yield on this bond greater or smaller than 6.5%? 7. A 6%, 6-year bond yields 12%, and a 10%, 6-year bond yields 8%. Calculate the 6-year spot rate r6 . (Assume annual coupon payments for both bonds.)
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1.3
The NPV Rule
1. In Figure 1, the sloping straight line represents the opportunities for investment in the capital market, and the solid curved line represents the opportunities for investment in plant and machinery (real assets). The company’s only asset at present is $2.6 million in cash.
Figure 1: This graph is used in problem #1
(a) What is the interest rate? (b) How much should the company invest in plant and machinery? (c) How much will this investment be worth next year? (d) What is the average rate of return on the investment? (e) What is the marginal rate of return? (f) What is the present value of this investment? (g) What is the net present value of this investment? (h) What is the total present value of the company? (i) How much will the individual consume today? (j) How much will he consume tomorrow? (k) Is he a borrower or a lender? 2. Casper Milktoast has $200,000 available to support consumption in periods 0 (now) and 1 (next year). He wants to consume exactly the same amount in each period. The interest rate is 8%. There is no risk. 10
(a) How much should he invest, and how much can he consume in each period? (b) Suppose Casper is given an opportunity to invest up to $200,000 at 10% risk-free. The interest rate stays at 8%. What should he do, and how much can he consume in each period i. if he is not allowed to borrow against future income? ii. if he is allowed to borrow against future income? (c) What is the NPV of each opportunity in (b)?
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1.4
Capital Budgeting under Certainty
1. The …rm Nalyd is considering an investment in equipment to produce a new product. The cost of the equipment is $150,000. This equipment falls into the 5-year asset class and thus would have to be capitalized and depreciated over 6 years at rates 20%, 32%, 19.2%, 11.52%, 11.52%, and 5.76%. Nalyd expects to use the equipment for three years and then to sell it for $60,000. For the three years of operation, the equipment will generate revenues of $40,000 per year and will have operating costs of $3,000 per year. If the opportunity cost of capital for Nalyd is 12% and its tax rate is 35%, should Nalyd purchase this equipment? 2. (Di¢cult): Conocococonut is considering the purchase of a new harvester. They are currently involved in deliberations with the manufacturer and as of yet the parties have not come to a settlement regarding the …nal purchase price. The management of Conocococonut has hired you, a high-priced consultant, to establish the maximum price it should be willing to pay for the harvester (i.e., the break-even price such that the NPV of the project would be zero). This will be of obvious use to Conocococonut in their haggling with the capital equipment manufacturer. You are given the following facts:
The new harvester would replace an existing one that has a current market value of $20,000.
The new harvester is not expected to a¤ect revenues, but before-tax operating costs will be reduced by $10,000 per year for ten years.
The old harvester is now …ve years old. It is expected to last for another ten years and to have a resale value of $1,000 at the end of those ten years.
The old harvester was purchased for $50,000 and is being depreciated to zero on a straight-line basis over ten years.
The new harvester will be depreciated straight-line over its 10 year life to a zero book value. Conocococonut expects to be able to sell the harvester for $5,000 at the end of the ten years.
The corporate tax rate is 34% and the …rm’s cost of capital is 15%. 3. The Scampini Supplies Company recently purchased a new delivery truck. The new truck costs $22,500, and it is expected to generate net after-tax operating cash ‡ows, including depreciation, of $6,250 per year. The truck has a 5-year expected life. The expected abandonment values (salvage values after tax adjustments) for the truck are given below. abandonment value at the end of year 0 1 2 3 4 5 22,500 17,500 14,000 11,000 5,000 0 12
The company’s opportunity cost of capital is 10%. Assume that the truck will be used inde…nitely – i.e. if the truck is sold early, it will be replaced with a new one. Should the …rm operate the truck until the end of its 5-year physical life or, if not, what is its optimal economic life? 4. The Baldwin Company is considering investing in a machine that produces bowling balls. The cost of the machine is $100,000 and production is expected to be 8,000 units per year during the …ve-year life of the machine. The expected resale value is $5,000 (in real terms). Since the interest in bowling is declining, the management believes that the nominal price of bowling balls will increase at only 2% per year. The nominal price of bowling balls in the …rst year will be $20. On the other hand, plastic used to produce bowling balls is rapidly becoming more expensive. Because of this, production costs are expected to grow at 10% (nominally) per year. First-year nominal production costs will be $10 per unit. The machine will be depreciated to zero on a straight-line basis over …ve years. The company’s tax rate is 34% and its nominal cost of capital is 15%. The rate of in‡ation is 5%. Should the project be undertaken? 5. A project requires use of spare computer capacity. If the project is not terminated, the company will need to buy an additional disk at the end of year 2. If it is terminated, the disk will not be required until the end of year 4. Disks cost $10,000 and last 5 years. The opportunity cost of capital is 10% (an annual rate compounded annually). (a) What is the present value of the cost of this extra usage if the project is terminated at the end of year 2? (b) What if the project continues inde…nitely? 6. Pilot Plus Pens is considering when to replace its old machine. The replacement costs $3 million now and requires maintenance costs of $750,000 at the end of each year during the machine’s economic life of …ve years. At the end of …ve years the new machine would have a resale value of $100,000. It will be fully depreciated (to zero after …ve years) by the straight-line method. The corporate tax rate is 34% and the appropriate discount rate is 12%. Maintenance cost, resale value, depreciation and year-end book value of the existing machine are given as follows. Year Maintenance Depreciation Book Value Resale 0 1,000,000 1,600,000 1 1,000,000 200,000 800,000 1,200,000 2 1,000,000 200,000 600,000 600,000 13
When should the company replace the machine (now, in one year, or in two years)? 7. (a) Construct an Excel spreadsheet that will calculate the IRR of any …ve-year project, and use it to calculate the IRR of the following two projects. cash ‡ow at the end of year project 0 1 2 3 4 5 A -100 10 20 60 25 15 B -100 -50 30 40 60 90 (b) How would you use your spreadsheet to calculate the IRR of a three-year project? Try it on the following project. cash ‡ow at the end of year project 0 1 2 3 C -100 20 40 60 (c) How would you use your spreadsheet to calculate the yield on a …ve-year bond with annual coupon payments? Try it on a …ve-year bond with a face value of $1,000 and a coupon rate of 8%, trading for $950. 8. An engineering …rm called Seltaeb is looking at three possible projects, knowing that it can undertake any (positive) fraction of each project. The yearly cash ‡ows for each project are given in the following table, along with the number of engineer-hours required for the projects over the next four years: cash ‡ow (in $millions) in year engineer-hours Project 0 1 2 3 4 (in thousand) A -4 2 2 1 0 25 B -6 -2 3 5 5 35 C -2 -3 0 7 4 60 Seltaeb is facing capital and labor constraints. In particular, the bank from which Seltaeb is borrowing money for the projects demands that the present value of the …rm’s total costs for the …rst year (i.e., PV of present and …rst year costs minus revenues) be smaller than $8 million. Also, there are currently 10 engineers working for Seltaeb, and the …rm does not want to hire any additional engineer for these projects (and does not expect any of these engineers to quit the …rm in the next four years). These engineers can work for up to 50 hours a week for 48 weeks a year. Assuming an opportunity cost of capital of 12%, set up the linear programming problem that the …rm will have to solve to choose what fraction of each project to undertake. Instructions :
You should not solve the linear programming problem; just set it up clearly for somebody else to solve it.
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9. Project A has a cost of $10,000 and is expected to produce bene…ts (cash ‡ows) of $3,000 per year for …ve years. Project B costs $25,000 and is expected to produce cash ‡ows of $7,400 per year for …ve years. Calculate the two projects’ net present values (NPV’s), internal rates of return (IRR’s), and pro…tability indexes (PI’s), assuming an opportunity cost of capital of 12%. (a) Which project would be selected assuming that they are mutually exclusive, using each ranking method? (b) Which should actually be selected? (c) How could you adjust the method(s) giving you wrong answers in order for it (them) to give you the right answer?
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2
Solutions to Problem Sets
This section contains the solutions to the problem sets of section 1. To optimize your learning and better prepare for the exam, it is highly recommended that you try solving the problems before looking at their solutions.
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2.1
Compounding and Discounting
(b) DF 0:5 =
1 (1:07)1=2 1 1:055 =
0:948, DF 1 =
(c) DF 0:5 =
1 (1:008)6
= 0:953, DF 1 =
(d) DF 0:5 =
1 1 = 0:956, DF 1 = e0:09(1) = 0:914, DF 5 e0:09(1=2) 1 1 = 0:930, DF 1 = (1:0004) 365(1) = (1:0004)365(1=2)
1. (a) DF 0:5 =
(e) DF 0:5 = 0:482.
1 1:07
= 0:935, DF 5 =
1 (1:07)5
1 (1:055)2
= 0:898, DF 5 =
1 (1:055)10
= 0:967, DF 1 =
1 (1:008)12
= 0:909, DF 5 = =
= 0:713. = 0:585.
1 (1:008)60 1 e0:09(5)
= 0:620.
= 0:638.
0:864, DF 5 =
1 (1:0004)365(5)
=
2. (a) (1:025)4 = 1 + ^r then r^ = (1:025)4
(b) (1:025)4 = 1 + (c) (1:025)4 = 1 + (d) (1:025)4 = 1 +
1 = 10:38%. then r^ = 2[(1:025) 1] = 10:13%. then r^ = 12 (1:025) 1 = 9:92%. 1 = 9:88%. then r^ = 365 (1:025)
r^ 2 2 r^ 12 12 r^ 365 365
2
1=3
4=365
(e) (1:025)4 = er^ then r^ = log [(1:025)4 ] = 4 log(1:025) = 9:88%. 3. (a) Alternative 2 because (b) Alternative 2 because (c) Alternative 1 because
10;000 1+0% 10;000 1:1 10;000 1:2
20;000 . (1+0%)5 12;418:43 = 20;000 (1:1)5 . 8;037:55 = 20;000 (1:2)5 .
= 10;000 < 20;000 = = 9;090:91 < = 8;333:33 >
(d) We seek to solve 10;000 20; 000 = : 1+r (1 + r)5
,
(1 + r)4 = 2
)
r = 2 1=4
1 = 18:92%:
4. There are many ways to attack this problem. One way is to calculate the present value of your deposits, and then take that amount forward to the end of year 7. The present value P V is 1;000 1 P V = 1;000a4j12% = 1 = 3; 037:35: 0:12 (1:12)4
At the end of seven years, this is worth F V = P V (1:12)7 = 6; 714:61: Another way to solve this problem is to notice the fact that the deposit made at the end of the …rst year will accrue interest for six years, the one made at the end of the second year will accrue interest for …ve years, and so on. Therefore, P V = 1;000(1:12)6 + 1;000(1:12)5 + 1;000(1:12)4 + 1;000(1:12)3 = 6; 714:61: Of course, the …rst method would be much faster if the deposits were made for 35 years instead of just four! 17
5. We need to solve 300(1 + r)2 + 200(1 + r) + 100 = 700
() ()
3(1 + r)2 + 2(1 + r)
p
This implies that
22 4(3)( 6) = 2(3)
2
1+r =
6=0
2 p 76 = 1 p 19 : 6
3
p p 19 4 1 + 19 1= r= = 11:96%;
3 3 since the other root implies r < 0, which does not make any economic sense. 6. We are looking for t which solves CF t =2. Since CF t = (1 + r)t , we have (1 + r)t = 2
t log(1 + r) = log 2
()
t=
()
log2 : log(1 + r)
7. Suppose …rst that the annual percentage rate is r. The interest credited to the account in year t is then 100(1 + r)t1 r. We are therefore looking for the smallest integer t which satis…es 100(1 + r)t 1 r > 100. First, let us solve for t which satis…es
100(1 + r)t1 r = 100
()
(1 + r)t1 r = 1
This implies t =1
()
(t
1) log(1 + r) + log(r) = 0:
log r log(1 ; + r)
and t is equal to the smallest integer greater than t. For r = 8%, we have t = 33:82, which means that year 34 is the …rst year in which the interest credited to the account will exceed the original investment. 8. We are looking for n which solves 100 100 + (1 + r)n (1 + r)2n (1 + r)2n = (1 + r)n + 1
() 1 = (1 +1 r) + (1 +1r) () (1 + r) (1 + r) 1 = 0:
100 =
() This means that
(1 + r)n =
1
p
n
2n
2n
n
p
12 4(1)( 1) 1 5 = : 2(1) 2
Since only the positive root makes economic sense, this is equivalent to n log(1 + r) = log 18
p ! 1+ 5 2
;
so that log n=
p p 5+1 2
log(1 + r)
=
log
5+1 log2 0:48121 = : log(1 + r) log(1 + r)
9. First, let us solve for the equivalent monthly rate r^: (1 + r^)12 = 1:12 then r^ = (1:12)1=12
1 = 0:9489%:
The present value of what you get is therefore given by 500 500 P V + = + + 1 + r^ (1 + r^)2
500 500 + = 1 (1 + r^)60 r^
1 = 22;793:90: (1 + r^)60
The present value of what you will have to pay back is given by
1 500 500 500 P V = + + + 60 2 (1 + r^) 1 + r^ (1 + r^) (1 + r^)120 1 500 1 = 1 = 20;272:89: 60 (1 + r^) r^ (1 + r^)120
Since the present value of the money you will get is larger than that you will have to pay back (P V + > P V ), you should accept the o¤er. 10. First, let us recall some notation from the lecture notes: let anjr (resp. anjr ) denote the present value of an n-year annuity payable at the end (resp. beginning) of each year when the annual rate of interest is r. From the lecture notes, we have
e
anjr anjr
e
1 1 = 1 ; and r (1 + r)n 1+r 1 = 1 : r (1 + r)n
(1) (2)
The amount of each yearly withdrawal, which we denote by W , must solve 1;000anjr =
e
1 W anjr : (1:04)25
e
Using (1) above, we have a25j4 = 16:25 and a15j3:5 = 11:92, so that
e
e
1;000(16:25)(1:04)25 W = = 3;633:38: 11:92 11. Let us solve this whole problem in real terms. The present value of what she wants to 500;000 accumulate is (1:05) 30 . Her income, which grows at 2% in real terms, has a present value
19
of
1:02 (1:02)2 40;000 + + 1:05 (1:05)2
(1:02)30 + : (1:05)30
This last expression is equal to 40;000a30jr^, where 1 + ^r = of her income, she wants to make sure that: 40;000a30jr^ =
500;000 (1:05)30
1:05 1:02 .
So, if she saves a fraction
then = 14:64%:
12. In what follows, we calculate the present value of each of these contracts. (a) The equivalent annual rate r^ must solve 1 + r^ = (1:04)2 , which implies r^ = 8:16%. Using the notation from the last problem, the present value of this contract is therefore 400;000 1 400;000a5j8:16% = 1 = 1;590;371:72: 0:0816 (1:0816)5
(b) The equivalent quarterly rate r^ must solve (1 + r^)4 = (1:04)2 , which implies r^ = 1:98%. The present value of this contract is therefore 95;000a4j1:98%
95;000 = 1 0:0198
1 = 1;556;329:80: (1:0198)20
(c) The equivalent monthly rate r^ must solve (1 + r^)12 = (1:04)2 , which implies r^ = 0:656%. The present value of this contract is therefore 31;000a12j0:656%
31;000(1:00656) = 1 0:00656
1 = 1;543;636:15: (1:00656)60
(d) The present value of this lump-sum contract is obviously 1,500,000. The player should accept contract (a). 13. (a) The value of the perpetuity at the end of the …rst year is P V 1 =
500 = 5;000: 0:10
Today, this is worth P V =
P V 1 500 1 = = 4;545:45: 1:10 0:10 1:10
(b) The value of the perpetuity at the end of the second year is P V 2 = 20
500 = 5;000: 0:10
Today, this is worth P V =
P V 2 500 1 = = 4;132:23: (1:10)2 0:10 (1:10)2
(c) The equivalent semiannual rate r^ must solve (1+^ r)2 = 1:1, which implies r^ = 4:88%. The present value of this perpetuity is therefore given by P V =
1 500(1:0488) = 7;338:33: (1:0488)8 0:0488
(d) Using the formula in the lecture notes, we have P V =
100(1:10) = 1;375:00: 0:10 0:02
14. For each question, let r^ denote the equivalent annual rate. (a) (1 + r^)2 = (1:04)2 then r^ = 4%. (b) (1 + r^)3 = (1:04)3 (1:02) then r^ = 4:69%. (c) (1 + r^)4 = (1:04)4 (1:02) then r^ = 4:52%. 15. The credit card company claims that the annual percentage rate is 12 1:583% = 18:996%. However, with monthly compounding, the e¤ective annual rate is (1:01583)12 1 = 20:74%. The rate of 18.996% corresponds to an annual rate compounded monthly.
16. (a) The removal cost will be 10;000;000(1:05)15 in 15 years, so that the present value is P V =
10;000;000(1:05)15 = 4;345;050: (1:11)15
(b) The (end-of-year) annual payments A must accumulate to 10;000;000(1:05)15 in 15 years. Since these payments are invested at 6.5%, equating present values yields: A a15j6:5
10;000;000(1:05)15 = (1:065)15
then A = 859;694:
17. Denote the present value of an annuity paying $1 at the beginning of every 6-month period during the next 10 years by P VA1 . Denote the present value of a 10-year deferred annuity paying $1 at the beginning of every 6-month period for 10 years by P VA2 . We have:
21
200P V A1 : + : 100P V A2 = 4;000 and 250P V A2 = 2;500: Solving for P V A1 and P V A2 yields P V A1 = 15 and P V A2 = 10. The desired present value is therefore given by 200P V A1 + 300P V A2 = 6;000: 18. (a) We are interested in calculating P V = Observe that
C C C + + + (1 + r)n (1 + r)2n (1 + r)3n
P V C C = + + (1 + r)n (1 + r)2n (1 + r)3n
so that P V
(1P+V r)
n
=
:
;
C (1 + r)n
which, after rearranging, yields P V =
C (1 + r)n
1:
(b) Let P VP t denote the present value of a t-year deferred perpetuity paying $1 at the beginning of every year. It can be shown that P VP t =
1 1+r 1=r = : (1 + r)t r (1 + r)t1
Also, notice that the perpetuity that we are interested in is simply the sum of
a 1-year deferred perpetuity paying $1 at the beginning of every year; a 2-year deferred perpetuity paying $1 at the beginning of every year; a 3-year deferred perpetuity paying $1 at the beginning of every year; Therefore, P V = P VP 1 + P VP 2 + P VP 3 +
22
= 1=r +
1=r 1=r + + 1 + r (1 + r)2
Observe that
P V 1=r 1=r = + + 1+r 1 + r (1 + r)2
so that P V
;
1P+V r = 1r :
After rearranging terms, we …nd that P V =
1+r : r2
19. Recall that the limit of a function is the function of the limit. Also, observe that
log 1 + = lim m!1 1=m
r lim log(CF 1 ) = lim m log 1 + m!1 m!1 m =
lim
m
!1
1 1+r=m
r m2 1=m2
r m
r = r; m!1 1 + r=m
= lim
where we used “L’Hôpital’s rule” to start the second line. We therefore have lim CF 1 = lim elog(CF 1) = elimm
!1
m
!1
m
!1
as desired.
23
log(CF 1 )
= er ;
2.2
Bond Pricing – Term Structure
1. The semi-annual interest payment is $1;000 0:08 2 = $40. There are a total of 40 periods of six months, i.e. two a year for 20 years. If the annual percentage rate is r, the appropriate equivalent 6-month discount rate is obtained by solving for r^ in
(1 + r^)2 = 1 + r
,
r^ =
p 1 + r 1:
The price of the bond is then
1;000 40 P = 40a40jr^ + = 1 (1 + r^)40 r^
1 1;000 + : (1 + r^)40 (1 + r^)40
(a) r^ = 3:923%; P = 1;015:41. (b) r^ = 4:881%; P = 846:35. (c) r^ = 2:956%; P = 1;242:96. 2. Let C denote the semiannual coupon. The bond price must satisfy 923:14 = Ca30j5 +
1;000 (1:05)30
The coupon rate is therefore
923:14 = C 15:37245 +
,
1;000 (1:05)30
)
C = 45:
45 2 1;000
= 9%.
1 3. Let DF = 1+y denote the annual discount rate implied by the yield y. The bond price must satisfy
960 = 100DF + 1;100DF 2
) )
1;100DF 2 + 100DF 960 = 0 100 + (100)2 4(1;100)( 960) DF = = 0:88984936: 2(1;100)
p
Note that we discarded the negative root, which does not make any economic sense. The 1 yield of the bond is therefore y = DF 1 = 12:3786%.
4. (a) First, let’s …nd the discount factors for 1, 2, and 3 years: DF 1 , DF 2 , and DF 3 . We know that the present value of a stream of cash ‡ows is the sum of the present values of each individual cash ‡ow: P V = (C 1
DF ) + (C DF ) + + (C DF ) : 1
2
24
2
T
T
Therefore, our three bonds must satisfy: 100DF 1
= 93:46
4DF 1 + 104DF 2
= 94:92
8DF 1 + 8DF 2 + 108DF 3 = 103:64 Solving for DF 1 , DF 2 , and DF 3 , we get DF 1 = 0:9346; DF 2 = 0:8767; DF 3 = 0:8255: We can now solve for r1 , r2 , and r3 using rt =
1 DF t
1=t
1;
t = 1; 2; 3:
We …nd r1 = 7:00%, r2 = 6:80%, and r3 = 6:60%. (b) The forward rates f 1 , f 2 , and f 3 are given by 1 DF 1 DF 1 = DF 2 DF 2 = DF 3
f 1 = f 2 f 3
1 = 7:00% = r ; 1 = 6:60%; 1 = 6:21%: 1
(c) The price of this bond is given by: P = 6DF 1 + 6DF 2 + 106DF 3 = 6(0:9346) + 6(0:8767) + 106(0:8255) = 98:37: (d) The price of this bond should be given by: P = 4DF 1 + 4DF 2 + 104DF 3 = 4(0:9346) + 4(0:8767) + 104(0:8255) = 93:09: Since this is smaller than 95.00, we could sell this bond for $95.00, and buy a portfolio of bonds A, B and C which would replicate it. Consider a portfolio consisting of nA of the A-bond, nB of the B-bond, and nC of the C-bond. This portfolio would pay 100nA + 4nB + 8nC
25
at the end of the …rst year. Since we would like to replicate a payo¤ 4 at the end of that year, we should choose nA , nB , and nC such that 100nA + 4nB + 8nC = 4:
(3)
Similar reasonings for the second and third years result in: 104nB + 8nC = 4; and
(4)
108nC = 104:
(5)
Solving (3), (4), and (5) for nA , nB and nC , we obtain nA = nB =
25 702 ;
nC =
26 : 27
So consider the following strategy: Strategy C0 C1 C2 C3 Sell 3-yr bond with 4% coupon 95.00 -4.00 -4.00 -104.00 25 Sell 702 A-bond 3.33 -3.56 0 0 25 Sell 702 B-bond 3.38 -0.14 -4.00 0 Buy 26 -99.80 7.70 7.70 104.00 27 C-bond Total 1.91 0 0 0 This $1.91 is an arbitrage opportunity. 5. (a) Using the de…nition of “yield to maturity,” the price of Bond A is given by 1; 000 = $952:381: 1:05 Similarly, the price of Bond B is given by 50 1;050 + = $984:383; 1:0585 (1:0585)2 and the price of Bond C is given by 100 1;100 + = $1;073:336: 1:06 (1:06)2 (b) We can use the …rst two bonds to solve for the discount factors:
(
952:381 = 1;000 DF 1 984:383 = 50 DF 1 + 1;050
26
DF
2
( )
=
DF 1 = 0:952 DF 2 = 0:892
This implies that the price of the third bond should equal 100
DF + 1;100 DF = 1;076:610: 1
2
Since this is not the case, there must be an arbitrage opportunity. In order to …nd the arbitrage opportunity we set up a portfolio of nA units of bond A and nB units of bond B so as to replicate the cash ‡ows of bond C:
(
100 = 1;000 nA + 50 nB 1;100 = 0 nA + 1;050 nB
cash ‡ow in 1 year cash ‡ow in 2 years
This gives nA = 1=21 and nB = 22=21. The portfolio therefore has a cost of nA (952:381) + nB (984:383) = $1;076:610: In order to obtain an arbitrage, we buy bond C and (short) sell the replicating portfolio (i.e., short sell nA units of bond A and nB units of bond B). This gives an immediate pro…t of 1;076:610 1;073:336 = $3:274
and zero cash ‡ows in years 1 and 2. Here is the complete arbitrage table: Strategy C 0 C 1 C 2 Buy 1 C-Bond -1,073.336 100.000 1,100 1 Sell 21 A-bond 45.351 -47.619 0 Sell 22 B-bond 1,031.258 -52.381 -1,100 21 Total 3.274 0 0 6. (a) We have r1 = f 1 = 6:0%; and (1 + r2 )2 f 2 = 1 + r1
1 = 6:50059%:
The 3-year spot rate r3 must solve: (1 + r3 )3 = (1 + r2 )2 (1 + f 3 ) = 1:2079297 then r3 = 6:49941%: Finally, we have (1 + r4 )4 f 4 = (1 + r3 )3
(1 + r4 )4 1= (1 + r2 )2 (1 + f 3 )
1 = 7:50530%:
(b) Since we have r1 < r 2 < r 3 < r4 , the term structure is increasing. 27
(c) The price of this bond is given by P =
90 90 90 1;090 + + + = 1;078:5094: 1 + r1 (1 + r2 )2 (1 + r3 )3 (1 + r4 )4
(d) All we need to do is to discount the bond’s cash ‡ows with a (‡at) rate of 6.50%, and see whether the result is greater than P (which implies a yield greater than 6.50%) or smaller than P (which implies a yield smaller than 6.50%): 90 90 90 1;090 + + + = 1;085:64: 1:065 (1:065)2 (1:065)3 (1:065)4 Since 1;085:64 > P , the yield on this bond is greater than 6.50%. 7. By the de…nition of “yield,” the price of the …rst bond is given by 60 60 60 1; 060 + + + + 1:12 (1:12)2 (1:12)5 (1:12)6 60 1 1;000 = 1 + = 753:32: 0:12 (1:12)6 (1:12)6
P 1 =
Similarly, the price of the second bond is given by 100 100 100 1; 100 + + + + 1:08 (1:08)2 (1:08)5 (1:08)6 100 1 1;000 = 1 + = 1;092:46: 0:08 (1:08)6 (1:08)6
P 2 =
Of course, we know that we can rewrite the expressions for these prices as 60DF 1 + 60DF 2 +
+ 1;060DF = 753:32;
(6)
+ 1;100DF = 1;092:46:
(7)
6
and 100DF 1 + 100DF 2 +
6
Now, let us subtract 0.6 (7) from (6) (observing that DF 1 , : : :, DF 5 will cancel out):
1; 060DF 6
0:6(1;100)DF = 753 0:6(1;092:46) () 400DF = 97:85 () DF = 0:2446 () (1 +1r ) = 0:2446 () r = 26:45%: 6
6
6
6
6
6
28
2.3
The NPV Rule
1. (a)
5 4
1 = 25%.
(b) $2.6 million – $1.6 million = $1.0 million. (c) $3 million. (d)
3 1
1 = 200%.
(e) 25%. (f) $4 million – $1.6 million = $2.4 million. (g) $2.4 million – $1 million = $4 million – $2.6 million = $1.4 million. (h) $4 million. (i) $1 million. (j) $3.75 million. (k) He is a lender (since he consumes less than $1.6 million today). 2. (a) If Casper invests I , he will consume 200;000 I today, and I wants 200;000 I = I 1:08;
1:08 next year. He
which implies an investment of I = 96;153:85. He can then consume 200;000 96;153:85 = 103;846:15 in each period. (b)
i. Casper must choose the amount I (to be invested at 10%) so that 200;000
I = I 1:10:
This implies that he invests I = 95;238:10, and consumes 200;000 104;761:90 in each period.
95;238:10 =
ii. If Casper is allowed to borrow against future income, he should invest the entire $200; 000 at 10%, and borrow B (at a rate of 8%) so that B = 200;000(1:10)
B(1:08)
This implies B = 105; 769:23, and this is the amount that Casper can consume in each period. (c)
i. The net present value of the opportunity in (b.i) is NP V =
95; 238:10 + 95; 238:10(1:10) = 1; 763:66: 1:08 29
ii. The net present value of the opportunity in (b.ii) is NP V =
200; 000 + 200; 000(1:10) = 3; 703:70: 1:08
30
2.4
Capital Budgeting under Certainty
1. The NPV of this project is given by NP V =
cost of equipment + P V (after-tax net operating pro…ts) + P V (depreciation tax shield) + P V (equipment sale) P V (tax on equipment sale):
These PVs are calculated as follows: P V (after-tax net operating pro…ts) = (1
" #
40;000 3;000 0:35) 1 0:12
P V (depreciation tax shield) = 0:35(150;000)
1 1:12
3
= 57;764:04;
0:2 0:32 0:192 + + = 29;942:60; 1:12 (1:12)2 (1:12)3
60;000 = 42;706:81; (1:12)3 0:35[60;000 (1 P V (tax on equipment sale) = P V (equipment sale) =
0:2 0:32 0:192)150;000] = 4;185:27:
This gives us NP V =
(1:12)3
23;771:81 < 0, so that Nalyd should not purchase this equipment.
2. The current book value of the old harvester is 50;000
50;000 10
5
= $25;000:
The incremental cash ‡ow at t = 0 if the new harvester is purchased is therefore C 0 = 20;000 + 0:34(25;000
20;000) P = 21;700 P
where we have have taken into account the tax e¤ect of the book loss on the sale of the old harvester. The incremental cash ‡ow between years 1 and 5 equals the after-tax cost savings plus the tax e¤ect of the incremental depreciation: C t = (1
P 0:34)10;000 + 0:34 10
5;000
= 4;900 + 0:034P (t = 1;:::; 5)
After year 5, the old harvester would have been completely depreciated, so that the incremental cash ‡ow between years 6 and 9 is: C t = (1
P 0:34)10;000 + 0:34 10 = 6;600 + 0:034P (t = 6;:::; 9)
Finally, the incremental cash ‡ow in year 10 re‡ects the incremental salvage value of the
31
new harvester and the consequent tax increase: C 10 = (1
P 0:34)10;000 + 0:34 10 + (5;000 1;000) 0:34(5;000 1;000)
= 9;240 + 0:034P
The NPV of the project is then: 10
NP V =
X t=0
C t 1:15t
= 21;700
6;600 + 0:034P + 1 0:15 (1:15)5 = 49;778
4;900 + 0:034P P + 1 0:15
0:829P:
1 (1:15)4
1 (1:15)5
+
9;240 + 0:034P (1:15)10
Setting NP V = 0 and solving for P gives P = $60;019. This is the maximum price that Conocococonut would be willing to pay for the new harvester. 3. If the truck is replaced after n years, its net present value is given by NP V n =
6;250 22;500 + 1 0:1
1 An + ; (1:1)n (1:1)n
where An is the abandonment value of the truck at the end of year n (these values are given in the problem’s table). These NPV’s are calculated in the table below.
NP V n
n 1 2 3 4 5 -909.09 -82.64 1,307.29 726.73 1,192.42
Since these …ve mutually exclusive projects have di¤erent lives, we need to calculate the equivalent annual cash ‡ow for each of them. In particular, we need to calculate EACF n which solves EACF n 1 1 = NP V n ; 0:1 (1:1)n
for n = 1; : : : ; 5. These values are given in the following table:
EACF n
n 1 2 3 4 5 -1,000 -47.62 525.68 229.26 314.56
Since the equivalent annual cash ‡ow for the three-year project is the highest equivalent annual cash ‡ow, the optimal economic life of the new delivery truck is three years.
32
4. Recall that the present value of a T -year growing annuity paying C at the end of the …rst year and growing at an annual rate g is given by C r
g
" # 1+g 1+r
1
T
;
where r is the annual interest rate. First, let us calculate the real rate of interest R: R=
1:15 1:05
1 = 9:52381%:
The NPV of this project is given by NP V =
cost of machine + P V (after-tax revenues) P V (after-tax production costs) + P V (depreciation tax shield) + P V (sale of machine) P V (tax on machine sale):
These PVs are calculated as follows: P V (after-tax revenues) = (1
" # " #
20 8;000 0:34) 1 0:15 0:02
P V (after-tax production costs) = (1 P V (depreciation tax shield) = 0:34
1:02 1:15
10 8;000 0:34) 1 0:15 0:1
100;000 1 1 5 0:15
5
= 366;413:08;
1:1 1:15
5
= 210;452:24;
1 = 22;794:65; (1:15)5
5;000 = 3;172:69; (1 + R)5 0:34 (5;000 0) P V (tax on machine sale) = = 1;078:71: (1 + R)5 P V (sale of machine) =
This gives us NP V = 80;849:47 > 0, so that the project should be undertaken. 5. Let us …rst calculate the equivalent annual cost C for the disk: C a5j10% = 10;000 then C =
10;000 = 2;637:9748: 3:7908
(a) If the project is terminated, there is no extra usage required, so the cost is zero. (b) If the project is terminated, the company only needs a new disk at the end of year 4. Assuming that this investment will have to be renewed every …ve years (the life of the 0 1 2 3 4 5 6
7
...
disk), the cash ‡ows will look as follows: 2,638 2,638 2,638 33
...
On the other hand, if the project is not terminated, the new disk has to be purchased at the end of two years. Assuming that the investment is renewed every …ve years, the 0
1
2
3
4
5
6
7
...
cash ‡ows will look as follows: 2,638 2,638 2,638 2,638 2,638 Comparing the last two …gures, we conclude that the cost of the extra usage is given by 2;637:9748 2;637:9748 + = 3;783:72: (1:1)3 (1:1)4 6. The P V of the cash out‡ows from the new machine is
750;000 P V = 3;000;000 + (1 0:34) 1 0:12 600;000 1 0:34 1 0:12 (1:12)5 = 4;011;539:70:
1 (1:12)5 (1
0:34)
100;000 (1:12)5
The equivalent annual cost (EAC ) for the new machine solves
EAC 4;011;539:70 = 1 0:12
1 (1:12)5
;
which implies EAC = 1;112;840:15: The P V of after-tax cash out‡ows over the next two year for the di¤erent replacement options are as follows.
Replace now: P V =
1;600;000 + 0:34(1;600;000
= 484;756:64:
EAC 1;000;000) + 1 0:12
1 (1:12)2
Replace in 1 year: P V = (1
200;000 0:34) 1;000;000 0:34 1:12 1:12 1;200;000 1;200;000 800;000 EAC 1:12 + 0:34 + 1:12 (1:12)
2
= 465;720:79: 34
...
Replace in 2 years:
1;000;000 P V = (1 0:34) 1 0:12 600;000 (1:12)2 = 522;193:88:
1 (1:12)2
200;000 0:34 1 0:12
1 (1:12)2
Thus, it is best to replace in 1 year. 7. (a) The spreadsheet is available on webCafé and is called IRR.xls. It contains a macro with instructions on how to run the macro. Alternatively, one can use the “Tools / Goal Seek” option to …nd the IRR. The three inputs (for this spreadsheet) are as follows:
Set cell: B18 To value: 0 By changing cell: B19 The IRR of project A is found to be 8.93%, and that of project B is 11.32%. (b) One can specify cash ‡ows of zero for years four and …ve. The IRR of project C is 8.21%. (c) The yield of a bond can be calculated as the IRR of a project which has an initial cash ‡ow equal to the (negative) price of the bond. In this case The yield of the bond is calculated to be 9.30%. 8. First, the net present value of each of these three projects can be calculated to be (in $millions): NP V A = 0:0918823, NP V B = 1:3423590, NP V C = 2:8459626. Also, the present values of their …rst year costs are given by PVC A = 4
2 1:12 = 2:2142857;
2 = 7:7857143; and 1:12 3 PVC C = 2 + = 4:6785714: 1:12 PVC B = 6 +
The total number of engineer-hours available to Seltaeb for these projects is 10 50 48 4 = 96;000. Letting xA , xB and xC denote the fractions of each project that will be undertaken, we can therefore write the linear programming problem as follows:
35
Maximize 0.0918823xA Subject to 2.2142857xA 25 0 xA 1;
+ 1.3423590xB + 7.7857143xB + 35xB 0 xB 1;
+ 2.8459626xC + 4.6785714xC + 60xC 0 xC 1
8 96
9. (a) First, let us calculate the present value of future bene…ts for these two projects:
3;000 P V A = 1 0:12 7;400 P V B = 1 0:12
1 = 10;814:33; (1:12)5 1 = 26;675:34: (1:12)5
The net present values of the two projects are thus given by N P V A =
10;000 + P V = 25;000 + P V
A
= 814:33; and
N P V B
B
= 1;675:34;
so project B would be chosen according to the NPV rule. Now, the internal rates of return for these projects must solve 0 =
0 =
3;000 10;000 + 1 IRRA 7;400 25;000 + 1 IRRB
1 ; and (1 + IRRA )5 1 : (1 + IRRB )5
Solving for the internal rates of return, we obtain IRRA = 15:24% > 14:67% = IRRB , so the IRR rule would select project A. Finally, the pro…tability indexes for these two projects are given by NP V A = 0:081; and 10;000 NP V B P I B = = 0:067: 25;000 P I A =
Since P I A > P I B , the pro…tability index rule would also select project A. (b) Project B, since this is the project that will increase the shareholders’ wealth the most. (c) Let us look at the incremental project B-A. This project costs $25;000 $10;000 = $15;000, and will produce bene…ts of $7;400 $3;000 = $4;000 per year for …ve years. The internal rate of return IRRB A must solve
0=
4;400 15;000 + 1 IRRB A 36
1 ; (1 + IRRB A )5
which implies IRRB A = 14:29%. Since this rate is greater than the opportunity cost of capital (12%), we should choose project B over project A. Also, the present value of this incremental project’s future bene…ts is given by
4;400 P V B A = 1 0:12 so that P I B A =
1 = 15;861:02; (1:12)5
NP V B A = 0:057: 15;000
Since this index is positive, we should again choose project B over project A.
37
3
Practice Placement Exams
This section contains two sample placement exams, along with their solutions.
38
3.1
Sample Placement Exam A: Questions Part I: Multiple Choice Questions
(Total points: 25) Instructions: A correct answer to each of these questions is worth 5 points. An incorrect answer is worth 0. Also, for each question that you choose not to answer, you get 1 point. If you do choose to answer, write your answer clearly on page 1 of your blue booklet. Your answers should be capital letters written with a pen. Only the …nal answer will be marked and there shall be no partial credit for the multiple choice questions. 1. A 10-year annuity paying $x at the beginning of every year (i.e. the …rst of ten payments is made today) is worth the same (today) as an annuity of $300 payable every 6 months for 10 years (20 payments), the …rst payment of which is due 66 months from now. If the annual interest rate (compounded annually) is 3%, …nd x. (a) 232.73 (b) 502.48 (c) 506.23 (d) 508.11 (e) 521.42 2. A machine costing $3,000 must be replaced at the end of 8 years. The resale value of the machine at the time of replacement is $600. At what annual discount rate (compounded annually) would it be equally economical to use a similar machine costing $4,000 with a life of 8 years and a resale value of $1,900? (Assume that there are no taxes.) (a) 2.4% (b) 2.7% (c) 3.0% (d) 3.3% (e) 3.6% 3. What is the present value of 15 payments of $100 each received every 18 months (the …rst one occurring in 18 months from now), if the annual discount rate (compounded annually) is 9%? (a) $620.43 (b) $875.56 39
(c) $930.61 (d) $951.28 (e) $1,209.10 4. Corporate managers can maximize shareholder wealth by choosing positive NPV projects because: (a) all investors have the same preferences. (b) the unhappy shareholders can sell o¤ their shares. (c) given the existence of …nancial markets, investors will be satis…ed with the same real investment decisions regardless of personal preferences. (d) managers are wiser than shareholders regarding investments. (e) none of the above. 5. In the …gure below, the sloping straight line represents the opportunities for investment in the capital market, and the solid curved line represents the opportunities for investment in plant and machinery (real assets). The company’s only asset at present is $21 million in cash.
(Note that the …gure is not drawn to scale, and that all the numbers are in millions) Let I denote the optimal amount that should be invested in real assets, and r the interest rate in capital markets. Calculate I=r. (a) 3:2 million (b) 12 million (c) 32 million (d) 40 million (e) 60 million
40
Part II: Essay Questions
(Total points: 75) Instructions : Each of the following questions is to be answered in the blue booklets. You can use a pen or a pencil. The number of points for each question is indicated in parentheses at the beginning of the question. In answering these questions, make sure to show all your calculations; in particular, no points will be given for calculator shortcuts. Finally, please keep in mind that I can’t grade what I can’t read. 1. (20 points) Every year, you receive your entire annual salary at the end of the year. This year, your end-of-year salary will be $50,000 (in nominal terms). In real terms, you expect your salary to increase at a rate of 2% per year in the future. You have decided to start saving for retirement by putting money in a savings account. You plan to retire in 35 years, and you expect to live for 25 years after that. You assess that a reasonable lifestyle during those 25 years will require you to have, at the end of every year, a disposable income of $25,000 in real terms (i.e. the same purchasing power as $25,000 today). The nominal interest rate on your savings account is 8%, and it is expected to stay at that rate forever. The real interest rate is also expected to stay at its current level of 3.5%. (a) What is the in‡ation rate? (b) How much money (in nominal terms) will you need to have in your savings account when you retire, in 35 years (end of year 35), in order to be able to enjoy the lifestyle that you …nd reasonable? Hint : First calculate the amount that you will need in real terms. (c) Suppose that you will start saving for retirement at the end of the current year. Suppose further that you plan to make 35 deposits (one at the end of every year). All deposits are a …xed fraction x of your salary. Find the fraction x that will allow you to reach your “reasonable lifestyle” objective. Hint : You will need to make use of the growing annuity formula. 2. (15 points) You are a …nancial analyst for a company that is considering a new project. If the project is accepted, it will use a fraction of a storage facility that the company already owns but currently does not use. The project is expected to last 10 years, and the annual discount rate is 10% (compounded annually). You research the possibilities, and …nd that the entire storage facility can be sold for $100,000 and a smaller (but big enough) facility can be acquired for $40,000. The book value of the existing facility is $60,000, and both the existing and the new facilities (if it 41
is acquired) would be depreciated straight line over 10 years (down to a zero book value). The corporate tax rate is 40%. What is the opportunity cost of using the existing storage capacity? Hint:
Think about what you would gain and lose if you did not.
3. (15 points) You own a rental building in the city and are interested in replacing the heating system. You are faced with the following alternatives: (a) A solar system, which will cost $12,000 to install and $500 at the end of every year to run, and will last forever (assume that your building will too). (b) A gas-heating system, which will cost $5,000 to install and $1,000 at the end of every year to run, and will last 20 years. (c) An oil-heating system, which will cost $3,500 to install and $1,200 at the end of every year to run, and will last 15 years. If your opportunity cost of capital (discount rate) is 10%, which of these three options is best for you? 4. (25 points) The following bonds are traded in a well functioning market: Face Bond Type Value Coupon Maturity Price A Zero Coupon Bond $100 — 1 year $92.00 B Coupon Bond $100 8% 2 years $101.32 (a) Assuming that the coupon bond (bond B) makes only annual payments, what discount factors (DF 1 , DF 2 ) are imbedded in these prices? Note :
Show all your calculations; no points will be given for answers found by a sophisticated calculator. (b) What are the 1-year, and 2-year spot rates (r1 and r2 )? (c) Suppose that you would like to purchase a two-year coupon bond with a face value of $10,000 and a coupon rate of 6% (with annual coupon payments). Since such a bond is not traded in this economy, what portfolio of bonds A and B could you form to satisfy your needs (i.e. how can you replicate this bond using the original two bonds). Note :
Make sure to describe that portfolio clearly, i.e. what you are buying/selling.
(d) What is the exact yield to maturity on i. bond A; 42
ii. bond B. Note : Again, show all your calculations; no points will be given for answers found by
a sophisticated calculator. In particular, you will need to use the following formula for the roots of ax2 + bx + c = 0:
p b b 4ac x= : 2
2a
Your answers should have at least two decimals, like 9.53%.
43
3.2
Sample Placement Exam A: Solutions PART I: Multiple Choice Questions
1. C. We must have xa ~10j3% =
300 a20jr^; (1:03)5
where the equivalent semiannual interest rate r^ must satisfy (1 + r^)2 = 1:03
r^ = 1:4889157%:
!
Since a ~10j3% = 8:7861089 and a20jr^ = 17:1874132, we …nd x = 506:23. 2. D. The interest rate r must satisfy:
3;000 + (1600 + r) () ()
1;000 = r=
=
8
4;000 + (11;900 + r)
8
1;300 (1 + r)8
1;300 1;000
1=8
1 = 3:33%:
3. A. First, let us …nd the equivalent 18-month rate r^: (1:09)3=2 = 1 + r^
r^ = 13:79934%:
)
The present value P V of the annuity is therefore
100 P V = 100a15jr^ = 1 0:1379934
1 = 620:43: (1:1379934)15
4. C. See Section I.3.1 of the lecture notes. 5. D. The amount invested in real assets is given by I = 21 million
15 million = 6 million:
The slope of the straight line (capital market investment opportunities) is is 34:5 (1 + r) = r = 15%: 30 Therefore, I=r = 40 million.
)
PART II: Essay Questions
44
(1 + r), that
1. (a) (4 points) The in‡ation rate is given by i=
1+r 1+R
1:08 1 = 1:035 1 = 4:3478261%:
(b) (8 points) At the end of year 35, the present value P V R in real terms of your retirement income is
25;000 = 1 0:035
P V R = 25;000a25j3:5%
1 = 412;037:86: (1:035)25
Since this amount is in real terms, we need to in‡ate it for 35 years. Therefore, the nominal amount P V n needed in the account in 35 years is P V n = 412;037:86(1:043478261)35 = 1;827;495:55: (c) (8 points) The present value at time 0 of the amount needed in the account in 35 years is 412;037:86 P V 0 = = 123;601:83: (1:035)35 Alternatively, we could do the calculations in nominal terms: P V 0 =
1;827;495:55 = 123;601:83: (1:08)35
The present value of your 35 contributions should be equal to this amount. In real terms:
" #
50;000x=(1:043478261) 123;601:83 = 1 0:035 0:02
1:02 1:035
35
)
x = 9:6713%:
Again, the calculations could have been done in nominal terms, in which case they grow at g = (1:02)(1 + i) 1 = 6:4347826%:
"
50;000x 123;601:83 = 1 0:08 0:064347826
1:064347826 1:08
# 35
2. (15 points) By selling the existing facility, the company would
gain $100,000 from the sale; pay a tax of
($100;000
$60;000)(40%) = $16;000
on the capital gain resulting from this sale; 45
)
x = 9:6713%:
lose the yearly depreciation tax shield of $60;000 (40%) = $2;400 10 for 10 years. By acquiring the new facility, the company would
pay $40,000 to buy the facility; gain a yearly depreciation tax shield of $40;000 (40%) = $1;600 10 for 10 years. The present value P V of all these gains and losses represents the opportunity cost of using the existing storage capacity: P V = 100;000 =
16;000 2;400a j 40;000 + 1;600a j 2;400 1 1;600 1 100;000 16;000 1 40;000 + 1 0:10 (1:10) 0:10 (1:10) 10 10%
= 39;084:35:
10
10 10%
10
3. (15 points) There are two equivalent approaches for solving this problem: (i) repeat the cash ‡ows to in…nity (which is already done for alternative A), and calculate and compare the net present values; (ii) Calculate and compare the equivalent annual costs of the three alternatives. Let us use the second approach. (a) The present value of the costs is P V A = 12;000 +
500 = 17;000: 0:10
The equivalent annual cost C A must solve 17;000 =
C A 0:10
) C
A
= 1;700:
(b) The present value of the costs is
1;000 P V B = 5;000 + 1 0:10
46
1 = 13;513:56: (1:10)20
The equivalent annual cost C B must solve
C B 13;513:56 = 1 0:10
1 (1:10)20
) C
B
= 1;587:30:
(c) The present value of the costs is
1;200 P V C = 3;500 + 1 0:10
1 = 12;627:30: (1:10)15
The equivalent annual cost C C must solve
C C 12;627:30 = 1 0:10
1 (1:10)15
) C
C
= 1;660:16:
Therefore, alternative B is the best alternative, since it involves the lowest costs. 4. (a) (6 points) Since the price of every bond must be the sum of its discounted cash ‡ows, the discount factors must solve: 100DF 1 + = 92:00 8DF 1 + 108DF 2 = 101:32
(1) (2)
Using (1), we have DF 1 = 0:92. Using this value for DF 1 in (2), we get DF 2 =
101:32
8(0:92) = 0:87:
108
(b) (4 points) The discount factors can be written as DF t =
1 : (1 + rt )t
Therefore, r1 = r2 =
1 DF 1 1
1 = 8:69565%; and 1 = 7:21125%:
1=2
DF 2
(8) (9)
(c) (7 points) The bond that you would like to purchase will pay 6%($10;000) = $600 at the end of the …rst year, and $10,600 at the end of the second year. Let us form a portfolio containing a quantity nA of bond A, and nB of bond B. We would like this portfolio to pay $600 at the end of the …rst year, and $10,600 at the end of the 47
second year. Mathematically we would like nA and nB to satisfy: 100 nA + 8 nB = 600 + 108 nB = 10;600
Using (2), we have nB =
10;600 108
nA =
(1) (2)
= 98:148148. Using this value for nB in (1), we get
600
8(98:148148) = 1:851852: 100
Therefore, the portfolio that would replicate the 6% coupon bond would consist in selling 1.851852 units of bond A, and buying 98.148148 units of bond B. (d)
i. (3 points) The yield on a zero-coupon bond with a maturity of t years is simply the t-year spot rate. Therefore the yield yA of bond A is yA = r1 = 8:69565%. ii. (5 points) The yield to maturity yB for bond B has to satisfy 101:32 =
8 108 + 1 + yB (1 + yB )2
()
108x2 +8x 101:32 = 0; where x =
1 : 1 + yB
Solving for x using the quadratic equation formula, we …nd 1 =x= 1 + yB
p 8
(8)2
4(108)(101:32) = 0:9327379:
2(108)
Solving for yB (ignoring the “minus” root, which has no economic meaning), we …nd yB = 7:26721%.
48
3.3
Sample Placement Exam B: Questions Part I: Multiple Choice Questions
(Total points: 25) Instructions: A correct answer to each of these questions is worth 5 points. An incorrect answer is worth 0. Also, for each question that you choose not to answer, you get 1 point. If you do choose to answer, write your answer clearly on page 1 of your blue booklet. Your answers should be capital letters written with a pen. Only the …nal answer will be marked and there shall be no partial credit for the multiple choice questions. 1. In the …gure below, the sloping straight line represents the opportunities for investment in the capital market, and the solid curved line represents the opportunities for investment in plant and machinery (real assets). The company’s only asset at present is $4.2 million in cash.
(Note that the …gure is not drawn to scale, and that all the numbers are in millions) Let A denote the average rate of return on the optimal real investment, and M the marginal rate of return on that real investment (i.e. the return on the last dollar invested in real assets). What is A M ?
A) 0
B) 30%
C) 60%
D) 80%
E) 160%
2. Which of the following should be treated as incremental cash ‡ows when deciding whether to invest in a new manufacturing plant? The site is already owned by the company, but existing buildings would need to be demolished. I. The market value of the site and existing buildings. II. Lost earnings on other products due to executive time spent on the new facility. III. Future depreciation of the new plant. 49
A) I and II B) I and III C) II and III D) I, II and III E) fewer than two should be considered as incremental cash ‡ows. 3. A man deposits $200 at the end of every 9 months to a fund earning an annual rate of 6% compounded semiannually. If the fund amounts to $4,500 after 6 years, what was the initial value of the fund? A) $1,177
B) $1,839
C) $1,878
D) $2,839
E) $3,849
4. In the …gure below, the sloping straight line represents the opportunities for investment in the capital market, and the solid curved line represents the opportunities for investment in plant and machinery (real assets). The company’s only asset at present is $6.3 million in cash.
(Note that the …gure is not drawn to scale, and that all the numbers are in millions) Let I denote the amount invested in real assets, and N the net present value of that investment. What is N I ? A)
2:4 million
B) 1:2 million
C) 0
D) 1:2 million
E) 2:4 million
5. A fund of $5,000 is set up to pay $123.50 at the end of a regular interval inde…nitely. If the equivalent semiannual rate on the fund is 5% (this is not the annual rate compounded semiannually), how frequently are the payments made? A) monthly B) every 2 months E) every 9 months
C) every 3 months
50
D) semiannually
Part II: Essay Questions
(Total points: 75) Instructions : Each of the following questions is to be answered in the blue booklets. You can use a pen or a pencil. The number of points for each question is indicated in parentheses at the beginning of the question. In answering these questions, make sure to show all your calculations; in particular, no points will be given for calculator shortcuts. Finally, please keep in mind that I can’t grade what I can’t read. 1. (20 points) The Baldwin Company is considering investing in a machine that produces bowling balls. The cost of the machine is $100,000 and production is expected to be 8,000 units per year during the …ve-year life of the machine. The expected resale value is $5,000 (in real terms). Since the interest in bowling is declining, the management believes that the nominal price of bowling balls will increase at only 2% per year. The nominal price of bowling balls in the …rst year will be $20. On the other hand, plastic used to produce bowling balls is rapidly becoming more expensive. Because of this, production costs are expected to grow at 10% (nominally) per year. First-year nominal production costs will be $10 per unit. As usual, assume that the revenues and costs are realized at the end of every year. The machine will be depreciated to zero on a straight-line basis over …ve years. The company’s tax rate is 34% and its nominal cost of capital is 15%. The rate of in‡ation is 5%. Should the project be undertaken? 2. (20 points) Anna is turning thirteen today. Her birthday resolution is to start saving towards the purchase of a sports car that she wants to buy on her 18th birthday. The car costs $16,000 today, and she expects that the price will grow at 2% per year. Anna has heard that a local bank o¤ers a savings accounts paying an interest rate of 6% per year, compounded monthly. She plans to make 60 monthly contributions of $100 each to the savings account (with the …rst contribution made today), and to use the funds in the account on her 18th birthday as a down payment for the car, …nancing the balance through the car dealer. She expects that the dealer will o¤er the following terms for …nancing: 48 equal monthly payments (with the …rst one due one month after she takes possession of the car) at an annual percentage rate (APR) of 8% (compounded annually). (a) What amount will need to be …nanced through the dealer? (b) Assuming that she is correct about the interest rate charged by the dealer, what will be the amount of her monthly payment? 51
3. (15 points) You You are a small-busi small-business ness owner and are considering considering two alternativ alternatives es for your phone system/equipment: Plan A Initial Cost $50,000 Ann Annual ual ma main inte tena nanc ncee cost cost $9,0 $9,000 00 System’s life 20 years Salvage value $10,000
Plan B $120,000 $6,0 $6,000 00 35 years $15,000
The appropriate annual discount rate is 8% (compounded annually). Also, for both plans, the system is depreciated annually on a straight line basis over its life down to its salvage value. alue. The compan company y will will be able able to resell resell the phone phone system system at the end of its life for the salvage value. The company’s tax rate is 35%. Which alternative would you pick? 4. (20 points) Today Today is your your 35th birthday birthday,, and you are considering considering your retiremen retirementt needs. You expect to retire at age 65 (on your 65th birthday), and your actuarial tables suggest that that you you will will live live to be 10 100. 0. You want want to move move to the Baham Bahamas as when you retire. retire. You estimate that it will cost you $300,000 to make the move (on your 65th birthday), and that your living expenses will be $30,000 a year after that (starting on your 66th birthday, and continuing until you die on your 100th birthday). You expect to earn an annual rate of 8% (compounded annually) on your money. (a) How How much will you need to have have saved by your your retirement retirement date to be able to a¤ord this course of action? (b) You already have have $50 $50,000 ,000 in savings savings.. Ho How w much much would you need to save save at the end of each of the next 30 years to be able to a¤ord this retirement plan? (c) If you did not have have any current current savings savings and did not expect to be able to start saving saving money for the next …ve years (i.e. …rst savings payment on your 41st birthday), how much would you have to set aside each year after that to be able to a¤ord this retirement plan?
52
3.4
Samp Sample le Place Placeme men nt Exam Exam B: Solut Solution ions s PART I: Multiple Choice Questions
1. C. The average rate of return on the real investment is A=
2:88 4:2 2:6
1 = 80%; 80%;
and the marginal rate of return on the real investment is equal to the interest rate, that is 6 M = 1 = 20%: 20%: 5 Therefore A M = 60%. 60%.
2. A. (I) TRUE. This is part of the opportunity cost of investing in a new manufacturing plant, as the site and existing buildings could be sold for their market value otherwise. (II) TRUE. These lost earnings on other products are incremental costs, as they are the direct result of the new manufacturing plant. (III) FALSE. Depreciation is never a cash ‡ow. 3. B. The equivalent 9-month interest rate r^ has to satisfy
0:06 (1 + r^) = 1 + 2 2
3
then r^ = 4:533583%: 533583%:
The present value of the fund today is 4;500 4;500 = = 3156: 3156:21: 21: (1 + r^)8 (1: (1:03)12 Let x denote the initial value of the fund. Using the annuity notation, we must have 3156: 3156:21 = 200a 200a8jr^ + x = 200(6: 200(6:5868) + x then x = 1; 1 ;838: 838:84: 84: 4. B. The amount invested in real assets is given by I = 6:3m
3:9m = 2:4m:
The net present value of the investment is N = 7: 7 :5m Therefore, N
6:3m = 1:1 :2m:
I = 1:2m.
5. C. The equiv equivale alent nt rate rate r for periods of 1=m year (m (m = 12: 12: mo mont nthly hly rate; rate; m = 6: 53
semiannual rate; etc.) must solve (1: (1:05)2 = (1 + r )m =
)
r = (1: (1:05)2=m
1:
Therefore, the present value of a perpetuity paying $123.50 at the end of every such period is given by 123: 123:50 123: 123:50 P V = = ; r (1: (1:05)2=m 1
which is equal to 5,000. Solving for m, we obtain m = 4, so that the payments are made every 3 months. PART II: Essay Questions
1. (20 points) points) Recall Recall that that the present present value value of a T -year T -year growing annuity paying C at the end of the …rst year and growing at an annual rate g is given by C r
g
" # 1+g 1+r
1
T
;
where r is the annual interest rate. First, let us calculate the real rate of interest R: R=
1:15 1:05
1 = 9:9 :52381%: 52381%:
The NPV of this project is given by N P V =
cost of machine + P V ( V (after-tax revenues) revenues) P V ( V (after-tax production costs) costs ) + P V ( V (depreciation tax shield) shield) + P V ( V (sale of machine) machine) P V ( V (tax on machine sale: sale:
These PVs are calculated as follows: P V ( V (after-tax revenues) revenues) = (1
" # " #
20 8;000 0:34) 1 0:15 0:02
P V ( V (after-tax production costs) costs ) = (1 P V ( V (depreciation tax shield) shield) = 0:34
1:02 1:15
10 8;000 0:34) 1 0:15 0:1
100; 100;000 1 1 5 0:15
5
= 366; 366;413: 413:08;
1:1 1:15
5
= 210; 210;452: 452:24;
1 = 22; 22 ;794: 794:65; (1: (1:15)5
5;000 = 3;172: 172:69; (1 + R)5 0:34 (5; (5;000 0) P V ( V (tax on machine sale) sale) = = 1; 1 ;078: 078:71: 71: (1 + R)5 P V ( V (sale of machine) machine) =
This gives us N P V = 80; 80 ;849: 849:47 > 0, so that the project should be undertaken. 54
2. (a) (10 points) points) The mont monthly hly interest interest rate rate on the the Savings Savings Accoun Accountt is r=
0:06 = 0: 0 :005: 005: 12
The present value of Anna’s contributions to the Savings Account is
100(1: 100(1:005) P V = 1 0:005
1 = $5; $5 ;198: 198:42: 42: (1: (1:005)60
The future value of her contributions on her 18th birthday will therefore be F V = P V (1 V (1::005)60 = $7; $7;011: 011:89: 89: On the other hand, the price of the car will be P = 16; 16 ;000(1: 000(1:02)5 = $17; $17;665: 665:29; 29; so that the amount that will need to be …nanced is 17; 17;665: 665:29
7;011: 011:89 = $10; $10;653: 653:40: 40:
(b) (10 points) points) The monthly monthly interest interest rate charged by the dealer is r = (1: (1:08)1=12
1 = 0:00643: 00643:
Therefore, the amount (C (C ) of Anna’s monthly payment will solve
C 10; 10;653: 653:40 = 1 0:00643
1 (1: (1:00643)48
)
C = $258: $258:69: 69:
3. (15 points) points) The present present value value of the cost of plan A is
P V A = 50; 50;000 + 9;000(1
0:35)
= 50; 50;000 + 9;000(0: 000(0:65)
50; 50;000
20 8%
j
10; 10;000 (1: (1:08)
40;000 40; (0: (0:35) 20
= 98; 98;417: 417:98: 98:
10; 10;000 (0: (0:35) a 20 1 1 0:08
1 (1: (1:08)20
20
10; 10;000 (1: (1:08)20
Since these costs are incurred over 20 years, the equivalent annual cost EAC A must satisfy: 98; 98;417: 417:98 = EAC A a20j8% = EAC A (9: (9:8181474)
55
)
EAC A = 10; 10;024: 024:09: 09:
Similarly, the present value of the cost of plan B is P V B
120;000 15;000 = 120;000 + 6;000(1 0:35) (0:35) a35j8% 35 105;000 1 1 = 120;000 + 6;000(0:65) (0:35) 1 35 0:08 (1:08)35 = 152;201:00:
15;000 (1:08)
35
15;000 (1:08)35
Since these costs are incurred over 35 years, the equivalent annual cost EAC B must satisfy: 152;201:00 = EAC B a35j8% = EAC B (11:6545682)
)
EAC B = 13;059:34:
Since EAC A < EAC B , plan A if the better plan. Note:
If you calculated the present values for the di¤erent types of cash ‡ows separately, you should have found the following after-tax values: Plan A Plan B Cost of system -50,000.00 -120,000.00 Annual maintenance costs -57,436.16 -45,452.82 System resale 2,145.48 1,014.52 Depreciation tax-shield 6,872.70 12,237.30 Total -98,417.98 -152,201.00 4. (a) (6 points) The cash ‡ows for the retirement plan are in the following …gure: 65
66
67
68
69
...
99
100
300,000 30,000 30,000 30,000 30,000 ... 30,000 30,000 PV The present value P V of these cash ‡ows at age 65 is P V = 300;000 + 30;000a35j8%
30;000 = 300;000 + 1 0:08
1 = 649;637:05: (1:08)35
(b) (7 points) The present value of savings needed for the retirement plan is given by 649;637:05=(1:08)30 = 64;559:20. Let x denote the annual amount saved at the end of each year. The following …gure shows the cash ‡ows for the money saved over the next 30 years:
56
35
36 37 38 39
...
64 65
50,000 x x x x ... x x PV The present value of these cash ‡ows must be equal to 64,559.20, so that x must satisfy x 1 64;559:20 = 50;000 + x a30j8% = 50;000 + 1 : 0:08 (1:08)30
Solving for x, we get x = 1;293:26. (c) (7 points) If you did not have any current savings and only started saving an annual amount y in …ve years (at age 41), the cash ‡ows would be as follows: 35
...
40 41 42
...
y
...
y
64 65
y
y
PV Again, the present value of these cash ‡ows must be equal to 64,559.02, so that y must satisfy
1 1 y 64;559:20 = y a = 1 25 8% j (1:08)5 (1:08)5 0:08 Solving for y, we get y = 8;886:24.
57
1 : (1:08)25
4
Additional Materials
This section contains some extra handouts that will help you go through some of the material covered in class and in the problem sets. Also included is a list of formulas that you are welcome to use to solve problems, and to bring to the placement exam.
58
4.1
Solution to DiMaggio’s Vow
An annuity growing at rate g per period pays C (1 + g)t1 at the end of every period t, for 0 1 2 ::: T T + 1 T + 2 ::: t = 1; : : : ; T : C
C (1 + g)T 1
C (1 + g)
:::
PV The cash ‡ows from this growing annuity equal the di¤erence between the cash ‡ows of two growing perpetuities, one starting at time 1 and the other starting at time T + 1: 0
1
2
:::
P V 1 P V 2
C
C (1 + g)
T
T + 1
T + 2
:::
C (1 + g)T 1
C (1 + g)T C (1 + g)T
C (1 + g)T +1 C (1 + g)T +1
::: :::
Assuming that the per-period rate of interest is r, the present value of the …rst perpetuity is P V 1 =
C
r
g;
whereas the present value of the second perpetuity is
C (1 + g)T 1 P V 2 = : r g (1 + r)T
The present value of the growing annuity is simply the di¤erence between the above two present values: T C 1+g P V = P V 1 P V 2 = 1 : r g 1+r
" #
To solve the problem, we …rst need to calculate the number of periods T , the equivalent monthly interest rate r^, and the equivalent monthly growth rate g^.
There are 52 weeks in each of the 30 years for which DiMaggio is planning to buy ‡owers, so that
T = 52
30 = 1;560:
The equivalent monthly (per-period) interest rate must satisfy: (1 + r^)52 = 1:06
)
r^ = (1:06)1=52
59
1 = 0:1121%:
Similarly, the equivalent monthly (per-period) growth rate must satisfy: (1 + g^)52 = 1:02
)
g^ = (1:02)1=52
1 = 0:0381%:
The present value of DiMaggio’s commitment is therefore P V =
4 r^
g^
" # 1
1 + ^g 1 + r^
60
T
= 3;699:21:
4.2
Addition to How Treasury Securities Are Quoted: Example
The “Ask yld” corresponds to the ask yield of the bond.
It assumes simple interest (no compounding or discounting) and a 365-day year. This yield y can be calculated as follows: 135 100;000 1+ y= y = 5:20%: 365 98;113:75
)
Simple interest: you convert y to a 135-day rate by simple multiplication and division to give y . If it were compound interest, we would convert it using (1 + y) 1 Note that the ask yield is just a notational convention. The yield that we will use in this 135 365
135 365
course is di¤erent and involves compound interest. It will be introduced shortly
61
4.3
Solution to BICC’s Toad Ranch
We will do the NPV calculations in nominal terms and in real terms. Of course, the results should be exactly the same.
Nominal Terms: First, we need to calculate the nominal interest rate r. The relationship between nominal and real rates is as follows: 1+R =
1+r 1+i
)
r = (1 + R)(1 + i)
1 = (1:05)(1:06) 1 = 11:3%:
The nominal cash ‡ows at the end of each of the …rst few years are shown in Panel A of the following table: Cash Flow at End of Year 1 2 3 4 ... Panel A: Nominal Terms Revenues 150,000 150,000(1 + i) 150,000(1 + i)2 150,000(1 + i)3 ... Other costs 40,000 40,000(1 + i) 40,000(1 + i)2 40,000(1 + i)3 ... Labor costs 80,000 80,000(1.01)(1 + i) 80,000(1.01) 2 (1 + i)2 80,000(1.01) 3 (1 + i)3 ... Lease 20,000 20,000 20,000 20,000 ... Panel B: Real Terms 150;000 150;000 150;000 150;000 Revenues ... (1+i) (1+i) (1+i) (1+i) Other costs Labor costs Lease
40;000 (1+i) 80;000 (1+i) 20;000 (1+i)
40;000 (1+i) 80;000 (1+i) (1.01) 20;000 (1+i)2
40;000 (1+i) 80;000 2 (1+i) (1.01) 20;000 (1+i)3
40;000 (1+i) 80;000 3 (1+i) (1.01) 20;000 (1+i)4
Using the present value formulas for perpetuities derived in section I.1, we can calculate the following present values of the nominal cash ‡ows (discounting at r): 150;000 150;000 = = 2;830;188:68; r i 0:113 0:06 40;000 40;000 P V (Other costs) = = = 754;716:98; r i 0:113 0:06 80;000 80;000 P V (Labor costs) = = = 1;886;792:45; r g 0:113 0:0706 20;000 20;000 P V (Lease) = = = 176;991:15: r 0:113 P V (Revenues) =
The growth rate g for the labor costs is calculated as follows: (1 + g) = (1:01)(1 + i)
)
g = (1:01)(1:06)
62
1 = 7:06%:
... ... ...
The NPV of the project is then easily calculated: NP V = P V (Revenues) =
P V (Other Costs) P V (Labor Costs) P V (Lease) 2;830;188:68 754;716:98 1;886;792:45 176;991:15
= 11;688:09:
Since this NPV is greater than zero, the project should be undertaken.
Real Terms: Panel B of the above table shows the real cash ‡ows for this project. Not surprisingly, the nominal year-t cash ‡ows (CF tn ) of Panel A and the real year-t cash ‡ows (CF tr ) of Panel B satisfy the following equation: CF tn = CF tr (1 + i)t : Again, using the present value formulas for perpetuities derived in section I.1, we can calculate the present values for the di¤erent categories of cash ‡ows (notice that we now discount at R): 150;000=(1 + i) 150;000=(1:06) = = 2;830;188:68; R 0:05 40;000=(1 + i) 40;000=(1:06) P V (Other Costs) = = = 754;716:98; R 0:05 80;000=(1 + i) 80;000=(1:06) P V (Labor Costs) = = = 1;886;792:45; R 0:01 0:05 0:01 20;000=(1 + i) 20;000=(1:06) P V (Lease) = = = 176;991:15: R g 0:05 ( 0:0566038) P V (Revenues) =
The growth rate g for the present value of the lease is calculated as follows: 1 (1 + g) =
1 1+i
)
g=
1 1:06
1 = 5:66038%:
As expected, the present values are exactly the same as before, so the NPV of the project is again 11,688.09.
1
The negative growth rate only re‡ects the fact that the cash ‡ows are decreasing through time. The same formulas as usual can still be applied.
63
4.4
Capital Budgeting Under Resource Constraints: A More Elaborate Technique
The appropriate way of dealing with capital budgeting in the presence of resource constraints is to look at it as a constrained maximization problem. Suppose that a $10 million budget limit applies to the cash ‡ows at times 0 and 1 (now and in one year), and the company faces the following investment opportunities. Cash Flows (millions) NPV Project C 0 C 1 C 2 at 10% A 10 30 5 21 B 5 5 20 16 C 5 5 15 12 D 0 40 60 13 Assume …rst that it is possible to accept fractional investments. Then the capital budgeting problem can be written as
Maximize 21xA + 16xB + 12xC + 13xD subject to 10xA + 5xB + 5xC + 0xD 10 30xA 5xB 5xC + 40xD 10; 0 xA 1, 0 xB 1, ...
where xA denotes the proportion accepted of project A, and so on. Computers equipped to handle linear programming can easily solve the above problem. When fractional projects are not feasible, integer programming should be used instead to account for the additional constraint that xA ; : : : ; xD should all be either 0 or 1. However, linear programming models are not widely used for a number of reasons. First, they are often expensive to use. Second, since obtaining accurate long-term data is often problematic in the …rst place, applying such sophisticated models may be overkill. Third, the NPV rule assumes that markets are perfect; in particular, it assumes that capital constraints are non-existent. The mere presence of these constraints suggests that (constrained) NPV maximization may not be the right objective.
64
4.5
Short Sales: An Overview
This handout is a reproduction of section 2.4.2 (pages 25-30) of Sharpe, William F., and Gordon J. Alexander, 1990, Investments , Prentice-Hall, New Jersey.
Introduction An old adage from Wall Street is to “buy low, sell high.” Most investors hope to do just that by buying securities …rst and selling them later. 2 However, with a short sale this process is reversed. The investor sells a security …rst and buys it back later. In this case the old adage about investor aspirations might be reworded as “sell high, buy low.” Short sales are accomplished by borrowing stock certi…cates for use in the initial trade and then repaying the loan with certi…cates obtained in a later trade. Note that the loan here involves certi…cates, not dollars and cents (although it is true that the certi…cates at any point in time have a certain monetary value). This means the borrower must repay the lender by returning certi…cates, not dollars and cents (although it is true than an equivalent monetary value, determined on the date the loan is repaid, can be remitted instead). It also means that there are no interest payments to be made by the borrower.
Rules Governing Short Sales Any order for a short sale must be identi…ed as such. The Securities and Exchange Commission has ruled that short sales may not be made when the market price for the security is falling, on the assumption that the short seller would exacerbate the situation, cause a panic, and pro…t therefrom— an assumption inappropriate for an e¢cient market with astute, alert traders. The precise rule is that a short sales must be made on an up-tick (for a price higher than that of the previous trade) or on a zero-plus tick (for a price equal to that of the previous trade but higher than that of the last trade at a di¤erent price). Within …ve business days after a short sale has been made, the short-seller’s broker must borrow and deliver the appropriate securities to the purchaser. The borrowed securities may come from the inventory of securities owned by the brokerage …rm itself or the inventory of another brokerage …rm. However, they are more likely to come from the inventory of securities held in street name by the brokerage …rm for investors that have margin accounts with the …rm. The life of the loan is inde…nite, meaning there is no time limit on it. 3 If the lender wants to sell the securities, then the short seller will not have to repay the loan if the brokerage …rm can borrow shares elsewhere, thereby transferring the loan from one source to another. However, if the brokerage …rm cannot …nd a place to borrow the shares, then the short seller will have to 2
After purchasing a security, an investor is said to have established a long position in the security. The New York Stock Exchange, the American Stock Exchange, and NASDAQ publish monthly list of the short interest in their stocks (short interest refers to the number of shares of a given company that have been sold short where, as of a given date, the loan remains outstanding). To be on the NYSE or AMEX list, either the total short interest must be equal to or greater than 100,000 shares or the change in the short interest from the previous month must be equal to or greater than 50,000 shares. The respective …gures for NASDAQ are 50,000 and 25,000. 3
65
repay the loan immediately. Interestingly, the identities of the borrower and lender are known only to the brokerage …rm— that is, the lender does not know who the borrower is and the borrower does not know who the lender is.
An Example An example of a short sale is indicated in the diagrams below. At the start of the day, Mr. Lane owns 100 shares of the XYZ Company, which are being held for him in a street name by Brock, Inc., his broker. During this particular day, Ms. Smith places an order with her broker at Brock to short sell 100 shares of XYZ (Mr. Lane believes that the price of XYZ will rise in the future, whereas Ms. Smith believes it is going to fall). In this situation, Brock takes the 100 shares of XYZ that they are holding in street name for Mr. Lane and sells them for Ms. Smith to some other investor, in this case Mr. Jones. At this point XYZ will receive notice that the ownership of 100 shares of its stock has changed hands, going from Brock (remember that Mr. Lane held his stock in a street name) to Mr. Jones. At some later date, Ms. Smith will tell her broker at Brock to purchase 100 shares of XYZ (perhaps from a Ms. Poole) and use these shares to pay o¤ her debt to Mr. Lane. At this point, XYZ will receive another notice that the ownership of 100 shares has changed hands, going from Ms. Poole to Brock, restoring Brock to their original position.
66
What happens when XYZ declares and subsequently pays a cash dividend to it stockholders? Before the short sale, Brock would receive a check for cash dividends on 100 shares of stock. After depositing this check in their own account at a bank, Brock would write a check for an identical amount and give it to Mr. Lane. Thus, neither Brock or Mr. Lane has been worse o¤ by having his shares held in the street name. After the short sales, XYZ will see that the owner of those 100 shares is not Brock any more but is now Mr. Jones. Thus, XYZ will now mail the dividend check to Mr. Jones, not Brock. However, Mr. Lane will still be expecting his dividend check from Brock. Indeed, if there was a risk that he would not receive it, he would have agreed to have his securities held in street name. Brock would like to mail him a check for the same amount of dividends that Mr. Jones received from XYZ— that is, for the amount of dividends Mr. Lane would have received from XYZ had he held his stock in his own name. If Brock does this, they will be losing an amount of cash equal to the amount of the dividends paid. In order to prevent themselves from experiencing this loss, they make Ms. Smith, the short seller, give them a check for an equivalent amount! Consider all the parties involved in the short sale now. Mr. Lane is content, since he has received his dividend check from his broker. Brock is content, since he has received his dividend check from his broker. Brock is content since their net cash out‡ow is still zero, just as it was 67
before the short sale. Mr. Jones is content, since he received his dividend check directly from XYZ. What about Ms. Smith? She should not be upset with having to reimburse Brock for the dividend check given by them to Mr. Lane, since the price of XYZ’s common stock can be expected to fall by an amount roughly equal to the amount of cash dividend, thereby reducing the dollar value of her loan from Brock by an equivalent amount. What about annual reports and voting rights? Before the short sale, these were sent to Brock, who then forwarded them to Mr. Lane. After the short sale, Brock no longer received them, so what happened? Annual reports are easily procured by brokerage …rms free of charge, so Brock probably got copies of them from XYZ and mailed a copy to Mr. Lane. However, voting rights are di¤erent. These are limited to the registered stockholders (in this case, Mr. Jones) and cannot be replicated in the manner of cash dividends by Ms. Smith, the short seller. Thus, when voting rights are issued, the brokerage …rm (Brock, Inc.) will try to …nd voting rights to give to Mr. Lane (perhaps Brock owns shares or manages a portfolio that owns shares of XYZ and will give these voting rights to Mr. Lane). Unless he is insistent, however, there is a chance he will not get his voting rights once his shares have been borrowed and used in a short sale. In all other matters, he will be treated just as if he were holding the shares of XYZ in his own name. As previously mentioned, a short sale involves a loan. Thus, there is a risk that the borrower (in the example, Ms. Smith) will not repay the loan. In this situation the broker would be left without the 100 shares that the short seller, Ms. Smith, owes him or her. Either the brokerage …rm, Brock, is going to lose money or else the lender, Mr. Lane, is going to lose money. To prevent this from happening, the cash proceeds from the short sale, paid by Mr. Jones are not given to the short seller, Ms. Smith. Instead, they are held in her account with Brock until she repays her loan. Unfortunately this will not assure the brokerage …rm that the loan will be repaid. In the example, assume the 100 shares of XYZ were sold at a price of $100 per share. In this case, the proceeds from the short sale of $10,000 are held in Ms. Smith’s account, but she is prohibited from withdrawing it until the loan is repaid. Now imagine that at some date after the short sale, XYZ stock rises by $20 per share. In this situation, Ms. Smith owes Brock 100 shares of XYZ with a current market value of 100 shares $120 per share = $12,000 but has only $10,000 in her account. If she skips town, Brock will have collateral of $10,000 (in cash) but a loan of $12,000, resulting in a loss of $2,000. However, Brock can use margin requirements to protect itself from experiencing losses from short sellers who do not repay their loans. In this example, Ms. Smith must not only leave the short-sale proceeds with her broker, but she must also give he broker initial margin applied to the amount of the short sale. Assuming the initial margin requirement is 60%, she must give her broker 0.6 $10,000 = $6,000 in cash. In this example, XYZ stock would have to rise in value to a price above $160 per share in order for Brock to be in jeopardy of not being paid. Thus, initial margin provides the brokerage
68
…rm with a certain degree of protection. However, this protection is not complete, since it is not unheard of for stock to rise in value by more than 60%. It is the maintenance margin that protects the brokerage …rm from losing money in such situations. In order to examine the use of maintenance margin in short sales, the actual margin in a short sale is de…ned as: Actual Margin =
Market Value of Assets - Loan : Loan
In this example, if XYZ stock rises to $130 per share, the actual margin in Ms. Smith’s account will be ($100 100)(1 + 0:6) ($130 100) = 23%: $130 100
Assuming the maintenance margin requirement is 30%, the account is undermargined, and Ms. Smith will receive a margin call. Just as in margin calls on margin purchases, she will be asked to put up more margin, meaning she will be asked to add cash or securities to her account. If, instead of rising, the stock price falls, then the short seller can take a bit more than the drop in the price of the account in the form of cash, since in this case the actual margin has risen above the initial margin requirement and the account is thus unrestricted. 4 Having discussed the cases for short sales where the stock price either (1) fell and the account was thereby unrestricted or (2) went up to such a degree that the maintenance margin requirement was violated and the account was thereby undermargined, there is one more case left to be considered. That is the case where the stock price goes up but not to such a degree that the maintenance margin is violated. In this case, the initial margin requirement has been violated, which means that the account is restricted. Here, restricted has a meaning similar to its meaning for margin purchases. That is, any transaction that has the e¤ect of further decreasing the actual margin in the account will be prohibited. An interesting question is: what happens to the cash in the short seller’s account? When the loan is repaid, the short seller will have access to the cash (actually, the cash is used to repay the loan). Before the loan is repaid, however, it may be that the short seller can earn interest on the portion of the cash balance that represents margin (some brokerage …rms will accept certain securities, such as Treasury bills, in lieu of cash for meeting margin requirements). In regard to the cash proceeds from the short sale, sometimes the securities may be lent only on the payment of a premium by the short seller, meaning the short seller not only does not earn interest on the cash proceeds but must pay a fee for the loan. At other times the lender may pay the short seller interest on the cash proceeds. Usually, however, securities are loaned “‡at”— the brokerage …rm keeps the cash proceeds from the short sale and enjoys the use of this money, and neither the short seller nor the investor who lent the securities receives any direct compensation. In this case, the brokerage …rm makes money not only from the commission 4
Alternatively, the short seller could short sell a second security and not have to put up all (or perhaps any) of the initial margin.
69
paid by the short seller but also on the cash proceeds from the sale (they may, for example, earn interest by purchasing Treasury bills with these proceeds.)
70
4.6
Mathematics and Statistics: A Reminder
Problems: 1. The only logarithm that we will use in this course (and in all other …nance courses as a matter of fact) is the natural logarithm. We will denote the natural logarithm of a positive number x by log x. The objective of this exercise is to remind you of some simple logarithm manipulations. (a) Are the following statements true of false. i. If y = log x, then x = ey . ii. log(x + y) = log x + log y. iii. log(xy) = log x + log y. log x . log y
iv. log(x=y) =
v. log(xy ) = y log x. (b) Solve yx = z for x (using natural logarithms only). (c) What is log (log ee )? 2. In this course, we will sometimes be faced with quadratic equations of the form ax2 + bx + c = 0. (a) What is the formula that gives the two roots (zeros) of this equation in terms of a, b, and c? (b) Solve
1 x
+1 =
2 x2
for x.
3. To simplify long equations, we will also make use of summation signs. (a) Calculate (b) Calculate
P P P 4 n=1
2n .
3 n=1
3 m=n
mn.
4. Random variables are crucial to the study of …nance. This course will often require calculations of the expected value and the variance of a random variable, as well as the covariance between two random variables. Suppose that the joint distribution (i.e. Pr x~ = x; y~ = y ) of two random variables x~ and y~ is given by:
f
g
x y
0 1 1 0.2 0.4 2 0.3 0.1
Calculate the following: 71
(a) Pr x~ = 0 and Pr x~ = 1 .
f g f g (b) Pr fy~ = 1g and Pr fy~ = 2g. (c) Pr(~ x) and Pr(~ y). (d) Pr(~ x) and Pr(~ y). (e) Pr(~ x; y~).
Solutions: 1. (a)
i. True. ii. False. iii. True. iv. False. v. True.
(b) Take the (natural) logarithm on both sides of yx = z to obtain: log y x = log z: Since the left-hand side of this last expression simpli…es to x log y using one of the rules from part (a), we have log z x= : log y (c) The (natural) logarithm of ee is simply e, and the (natural) logarithm of e is 1. So log (log ee ) = log(e) = 1: (d) The two roots (if they both exist) are given by
p b b 4ac x= : 2
2a
(e) First multiply both sides of the equation by x2 to obtain x + x2 = 2; which we can rewrite as x2 + x
2 = 0:
We can now use the above formula to get
1 x=
p
(1)2 4(1)( 2) = 1 or 2(1) 72
2:
(f)
P
4 n=1
2n = 2 1 + 22 + 23 + 24 = 2 + 4 + 8 + 16 = 30.
(g) This one can be calculated step by step: 3
3
XX
"X # "X # "X # 3
mn =
n=1 m=n
3
m(1) +
3
m(2) +
m=1
m=2
m(3)
m=3
= [1(1) + 2(1) + 3(1)] + [2(2) + 3(2)] + [3(3)] = 6 + 10 + 9 :=: 25:
(h) Pr x~ = 0 = 0:2 + 0:3 = 0:5, and Pr x~ = 1 = 0:4 + 0:1 = 0:5
f g f g (i) Pr fy~ = 1g = 0:2 + 0:4 = 0:6, and Pr fy~ = 2g = 0:3 + 0:1 = 0:4 (j) E (x ~ ) = 0 Pr fx~ = 0g + 1 Pr fx~ = 1g = 0:5. E (y~) = 1 Pr fy~ = 1g + 2 Pr fy~ = 2 g = 1:4. (k) V ar(~x) = (0 0:5) Pr fx~ = 0 g + (1 0:5) Pr fx~ = 1g = 0:25. V ar(~y) = (1 1:4) Pr fy~ = 1g + (2 1:4) Pr fy~ = 2 g = 0:24. 2
2
2
2
(l) Let us solve this one in more details: Cov(~x; y~)
= =
XX
E (~x)] [y E (~y)]Pr fx~ = x; y~ = yg [0 0:5][1 1:4](0:2) + [0 0:5][2 1:4](0:3) +[1 0:5][1 1:4](0:4) + [1 0:5][2 1:4](0:1) 0:04 0:09 0:08 + 0:03 :=: 0:1 x
[x
y
73
4.7
Linear Algebra: Simple Methods
As seen in section I.2 of my lecture notes, knowing how to solve a system of linear equations can often be useful in this course. In this handout, I will try to show you three approaches for solving systems of linear equations. My goal is not to give you a rigorous exposition of linear algebra, but rather to give you a few simple methods that (I hope) will help you in your homeworks and exams. Let’s start with a concrete example: suppose you want to solve the following system of linear equations, i.e. you want to solve for x;y;z in 2x + y + z =
9
(10)
x + y + 4z = 19
(11)
4x + 3y + z = 15
(12)
First Method: Substitution The idea of this …rst method is to substitute one equation into the others in order to reduce the 3 3 (3 equations and 3 unknowns) system to a 2 2 system …rst, and eventually to a single equation with one unknown. For example, let’s substitute equation (10) into equations (11) and (12), using the y variable.5 By this, I mean that we …rst write an expression for y in terms of x and z using equation (10):
y=9
2x z:
(13)
Then we use this equation to replace y in each of equations (11) and (12): x + (9
2x z) + 4z = 19 () x + 3z = 10 () 2x 2z = 12 4x + 3(9 2x z) + z = 15
(14) (15)
Observe that we have reduced our original 3 3 system of equations to a 2 2 system of equations given by (14) and (15). We can now repeat the same strategy to reduce this system to a single equation with a single unknown. For example, let’s substitute equation (14) into equation (15) using the variable x. From (14), we have
x = 3z 5
10:
Note that we could use any of the three equations and any of the three variables.
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(16)
Replacing x with this expression in (15), we get
2(3z 10) 2z = 12 () 6z + 20 2z = 12 () 8z = 32 () z = 4: Now that we have found z, we can use it in (16) to obtain x: x = 3(4)
10 = 2:
Finally, we can use the values of x and z in (13) to obtain y: y =9
2(2) 4 = 1:
So the solution to the system of equations (10)-(12) is given by x = 2; y = 1; z = 4. In fact, you can plug these values in (10), (11), (12), and verify that these values of x;y;z do indeed solve the equations.
Second Method: Diagonal Reduction This second method requires a little more time to get used to than the …rst method. However, with a little practice, it becomes much faster than the …rst method. To use this method, we …rst need to rewrite our original system of equations (10)-(12) in matrix form. This is done by writing in the …rst column the coe¢cients of x in the three equations. Similarly, the second and third column contain the y and z coe¢cients. Finally, the last column, which should be separated from the other three, contains the right-hand sides of each equation. So in this example, we write:
26 4
2 1 1 1 1 4 4 3 1
37 5 9 19 15
In order to solve our problem, we are allowed three kinds of operations on this matrix:
We can multiply (divide) any line by any nonzero constant. We can interchange any two lines. We can add (subtract) any multiple of a line to any other line.
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Suppose for example that we want to interchange line 1 and line 2 above. Then we write 6
26 4 Suppose that we now want write7 1 1 2 1 4 3
26 4
2 1 1 1 1 4 4 3 1
37 26 5 4 37 26 5 4 26 37 4 5 9 19 15
L1
$L2
37 5 37 5
1 1 4 2 1 1 4 3 1
19 9 15
to subtract 2 times line 1 to line 2 of this last matrix. We then 4 1 1
19 9 15
L2 2L1
!L2
1 0 4
1 1 3
4 7 1
19 29 15
The general goal of this method is to eventually obtain an identity matrix, 1 0 0 0 1 0 0 0 1
;
in the 3 3 part of the matrix. The rightmost column will then give us the values of x;y;z that solve our system of equations. This can be achieved by the following steps:
1. Reduce the numbers below the diagonal to zeros. 2. Reduce the numbers above the diagonal to zeros. 6
The symbol ‘ ’ means that the two systems are equivalent. The left-right arrow ‘ ’ means that we interchange two lines. 7 The right arrow ‘ ’ means that the expression on the left-hand side will replace the line on the right-hand side.
$
!
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The complete solution to our example is then 8
26 4
26 4 26 4 26 4
2 1 1 1 1 4 4 3 1
L3 4L1
L3 L2
!L3
!L3
L2 7L3
!L2
9 19 15 1 0 0
37 26 37 26 5 4 5 4 37 26 5 4 3 2 3 75 64 75 3 2 3 75 64 75 L1
$L2
1 1 1
4 7 15
1 1 4 0 1 7 0 0 8
19 29 32
1 1 0 0 1 0 0 0 1
3 1 4
1 1 4 2 1 1 4 3 1 19 29 61
1 L 8 3
19 9 15
L2 2L1
!L2
1 0 4
1 0 0
1 1 1
4 7 15
L2!L2
!L3
1 1 4 0 1 7 0 0 1
L1 L2
!L1
19 29 4
1 0 0 0 1 0 0 0 1
37 5 26 4 1 1 3
19 29 61
L1 4L3
2 1 4
4 7 1
!L1
19 29 15
L3!L3
1 1 0 0 1 7 0 0 1
37 5 26 4 37 5
1 1 4 0 1 7 0 1 15
3 29 4
37 5 19 29 61
The solution to our original system of equations (10)-(12) is therefore given by x = 2; y = 1; z = 4.
Third Method: Matrix Inversion This method is the quickest of the three methods presented in this handout. However, it requires a computer or a calculator that performs matrix inversions and multiplications. 9 Also, I will only present an outline of the method since this method requires more knowledge of linear algebra. The systems of equations (10)-(12) can be written in matrix form as
26 4
2 1 1 1 1 4 4 3 1
37 26 37 26 37 54 5 4 5 x y z
=
9 19 15
:
The solution is then given by
26 37 26 454 x y z
=
2 1 1 1 1 4 4 3 1
37 26 37 54 5 1
9 19 15
:
Therefore, the solution to our system of equations can be found by applying the following 8
Note that, once you get used to this method, you don’t really need to write anything above ‘ ’ at every step. However, it is a good idea to do so at the beginning so that you can trace back your mistakes. 9 It is possible to perform matrix inversions and multiplications by hand, but it is tedious and not very important for this course. In any case, I show in the next section how to use a spreadsheet software to perform such calculations.
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two steps:
37 5 26 37 4 5 37 26 37 5 4 5 26 37 4 5 26 37 26 37 26 37 4 54 5 4 5
1. Find the (matrix) inverse of
26 4
2 1 1 1 1 4 4 3 1
.
9 19 15
2. (Matrix) Multiply that inverse by
.
Using either a computer (see next section) or a calculator, we …rst …nd that
26 4
2 1 1 1 1 4 4 3 1
1
11 8 15 8 1 8
=
9 19 15
Finally, by (matrix) multiplying this inverse by of equations:
26 37 45 x y z
=
11 8 15 8 1 8
1 4 1 4 1 4
3 8 7 8 1 8
1 4 1 4 1 4
3 8 7 8 1 8
:
, we obtain the solution to our system
9 19 15
=
2 1 4
:
Matrix Manipulations Using a Spreadsheet Software In this section, I show how to use Microsoft Excel to manipulate matrices. The steps to perform the same manipulations on other spreadsheet software should be similar. The table below shows the spreadsheet on which I did my calculations. First in A1:C3, I enter the matrix I want to invert. Then I position the cursor in cell A5, highlight the range A5:C7, type = MINVERSE (A1 : C3) and press Ctrl-Shift-Enter.10 The inverted matrix appears in A5:C7. To perform the matrix multiplication, I …rst need to enter my second matrix; this is what I do in E5:E7. Then I position the cursor in cell G5, highlight the range G5:G7, type = MMULT(A5 : C7; E5 : E7) and press Ctrl-Shift-Enter. The solution to my system of linear equations then appears in G5:G7. 10
If you only press Enter, only the top-left cell (cell A5) will appear. The Ctrl-Shift-Enter tells Excel that you want a matrix as a result.
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A Few Additional Problems For practice, you can solve the following systems of linear equations using each of the three methods. The solutions are also provided so that you can check your answers. System 1: x+ y 2x + 4y
= 1
z = 6
y
z = 0
y + 2z = 2 System
x 2x
Solution :
x=
1;
y = 2; z = 0:
Solution :
1 1 1 x= ; y= ; z= : 2 4 4
2:
+ 4z = 2
2y + 2z = 1
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