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Dr. Patrick Convery, Econ 106F Corporate Finance, Department of Economics, UCLA Spring, 2023 Midterm Exam May 15, 2023 First Name Last Name UCLA ID #

Multiple Choice Select the correct letter. No partial credit. Question 1 (10 points) You are saving for retirement. To live comfortably, you decide that you will need $2.5 million dollars by the time you are 45. If today is your 30th birthday, and you decide, starting today (t=0), and on every birthday up to and including your 45th birthday, that you will deposit the same amount into your savings account. Assum- ing the interest rate is 5.0%, the amount that you must set aside today plus every year on your birthday up to and including your 45th birthday is closest to: A) $105,674 B) $88,261 C) $72,830 D) $26,086 Answer: A According to this question, the t = 0 is set on your 30th birthday. And note that the first savings comes at t = 0. To this we also add the same savings amount (C) at every additional birthday from your 31st (which is t = 1) to the 45th which is t = 15. Thus we have two contributions to consider, the C at t = 0, and the annuity from t = 1 to t = 15. Per the assumption in the question, we want $2.5M at retirement. Thus, PV(age 30) = $2,500,000 / (1.05) 15 = $1,202,542. We will make 16 (not 15) deposits, which include 1 right now at t = 0, and 15 more from t =1 to t = 35. Thus, $1,202,542 = Deposit + (Deposit)* [ (1 / 0.05) )* (1 - 1/(1.05) 15 ) ] From look up table 4, we see the [ brackets ] = 10.38 for 15 years at 5% Solve for Deposit = $105,674.

Question 2 (10 points) You are deciding between two mutually exclusive investment opportunities. Both require the same initial investment of $10 million. Investment A will generate $2 million per year (starting at the end of the first year) in perpetuity. Investment B will generate $1.5 million at the end of the first year and its revenues will grow at 2% per year for every year after that. Which investment should you choose if the discount rate for both investments is 7%? A) Investment A B) Investment B C) They have the same NPV D) Not enough information Answer: B NPV A = (2 / r) - 10M NPV B = [ (1.5) / (r – 0.02) ] - 10M Substituting r = 0.07 into the NPV formulas gives NPV A = $18.5714 million, NPV B = $20 million. So the NPV says chose investment B.

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Question 3 (10 points) You can earn $50 in interest on a $1000 deposit for six months. If the EAR is con- stant over the length of the investment, the ($) amount of interest you will earn on a $1000 deposit over two years is closest to: A) $427. B) $110. C) $100. D) $215. Answer: D Because we earn $50 on $1000 over 6 months (i.e. 5% twice per year), we can cal- culate the EAR (which is for 12 months) as (1 + EAR) = (1+5%)*(1+5%), and thus EAR = 1.05^2 -1 = 10.25% For 2 years, we earn $1000*(1.1025)*(1.1025) = $1,215.51, which is $215.51 of interest.

Question 4 (10 points) Your friend has invented a money machine. The main drawback of the machine is that it is slow. It takes one year to manufacture $100. However, once built, the ma- chine will last forever and will require no maintenance. The machine will cost $1,000 today to build and it will be ready to turn it on and use, one year from to- day. ( Hint: draw a timeline ) Your friend wants to know if he should invest the money to construct it. You tell him he must calculate the NPV to answer this question. If the interest rate is 9.5% per year, the NPV of this investment is closest to? A) -$12.27 B) -$38.69 C) +$91.48 D) Not enough information. Answer : B Solution (not needed for full credit): To decide whether to build the machine, you need to calculate the NPV: The cash flows the machine generates are a perpetuity with first payment starting at year 2. Computing the PV at year 1 gives: PV 1 = 100/0.095 = 1.052.63 The value today is PV 0 = 1,052.63 / (1+0.095) = $961.31. Thus, NPV = 961.31 – 1,000 = - 38.69 The NPV is negative, thus your friend should not build the machine.

Question 5 (10 points) During the chapter on bonds we discussed the possibility of defaults by both corpo- rations and nations. We showed a graph discussing the percent of countries in de- fault or restructuring debt over the past 200 years. The percent of countries in default during this time, roughly on average, is closest to: A) it varies approximately between 95% - 99% in most years B) it varies approximately between 10% - 40% in most years C) it varies approximately between 1% - 4% in most years D) countries do not default on their loans because they can simply print money if needed, to pay off their loans. Answer : B See page 57 of Chapter 6 lecture notes.

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Question 6 (10 points) Use the table for the question(s) below. There are two possible states of the Economy, either poor or good. Each is equally likely to occur. Use the law of one price to answer this question: What is the price of Omega today if there are no arbitrage opportunities? ( Hint: The law of one price implies that if the cash flows are the same, then the price must be the same.) A) $1,800 B) $1,650 C) $1,000 D) $8,700 Answer: C Solution (not needed for full credit): Omega = $1000 is the answer. We know, by the law of one price, that Omega = Kappa/2 + 2*Delta = $200 + $800 = $1,000. OR Omega = Kappa/2 + Gamma = $200 + $800 = $1,000. Market Price Cash Flow in One Year Security Today if Poor Economy if Good Economy Kappa 400 1250 0 Gamma 800 0 12 50 Delta 400 0 625 Omega ??? 625 1250

Question 7 (10 points) The owner of the Dave’s Diner is considering selling his restaurant and retiring. An investor has offered to buy Dave’s Diner for $350,000 whenever the owner is ready for retirement. The owner is considering the following three alternatives: 1. Sell the restaurant now and retire. 2. Hire someone to manage the restaurant for the next year and retire. This will require the owner to spend $50,000 now, but will generate $100,000 in profit next year. In one year the owner will sell the restaurant for $350,000. 3. Scale back the restaurant's hours and ease into retirement over the next year. This will re- quire the owner to spend $40,000 on expenses now, but will generate $75,000 in profit at the end of the year. In one year the owner will sell the restaurant for $350,000. If the interest rate is 7%, the alternative with the highest NPV is: A) Alternative #1 with an NPV of approximately $350,000 B) Alternative #2 with an NPV of approximately $370,561 C) Alternative #3 with an NPV of approximately $357,196 D) Alternative #2 with an NPV of approximately $383,561 Answer: B Alternative #1: NPV = $350,000. Alternative #2: NPV = -50,000 + (100,000 + 350,000)/1.07 = $370,561 Alternative #3: NPV = -40,000 + (75,000 + 350,000)/1.07 = $357,19

Question 8 (10 points) A local bank is running the following advertisement in the newspaper: “ For just $1000 we will pay you $100 forever! ” The details of the ad says that for a $1000 deposit, the bank will pay you $100 every year in perpetuity, starting one year after the $1,000 deposit is made. The interest rate that the bank is advertising with this deal is closest to? A) 4.1% B) 6.0% C) 7.8% D) 10.0% Answer: D The timeline is: These payments are simply a perpetuity, so: PV = 100 / r Setting the NPV of the cash flow stream equal to zero and solving for r gives the answer: NPV = 0 = - 1000 + 100 / r ==> r = 10% 0 1 2 3 … infinity -$1,000 $100 $100 $100 $100

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Question 9 (10 points) We are considering a project to build a new football stadium. It is expected to take 4 years to build (costing $100 million at the end of each year for the first 4 years), with the construction costs being evenly distributed over these four years (all costs incurred at the end of each year, so the first cost in the timeline is one year from now). Upon completion, the stadium will yield benefits of $40 million per year for 100 years (with the first benefit in the timeline being FIVE years from today). At an interest rate of 6%, the present value of this project is closest to: A) +$514 million B) +$167 million C) - $62 million D) - $347 million Answer: B First, find the PV of the construction costs: PV = 100/(1.06) + 100 / (1.06^2) + 100 / (1.06^3) + 100 / (1.06^4) = $347M We must compare this figure with the PV of the benefit stream. Note that the benefit stream starts FIVE years from now, so we cannot directly use the formula. One way around this is to first calculate the PV of the benefit stream as if it is started one year from today and then subtract off the PV of a four-year series starting one year from today: ($40M / 0.06) – {$40M*{1 / (1.06) + 1/ (1.06)^2 + 1 / (1.06)^3 + 1 / (1.06)^4 } = $514 million. Thus, the answer is 514M - 347M = $167M Notice the use of the perpetuity formula in the first term. This is an acceptable approximation because it involves a large number of years (N=100). But students can use an annuity formula instead, and they will also receive full credit.

Question 10 (10 points) You are in the process of purchasing a new automobile that will cost you $25,000. The dealership is offering you EITHER a $1,000 rebate if you pay today (the re- bate is applied toward the purchase price, so that the net purchase price becomes $24,000) OR getting a loan with an interest rate of 3.9% APR with monthly com- pounding for 4 months and no rebate (with each payment made at the end of the month and the first payment occurs one month from today). You have been pre-approved for an auto loan through your local credit union at an interest rate of 7.5% APR with monthly compounding for 4 months – this would allow you to pay the dealership today and get the $1000 rebate. Which option has the higher monthly payment - financing through your credit union at 7.5% APR and getting the $1000 rebate OR skipping the rebate and fi- nancing the full $25,000 through the dealership at the lower 3.9% APR? The answer is closest to: A) The credit union has a lower monthly payment, equal to about $348./month. B) The credit union has a lower monthly payment, equal to about $6,094./month. C) The dealership has a lower monthly payment, equal to about $1288./month. D) The dealership has a lower monthly payment, equal to about $4171./month. Answer: B The credit union has a lower monthly payment. You must show work for full credit. Here are the details. Credit Union: First we need the monthly interest rate = APR/k = .075/12 = 0.00625 or 0.625%. Now: PV = $25,000 - $1000 rebate = $24000 Monthly interest rate r = 0.625% FV = 0 and N = 4. And thus, PMT = PV / { (1/r)*(1 – 1/(1+r)^N) } = (24,000) / {(1/.00625)*(1 – 1/(1+ 1/0.00625)^4)} = $6094.04 Dealership: First we need the monthly interest rate = APR/k = .039/12 = 0.00325 or 0.325%.

Now: PV = $25,000 (no rebate) Monthly interest rate r = 0.325% FV = 0 and N = 4. And thus, PMT = PV / { (1/R)*(1 – 1/(1+r)^N) } = (25,000) / {(1/.00325)*(1 – 1/(1+ 1/0.00325)^4)} = $6300.86

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Question 11 (10 points) A delivery company wants to buy a new more efficient van. The van will cost $30,000 to buy. The new van will replace the existing one and will save the firm $7,500 a year in fuel and maintenance costs for three years. After three years the firm can sell the van for $15,000. Using a discount rate of 5% EAR, what is the NPV of this project? The answer is closest to: A) - $2,714 B) +$1,298 C) +$3,386. D) +$7,281 Answer: C We have a r = 5% discount rate. NPV = -$30,000 + 7,500 / (1+r) + 7,500 / (1+r)^2 + 7,500 / (1+r)^3 + 15,000 / (1+r)^3 = − 30,000 +7142.857 + 6,802.72 + 6,478.78 + 12,957.56 = + $3,381.70

Question 12 (10 points) We have had two guest speakers this quarter. This question is about the first one. The first speaker has an engineering background and has worked at SpaceX and an electric vehicle company. What else did we learn during his visit with our class? Select the most complete answer. A) His current start-up involves selling pizza out of a truck B) His current start-up involves robots. C) He currently works at Goldman Sachs as an equity analyst, applying machine learning formulas he learned as an engineer to evaluate stocks. D) Both A and B are true. E) Both A and C are true. F) A, B, and C are all true. Answer: D You can see the lab lecture video from April 21 for more information. Also, see this link: https://www.cbsnews.com/losangeles/news/former-spacex-scientists-turned-restauranteurs-crafting-full-automated- mobile-pizza-maker/

Question 13 (10 points) Your son has been accepted into college. This college guarantees that your son’s tuition will not increase for the four years he attends college. The first $10,000 tu- ition payment is due six months from now. After that, the same payment is due every six months until you have made a total of eight payments. The college offers a bank account that allows you to withdraw money every six months and has a fixed APR of 4% (semi-annual compounding) guaranteed to re- main the same rate over the next four years. How much money must you deposit today in to this account if you intend to make no further deposits and would like to make all the tuition payments from this ac- count, leaving the account empty when the last payment is made? The answer is closest to: A) +$119,374, B) +$73,254. C) +$52,291. D) +$48,833. Answer: B Timeline (not needed for credit, but its helpful to see the cash flows): A 4% APR (semi-annual) implies a semi-annual discount rate of 4%/2 = 2.0%. Then, the PV = ($10,000 / 0.02)*(1 – 1 / (1.02^8) = $73,254. Period 0 1 2 3 4 5 6 7 8 Payment $0. $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 Time (years) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

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Question 14 (10 points) Bruin Corporation currently pays a dividend and it will continue to pay this divi- dend forever. Assume the discount rate is 15% per year. Now consider the payment timeline of the dividends. In situation A, they pay an- nual dividends of $2.00, once per year, at the end of each year. In situation B, they pay quarterly dividends of $0.50 per quarter, at the end of each quarter. ( Hint: Use the correct cash flows timeline and periodic interest rate in each situa- tion. ) Choose the best answer. A) Situation A has a higher stock price. B) Situation B has a higher stock price. C) Situation A and B both have the same stock price because in each situation, there are a total of $2.00 in dividends per year. D) There is not enough information to answer this question. Answer: B With the simplifying assumption that dividends are paid at the end of the year, then in situation A, the stock pays a total of $2.00 in dividends per year. Valuing this dividend as a perpetuity, we have, P=$2.00/0.15= $13.33. Alternatively, if the dividends are paid quarterly as in situation B, we can value them as a perpetuity as well, but we need to be certain to use the quarterly cash flows and the quarterly periodic rate to property align our formula with the dividend timeline. Thus, the numerator in situation B is $0.50 per quarter, and the quarterly discount rate is given by: (1.15) 4 − 1 = 3.556%. Then P = ($0.50) / 0.03556 = $14.06. This question could be answered without using the formula by simply understanding that in situation B, the cash flows are arriving soon in time than in situation A, and thus the price (i.e. the present value) of those cash flows is greater in B compared to A. In other words, the reason P is larger for quarterly dividends compared to P for annual dividends is because with quarterly dividends we get the cash flow sooner and thus (due to the time value of money) the sooner cash flows are worth more.

Question 15 (10 points) Coke Cola will pay an annual dividend of $0.65 one year from now. All dividends are paid at the end of each year. Financial analysts expect this dividend to grow at 12% per year thereafter until, and including, the dividend at the end of the fifth year. After then (for year 6 and beyond), the dividend growth will level off at 2% per year. According to the dividend-discount model, what is the value at the end of year 5 of all the dividend payments which come in year 6 and beyond? Use a discount rate of 8.0%. The answer is closest to: A) $87.62 B) $29.31 C) $17.39 D) $4.23 Answer: C PV5 = { 0.65* ( 1.12 ) 4 *(1.02) } / (0.08 - 0.02) = 17.39

Question 16 (10 points) Joe Bruin has just purchased a home and taken out a $400,000 mortgage. The mortgage has a 30-year term with monthly payments of $2,246.12 and has an APR of 5.4% with monthly compounding. What is the total amount of principal that Mr. Bruin will pay during the first three months of his mortgage? The answer is closest to: A) $446. B) $3,714. C) $2,239. D) $1,344. Answer: D Monthly payment = $2,246.12 First Month's Interest = $400,000*(0.054/12) = $1,800 First Month's Principal = $2,246.12 - $1,800 = $446.12 Second Month's Interest = ($400,000 - 446.12)*(.054/12) = $1,797.99 Second Month's Principal = $2,246.12 - $1,797.99 = $448.13 Third Month's Interest = (400,000 - 446.12 - 448.13)*(.054/12) = $1,795.98 Third Month's Principal = $2,246.12 - $1,795.98 = $450.14 Total Interest = $1,800 + $1,797.99 + $1,795.98 = $5,393.97 Total Principal over first 3 months = $446.12 + $448.13 + $450.14 = $1,344.39 END OF EXAM (Look up tables given on the following pages)

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