Econ 422 Eric Zivot Fall 2005 Midterm Exam Solutions. I. Intertemporal Consumption and Investment Decisions (25 points, 5 points each)

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1 Econ 422 Eic Zivot Fall 2005 Midtem Exam Solutions his is a closed book exam. Howeve, you ae allowed one page of notes (double-sided). Answe all questions. Fo the numeical poblems, if you make a computational eo you may still eceive full cedit if you povide the coect fomula fo the poblem. hee ae 25 questions, and each question is woth 4 points. otal points = 100. You have 1 hou and 50 minutes to complete the exam. Good luck. I. Intetempoal Consumption and Investment Decisions (25 points, 5 points each) In the figue below, the downwad sloping staight line connecting the points (0,5) and (4,0) epesents the oppotunities fo investment in the capital maket (money maket), and the downwad sloping cuved line connecting the points (0,4) and (2.6,0) epesents the oppotunities fo investment in physical capital (e.g. plant and machiney). he only asset at time 0 is $2.6 million in cash (initial endowment). hee is no additional endowment pesent at time 1. Dollas, yea IC Dollas, yea 0

2 Please answe the following questions elated to the figue: a. What is the inteest ate,, and what is the slope of the capital maket line? o detemine the inteest ate,, we use the money maket line and solve 4(1 + ) = 5 => 1 + = 5/4 = 1.25 => = 0.25 o 25% he slope of the capital maket line (intetempoal budget constaint) is (1+) = b. How much should be invested in physical capital (plant and equipment), and how much will this investment be woth next yea? Optimal investment occus whee the slope of poduction function (MR) = slope of budget constaint = -(1 + ). his tangency point occus when = 1 M is invested. he investment of 1 M etuns 3 M next yea. c. What is the pesent value (PV) and net pesent value (NPV) of this investment? he PV and NPV of the investment poject is PV = 3/1.25 = 2.4 M NPV = = 1.4 M d. What is the optimal consumption at times 0 and 1? he optimal allocation of consumption at times 0 and 1 is detemined by the tangency point of the IC with the budget constaint. Fom the figue, we see that optimal consumption at time 0 is 1 M and optimal consumption at time 1 is 3.75 M. e. How much is boowed o lent at time 0? he owne lends = 0.6 M (at = 25%) today and eceives 0.6*(1.25) = 0.75 M next peiod. Consumption next peiod is then consists of the 3M investment etun plus the 0.75 M money maket etun (= 3.75 M). II. Pesent Value Computations (25 points, 5 points each) 1. You have $1,000 to invest today in a money maket account that pays an annual inteest ate of. a. If you invest the $1000 fo yeas, what is the futue value of the investment as a function of? he FV of the investment is FV = $1,000(1 + )

3 b. Using you answe fom pat a above, detemine how many yeas it will take fo you investment to double as a function of. How many yeas will it take fo you investment to double if = 2%, 5% and 10%. o find out how many yeas it will take fo the investment of $1000 to double, we solve the equation FV = $2,000 = $1,000(1 + ) fo. Dividing both sides by $1,000, taking logs of both sides and solving fo gives ln(2) ln(2) = ln(1 + ) = = ln(1 + ) ln(1 + ) he appoximation above uses the fact that ln(1 + ) if is close to zeo. If we expess the inteest ate in pecent (e.g., multiply by 100), then this esult is known as the ule of 70. hat is, Using the ule of 70, it is easy to detemine the numbe of yeas fo an investment to double. Just divide 70 by the annual gowth ate as a pecent. Using the ule of 70, if =2%, 5% and 10% then the coesponding values of ae 70 = = = At the Univesity of Washington, employees have the ability to save fo etiement in a tax defeed 403B pogam administeed by IAA-CREF. his pogam is a defined benefit pogam in that employees put aside a pecentage of thei salay in specific mutual fund administeed by IAA-CREF, and these funds gow tax fee until the employee eties. Upon etiement the employee has accumulated savings fom the pogam and is given the option of puchasing a vaious annuities fom IAA-CREF to eplace the lost monthly salay. Fo this question, suppose a etied pofesso has accumulated savings of $1 million upon etiement. a. Suppose IAA-CREF offes a 20 yea annuity making annual payments stating next yea. If the discount ate is 3 pecent pe yea, what is the annual annuity payment?

4 Hee we use the finite annuity fomula: PV = C PVA(, ) 1 1 PVA(, ) = 1 ( 1+ ) and solve fo the annuity payment C: $1,000,000 $1,000,000 = C PVA(0.03, 20) C = = $67, b. Suppose IAA-CREF offes a 20 yea annuity making annual payments stating next yea, and that the payments incease with the inflation ate. If the inflation ate is 2 pecent pe yea and the discount ate is 3 pecent pe yea, what is the fist annuity payment? Now we use the finite gowing annuity fomula: PV = C PVGA(, g, ) 1 (1 + g) PVGA(, g, ) = 1 g ( 1+ ) and solve fo the fist annuity payment C: $1,000,000 $1,000,000 = C PVGA(0.03,0.02, 20) C = = $56, c. Suppose IAA-CREF offes a 20 yea annuity making monthly payments stating next month. If the discount ate is 3 pecent pe yea, what is the monthly annuity payment assuming monthly compounding? his is the same as pat a, except now we must use monthly compounding: PV = C PVA( /12, 12) 1 1 PVA( /12, 12) = 1 /12 ( 1 + /12) Solving fo the monthly payment C gives $1,000,000 $1,000,000 = C PVA(0.03/12, 20 12) C = = $5,

5 III. Bond Picing and the em Stuctue of Inteest Rates (25 points, 5 points each) he following is a list of pices fo zeo coupon bonds (SRIPS) of vaious matuities (taken fom finance.yahoo.com): Matuity (yeas) Pice of zeo coupon bond he bond pices ae quoted as a pecentage of pa (face) value, and the pa value is $1,000. Please answe the following questions: a. Calculate the spot ates associated with each bond, and plot the tem stuctue of inteest ates. he spot ates ae computed using 0,1 0,2 0,3 0, = 1 = / = 1 = / = 1 = / = 1 = he tem stuctue is plotted below.

6 em Stuctue Spot ate Matuity (yeas) b. Calculate the implied 1-yea fowad ates, ft 1, t, fo t=2, 3, 4 he implied fowad ates ae computed using (1 + ) 2 0,2 1,2 = = 1+ 0,1 f f f 3 0,3 2,3 2 ( 1+ 0,2 ) 4 0,4 3,4 3 ( 1+ 0,1 ) (1 + ) = 1 = (1 + ) = 1 = c. If the expectations hypothesis of the tem stuctue holds, what does the infomation in the yield cuve say about the couse of futue inteest ates? he expectations hypothesis states that the implied fowad ates ae the best foecasts of the futue spot ates. Using the implied fowad ates fom pat b, we see that the expectations hypotheses pedicts slightly ising one yea spot ates fo two yeas and then a slight decease. d. What is the pice of a 2 yea coupon bond making annual coupon payments with a annual coupon ate of 3% and a face value of $1,000?

7 C PV = C + + C ,1 1+ 0,2 ( ) $30 $1,030 = + = $ Exta cedit (5 points). Compute the value of the yield-to-matuity on the 2 yea coupon bond fom pat d above. o find the yield-to-matuity on the bond, solve the following equation: $30 $1030 $ (1 + ) = his is a quadatic equation. he positive oot is given by =

8 IV. Valuing Stocks (25 points, 5 points each) 1. Micosoft stock is expected to pay an annual dividend of $0.32 at the end of the yea, and then the dividend is expected to gow at 5% pe yea foeve. he equied ate of etun (maket capitalization ate) on Micosoft stock is 10% pe yea. a. What is the cuent pice of Micosoft stock? Hee we use the gowing pepetuity fomula: P = D $0.32 $6.40 = g = b. Suppose the expected gowth ate of the dividend changes fom 5% to 6%. What is the expected pecentage change in the stock pice afte this incease in the dividend gowth ate? his poblem can be solved in two ways. he exact answe follows fom valuing the stock at the new gowth ate and then computing the pecentage change in pice new DIV $0.32 $0.32 P = = = = $8 g new % P = ( P P) / P = (8 6.4) / 6.4 = 0.25 = 25% An appoximate answe uses the deivative of P with espect to g: dp 1 dp 1 = P ( g) = ( g) d d P = = = = 1 ( ) ( ) 20(0.01) % Notice that the appoximate answe is a little off. 2. Google stock is cuently selling fo $381.89, its eaning pe shae (EPS) is $4.51 and its pice-eanings atio (P/E) is Assume that Google has an annual maket capitalization ate of 20%. a. What is the pesent value of gowth oppotunities (PVGO) fo Google? What pecentage of the cuent stock pice does PVGO epesent? Hee we use the fomula EPS EPS $4.51 P = PVGO PVGO P $ $ = = 0.20 =

9 3. You ae a financial analyst at a lage investment fim and you job is to foecast the pice of Boondogle stock. Cuently Boondogle does not pay any dividend, but you foecast that Boondogle will pay a dividend of $2 stating 3 yeas fom today and that the dividend will gow at 5% pe yea foeve aftewads. You boss tells you to assume a maket capitalization ate of 8% pe yea fo Boondogle stock. Given this infomation, what is you foecast of the cuent pice of Boondogle stock? At time 2, the stock may be viewed as a gowing pepetuity. he pice at time 2 is theefoe D $2 P = 2 $66.67 g = = Howeve, we want the pice today. hat pice is P2 $66.67 P0 = = = $ (1 + ) (1.08) V. Capital budgeting (10 points) Stabucks has to choose between two industial stength espesso machines which do the same job but have diffeent lives: the Italian machine povides sevice fo two yeas, while the Fench machine povides sevice fo thee yeas. he two machines have the following costs, expessed in eal tems: (Neithe has a scap value at the end of its life.) Yea Italian Machine Fench Machine 0 $100,000 $125,000 1 $6,000 $4,000 2 $6,000 $4,000 3 $4,000 Assuming that the oppotunity cost of capital is 5 pecent in eal tems and ignoing taxes, which machine would you choose? Explain and show you wok. We discussed two solutions to this type of poblem. he fist solution is based on computing the equivalent annual cost (EAC) fo each poject. his is based on fist computing the PV of the costs fo the two pojects, and then computing the fixed annual payment (EAC) that has the same PV of cost. Fo PV of the costs fo the two pojects ae PVC PVC Italian Fench $6,000 $6,000 = $100, = $111, ( 1.05) $4,000 $4,000 $4,000 = $125, = $135, ( ) ( )

10 he EAC ae computed using the finite annuity fomula PVC 1 1 EAC =, PVA(, ) = 1 PVA(, ) 1 ( + ) Fo the Italian machine, PVA(, ) is 1 1 PVA(0.05,2) = 1 = $ ( 1.05) 2 Fo the Fench machine, PVA(, ) is 1 1 PVA(0.05, 4) = 1 = $ ( 1.05) 3 heefoe, the EAC values fo the Italian and Fench machines ae EAC EAC Italian Fench $111, = = $1.86 $59, $135, = = $2.72 $49, Since the EAC fo the Fench machine is less, choose it ove the Italian machine.. he second method is based on eplicating the pojects enough times so that they cove the same numbe of peiods and then computing the PV of each poject. Hee, the Italian machine must be eplaced 2 times (in yeas 3, 6) and the Fench machine must be eplaced once (in yea 3). Unde this appoach the PV of costs ae computed using PV PV Italian Fench $111, $111, = $111, = $303, (1.05) (1.05) $135, = $135, = $253, (1.05) As with the equivalent annual cost appoach, the Fench machine is cheape.