Unformatted text preview: ARE 171a Fall 2011 Homework 2 Due Monday, Oct. 24, in class M. Whitney 1. For each set of cash flows below, find the present value given the following annual discount rates: 2%, 6%, and 10% a. An ordinary annuity that pays $400 per month, for 5 years. PMT x PVIFA At 2% PV = $400 (PVIFA .02/12, 60)= 400 57.052356 At 6% PV = $400 (PVIFA .06/12, 60)= 400 51.725561 At 10% PV = $400 (PVIFA .10/12, 60)= 400 47.065369 Here, PVIFA= = 22820.942 20690.224 18826.148 (11/(1+(r/12))^60)/(r/12) No_ce that when the discount rate is low, the PVIFA value approaches the number of payments, in this case 60. It reaches its maximum of 60 if the discount rate is zero. One issue in the news today is that many pension funds have become underfunded, due largely to the present verylow interest rate environment. A firm that has promised to pay its re_rees a series of future payments has to set aside much more money today to fund these future benefits than would have been expected a few years ago, when interest rates were higher. For instance here, you can see that if you promise to pay someone $400/mo for the next 5 years, you would only need $18826 to meet this obliga_on if funds earned 10%, but at 2% you need $22821. Even prudent business managers probably didn't expect interest rates would fall to nearzero, back at the _me their pension fund policies and payments were established. b. A lump sum of $400,000, that you will inherit at the end of 20 years. At 2% At 6% At 10% PV=400000/(1+.02)^20 = PV=400000/(1+.06)^20 = PV=400000/(1+.10)^20 = 269188.53 124721.89 59457.451 Note how much cheaper it would be for your grandmother to promise you this, if rates were high! c. An apartment building that is expected to generate the following annual cash flows (at the end of each year) The first year shows a loss, due to costs of renova_ng the property. 1 2 3 4 5 6 7 60000 20000 20800 21600 22500 23400 700000 Here we have to discount each payment separately, since they are not level payments This is typical of most realworld investment projects....due to factors such as deprecia_on schedules, cash flows tend to vary by year. Method: divide each CF by (1+r)^n, where n is the year received, then sum the answers 0.02 58823.53 19223.376 19600.305 19955.061 20378.943 20778.53 609392.13 Totl PV 650504.81 494618.82 0 0.06 56603.77 17799.929 17464.081 17109.223 16813.309 16496.077 465539.98 0.1 54545.45 16528.926 15627.348 14753.091 13970.73 13208.69 359210.68 378754.01 Again, we can see the huge impact the level of interest rates has on values. No_ce how today's value of this property is tremendously enhanced by very low interest rates, assuming cash flows are as given. In today's weak economic environment, low mortgage rates are helping to support property values, but other factors such as strict credit standards for borrowers, and weakness in property cash flows, are working in the opposite direc_on. d. A Bri_sh consol bond (a perpetuity) that pays $25 per year, at the end of each year, forever. PV = PMT/r 0.02 1250 0.06 0.1 416.66667 250 2. Margaret plans to buy a new car a year from now. Its price will be $20,000 at that _me. She has $6000 in cash on hand today. How much will she need to deposit at the end of each month in order to afford this car? First assume that her savings earn a 3% annual rate, then assume they earn a 12% annual rate. First, find how much her lump sum will grow to....deduct from the total car price to find her remaining $ need at _me 12 (end of month 12, 1 year from now) mn=12 periods, r/12 = rate per period Then, find a PMT such that an ordinary annuity will have a future value equal to this remaining need. stated rate rate per period FV of lump sum FV of lump sum remaining need 0.03 0.0025 =6000*(1+.0025)^12 6182.4957 13817.504 0.12 0.01 =6000*(1+.01)^12 6760.9502 13239.05 Req'd payment 1135.7118 1043.883 0.03 0.12 FV = 13817.5 = PMT(FVIFA .0025, 12) FV = 13239.5 = PMT(FVIFA .01, 12) solve for PMT= solve for PMT= As you would expect, it's easier to save up a certain sum of money to be available on a future date, when rates are high. FVIFA formula is: [(1+r/12)^12  1]/(r/12) 3. Donte plans to purchase a home that costs $400,000. He will pay 20% in cash as a downpayment, and finance the rest. a. One op_on he is considering is a 30 year fully amor_zed mortgage, with an annual rate of 4.8% How much is his payment? Show the first 2 lines of his amor_za_on schedule. If he buys the home late in the year, such that only Nov and Dec.'s payments are made in 2010, how much interest will he be able to claim as a deduc_on on his 2010 taxes? What will his ending balance be aner these first 2 payments? First find his payment (360 periods, rate per period .048/12 = .004) he needs to borrow 400000x.8 So PV = 320000= 320000 PVIFA PMT = PV/PVIFA 190.59768 1678.9291 PMT*(PVIFA.004,360) In the table below, interest = start balance x monthly int. rate; principal = pmt  interest; ending balance = star_ng balance  principal Star_ng balance 320000 319601.07 PMT Interest Principal Remaining balance 1678.9291 1280 398.92913 319601.07 1678.9291 1278.4043 400.52485 319200.55 2558.4043 (add Nov and Dec interest) Nov Dec total interest paid in 2010 In future years, he will be able to deduct 12 months' worth of interest, he was limited here by purchasing the house late in the calendar year, thus making only 2 payments in 2010. His ending balance at end of the first 2 payments is $319200.55, as shown in the table. This amount would need to be paid to the lender, if he were to pay off or refinance his loan. b. Repeat part a, but assume he instead chooses a 15 year mortgage with an annual rate of 4.56% First find his payment (180 periods, rate per period .0456/12 = .0038) he needs to borrow 400000x.8 320000 So PV = Star_ng balance 320000 318758.2 320000= PMT*(PVIFA.0038,180) 0.0038 PVIFA PMT = PV/PVIFA 130.1976 2457.8026 Nov Dec PMT Interest Principal Remaining balance 2457.8026 1216 1241.8026 318758.2 1678.9291 1211.2812 467.64798 318290.55 2427.2812 (add Nov and Dec interest) total interest paid in 2010 His ending balance at end of the first 2 payments is $318290.55, as shown in the table. 4. Mrs. Sheridan wants to make a gin of $800,000 today (Fall 2011), to be shared by her two grown children, Thomas and Brad. However, she wants to take into account the value of other gins she has given each of them in the past, so that the total value of the two boy's gins is equal. uch of today's $800,000 cash gin should each son receive? Assume the discount rate is 6% How m Thomas received an annual payment of $30,000 in the fall of 2003, 2004, and 2005, to apend graduate school He also received $50,000 in cash in the fall of 2009, to help with the purchase of a home. Brad was given a new car with a value of $25,000 in the fall of 2008. Step 1, find value of each kid's past gins in 2011 dollars: PVIFA 6%, 3 = 2.6730119 (note, there are several ways to set this up, I found the PV of an annuity due but you could use a different approach) Thomas Brad Total V2011 = V2011 = 30000*(PVIFA 6%, 3)*(1.06)*(1.06^8) + 50000*1.06^2 = 25000*(1.06)^3= 191659.92 29775.4 221435.32 Step 2: find total value of all gins (past and new) and divide fairly: total gins: 221435.3+800000 Each child should receive 1/2 = Step 3: deduct gin amounts already received to get the NEW gin amount for each: Thomas gets 510717.7191659.0 Brad gets 510717.729775.4 total 1021435.3 510717.66 319057.74 480942.26 800000 5. Suppose that a 30 year treasury bond purchased in Oct. 1991 had a coupon rate of 8%, semiannual payments and a par value of $1000. Today, the bond has 10 years remaining _ll maturity. a. What is its value today, if new 10 year treasuries pay 2%? 18.0456 40 (PVIFA 1%, 20) + 1000/(1.01)^20 20 payments of $40 each, 1000 at end. PV 0 = 1541.3685 b. What would it have been worth today, if instead of falling to their current low levels, rates had risen sharply, such that new 10 year Tbonds yielded 14%? 10.594 40 (PVIFA 7%, 20) + 1000/(1.07)^20 PV 0 = 682.179 You can see the possible risk/reward of inves_ng in bonds...possibility of major change in value. If you do not want to be exposed to this type of risk, you can either use TIPS, or choose bonds with short maturi_es and low dura_ons...these are much less vola_le. Of course, they have a lower average return as well. 6. A corporate bond has 3 years remaining to maturity. It has a par value of $1000 and a coupon rate of 9% a. Find its present value, if the current required rate of return is 4% 5.6014 PV 0 = 45 (PVIFA 2%, 6) + 1000/(1.02)^6 1140.0344 b. How might your answer to a. differ, if the bond is callable? (verbal answer only) The bond's price might be limited by the amount it can be called for. Example: if the bond's current call price is $1060, investors might fear it will be called any _me soon. So its price might end up at $1060 or a liple above, rather than rising all the way to $1140 You wouldn't want to pay $1140, then have it called for less $ a month later. c. What is this bond's dura_on, if purchased today at the value in part a? Here we have to computer the PV of each cash flow, separately, so we can see what % of the bond's total PV each payment contributes" Years CF PV 0.5 1 1.5 2 2.5 3 45 45 45 45 45 1045 44.117647 43.252595 42.404505 41.573044 40.757886 927.93009 SUM OF PV 1140.0358 weight 0.0386985 0.0379397 0.0371958 0.0364664 0.0357514 0.8139482 =PV/sum of PV's mul_ply each weight x its years to receipt, and sum .0387x0.5 + .0379x1.0 + .....etc. 2.7172387 years This implies that, if the required rate of return were to rise from its current level of 4%, to a 1% higher level of 5%, the bond's PV would decline by roughly 2.72 % dura_on = ...
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