Problem Set III: Foundations of Finance Solutions1Foundations of FinanceProf. Alex Shapiro1. a. The (intrinsic) price of a stock can be determined using the dividend growth model:Price = E1× (1-b) / [k-b×ROE] = [5 × (1-0.5) / [0.1-0.5×16] = $125.b. Recall from the formula above, that a higher required rate of return implies a lowerintrinsic price of the stock. On the other hand, as the growth rate (which is a function ofROE) decreases, the price of the stock decreases. Thus, a drop in the observed equilibriumstock price could indicate that either the required rate of return is higher thanoriginally expected, and/or the ROE on funds plowed back is less than originallyestimated.c. To solve for the expected return on equity, use the dividend growth model as follows:50 = [5 × (1-0.5) / [0.1-0.5×ROE]Solving for ROE, you get: 50 = 2.5/[0.1-0.5×ROE].So, 5 - 25×ROE = 2.5. Therefore, ROE=0.1 or 10%Note: when the ROE equals the required rate of return, plowing back earnings neitherincreases or decreases the price of the stock. In addition, the price-earnings ratio (P0/E1)is now equal to the inverse of the expected return on equity.2. a. P0 = D1/(k-g) = 5/(.14-.06) = 62.5b. The retention ratio b=3/8. g = b × ROE ⇒ .06 = 3/8 × ROE ⇒ ROE =.16c. If the firm were valued as a perpetuity, it would be worth E1/k = 8/.14 = 57.14.The excess, 62.50-57.14 = 5.36, is due to future growth.3. See BKM solutions manual.4. See BKM solutions manual.5. See BKM solutions manual.6. a. Use your financial calculator to get PA=859.53; PB=1,116.54; PC=816.30; PD=258.42. b. As we discussed in class, the total holding one-year return will be 7% on all bonds,although the components (current yield and capital gains/losses) will vary.c. One year from now, use your financial calculator to getPA=869.70; PB=1,114.69; PC=873.44; PD=276.51. As stated in the question, the after-taxrates of return are computed as simple returns, except we subtract the tax on capital gainsand coupon income (zeros have only capital gains):(A) [(869.70-859.53)(1-.30) + 50(1-.35)]/859.53 = 4.61%(B) [(1,114.69-1,116.54)(.7) + 80(.65)]/1,116,54 = 4.54%(C) [(873.44-816.30)(.7)]/816.30 = 4.9%(D) [(276.51-258.42)(.7)]/258.42 = 4.9%d. At y = 6%, the end-of-year price of bond A is 931.98, so the after-tax return is (A) [(931.98-859.53)(1-.30) + 50(1-.35)]/859.53 = 9.68%Problem Set III: Foundations of Finance Solutions27. a. (1-1/1.0720)/0.07 = 10.6Mb. Using the formula for the duration of annuity: D = 1.07/.07 - 20/(1.0720 - 1) = 8.32 yearsc. Let x be the fraction in bond C and (1-x) in bond D. Then 3x + (1-x)20 = 8.32⇒ x = .69. So 69%(10.6M) = 7.3M in C and 3.3M in Dd. Not unique, you could form an immunized portfolio out of any set of bonds as long as one of the bonds has a duration less than 8.32 years and one has a duration greater than 8.32 years. Use of zeroes minimizes the reinvestment and rebalancing requirements.e. Immunization will ensure that the nominal CF requirements are met. In thepresent situation, this is an appropriate requirement.8. Real Time ExerciseUsing collected data: (Your answer will differ from the below depending on when you collected your data. Butyou should follow Lines 1-10 to determine whether your answer is correct.)U.K. Japan11-year spot rate (%)3.828 0.02022-year yield of a zero (%)4.009 0.0503Expected 1-yr Rate (in 1 year)3.828 0.0204Implied forward rate4.190 0.080Today:5Buy 2-yr zero (100 par value)92.440 99.9006Enter into 1 year repo-92.440 -99.9007Total cash outlay today0.000 0.000Expected for End of Year 1:8Sell 2-yr zero after 1 year96.313 99.9809Complete the Repo-95.978 -99.92010Net proceeds0.335 0.060Explanation:Line 1: collected from BloombergLine 2: collected from BloombergLine 3: the assumption in the exercise (hence same as Line 1)Line 4: using the definition of the forward rate between year 1 and year 2, and Lines 1,2Line 5: 100 discounted over two years at the rate in Line 2Line 6: Sell a two-year zero on a one year repo (the rate for repurchase in a year is in Line 1)Line 7: Summarizes cash outflow (=0)Line 8: 100 discounted at the rate in Line 3Line 9: The cost of funds in Line 6 is [Line 6 × (1 + (Line 1 / 100))]Line10: Projected profit (not guaranteed)Problem Set III: Foundations of Finance Solutions3Remarks:Note that in both countries you can expect to capture the liquidity premium, because theforward rate, f2, is higher than the expected spot rate, E[r2], but U.K. produces the higherpremium. (The short-end of the yield curve in Japan is relatively less upward sloping.)You expect higher profits in the UK, because the bond market there, according to the data, isdominated to a larger extent by the one-year-horizon investors who may need money in oneyear. These investors are requiring a higher premium to hold the longer maturity 2-year zero,because of the interest rate uncertainty in one year’s time. In Japan, on the other hand, thedata indicates that investors are more willing to hold longer maturity bonds, not planning tosell them within a year.However, this is not a riskless transaction (not an arbitrage). The net proceeds at the end ofone year will depend on the price at which you will be able to sell the remaining 1 year of the2-year zero.If your expectations are correct, the net proceeds will be as shown. But if 1-year interestrates increase so that the actual spot rate is higher next year than you expect (and therefore inLine 8 you will actually have lower values), the transaction could lose money. This isexactly the “liquidity problem” that the one-year-horizon investors fear (they know they mayneed money, i.e. liquidity, in one year, but if they buy the two-year zero and next year ratesincrease, the value of their zero will decrease, and these investors will face a liquiditysqueeze).If you follow the above strategy you may indeed sometime lose money, but on average if youform your expectations correctly, you will be making money. Your average reward is thecompensation for the risk you bear. Individual investors may not be able to afford occasionallosses. However, financial institutions performing such transactions repeatedly, will makemoney on average.Problem Set III: Foundations of Finance Solutions4Answers to Suggested ProblemsS1. a. Price = 90 × APVF(7.5%, 10) + 1,000 × PVIF(7.5%, 10) = 1,102.96.b. The time line for the coupon payments (in bold) is:0 1 2 3 4 5 6 7 8 9 1090 90 90 90 90*507 **71190 90 90 90