CHAPTER 2 How to Calculate Present Values

Transcription

1 CHAPTER How to Calculate Present Values 0. Mr. Basset is buying a security worth $0,000 now, which is its present value. The unknown is the annual payment. Using the present value of an annuity formula, we have: PV C r r ( r) t $0,000 C (.08) C $0, (.08) $, a. Each installment is: $9,40,73/9 = $495,87. PV $495,87 $4,76, (.08) b. If ERC is willing to pay $4. million, then: $4,00,000 $495,87 r r ( r) 9 Using Excel or a financial calculator, we find that r = 9.8%. 3. This is an annuity problem with the present value of the annuity equal to $ million (as of your retirement date), and the interest rate equal to 8% with 5 time periods. Thus, your annual level of expenditure (C) is determined as follows:

2 PV C r r ( r) t $,000,000 C (.08) 5 C $,000, (.08) 5 $33,659 With an inflation rate of 4% per year, we will still accumulate $ million as of our retirement date. However, because we want to spend a constant amount per year in real terms (R, constant for all t), the nominal amount (C t ) must increase each year. For each year t: R = C t /( + inflation rate) t Therefore: PV [all C t ] = PV [all R ( + inflation rate) t ] = $,000,000 (0.04) ( 0.08) ( 0.04) (0.08) ( 0.04)... ( 0.08) 5 R 5 R [ ] = $,000,000 R.390 = $,000,000 R = $77,95 $,000,000 ( 0.08) Alternatively, consider that the real rate is Then, redoing ( 0.04) the steps above using the real rate gives a real cash flow equal to: C $,000, (.03846) 5 $77,95 Thus C = ($77,95.04) = $85,070, C = $9,473, etc. 36. a. First we must determine the 0-year annuity factor at a 6% interest rate. b. 0-year annuity factor = [/.06 /.06(.06) 0 ) = Once we have the annuity factor, we can determine the mortgage payment. Mortgage payment = $00,000/.4699 = $7,436.9.

3 Year Beginning Balance ($) Year-End Interest ($) Total Year- End Payment ($) Amortization of Loan ($) End-of-Year Balance ($) 00,000.00, , , , ,563.09, , , , ,799.96, , ,08.9 8, , , , , , ,5.60 0, , , , , ,6.0 7, ,75.8 6, , , , , , , ,6.8 7, , , , , , , , ,5.73 8,5.36 7, , , , , , , , , ,6.03 7, , , ,79.6 6, , , , , , ,436.9, , , , ,436.9, , , , , , , , ,65.4 7, , , ,609.07, , , , ,968.7,98. 7, , , , , , b. Nearly 69% of the initial loan payment goes toward interest ($,000/$7, =.688). Of the last payment, only 6% goes toward interest (987.4/7, =.06). After 0 years, $7,66. has been paid off ($00,000 remaining balance of $8,338.79). This represents only 36% of the loan. The reason that less than half of the loan has paid off during half of its life is due to compound interest a. This calls for the growing perpetuity formula with a negative growth rate (g = 0.04): $million PV 0.0 ( 0.04) $million 0.4 $4.9 million b. The pipeline s value at year 0 (i.e., at t = 0), assuming its cash flows last forever, is: PV 0 C C ( g) r g r g 0

4 With C = $ million, g = 0.04, and r = 0.0: PV 0 ($ million) ( 0.04) $0.884 million $6.34 million 0.4 Next, we convert this amount to PV today, and subtract it from the answer to Part (a): $6.34million $4.9million $3.35million (.0) PV 0 CHAPTER 3 Valuing Bonds 0. a. Price today is $08.45; price after one year is $ b. Return = ( )/ =.06, or 6%. b. If a bond s yield to maturity is unchanged, the return to the bondholder is equal to the yield.. a. False. Duration depends on the coupon as well as the maturity. b. False. Given the yield to maturity, volatility is proportional to duration. c. True. A lower coupon rate means longer duration and therefore higher volatility. d. False. A higher interest rate reduces the relative present value of (distant) principal repayments.. To calculate the duration, consider the following table similar to Table 3.4: Year Totals Payment ($) ,030 PV(C t ) at 4% ($) Fraction of total value [PV(C t )/PV] Year fraction of total value

5 Duration (Years) The duration is the sum of the year fraction of total value column, or years. The modified duration, or volatility, is 6.395/( +.04) = 6.5. The price of the 3% coupon bond at 3.5%, and 4.5% equals $ and $9.64, respectively. This price difference ($57.8) is 5.93% of the original price, which is very close to the modified duration.. a. If the bond coupon payment changes from 9% as listed in Table 3.4 to 8%, then the following calculation for duration can be made: Year a. 7 Totals Payment ($) ,080 PV(C t ) at 4% ($) ,40.08 Fraction of total value [PV(C t )/PV] Year fraction of total value Duration (years) A decrease in the coupon payment will increase the duration of the bond, as the duration at an 8% coupon payment is years. The volatility for the bond in Table 3.4 with an 8% coupon payment is: 5.778/(.04) = The bond therefore becomes less volatile if the coupon payment decreases. b. For a 9% bond whose yield increases from 4% to 6%, the duration can be calculated as follows: Year Totals Payment ($) PV(C t ) at 6% ($) ,67.47 Fraction of total value [PV(C t )/PV] Year fraction of total value Duration (years) There is an inverse relationship between the yield to maturity and the duration. When the yield goes up from 4% to 6%, the duration decreases slightly. The volatility can be calculated as follows: 5.595/(.06) = This shows that the volatility decreases as well when the yield increases.

6 3. The duration of a perpetual bond is: [( + yield)/yield]. The duration of a perpetual bond with a yield of 5% is: D 5 =.05/0.05 = years The duration of a perpetual bond yielding 0% is: D 0 =.0/0.0 = years Because the duration of a zero-coupon bond is equal to its maturity, the 5-year zero-coupon bond has a duration of 5 years. Thus, comparing the 5% perpetual bond and the zero-coupon bond, the 5% perpetual bond has the longer duration. Comparing the 0% perpetual bond and the 5-year zero, the zero has a longer duration. CHAPTER 4 The Value of Common Stocks DIV $0 8. PA $ r 0.0 DIV r g PB P C $5 $ DIV DIV DIV DIV DIV DIV DIV PC 6 At a capitalization rate of 0%, Stock C is the most valuable. For a capitalization rate of 7%, The results are: P A = $4.86 P B = $66.67 P C = $56.48 Therefore, Stock B is the most valuable at a 7% capitalization rate. $ DIV $ a. P0 DIV0 $.35 $.90. r g

7 b. First, compute the real discount rate as follows: ( + r nominal ) = ( + r real ) ( + inflation rate).095 = ( + r real ).075 ( + r real ) = (.095/.075) =.0657 = 6.57% In real terms, g = 0. Therefore: DIV $.35 P0 DIV0 $.35 $.90 r g a. Plowback ratio = payout ratio = = 0.5. Dividend growth rate = g = plowback ratio ROE = = Next, compute EPS 0 as follows: ROE = EPS 0 /Book equity per share 0.4 = EPS 0 /$50 EPS 0 = $7.00 Therefore: DIV 0 = payout ratio EPS 0 = 0.5 $7.00 = $3.50. EPS and dividends for subsequent years are: Year EPS DIV 0 $7.00 $ = $3.50 $ = $ $ = $ = $ $ = $8.043 $ = $ = $ $ = $ $ = $ = $ $ = $9.756 $ = $ = $ $ = $ $ = $ = $ EPS and dividends for year 5 and subsequent years grow at.3% per year, as indicated by the following calculation: Dividend growth rate = g = plowback ratio ROE = ( 0.8) 0.5 = b. DIV DIV DIV DIV DIV5 P $45.65 The last term in the above calculation is dependent on the payout ratio and the growth rate after year 4.

8 . EPS Share price r NPV r g Therefore: Ρ EPS NPV (r 0.5) α α rα α Ρ β EPS r β β NPV (r 0.08) β β The statement in the question implies the following: NPV β EPSβ NPVβ NPVα EPSα NPVα (r β 0.08) rβ (rβ 0.08) (rα 0.5) rα (rα 0.5) Rearranging, we have: NPVα rα (r 0.5) EPS α α NPVβ rβ (r 0.08) EPS β a. NPV < NPV ; everything else equal. b. (r - 0.5) > (r ); everything else equal. β c. NPV NPV α β (r 0.5) (r 0.08) α β ; everything else equal. d. rα EPS α rβ EPS β ; everything else equal. 3. a. Growth-Tech s stock price should be: $0.50 $0.60 (.) (.) $.5 3 (.) (.) $.4 ( ) P $ 3 b. The horizon value P 3 contributes:.4 PV(P H) $.07 3 (.) ( ) 3.8 c. Without PVGO, P 3 would equal earnings for year 4 capitalized at %: P 3 = EPS 3 /r = $ $0.75

9 $.4 (P H ) $3 ( ) Therefore: PVGO = $3.00 $0.75 = $0.5 d. The PVGO of $0.5 is lost at year 3. Therefore, the current stock price of $3.8 will decrease by: $0.5 3 (.) $7.30 The new stock price will be: $3.8 $7.30 = $6.5. CHAPTER 6 Making Investment Decisions with the Net Present Value Rule. The following spreadsheet calculates a NPV of $47,50 (in nominal terms): Nominal Calculation YEAR Capital Investment 500,000 Accumulated Depreciation 00,000 00, , , ,000 Year-End Book Value 500, , ,000 00,000 00,000 0 Working Capital 40,000 44,000 48,400 53,40 58,564 0 Total Book Value 540, , ,400 53,40 58,564 0 Revenues 00,000 0,000 4,000 66,00 9,80 Costs 00,000 0,000,000 33,00 46,40 Depreciation 00,000 00,000 00,000 00,000 00,000 Pretax Profit 0 0,000,000 33,00 46,40 Taxes at 35% 0 3,500 7,350,585 6,44 Profit after Tax 0 6,500 3,650,55 30,67

10 Revenues 00,000 0,000 4,000 66,00 9,80 Costs 00,000 0,000,000 33,00 46,40 Tax on Operations 0 3,500 7,350,585 6,44 Cash Flow from Operations 00,000 06,500 3,650,55 30,67 Change in Working Capital -40,000-4,000-4,400-4,840-5,34 58,564 Capital Investment -500,000 Net Cash Flows -540,000 96,000 0,00 08,80 6,9 88,73 Discount 5% Present Value -540,000 83,478 77,0 7,544 66,433 93,83 NPV -47,50 Since the nominal rate is 5% and the expected inflation rate is 0%, the real rate is given by the following: ( + r nominal ) = ( + r real ) ( + inflation rate).5 = ( + r real ) (.0) r real = = 4.545% Adjusting the cash flows to real dollars and using this real rate gives us the same result for NPV (with a slight rounding error). YEAR Net Cash Flows (nominal) -540,000 96,000 0,00 08,80 6,9 88,73 Adjustment Factor for Real CF Net Cash Flows (real) -540,000 87,73 84,380 8,75 79,360 7,87 Discount 4.545% Present Value -540,000 83,479 77,03 7,545 66,434 93,834 NPV -47,505. No, this is not the correct procedure. The opportunity cost of the land is its value in its best use, so Mr. North should consider the $45,000 value of the land as an outlay in his NPV analysis of the funeral home. Est. Time: 0-05

11 3. a. Investment in net working capital arises as a forecasting issue only because accrual accounting recognizes sales when made, not when cash is received (and costs when incurred, not when cash payment is made). b. If cash flow forecasts recognize the exact timing of the cash flows, then there is no need to also include investment in net working capital. Est. Time: If the $50,000 is expensed at the end of year, the value of the tax shield is: 0.35$50,000 $6, If the $50,000 expenditure is capitalized and then depreciated using a five-year MACRS depreciation schedule, the value of the tax shield is: [ $50,000] $5,306 If the cost can be expensed, then the tax shield is larger, so that the after-tax cost is smaller. Est. Time: Note: This answer assumes that the $3 million initial research costs are sunk and excludes this from the NPV calculation. It also assumes that working capital needs begin to accrue in year 0. The following spreadsheet calculates a project NPV of $465,000. Figures in 000's YEAR Capital Investment 6, Accumulated Depreciation,00,400 3,600 4,800 6,000 Year-End Book Value 6,000 4,800 3,600,400,00 0 Working Capital Total Book Value 6,00 5,040 4,000,800,440 0 Unit Sales ,000, Revenues,000,400 4,000 4,000,400 Costs ,500, Depreciation,00,00,00,00,00

12 Pretax Profit (includes salvage in Year 5) ,300, Taxes at 35% Profit after Tax Revenues,000,400 4,000 4,000,400 Costs ,500, Tax on Operations Cash Flow from Operations,33,395,045,045,0 Change in Working Capital Capital Investment -6,000 Net Cash Flows -6,00,93,35,045,05,460 Discount % Present Value -6,00, ,456,40 88 NPV -465 Est. Time: Assume the following: a. The firm will manufacture widgets for at least 0 years. b. There will be no inflation or technological change. c. The 5% cost of capital is appropriate for all cash flows and is a real, after-tax rate of return. d. All operating cash flows occur at the end of the year. e. We cannot ignore incremental working capital costs and recovery. Note: Since purchasing the lids can be considered a one-year project, the two projects have a common chain life of 0 years. Compute NPV for each project as follows: 0 ($ 00,000) ( 0.35) NPV(purchase) = $,304,880 t.5 t NPV(make) = $ 50,000 $30,000 0 t ($.50 00,000) ( 0.35) t $ 50, $30,000 $,8, Thus, the widget manufacturer should make the lids.

13 . All numbers are in thousands: t = 0 t = t = t = 3 t = 4 t = 5 t = 6 t = 7 t = 8 Sales 4,00.0 4,40.0 4, ,86.0 5,05. 5, ,68.4 5,909.8 Manufacturing Costs 3, , ,67.5 4, , ,84.4 5, ,38.8 Depreciation Rent Earnings before Taxes Taxes Cash Flow Operations Working Capital Increase in W.C Initial Investment,00.0 Sale of Plant Tax on Sale 56.0 Net Cash Flow -, ,47.4 NPV (at %) = $85.8 Est. Time: In order to solve this problem, we calculate the equivalent annual cost for each of the two alternatives. (All cash flows are in thousands.) Alternative Sell the new machine: If we sell the new machine, we receive the cash flow from the sale, pay taxes on the gain, and pay the costs associated with keeping the old machine. The present value of this alternative is: PV 50 [0.35(50 0)] (5 0) $ The equivalent annual cost for the five-year period is computed as follows: PV = EAC [annuity factor, 5 time periods, %] = EAC [3.605] EAC = 6.0, or an equivalent annual cost of $6,00 Alternative Sell the old machine: If we sell the old machine, we receive the cash flow from the sale, pay taxes on the gain, and pay the costs associated with keeping the new machine. The present value of this alternative is: PV 5 [0.35(5 0)]

14 (5 0) $ The equivalent annual cost for the 0-year period is computed as follows: PV = EAC [annuity factor, 0 time periods, %] 7.5 = EAC [5.650] EAC =.57, or an equivalent annual cost of $,570 Thus, the least-expensive alternative is to sell the old machine because this alternative has the lowest equivalent annual cost. One key assumption underlying this result is that, whenever the machines have to be replaced, the replacement will be a machine that is as efficient to operate as the new machine being replaced. Est. Time: Assuming that the light bulb purchases occur at year 0 (for use during the following year or years), the cost structure and PV of each option is: YEAR % Low Energy Conventional The equivalent annual cost for the low-energy bulb is computed as follows: PV LE = EAC LE [annuity factor, 9 time periods, 4%] 8.47 = EAC LE [7.435] EAC LE = $.48, which is much cheaper than the $7.33 cost of using a conventional light bulb for the year. 6. The current copiers have net cost cash flows as follows: Year Before- Tax Cash Flow Net Cash Flow After-Tax Cash Flow -,000 (-,000.65) + ( ,000) ,000 (-,000.65) + ( ,000) ,000 (-8,000.65) + ( ,000) -4, ,000 (-8,000.65) + ( ,000) -4, ,000 (-8,000.65) -5,00.0

15 6-8,000 (-8,000.65) -5,00.0 These cash flows have a present value, discounted at 7%, of $5,857. Using the annuity factor for 6 time periods at 7% (4.767), we find an equivalent annual cost of $3,36 (calculated by taking the NPV/annuity factor: $5,857/4.767). Therefore, the copiers should be replaced only when the equivalent annual cost of the replacements is less than $3,36. When purchased, the new copiers will have net cost cash flows as follows: Year Before- Tax Cash Flow Net Cash Flow After-Tax Cash Flow 0-5,000-5,000-5, ,000 (-,000.65) + ( ,000) ,000 (-,000.65) + ( ,000), ,000 (-,000.65) + ( ,000) ,000 (-,000.65) + ( ,000) ,000 (-,000.65) + ( ,000) ,000 (-,000.65) + ( ,000) ,000 (-,000.65) + ( ,000) ,000 (-,000.65) + ( ,000) These cash flows have a present value, discounted at 7%, of $,967. The decision to replace must also take into account the resale value of the machine, as well as the associated tax on the resulting gain (or loss). Consider three cases: a. The book (depreciated) value of the existing copiers is now $6,48. If the existing copiers are replaced now, then the present value of the cash flows is:, ,000 [0.35 (8,000 6,48)] = $4,580 Using the annuity factor for 8 time periods at 7% (5.97), we find that the equivalent annual cost is $,44. b. Two years from now, the book (depreciated) value of the existing copiers will be $,678. If the existing copiers are replaced two years from now, then the present value of the cash flows is: ( 674.9/.07 ) + ( 675.6/.07 ) + (,967/.07 ) + {3,500 [0.35 (3,500,678)]}/.07 = $7,60 Using the annuity factor for 0 time periods at 7% (7.04), we find that the equivalent annual cost is $,506. c. Six years from now, both the book value and the resale value of the existing copiers will be zero. If the existing copiers are replaced six years from now, then the present value of the cash flows is: 5,857+ (,967/.07 6 ) = $30,495

16 Using the annuity factor for 4 time periods at 7% (8.745), we find that the equivalent annual cost is $3,487. The copiers should be replaced immediately. CHAPTER 7 Introduction to Risk and Return 7. a. 6% b. Zero c..75 d. Less than.0 (the portfolio s risk is the same as the market, but some of this risk is unique risk) Est. Time: 0 05 Est. Time: x I = 0.60 I = 0.0 x J = 0.40 J = 0.0 a. ρ IJ b. σ p [ xi σi xj σj (xix JρIJσIσ J)] [(0.60) (0.0) (0.40) (0.0) ρ IJ 0.50 (0.60)(0.40)()(0.0)(0.0)] σ p [ x I σ I x J σ J [(0.60) (0.0) (x x ρ I J IJ σ σ I J )] (0.40) (0.0) (0.60)(0.40)(0.50)(0.0)(0.0)] c. ρ ij 0 σ p [ x I σ I x J σ J (x x ρ I J IJ σ σ I J )] Est. Time: 06 0 [(0.60) (0.0) (0.40) (0.0) (0.60)(0.40)(0)(0.0)(0.0)] 0.000

17 5. a. Refer to Figure 7.3 in the text. With 00 securities, the box is 00 by 00. The variance terms are the diagonal terms, and thus there are 00 variance terms. The rest are the covariance terms. Because the box has (00 times 00) terms altogether, the number of covariance terms is: = 9,900 Half of these terms (i.e., 4,950) are different. b. Once again, it is easiest to think of this in terms of Figure 7.3. With 50 stocks, all with the same standard deviation (0.30), the same weight in the portfolio (0.0), and all pairs having the same correlation coefficient (0.40), the portfolio variance is: σ = 50(0.0) (0.30) + [(50) 50](0.0) (0.40)(0.30) = σ = 0.93 = 9.3% c. For a fully diversified portfolio, portfolio variance equals the average covariance: Est. Time: 06 0 σ = (0.30)(0.30)(0.40) = σ = 0.90 = 9.0% 6. a. Refer to Figure 7.3 in the text. For each different portfolio, the relative weight of each share is [one divided by the number of shares (n) in the portfolio], the standard deviation of each share is 0.40, and the correlation between pairs is Thus, for each portfolio, the diagonal terms are the same, and the off-diagonal terms are the same. There are n diagonal terms and (n n) off-diagonal terms. In general, we have: Variance = n(/n) (0.4) + (n n)(/n) (0.3)(0.4)(0.4) For one share: Variance = () (0.4) + 0 = For two shares: Variance = (0.5) (0.4) + (0.5) (0.3)(0.4)(0.4) = The results are summarized in the second and third columns of the table below. b. (Graphs are on the next page.) The underlying market risk that cannot be diversified away is the second term in the formula for variance above: Underlying market risk = (n - n)(/n) (0.3)(0.4)(0.4) As n increases, [(n - n)(/n) ] = [(n-)/n] becomes close to, so that the underlying market risk is: [(0.3)(0.4)(0.4)] =

18 Variance Standard Deviation Variance Standard Deviation c. This is the same as Part (a), except that all of the off-diagonal terms are now equal to zero. The results are summarized in the fourth and fifth columns of the table below. (Part a) (Part a) (Part c) (Part c) No. of Standard Standard Shares Variance Deviation Variance Deviation Graphs for Part (a): Portfolio Variance Portfolio Standard Deviation Number of Securities Number of Securities Graphs for Part (c): Portfolio Variance Portfolio Standard Deviation Expected portfolio return = x A E[R A ] + x B E[R B ] = % = Let x B = ( x A ) Number of Securities Number of Securities

19 x A (0.0) + ( x A ) (0.5) = 0. x A = 0.60 and x B = x A = 0.40 Portfolio variance = x A σ A + x B σ B +(x A x B ρ AB σ A σ B ) = (0.60 ) (0 ) + (0.40 ) (40 ) + (0.60)(0.40)(0.50)(0)(40) = 59 Standard deviation = σ % Est. Time: Expected portfolio return = x A E[R A ] + x B E[R B ] = % = 0. Let x B = ( x A ) x A (0.0) + ( x A ) (0.5) = 0. x A = 0.60 and x B = x A = 0.40 Portfolio variance = x A σ A + x B σ B +(x A x B ρ AB σ A σ B ) = (0.60 ) (0 ) + (0.40 ) (40 ) + (0.60)(0.40)(0.50)(0)(40) = 59 Standard deviation = σ % Est. Time: a. In general: Thus: Portfolio variance = P = x + x + x x P = (0.5 )(3.4 )+(0.5 )(4 )+(0.5)(0.5)(0.48)(3.4)(4) P = Standard deviation = P = 0.50% b. We can think of this in terms of Figure 7.3 in the text, with three securities. One of these securities, T-bills, has zero risk and, hence, zero standard deviation. Thus: P = (/3) (3.4 )+(/3) (4 )+(/3)(/3)(0.48)(3.4)(4) P = Standard deviation = P = 3.67% Another way to think of this portfolio is that it is comprised of one-third T-Bills and two-thirds a portfolio which is half Dell and half McDonalds. Because the risk of T-bills is zero, the portfolio standard deviation is twothirds of the standard deviation computed in Part (a) above: Standard deviation = (/3)(0.50) = 3.67%

20 c. With 50% margin, the investor invests twice as much money in the portfolio as he had to begin with. Thus, the risk is twice that found in Part (a) when the investor is investing only his own money: Standard deviation = 0.5% = 4% d. With 00 stocks, the portfolio is well diversified, and hence the portfolio standard deviation depends almost entirely on the average covariance of the securities in the portfolio (measured by beta) and on the standard deviation of the market portfolio. Thus, for a portfolio made up of 00 stocks, each with beta =.5, the portfolio standard deviation is approximately:.5 9.5% = 4.38%. For stocks like McDonalds, it is: % = 8.775%. Est. Time: 5 CHAPTER 8 Portfolio Theory and the Capital Asset Pricing Model 7. a. True b. False (It offers twice the market risk premium.) c. False (Stocks that lie below the market line generally are those with higher betas.) Est. Time: a. False. investors demand higher expected rates of return on stocks with more nondiversifiable risk. b. False. a security with a beta of zero will offer the risk-free rate of return. c. False. Treasury bills have a beta of 0 and the market has a beta of. Therefore, with /3 of the investor s money in T bills and /3 of his or her money in the market, the beta will be: (/3 0) + (/3 ) = d. True e. True

21 Est. Time: In the following solution, security one is Heinz and security two is Exxon. Then: r =.06 =.46 r =.0 =.9 Further, we know that for a two-security portfolio: r p = x r + x r p = x + x x + x Therefore, we have the following results: p p x x r p when = 0 when = % 4.60% 4.60% % 3.3% 3.85% %.47% 3.46% %.5% 3.46% %.39% 3.85% % 3.6% 4.60% % 4.38% 5.66% % 5.94% 6.96% % 7.76% 8.47% % 9.76% 0.% %.90%.90%

22 Est. Time: - 5. a. The below chart uses the following formulas: r p = x r + x r p = x + x x + x Where r =. and r =.08; =.08 and =.05. Portfolio x x r p p when = 0.0

23 X % % 5.% % 4.59% % 6.37% Y % 5.00% b. See the figure below. The set of portfolios is represented by the curved line. The five points are the three portfolios from Part (a) plus the following two portfolios: one consists of 00% invested in X and the other consists of 00% invested in Y. c. See the figure below. The best opportunities lie along the straight line. From the diagram, the optimal portfolio of risky assets is portfolio, and so Mr. Harrywitz should invest 50% in X and 50% in Y. 4.00%.00% 0.00% 8.00% 6.00% 4.00%.00% 0.00% 0.00%.00% 4.00% 6.00% 8.00% 0.00%. a. Expected return = (0.6 5) + (0.4 0) = 7%. Variance = (0.6 0 ) + (0.4 ) + (0.6)(0.4)(0.5)(0)() = Standard deviation = (/) = 8.08%. b. Correlation coefficient = 0 standard deviation = 4.88%. Correlation coefficient = 0.5 standard deviation = 0.76%. c. His portfolio is better. The portfolio has a higher expected return and a lower standard deviation.

24 Expected Return Est. Time: a Beta b. Market risk premium = r m r f = = 0.08 = 8.0%. c. Use the security market line: r = r f + (r m r f ) r = [.5 ( )] = 0.6 = 6.0% d. For any investment, we can find the opportunity cost of capital using the security market line. With = 0.8, the opportunity cost of capital is: r = r f + (r m r f ) r = [0.8 ( )] = 0.04 = 0.4% The opportunity cost of capital is 0.4% and the investment is expected to earn 9.8%. Therefore, the investment has a negative NPV. e. Again, we use the security market line: r = r f + (r m r f ) 0. = ( ) = 0.9 Est. Time: a. Percival s current portfolio provides an expected return of 9% with an annual standard deviation of 0%. First we find the portfolio weights for a combination of Treasury bills (security : standard deviation = 0%) and the index fund (security : standard deviation = 6%) such that portfolio standard deviation is 0%. In general, for a two security portfolio: P = x + x x + x

25 Further: (0.0) = x (0.6) x = 0.65 x = r p = x r + x r r p = ( ) + ( ) = 0. =.0% Therefore, he can improve his expected rate of return without changing the risk of his portfolio. b. With equal amounts in the corporate bond portfolio (security ) and the index fund (security ), the expected return is: Est. Time: 06-0 r p = x r + x r r p = ( ) + ( ) = 0.5 =.5% P = x + x x + x P = (0.5) (0.0) + (0.5)(0.5)(0.0)(0.6)(0.0) + (0.5) (0.6) P = P = = 9.85% Therefore, he can do even better by investing equal amounts in the corporate bond portfolio and the index fund. His expected return increases to.5% and the standard deviation of his portfolio decreases to 9.85%. CHAPTER 9 Risk and the Cost of Capital. a. r equity = r f + (r m r f ) = ( ) = 0.3 = 3%. D E $4million $6million b. r rdebt requity V V $0million $0million assets. r assets = = 9.4%. c. The cost of capital depends on the risk of the project being evaluated. If the risk of the project is similar to the risk of the other assets of the

26 company, then the appropriate rate of return is the company cost of capital. Here, the appropriate discount rate is 9.4%. d. r equity = r f + (r m r f ) = (. 0.06) = 0. =.%. r Est. Time: 06-0 D E $4million $6million rdebt requity V V $0million $0million assets. r assets = = 8.3%. 6. Financial analysts or investors working with portfolios of firms may use industry betas. To calculate an industry beta we would construct a series of industry portfolio investments and evaluate how the returns generated by this portfolio relate to historical market movements. Est. Time: We should use the market value of the stock, not the book value shown on the annual report. This gives us an equity value of 500,000 shares times $8 = $9 million. Therefore the total value of the company is $9 million + $5 million = $4 million. So Binomial Tree Farm has a debt/value ratio of 5/4 = 0.36 and an equity /value ratio of 9/4 = Est. Time: a. The threat of a coup d état means that the expected cash flow is less than $50,000. The threat could also increase the discount rate, but only if it increases market risk. b. The expected cash flow can be found by finding an unbiased forecast. The probability of having a $0 cash flow is 5%, and the probability of having a $50,000 cash flow is therefore (-.5) =.75, or 75%.Therefore, the unbiased forecast is: (0.5 0) + ( ,000) = $87,500. Assuming that the cash flow is about as risky as the rest of the company s business: PV = $87,500/. = $67,4 Est. Time: 0-05

27 0. a. Expected daily production = (0. 0) [(0.4 x,000) + (0.6 x 5,000)] =,70 barrels Expected annual cash revenues =,70 x 365 x $00 = $99,80,000. b. The possibility of a dry hole is a diversifiable risk and should not affect the discount rate. This possibility should affect forecasted cash flows, however. See Part a. Est. Time: 06-0 CHAPTER Agency Problems, Compensation, and Performance Measurement 4. a. EVA = income earned (cost of capital x investment) = $8.03m (0.09 x $55.40m) = $3.04m. b. EVA = $8.03m (0.09 x $95m) = $0.5m. The market value of the assets should be used to capture the true opportunity cost of capital. Est. Time: EVA = income earned (cost of capital x investment) Est. Time: 0-05 = $.m [0.5 x ($4m + $m + $8m)] = $.m $.m = $0.9m. CHAPTER 6 Payout Policy 6. To make a $ per share cash dividend Rational needs an additional $ million in cash. If the firm does not want to borrow more money it must finance the extra cash dividend by selling more shares. This results in a transfer of value from the old to the new shareholders. The capital loss borne by the old shareholders offsets the extra cash dividend received by the new shareholders.

28 With a $ dividend the ex-dividend stock price is $ = $9 per share. The company then issues, shares ($,000,000/9) to raise $ million. Rational s equity market capitalization remains $0 million (,, 9 = $0 million). Est. Time: To generate $5,000 in living expenses Mr. Milquetoast will have to sell a fraction of his investment or borrow against his holdings with the hope that Mr. Buffet will generate a higher return than the interest cost on the loan. Est. Time: a. At t = 0 each share is worth $0. This value is based on the expected stream of dividends: $ at t =, and increasing by 5% in each subsequent year. Thus, we can find the appropriate discount rate for this company as follows: DIV r g P0 $0 r = 0.0 = 0.0% r 0.05 Beginning at t =, each share in the company will enjoy a perpetual stream of growing dividends: $.05 at t =, and increasing by 5% in each subsequent year. Thus, the total value of the shares at t = (after the t = dividend is paid and after N new shares have been issued) is given by: V $.05 million $million If P is the price per share at t =, then: and: V = P (,000,000 + N) = $,000,000 P N = $,000,000 From the first equation: (,000,000 P ) + (N P ) = $,000,000 Substituting from the second equation: (,000,000 P ) + $,000,000 = $,000,000 so that P = $0.00.

29 b. With P equal to $0 the firm will need to sell 50,000 new shares to raise $,000,000. c. The expected dividends paid at t = are $,050,000, increasing by 5% in each subsequent year. With,050,000 shares outstanding, dividends per share are: $ at t =, increasing by 5% in each subsequent year. Thus, total dividends paid to old shareholders are: $,000,000 at t =, increasing by 5% in each subsequent year. d. For the current shareholders: PV( t 0) $,000,000.0 $,000,000 $0,000,000 ( ) (.0) Est. Time: - 5. a. $30.75 = $7.5. b. Nothing. The stock price will stay at $ ,54 shares will be repurchased. c. The with-dividend price stays at $30. Ex-dividend drops to $4.50; 883,534 shares will be issued. Est. Time: 06-0 CHAPTER 7 Does Debt Policy Matter? 9. We begin with r E and the capital asset pricing model: r E = r f + E (r m r f ) = ( ) = 0. =.0% Similarly for debt: r D = r f + D (r m r f ) 0. = D ( ) D = 0.5 Also, we know that:

30 D r r A D E D E r D E To solve for A, use the following: β A D β D E Est. Time: 06-0 D E E βe D E (0.5 0.) (0.5 0.) % ( ) (0.5.5) We know from Proposition I that the value of the firm will not change. Also, because the expected operating income is unaffected by changes in leverage, the firm s overall cost of capital will not change. In other words, r A remains equal to 7% and A remains equal to However, risk and, hence, the expected return for equity and for debt, will change. We know that r D is %, so that, for debt: For equity: Also: Est. Time: - 5 r D = r f + D (r m r f ) 0. = D ( ) D = 0.5 D r r A D E D E r D E 0.7 = (0.3 0.) + (0.7 r E ) r E = 0.96 = 9.6% r E = r f + E (r m r f ) 0.96 = E ( ) E =.0 E. [Note: In the following solution, we have assumed that $00 million of long-term bonds have been issued.] a. E = $55 0 million = $550 million. V = D + E = $00 million + $550 million = $750 million. D V $00million 0.67 $750million

31 E V $550million $750million After-tax WACC = r D ( T C D ) r V 0.07 ( 0.35) 0.67 ( ) %. E b. The after-tax WACC would increase to the extent of the loss of the tax deductibility of the interest on debt. Therefore, the after-tax WACC would equal the opportunity cost of capital, computed from the WACC formula without the tax-deductibility of interest: WACC r D D r V E E V E V ( ) ( ) % CHAPTER 8 How Much Should a Corporation Borrow? TC (rdd) 0.35(0.08 $,000). a. PV(tax shield) $5.93. r (0.08$,000) b. PV(tax shield) $.80 (.08 c. PV(tax shield) = T C D = $350. Est. Time: For $ of debt income: D. t t ) Corporate tax = $0 Personal tax = 0.35 $ = $0.350 Total = $0.350 For $ of equity income, with all capital gains realized immediately: Corporate tax = 0.35 $ = $0.350 Personal tax = [$ (0.35$)] [$ (0.35$)] = $0.63

32 Total = $0.53 For $ of equity income, with all capital gains deferred forever: Est. Time: 06 0 Corporate tax = 0.35 $ = $0.350 Personal tax = [$ (0.35$)] = $0.4 Total = $ a. Stockholders win. Bond value falls since the value of assets securing the bond has fallen. b. Bondholder wins if we assume the cash is left invested in Treasury bills. The bondholder is sure to get $6 plus interest. Stock value is zero because there is no chance that the firm value can rise above $50. c. The bondholders lose. The firm adds assets worth $0 and debt worth $0. This would increase Circular s debt ratio, leaving the old bondholders more exposed. The old bondholders loss is the stockholders gain. d. Both bondholders and stockholders win. They share the (net) increase in firm value. The bondholders position is not eroded by the issue of a junior security. (We assume that the preferred does not lead to still more game playing and that the new investment does not make the firm s assets safer or riskier.) e. Bondholders lose because they are at risk for a longer time. Stockholders win. Est. Time: 06 0 CHAPTER 9 Financing and Valuation 8. a. Base-case NPV = -, /.0 = 0. b. PV tax shield = (.35 x. x.3(000))/. = APV = = $9.55.

33 Est. Time: If the bank debt is treated as permanent financing, the capital structure proportions are: Est. Time: 0 05 Bank debt (r D = 0 percent) $80 9.4% Long-term debt (r D = 9 percent) Equity (r E = 8 percent, 90 x 0 million shares) $, % WACC* = [0.0 ( ) 0.094] + [0.09 ( ) 0.604] + [ ] = = 9.6%. 6. Base case NPV = $,000 + ($600/.) + ($700/. ) = $93.75 or $93,750. Year Debt Outstanding at Start of Year Interest Interest Tax Shield PV (Tax Shield) APV = $ $ $3.09 = 03.5 or $03,500. Est. Time: a. Base-case NPV = $,000,000 + ($95,000/0.0) = $50,000. PV(tax shields) = 0.35 $400,000 = $40,000. APV = $50,000 + $40,000 = $90,000. b. PV(tax shields, approximate) = ( $400,000)/0.0 = $98,000. APV = $50,000 + $98,000 = $48,000. The present value of the tax shield is higher when the debt is fixed and therefore the tax shield is certain. When borrowing a constant proportion of the market value of the project, the interest tax shields are as uncertain as the value of the project, and therefore must be discounted at the project s opportunity cost of capital. Est. Time: 06 0

34 0 $ a. Base-case NPV $ 0 $0. or $,0. t. t APV = base-case NPV + PV(tax shield). PV(tax shield) is computed from the following table: Debt Outstanding Interest Present Value Year Interest at Start of Year Tax Shield of Tax Shield $5,000 $400 $40 $9.63 4, , , , , , , , Total APV = $,0 + $575,740 = $463,630. b. APV = base-case NPV + PV(tax shield) equity issue costs = $,0 + $575,740 $400,000 = $63,630. Est. Time: 06 0

35 CHAPTER Valuing Options 5. a. Delta = 00/(00 50) =.667. b. p, p. 4 c. 00 p a. Value of call = e. No. The true probability of a price rise is almost certainly higher than the risk-neutral probability, but it does not help to value the option. Est. Time: a. Let p equal the probability of a rise in the stock price. Then, if investors are risk neutral: (p 0.5) + ( p) ( 0.3) = 0.0 p = 0.8 The possible stock prices next period are: $60.5 = $69.00 $ = $5.0 Let X equal the break-even exercise price. Then, the following must be true: X 60 = [(p)($0) + ( p)(x 5.0)]/.0 That is, the value of the put if exercised immediately equals the value of the put if it is held to next period. Solving for X, we find that the break-even exercise price is $6.5. b. If the interest rate is increased, the value of the put option decreases. Est. Time: 06-0

36 CHAPTER 5 Leasing. If the cost of new limos decreases by 5% per year, then the lease payments also decrease by 5% per year. In terms of Table 5., the only change is in the breakeven rent. t = 0 t = t = t = 3 t = 4 t = 5 t = 6 Total Break-Even Rent Tax Break-Even Rent After Tax Cash Flow NPV (at 7%) = 0.00 Here, when we solve for the first after-tax payment, use a discount rate of: Est. Time: 06 0 (.07/0.95) = 0.63 =.63% 3. a. t = 0 t = t = t = 3 t = 4 t = 5 t = 6 t = 7 Cost of New Bus Lost Depreciation Tax Shield Lease Payment Tax Shield of Lease Payment Cash Flow of Lease NPV (at 8.0%) = -$40 b. Assume the straight-line depreciation is figured on the same basis as the ACRS depreciation, namely five years, beginning halfway through the first year. t = 0 t = t = t = 3 t = 4 t = 5 t = 6 t = 7 Cost of New Bus Lost Depreciation Tax Shield Lease Payment Tax Shield of Lease Payment Cash Flow of Lease NPV (at 6.5%) = $566 Est. Time: 06 0

37 4. The net present value of the lessor s cash flows consists of the cost of the bus ($00), the present value of the depreciation tax shield ($9.469), and the present value of the after-tax lease payments: To find the minimum rental, set NPV = 0 and solve for P: 4.494P = P = 6.73, or $6,730 Greymare should take the lease as long as the NPV of the lease is greater than or equal to zero. The net present value of the cash flows is the cost of the bus saved ($00) less the present value of the lease payments: (Note that, because Greymare pays no taxes, the appropriate discount rate is 0%.) Setting this expression equal to zero and solving for P, we find: P = 7.04, or $7,040 This is the maximum amount that Greymare could pay. Thus, the lease payment will be between $6,730 and $7,040. Est. Time: ( 0.35) P P a. Because Nodhead pays no taxes: NPV 50 5 t 0 b. The cash flows to Compulease are as follows (assume five-year ACRS beginning at t = 0): t = 0 t = t = t = 3 t = 4 t = 5 t = 6 Cost of Computer Depreciation Depreciation Tax Shield Lease Payment Tax on Lease Payment Net Cash Flow The after-tax interest rate is: ( 0.35) 0.08 = 0.05 = 5.%. The NPV of the cash flows for Compulease is: 40.0, or $39, or $59,500 t.08

38 c. The overall gain from leasing is: $39,954 $59,600 = $9,646. Est. Time: 06 0 CHAPTER 3 Mergers 3. a. The gain from the merger is equal to the present value of the savings. It is estimated that $500,000 per year will be saved in perpetuity. Therefore, the PV of the savings is $500,000/0% = $5 million (we assume that the $500,000 saving is an after-tax figure). b. Cost = cash paid PV(Pogo). Therefore, the cost = $4 million - $0 million = $4 million. c. Cost = xpv(ab) PV(B). The PV of (AB) is $35 million. Therefore, cost = (.5 x $35 million) - $0 million = $7.5 million. d. NPV = gain cost. NPV = $5 million - $4 million = +$ million. e. NPV = gain cost. NPV = $5 million - $7.5 million = -$.5 million. Est. Time: a. Use the perpetual growth model of stock valuation to find the appropriate discount rate (r) for the common stock of Plastitoys (Company B): Under new management, the value of the combination (AB) would be the value of Leisure Products (Company A) before the merger (because Company A s value is unchanged by the merger) plus the value of Plastitoys after the merger, or: PV AB $0.80 (,000,000 $ 90) 600,000 $4,000, We now calculate the gain from the acquisition: Gain PV r 0.06 AB (PV A PV B ) r % Gain $4,000,000 ($90,000,000 $,000,000) $,000,000 b. Because this is a cash acquisition: Cost = cash paid PV B = ($5 600,000) $,000,000 = $3,000,000

39 c. Because this acquisition is financed with stock, we have to take into consideration the effect of the merger on the stock price of Leisure Products. After the merger, there will be,00,000 shares outstanding. This is determined by adding the outstanding shares of Leisure Products (,000,000) to the number of shares offered for Plastitoys existing stock (/3 x 600,000 = 00,000). Hence, the share price will be: Therefore: $4,000,000/,00,000 = $95.00 Cost = ($95 00,000) ($0 600,000) = $7,000,000 d. If the acquisition is for cash, the cost is the same as in Part b above: Cost = $3,000,000 Est. Time: 5 If the acquisition is for stock, the cost is different from that calculated in Part c. This is because the new growth rate affects the value of the merged company. This, in turn, affects the stock price of the merged company and, hence, the cost of the merger. It follows that: PV AB = ($90,000,000) + ($0 600,000) = $0,000,000 The new share price will be: Therefore: $0,000,000/,00,000 = $85.00 Cost = ($85 00,000) ($0 600,000) = $5,000,000