Unformatted text preview: Chapter 4 problems 1. We need to find the annuity payment in retirement. Our retirement savings ends at the same time the retirement withdrawals begin, so the PV of the retirement withdrawals will be the FV of the retirement savings. So, we find the FV of the stock account and the FV of the bond account and add the two FVs. Stock account: FVA = $700[{[1 + (.10/12) ]360 – 1} / (.10/12)] = $1,582,341.55 Bond account: FVA = $300[{[1 + (.06/12) ]360 – 1} / (.06/12)] = $301,354.51 So, the total amount saved at retirement is: $1,582,341.55 + 301,354.51 = $1,883,696.06 Solving for the withdrawal amount in retirement using the PVA equation gives us: PVA = $1,883,696.06 = C[1 – {1 / [1 + (.08/12)]300} / (.08/12)] C = $1,883,696.06 / 129.5645 = $14,538.67 withdrawal per month 2. Here, we are trying to find the interest rate when we know the PV and FV. Using the FV equation: FV = PV(1 + r) $4 = $3(1 + r) r = 4/3 – 1 = 33.33% per week The interest rate is 33.33% per week. To find the APR, we multiply this rate by the number of weeks in a year, so: APR = (52)33.33% = 1,733.33% And using the equation to find the EAR: EAR = [1 + (APR / m)]m – 1 EAR = [1 + .3333]52 – 1 = 313,916,515.69% 3. The amount of principal paid on the loan is the PV of the monthly payments you make. So, the present value of the $1,200 monthly payments is: PVA = $1,200[(1 – {1 / [1 + (.068/12)]}360) / (.068/12)] = $184,070.20 The monthly payments of $1,200 will amount to a principal payment of $184,070.20. The amount of principal you will still owe is: $250,000 – 184,070.20 = $65,929.80 This remaining principal amount will increase at the interest rate on the loan until the end of the loan period. So the balloon payment in 30 years, which is the FV of the remaining principal will be: Balloon payment = $65,929.80[1 + (.068/12)]360 = $504,129.05 Chapter 5 problems 4. a. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR for each project is: Deepwater Fishing IRR:
0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3
0 = –$750,000 + $310,000 / (1 + IRR) + $430,000 / (1 + IRR)2 + $330,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 19.83% Submarine Ride IRR:
0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3
0 = –$2,100,000 + $1,200,000 / (1 + IRR) + $760,000 / (1 + IRR)2 + $850,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 17.36% Based on the IRR rule, the deepwater fishing project should be chosen because it has the higher
IRR.
b. To calculate the incremental IRR, we subtract the smaller project’s cash flows from the larger
project’s cash flows. In this case, we subtract the deepwater fishing cash flows from the submarine
ride cash flows. The incremental IRR is the IRR of these incremental cash flows. So, the
incremental cash flows of the submarine ride are:
Submarine Ride
Deepwater Fishing
Submarine – Fishing Year 0
–$2,100,000
–750,000
–$1,350,000 Year 1
$1,200,000
310,000
$890,000 Year 2
$760,000
430,000
$330,000 Year 3
$850,000
330,000
$520,000 Setting the present value of these incremental cash flows equal to zero, we find the incremental
IRR is:
0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3
0 = –$1,350,000 + $890,000 / (1 + IRR) + $330,000 / (1 + IRR)2 + $520,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: Incremental IRR = 15.78% For investingtype projects, accept the larger project when the incremental IRR is greater than the
discount rate. Since the incremental IRR, 15.78%, is greater than the required rate of return of 14
percent, choose the submarine ride project. Note that this is not the choice when evaluating only
the IRR of each project. The IRR decision rule is flawed because there is a scale problem. That is,
the submarine ride has a greater initial investment than does the deepwater fishing project. This
problem is corrected by calculating the IRR of the incremental cash flows, or by evaluating the
NPV of each project. c. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each
project will be:
Deepwater fishing:
NPV = –$750,000 + $310,000 / 1.14 + $430,000 / 1.142 + $330,000 / 1.143
NPV = $75,541.46
Submarine ride:
NPV = –$2,100,000 + $1,200,000 / 1.14 + $760,000 / 1.142 + $850,000 / 1.143
NPV = $111,152.69
Since the NPV of the submarine ride project is greater than the NPV of the deepwater fishing
project, choose the submarine ride project. The incremental IRR rule is always consistent with the
NPV rule. Chapter 6 5. We need to find the bid price for a project, but the project has extra cash flows. Since we don’t already produce the keyboard, the sales of the keyboard outside the contract are relevant cash flows. Since we know the extra sales number and price, we can calculate the cash flows generated by these sales. The cash flow generated from the sale of the keyboard outside the contract is: Year 1 Year 2 Year 3 Year 4 Sales $1,100,000 $3,300,000 $3,850,000 $1,925,000 Variable costs 660,000 1,980,000 2,310,000 1,155,000 EBT $440,000 $1,320,000 $1,540,000 $770,000 Tax 176,000 528,000 616,000 308,000 Net income (and OCF) $264,000 $792,000 $924,000 $462,000 So, the addition to NPV of these market sales is: NPV of market sales = $264,000/1.13 + $792,000/1.132 + $924,000/1.133 + $462,000/1.134 NPV of market sales = $1,777,612.09 You may have noticed that we did not include the initial cash outlay, depreciation, or fixed costs in the calculation of cash flows from the market sales. The reason is that it is irrelevant whether we include these here. Remember that we are not only trying to determine the bid price, but we are also determining whether the project is feasible. In other words, we are trying to calculate the NPV of the project, not just the NPV of the bid price. We will include these cash flows in the bid price calculation. The reason we stated earlier that whether we included these costs in this initial calculation was irrelevant is that you will come up with the same bid price if you include these costs in this calculation, or if you include them in the bid price calculation. Next, we need to calculate the aftertax salvage value, which is: Aftertax salvage value = $200,000(1 – .40) = $120,000 Instead of solving for a zero NPV as is usual in setting a bid price, the company president requires an NPV of $100,000, so we will solve for a NPV of that amount. The NPV equation for this project is (remember to include the NWC cash flow at the beginning of the project, and the NWC recovery at the end): NPV = $100,000 = –$3,200,000 – 75,000 + 1,777,612.09 + OCF (PVIFA13%,4) + [($120,000 + 75,000) / 1.134] Solving for the OCF, we get: OCF = $1,477,790.75 / PVIFA13%,4 = $496,824.68 Now we can solve for the bid price as follows: OCF = $496,824.68 = [(P – v)Q – FC ](1 – tC) + tCD $471,253.44 = [(P – $165)(9,000) – $600,000](1 – 0.40) + 0.40($3,200,000/4) P = $264.41 Chapter 7 a. To calculate the accounting breakeven, we first need to find the depreciation for each year. The
depreciation is:
Depreciation = $724,000/8
Depreciation = $90,500 per year
And the accounting breakeven is:
QA = ($850,000 + 90,500)/($39 – 23)
QA = 58,781 units
b. We will use the tax shield approach to calculate the OCF. The OCF is:
OCFbase = [(P – v)Q – FC](1 – tc) + tcD
OCFbase = [($39 – 23)(75,000) – $850,000](0.65) + 0.35($90,500)
OCFbase = $259,175
Now we can calculate the NPV using our basecase projections. There is no salvage value or
NWC, so the NPV is:
NPVbase = –$724,000 + $259,175(PVIFA15%,8)
NPVbase = $439,001.55
To calculate the sensitivity of the NPV to changes in the quantity sold, we will calculate the NPV
at a different quantity. We will use sales of 80,000 units. The NPV at this sales level is:
OCFnew = [($39 – 23)(80,000) – $850,000](0.65) + 0.35($90,500)
OCFnew = $311,175
And the NPV is:
NPVnew = –$724,000 + $311,175(PVIFA15%,8)
NPVnew = $672,342.27 So, the change in NPV for every unit change in sales is:
ΔNPV/ΔS = ($439,001.55 – 672,342.27)/(75,000 – 80,000)
ΔNPV/ΔS = +$46.668
If sales were to drop by 500 units, then NPV would drop by:
NPV drop = $46.668(500) = $23,334.07
You may wonder why we chose 80,000 units. Because it doesn’t matter! Whatever sales number
we use, when we calculate the change in NPV per unit sold, the ratio will be the same.
c. To find out how sensitive OCF is to a change in variable costs, we will compute the OCF at a
variable cost of $24. Again, the number we choose to use here is irrelevant: We will get the same
ratio of OCF to a one dollar change in variable cost no matter what variable cost we use. So, using
the tax shield approach, the OCF at a variable cost of $24 is:
OCFnew = [($39 – 24)(75,000) – 850,000](0.65) + 0.35($90,500)
OCFnew = $210,425
So, the change in OCF for a $1 change in variable costs is:
ΔOCF/Δv = ($259,175 – 210,425)/($23 – 24)
ΔOCF/Δv = –$48,750
If variable costs decrease by $1 then, OCF would increase by $48,750 Chapter 11 7. First, we need to find the standard deviation of the market and the portfolio, which are:
σM = (.0429)1/2
σM = .2071 or 20.71%
σZ = (.1783)1/2
σZ = .4223 or 42.23%
Now we can use the equation for beta to find the beta of the portfolio, which is:
βZ = (ρZ,M)(σZ) / σM
βZ = (.39)(.4223) / .2071
βZ = .80
Now, we can use the CAPM to find the expected return of the portfolio, which is: E(RZ) = Rf + βZ[E(RM) – Rf] E(RZ) = .048 + .80(.114 – .048) E(RZ) = .1005 or 10.05% Chapter 13 8. We will begin by finding the market value of each type of financing. We find:
MVD = 5,000($1,000)(1.03) = $5,150,000
MVE = 160,000($57) = $9,120,000 And the total market value of the firm is: V = $5,150,000 + 9,120,000 = $14,270,000 Now, we can find the cost of equity using the CAPM. The cost of equity is: RE = .06 + 1.10(.07) = .1370 or 13.70% The cost of debt is the YTM of the bonds, so: P0 = $1,030 = $40(PVIFAR%,40) + $1,000(PVIFR%,40) R = 3.851% YTM = 3.851% × 2 = 7.70% And the aftertax cost of debt is: RD = (1 – .35)(.0770) = .0501 or 5.01% Now we have all of the components to calculate the WACC. The WACC is: WACC = .0501(5.15/14.27) + .1370(9.12/14.27) = .1056 or 10.56% Notice that we didn’t include the (1 – tC) term in the WACC equation. We simply used the aftertax cost
of debt in the equation, so the term is not needed here. Chapter 16 9. a. In a world with corporate taxes, a firm’s weighted average cost of capital is equal to: RWACC = [B / (B+S)](1 – tC)RB + [S / (B+S)]RS We do not have the company’s debt
to
value ratio or the equity
to
value ratio, but we can calculate either from the debt
to
equity ratio. With the given debt
equity ratio, we know the company has 2.5 dollars of debt for every dollar of equity. Since we only need the ratio of debt
to
value and equity
to
value, we can say: B / (B+S) = 2.5 / (2.5 + 1) = .7143 S / (B+S) = 1 / (2.5 + 1) = .2857 We can now use the weighted average cost of capital equation to find the cost of equity, which is: .15 = (.7143)(1 – 0.35)(.10) + (.2857)(RS) RS = .3625 or 36.25% b. We can use Modigliani
Miller Proposition II with corporate taxes to find the unlevered cost of equity. Doing so, we find: c. RS = R0 + (B/S)(R0 – RB)(1 – tC) .3625 = R0 + (2.5)(R0 – .10)(1 – .35) R0 = .2000 or 20.00% We first need to find the debt
to
value ratio and the equity
to
value ratio. We can then use the cost of levered equity equation with taxes, and finally the weighted average cost of capital equation. So: If debt
equity = .75 B / (B+S) = .75 / (.75 + 1) = .4286 S / (B+S) = 1 / (.75 + 1) = .5714 The cost of levered equity will be: RS = R0 + (B/S)(R0 – RB)(1 – tC) RS = .20 + (.75)(.20 – .10)(1 – .35) RS = .2488 or 24.88% And the weighted average cost of capital will be: RWACC = [B / (B+S)](1 – tC)RB + [S / (B+S)]RS RWACC = (.4286)(1 – .35)(.10) + (.5714)(.2488) RWACC = .17 If debt
equity =1.50 B / (B+S) = 1.50 / (1.50 + 1) = .6000 E / (B+S) = 1 / (1.50 + 1) = .4000 The cost of levered equity will be: RS = R0 + (B/S)(R0 – RB)(1 – tC) RS = .20 + (1.50)(.20 – .10)(1 – .35) RS = .2975 or 29.75% And the weighted average cost of capital will be: RWACC = [B / (B+S)](1 – tC)RB + [S / (B+S)]RS RWACC = (.6000)(1 – .35)(.10) + (.4000)(.2975) RWACC = .1580 or 15.80% Chapter 17 10. a. The total value of a firm’s equity is the discounted expected cash flow to the firm’s stockholders. If the expansion continues, each firm will generate earnings before interest and taxes of $2.4 million. If there is a recession, each firm will generate earnings before interest and taxes of only $900,000. Since Steinberg owes its bondholders $800,000 at the end of the year, its stockholders will receive $1.6 million (= $2,400,000 – 800,000) if the expansion continues. If there is a recession, its stockholders will only receive $100,000 (= $900,000 – 800,000). So, assuming a discount rate of 15 percent, the market value of Steinberg’s equity is: SSteinberg = [.80($1,600,000) + .20($100,000)] / 1.15 = $1,130,435 Steinberg’s bondholders will receive $800,000 whether there is a recession or a continuation of the expansion. So, the market value of Steinberg’s debt is: BSteinberg = [.80($800,000) + .20($800,000)] / 1.15 = $695,652 Since Dietrich owes its bondholders $1.1 million at the end of the year, its stockholders will receive $1.3 million (= $2.4 million – 1.1 million) if the expansion continues. If there is a recession, its stockholders will receive nothing since the firm’s bondholders have a more senior claim on all $800,000 of the firm’s earnings. So, the market value of Dietrich’s equity is: SDietrich = [.80($1,300,000) + .20($0)] / 1.15 = $904,348 Dietrich’s bondholders will receive $1.1 million if the expansion continues and $900,000 if there is a recession. So, the market value of Dietrich’s debt is: BDietrich = [.80($1,100,000) + .20($900,000)] / 1.15 = $921,739 b. The value of company is the sum of the value of the firm’s debt and equity. So, the value of Steinberg is: VSteinberg = B + S VSteinberg = $1,130,435 + $695,652 VSteinberg = $1,826,087 And value of Dietrich is: VDietrich = B + S VDietrich = $904,348 + 921,739 VDietrich = $1,826,087 You should disagree with the CEO’s statement. The risk of bankruptcy per se does not affect a firm’s value. It is the actual costs of bankruptcy that decrease the value of a firm. Note that this problem assumes that there are no bankruptcy costs. ...
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 Fall '11
 WilliamHandorf
 Annuity, Corporate Finance, Internal rate of return, the00

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