OPPORTUNITY COST OF CAPITAL AND CAPITAL BUDGETING

P 3-1: Solution to IRR Problem (5 minutes)

[Simple IRR calculation]

0 = ($50,000) + 8,330 ×Annuity Factor (IRR = ?, t = 7)

6.002 = Annuity Factor (IRR = ?, t = 7)

IRR = 4%

P 3-2: Solution to Accelerated Depreciation (10 minutes)

[Depreciation tax shields]

As long as the firm has positive taxable income and tax rates remain constant, accelerated depreciation is preferable to the straight-line method. The accelerated depreciation method allows for earlier write offs of the original cost. Since larger and earlier write offs shield more income from taxes in earlier time periods, the present value of the tax shield from depreciation is greater under the accelerated method.

P 3-3: Solution to Jasper, Inc. (10 minutes)

[Internal rate of return]

a. The internal rate of return is the discount rate that equates the present values of the inflows to the outflows. Or, for project A:

0 = –$300,000 + $100,300 x Annuity Factor (IRRa = ?, t = 5)

$300,000 = $100,300 × Annuity Factor (IRRa = ?, t = 5)

2.991 = Annuity Factor (IRRa = ?, t = 5)

Looking in the present value of an annuity table, IRRa = 20%.

Project B:

0 = –$150,000 + $55,783 × Annuity Factor (IRRb = ?, t = 5)

$150,000 = $55,783 × Annuity Factor (IRRb = ?, t = 5)

2.689 = Annuity Factor (IRRb = ?, t = 5)

Looking in the annuity tables, IRRb = 25%.

b. Both projects have IRRs greater than the required rate of return. Using the IRRs as a measure then, we should take alternative B since it has the highest net present value.

Annual cash flows

$100,300

$55,783

NPV @ 16%

328,382

182,634

Less: initial outlay

300,000

150,000

NPV

$28,382

$32,634

P 3-4: Solution to Just One, Inc. (10 minutes)

[IRR vs. NPV]

The original IRR ranking is misleading because the scale of each project is different. While Q has the higher IRR, it is a smaller project and yields fewer dollars of net present value.

P 3-5: Solution to Equity Corp. (10 minutes)

[Determining relevant cash flows for valuing an investment]

The consulting fees are a sunk cost and are not used in this decision. Let’s look at the cash flows.

PV operating revenues (after tax)
$90,000
PV operating expense (after tax)
($20,000)
PV tax savings from depreciation .4 × $87,500
$35,000
Cost of machine
($100,000)
Net present value
$ 5,000

Equity Corporation should install the equipment.

P 3-6: Solution to Declining Market, Inc. (10 minutes)

[When to discontinue a product line]

There are two things wrong. First, depreciation shields some income from tax. This shield must be taken into account in the contribution equation. A better equation is:

(1 – t)(sales – variable cost) + t × depreciation

Second, one should compare the present value of the contribution to earnings to the price the company can get for selling the equipment.

P 3-7: Solution to Northern Sun, Inc. (15 minutes)

[Net present value and payback]

a. Note: all cash inflows and outflows are at the end of the year.

Net Cash Inflow (Outflow)
Year
Discount Factor @ 10%

Present Value

$(355)

.909

($323)
55

.826

45
190

.751

143
240

.683

164
190

.621

118
Net Present Value

$147

Year

Inflow

Cumulative Inflow

Investment to be Recovered

0
0
355

55
55
300

190
245
110

240
485
–

190
675
–

3 years + (110 ÷ 240) = 3.46 years

P 3-8: Solution to Ab Landlord (15 minutes)

[Valuing rental problem with inflation]

a. Ab Landlord needs to weigh the purchase offer against the present value of the net cash flows from his current operation. Mr. Landlord expects the net cash flow in real dollars to remain constant at $200,000/year. Since we have the cash flow in real dollars, we can use the real interest rate of 5% to calculate the present value.

PV = $200,000 × Annuity Factor (i = .05, t = 10)

= $200,000 × 7.722

= $1,544,000

Since the present value of the current operation is greater than the purchase offer, Mr. Landlord should keep the property.

b. In this case, the net cash flows remain constant in nominal dollars.

PV = $200,000 × Annuity Factor (i = .16, t = 10)

= $200,000 × 4.833

= $966,600

If rent control is imposed, then Mr. Landlord should sell the building now for $1,500,000.

P 3-9: Solution to Lottery (15 minutes)

[Present value and future value of annuity]

a. The minimum lump sum you should take is the present value of the cash payments.

PV = $100,000 × Annuity Factor (i = .10, t = 10)

= $100,000 × 6.145

= $614,500

b. This question is essentially (a) in reverse. You are looking for the future value of the cash payments. Looking in the future value in arrears table, the annuity factor is 15.937.

FV = $100,000 × 15.937

= $1,593,700

c. This is similar to (a). This time, t = 7.

PV = $100,000 × Annuity Factor (i = .10, t = 7)

= $100,000 × 4.868

= $486,800

d. To convert an end-of-year payment schedule to a beginning-of-year schedule, we need only multiply by 1 + r. The minimum payment is $614,500 × 1.10 = $675,900.

P 3-10: Solution to Mr. Jones’s Retirement (15 minutes)

[Saving for retirement]

This problem must be broken into two parts to solve. First, the present value of the retirement annuity must be calculated.

PV = $30,000 × Annuity Factor (r = .04, t = 15)

= $30,000 × 11.118

= $333,540

Now we need to calculate the annual savings required that will grow to this retirement amount using the future value of an annuity table.

$333,540 = Payment × Future Annuity Factor (r = .04, t = 20)

= Payment × 29.778

$11,200 = Payment

P 3-11: Solution to NPV vs. Payback (15 minutes)

[NPV and payback]

The investment generates cash flows of $1,200 in its first five years. The worst NPV would be generated when the entire $1,200 is received at the end of the fifth year. In that case the NPV is:

NPV = –1,200 +

= –1,200 + 482

= –$718

P 3-12: Solution to Clean Tooth (15 minutes)

[Evaluating a divestiture]

PV = $50,000 × PV of annuity for 10 yrs. @ r

PV depends on r.

Point of indifference: $250,000 = $50,000 × PV of annuity for 10 yrs. @ r

PV of annuity @ r% for 10 yrs. = = 5.000

PV of annuity for 10 yrs. @ 14% = 5.216

PV of annuity for 10 yrs. @ 16% = 4.833

If market rate of interest > 15% Keep

If market rate of interest < 15% Sell

P 3-13: Solution to New Car (15 minutes)

[Computing effective interest rates]

Let A = annuity factor

$50,000 = $2,233.33 × A

A = 22.39 Þ i = 2% month Þ 24% annual rate compounded monthly

Or,

(1.02)12 – 1 = 26.8% = annual rate compounded annually.

P 3-14: Solution to National Taxpayers Union (15 minutes)

[Relation between interest rates and inflation]

If nominal interest rates are determined in a competitive market, then

rn = (1 + rR) (1 + I) – 1

where

rn = the nominal interest rate in year t

rR = real rate of interest

I = expected rate of inflation in year t.

Since the interest rate on Treasury bills is determined in a competitive market, the observed rate of interest on Treasury bills incorporates inflation. As the rate of inflation increases, the nominal rate of return increases. Thus, the real rate of return is unaffected by inflation. The author of the letter ignores this link between the rate of inflation and nominal interest rates.

The author also ignores the fact that the real value of the $1,000 annual savings depends on the rate of inflation. If the real value of $1,000 was saved every year, the present value of the amount available for retirement would be unaffected by inflation.

Present Value of Amount Available for Retirement

P 3-15: Solution to Federal Dam Project (15 minutes)

[Present value of a government subsidy]

Assume the farmer values the dam at the construction cost, or $300 per acre.

Subsidy = $30,000 – PV (payments)

For 10 percent:

PV (payments) = ()5

= (.6209) (9.427) ($1,000)

= $5,853.22

Subsidy = $24,146.78

P 3-16: Solution to South American Mining (20 minutes)

[IRR of an additional investment]

a. The IRR is calculated by looking at the incremental differences (000s).

Period 0

Period 1

Period 2

Current situation
–$8
$ 0
$10
Additional expenditure
–$9
$10
$ 0
Difference
–$1
$10
–$10

Let X = 1 ÷ (1 + IRR).

Using the equation,

0 = –$1 + $10X – $10X2

IRR can be calculated using the quadratic formula.

x =

where: a = -$10

b = $10

c = -$1

x =

Or, x1 = .8873 and

x2 = .1127

Since, x =

then irr = – 1

And, irr1 = – 1 = .127

irr2 = – 1 = 7.87

The IRRs are 12.7 percent and 787 percent.

b. The bond rate of 15 percent lies between the two IRRs, making the additional outlay a positive NPV project. The company should invest the extra $1 million.

P 3-17: Solution to House Mortgage (20 minutes)

[Simple discounting]

a. The present value of a mortgage equals the period payment times the annuity factor

$200,000 = Payment x Annuity Factor (r = .10, n = 30)

Payment =

Payment = $21,216

b. After one year: Principal = $21,216 × Annuity Factor (r = .10, n = 29)

= $21,216 × 9.370 = $198,793

After ten years: Principal = $21,216 × Annuity Factor (r = .10, n = 20)

= $21,216 × 8.514 = $180,633

c. The remaining principal after year 4 is

Principal = $21,216 × Annuity Factor (r = .10, n = 26)

= $21,216 × 9.161 = $194,360

If the house is remortgaged at 8 percent, the payments are

$194,360 = Payment × Annuity Factor (r=.08, n=30)

Payment = $194,360 ÷ 11.258

Payment = $17,264

The difference between the two payments is $3,952 ($21,216 – $17,264). The present value of the incremental difference over five years is

PV of Savings = $3,952 × Annuity Factor (r = .06, n = 5)

= $3,952 × 4.212 = $16,646

P 3-18: Solution to Flower City Grocery (20 minutes)

[Replacement problem without taxes]

Old Machine

Outflow

Repair ($ 1,000)

Inflow

PV of Profits @ $5,000 per year for 5 years, r = .09 (3.890) $19,450

Net Present Value of keeping old machine $18,450

New Machine

Outflow

Purchase ($ 5,000)

Inflow

Sale of old machine $25,285

$500 Why or why not? r for 5 years, r =.09 (3.890)

Net Present Value of purchasing new machine $20,785

NPVnew – NPVold $ 2,335

Purchasing the new machine is the better choice.

P 3-19: Solution to Toledo Stadium (20 minutes)

[Simple capital budget]

a. Another way of asking this question is: Does this project have a positive NPV?

Annual Cash Flows:

Maintenance
$(250,000)

Cash Inflows:

Lease payments
650,000

Concerts
600,000

Other sports events
50,000

Annual Net Cash Flows
$1,050,000

× Annuity Factor
9.818

Less: Original Outlay

$10,308,900
Net Present Value

12,000,000

$(1,691,100)

The city should not build the stadium.

b. In this case, we need to look at which choice costs the city more. The present value of the lost revenues is

PV = ($350,000) × Annuity Factor (r = .08, t = 10)

= (2,348,500)

The negative present value of the lost revenues is greater than the negative present value of the stadium. It is now in the interest of the city to build the stadium.

P 3-20: Solution to PQR Coal (20 minutes)

[Modify or sell obsolete equipment]

The firm has two choices:

(A) Sell the old equipment for $500,000 and purchase new equipment for $800,000. Operating costs are unchanged.

(B) Refurbish the old equipment for $250,000 and incur an additional $20,000 per year operating expense for ten years.

We need to examine the present value of each alternative in order to determine which alternative is the better choice.

Choice (A) is simple. The additional cost to the firm is:

$500,000 – $800,000 = ($300,000)

Choice (B) is a bit more complicated. PQR Coal will incur costs of $250,000 to adapt the machinery and $20,000 per year increase in operating costs.

PVB = ($250,000) + (20,000) × Annuity Factor (r = .10, t = 10)

= ($250,000) + (20,000) × 6.145

= ($372,900)

Ms. Big is right. Choice (A) is the correct choice because it has the smallest present value of cash outlays.

P 3-21: Solution to Student Loan Program (20 minutes)

[Calculating the government’s subsidy]

PV = Payment x Annuity Factor (r = .04, t = 10)

$10,000 = Payment x 8.111

$1,233 = Payment

PV = Payment x Annuity Factor (r = .06, t = 10)

= $1,233 × 7.360

= $9,075

c. Yes, the NDSL recipients are receiving a subsidy. Recall that five years have passed since the student received the $10,000 loan. The present value of the annual loan repayments ($1,233) amounts to:

Present value of repayments = $9,075 × PV(r = .06, t = 5)

= $9,075 × .747

= $6,779

Present value of subsidy = $10,000 – $6,779

= $3,221

P 3-22: Solution to Geico (20 minutes)

[Valuing land]

Yes. The land should be considered as part of the cost of the expansion project. Just because the land will not be sold does not mean it is “free” to this project. Accepting this expansion project restricts the firm’s future use of the land. It cannot be sold or used for some other expansion.

The forgone receipts that could have been received from investing the proceeds of selling the land should be included in the cost of the expansion project. If the land were sold for $820,000 and the proceeds invested in U.S. government bonds paying 8% then $65,600 ($820,000 × 8%) is the annual opportunity cost of using this land in this investment project.

P 3-23: Solution to Depreciation Tax Shield (20 minutes)

[Depreciation tax shield, real and nominal interest rates]

Depreciation per year = = $20, so tax shield per year = $20 ´ .40 = $8 million

The tax shield is nominal. Nominal interest rate = (1 + real rate)(1 + inflation) -1

= (1.05)2 – 1 = .1025

PV of tax shield = $8 ´ (Present Value of annuity for 5 years @ 10.25%)

= $30,134 million.

P 3-24: Solution to Housing Markets (20 minutes)

[Valuing real estate]

a. The question is, how much is the assumable mortgage on your house worth?

PV of annuity at 8% = 11.258

PV of annuity at 15% = 6.566

= Annual mortgage payment @ 8% = 10,659 x 6.566

= $69,987 PV of payments

Gain on loan: $120,000

– 69,987

$ 50,013

+ sale price of identical house 150,000

Price of house $200,013

b. Difference in property taxes is $1,000/year for perpetuity.

PV = = $6,667

Third home should sell for $156,667.

P 3-25: Solution to Mortgage Department (20 minutes)

[Market rates of interest and usury laws]

a. Payments = = $5,303.91

b. $50,000 – ($5,304 × 8.055) = $50,000 – 42,724 = $7,276

c. How many points the bank has to charge to eliminate the loss of part (b) is:

P 3-26: Solution to Electric Generator (20 minutes)

[Present value of alternative energy sources]

a. Investment: $140,000

Annual Savings = ($42,000 – $22,000) = $20,000

NPV = –$140,000 + (6.145) ($20,000)

= –$17,100

The generator should not be purchased because it has a negative net present value.

b. The NPV of using the generator for both electric power and heating evaluated:

Total investment: $140,000 + $40,000 = $180,000

Total Annual Savings: ($42,000 + $21,000) – ($22,000 + $10,000) = $31,000

NPV = –$180,000 + (6.145) ($31,000)

= $10,495.

The generator should be purchased and used for both electrical generation and heating.

P 3-27: Solution to Dakota Mining (30 minutes)

[Multiple internal rates of return]

a. Let x = 1÷(1+ IRR)

IRR = -4.4 + 27.7x – 25×2

x =

= 0.192 or 0.916

IRR = –1 + 1/x

= –1 + 5.21 or -1 + 1.09

= 4.21 or 0.09

There are multiple internal rates of return in this case, so we have to know whether the NPV curve cuts the axis from above or below. One way is to calculate the NPV using 8 percent.

NPV (r = 8%) = –4.4 + –

= –4.4 + 25.6 – 21.4

= –0.2

So the curve cuts the axis from below and NPV is negative for market rates below 9 percent. The project should not be accepted.

b. For a market rate of return of 14 percent the NPV function is positive, so we should accept the project.

P 3-28: Solution to Overland Steel (30 minutes)

[Evaluating pollution fines and shut down]

PV of Annuity 30 year 14% 7.003

PV of $1 in 30 years 14% 0.020

Alternatives

(1) Keep running and pay fine:

NPV (–$365,000 + $450,000) × 7.003 $595,255

+ Salvage 0.020 × 2,000,000 40,000

NPV $635,255

(2) Pollution devices installed:

NPV ($450,000 + 25,000) × 7.003 $3,326,425

+ Salvage 0.020 × 2,000,000 40,000

$3,366,425

– initial outlay 2,750,000

NPV $ 616,425

(3) Sell it: Purchase Price $1,000,000

Demolition Costs – 650,000

NPV $ 350,000

Alternative (1) is best if all the costs the company incurs are confined to the fine and there are no adverse publicity costs, lawsuits, or reduction in firm brand-name capital.