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13. LEASING

Objectives: After reading this chapter, you will be able to

1. Make the optimal choice between leasing and buying an asset.

2. Calculate the lease payments, or the purchase price, which will make leasing equivalent to buying.

3. Discern some of the advantages and disadvantages of leasing.

13.1 Leasing

A company may be in need of a piece of equipment. It can either lease the equipment or buy it outright. In recent years, leasing has become a common source of financing for corporations.

There are several reasons why companies want to lease an asset whether it is a computer, an airplane, or a warehouse. First, it gives them some flexibility. If it is a startup company, they may want to lease some office space and start their operations. They do not want to invest in a building, for instance, where they have to tie up a large amount of capital for a long of time. Second, a company may not have the cash needed to buy an asset. They may be trying to conserve cash and invest in some profitable projects. Third, they may simply need the asset for a short time and there is no point in buying it. Fourth, it is perhaps cheaper for them to lease the equipment, as indicated by the NPV analysis of the lease.

The companies may still want to buy certain assets, especially the ones that actually increase in value, such as buildings and land. In many instances, they have to buy an asset because it is not possible for them to lease it. A company may want to build a factory to their own specifications and needs.

In some cases, a company may sell an asset and lease it back. For example, a bank may sell the building that it owns, generate a substantial amount of cash, and also make a large profit because it bought the building a long time ago. Then it simply leases the building for its own use. It does not have to move out of its quarters. This is a sale-and-leaseback arrangement.

( 221 )

IBM

Corporation (Lessor)

Computer (asset)





Lease payments

Mercy Hospital (Lessee)

Figure 13.1. In a direct lease, only two parties are involved

A lease may be direct lease, or a leveraged lease. In a direct lease, only two parties are involved, the owner of the asset and the end user of the asset. The owner of the asset is known as the lessor, and the end user, the lessee. An example may be IBM Corporation leasing a computer to Mercy Hospital. The two parties negotiate various terms of the

lease, such as the amount of lease payments, when are they due, who will take care of the maintenance costs, and so on. The lease arrangement is shown in Figure 13.1.

A leveraged lease involves three parties. The owner of the asset buys the asset by borrowing the money from a bank, and then rents the asset out to the end user. The owner of the asset is leveraging the lease and he does not want to tie up his personal capital in the asset. For example, a property owner may buy an apartment building by borrowing the money from the bank and then lease the apartments to individuals or families. The arrangement for a leveraged lease is as shown below.

( A n al y t i ca l T ech n i q u e s ) ( 1 3 . L e as i n g )

Bank

Mortgage loan





Loan payments

Landlord

Apartment house





Rent payments

Tenants

Fig. 13.2. In a leveraged lease, three or more parties may be involved.

In the above situation, it is possible to carry out the financial analysis from the point of view of the bank, or the proprietor, or the tenant. The loan officer at the bank may want to make sure that the loan is properly secured by examining the physical condition of the building, the financial strength of the property owner, and assessing his managerial ability. He should also look at the actual lease contracts, if available.

The property owner is an entrepreneur. He wants to analyze the lease and see if it is profitable for him. He has to look at all the cash flows: mortgage loan payments, rents, the maintenance expenses, the real estate taxes, his income tax rate, and the tax benefits of depreciation of the building. The tenant wants to examine the advantages of renting the apartment: convenience, predictable rent payments, and his personal preference. He may also consider the advantages of borrowing the money and buying a house: the deductibility of interest on the mortgage loan, the freedom to remodel the house, the needs of his family, and so on. We are interested in the financial analysis only.

13.2 Capital Leases

Frequently corporations sign up long-term leases to make sure that they have access to a particular asset without interruption. Wal-Mart may lease a store for several years, or a corporation may lease a computer until it becomes obsolete. A long-term lease is called a capital lease, or a financial lease, if it satisfies at least one of the following four conditions:

1. The lessee will become the owner of the asset when the lease expires.

2. The lessee will have the right to purchase the asset at below its then market value at the expiration of the lease.

3. The term of the lease exceeds 75% of the economic life of the asset.

4. The present value of the lease payments is more than 90% of the initial value of the asset.

If the lease does not satisfy any of the above terms, it is then an operating lease. A capital lease is essentially similar to buying the asset. From the accounting point of view, capital leases are shown as capital assets on the balance sheet of a corporation.

Occasionally, a company may sell an asset and lease it back. This type of transaction is called a sale-and-leaseback arrangement, or simply a leaseback . A company may own a valuable asset, such as a large office building, but it is not utilizing it fully. It can sell the asset to generate much needed cash and then, instead of vacating that property, it can lease some of that space for its own use. The best example of this arrangement is the sale- and-leaseback of Pan Am Building by Pan American World Airways in 1981.

13.3 Lease Analysis

After calculating the NPV of all cash flows of buying or leasing, one can determine which method is less expensive. The answer may very well depend upon the corporate tax rate, the cost of capital, the depreciation schedule, whether or not outside financing is involved. The lease payments are generally due at the beginning of each period and the taxes are paid at the end of each period. The lease payments, the depreciation, and the interest payments are all deductible business expenses for tax purposes.

It is generally accepted that the discount rate used to find the NPV is the after-tax cost of debt, (1 − t)kd. There are several reasons for it. When a corporation is leasing an asset, it has to sign a lease contract that requires regular cash payments, and if the corporation is not making the lease payments on time, the owner has the right to repossess the asset. It is quite similar to borrowing the money to buy the asset. If the borrower does not pay the regular installment payments on time, the lender has a right to repossess the asset.

When a company is leasing an asset, it is simply borrowing it. The company has no ownership right, or an equity interest in the asset. It has to return the asset to the owner at the end of the lease in good condition. This is similar to borrowing money by selling a bond where you have to pay the interest on the principal, and pay back the full amount when the bond matures. The after-tax of cost of debt, the cost associated with a bond, should also apply to a lease.

The corporations tend to use lease financing an alternative to debt financing. Although they are not exact substitutes of one another, they play essentially similar roles. Therefore, for lease analysis, we shall use the after-tax cost of debt, (1 − t)kd as the proper discount rate.

Examples

Video 13.1 13.1. Alpha Power Corporation needs 25 new vans for their maintenance crews. The vans cost $25,000 each, and an investment tax credit of 10% is available. Alpha will depreciate the vans over a 4-year period on a straight-line basis and then sell them for $5,000 each on the average. The company can borrow money at 9% interest by making the loan payments at the end of each year for four years. The company expects to

pay income tax at 30% rate for the next 4 years. Beta Leasing Co leases the same vans at

$5,000 each annually for a 4-year period, with the payments made in advance for each year. Alpha can take the tax benefits of lease payments at the end of each year. Using NPV method, find the cheaper way to acquire the vehicles.

The best way to do the problem is to calculate the NPV of leasing and the NPV of buying a single van, compare the two, and select the cheaper one. The cheaper method is applicable to one van or 25.

In all leasing problems, we will use the after-tax cost of debt, (1 – t)kd as the proper discount rate. In this case, it is (1 – .3)(.09) = .063.

Because of the 10% investment credit, the vans will effectively cost 90% of their purchase price, or, .9(25,000) = $22,500 each. Thus,

Initial investment = −$22,500

Since the company spends only $22,500 on a van, it is able to depreciate that amount over 4 years. Thus, the depreciation per year is 22,500/4 = $5625. With the tax rate at 30%, the annual tax benefit of depreciation is .3(5625) = $1687.50. The present value of the tax benefit of depreciation for four years is

4 1687.50

1687.50(1 − 1.063−4)

PV(tax benefit of depreciation) = 

i=1

1.063i =

.063 = $5807.42

WRA Sum[.9*25000/4*.3/1.063^i,{i,1,4}]

The company will sell the van for $5000 after 4 years. Since the asset is depreciated fully, the company will pay taxes on the whole amount. The after-tax value is (1 – .3)(5000) =

$3500. Its present value of this amount is

3500

PV(resale value of equipment) = 1.0634 = $2741.16

WRA (1-.3)*5000/1.063^4

To find the NPV of buying a van, add the initial investment, the present value of the tax benefits of depreciation, and the present value of after-tax final sales value. This gives

NPV(buying) = −22,500 + 5807.42 + 2741.16 = −$13,951.42

WRA -.9*25000+ Sum[.9*25000/4*.3/1.063^i,{i,1,4}]+(1-.3)*5000/1.063^4

Now consider the cash flows due to leasing a van. The company has to make four payments at the beginning of each of the four years. The present value of this is

3 5000

PV(lease payments) = − 5000 − 1.063i = − 5000 –

i=1

5000(1 −1.063−3)

.063 = −$18,291.22

WRA -Sum[5000/1.063^i,{i,0,3}]

The tax benefit of a lease payment, tL, is .3(5000) = $1500 per year, at the end of each of the next four years. Their present value is

4 1500

PV(tax benefit of lease payments) = 1.063i =

i=1

1500(1 − 1.063−4)

.063 = $5162.15

WRA Sum[.3*5000/1.063^i,{i,1,4}]

Combining the present value of lease payments and their tax benefits, we get

NPV(leasing) = −18,291.22 + 5162.15 = −$13,129.07

WRA -Sum[5000/1.063^i,{i,0,3}]+Sum[.3*5000/1.063^i,{i,1,4}]

Comparing the NPV(buying) = −$13,951.42 with NPV(leasing) = −$13,129.07, we notice that leasing is cheaper. ♥

To set it up in Excel, create the following spreadsheet.

	A	B
1	Purchase price =	25000
2	Investment tax credit =	.1
3	Net cost =	=B1*(1-B2)
4	Cost of debt =	.09
5	Income tax rate =	.3
6	Discount rate =	=(1-B5)*B4
7	Sale price =	5000
8	Number of years =	4
9	PV of tax benefit of depreciation =	=B5*B3/B8*(1-1/(1+B6)^B8)/B6
10	PV of after-tax sales price =	=B7*(1-B5)/(1+B6)^B8
11	NPV(buy) =	=-B3+B9+B10
12	Lease payment, in advance =	5000
13	PV of lease payments =	=-B12-B12*(1-1/(1+B6)^(B8-1))/B6
14	Tax benefit of lease payments =	=B5*B12*(1-1/(1+B6)^B8)/B6
15	NPV(lease) =	=B13+B14

Video 13.2 13.2. Campbell Company has to decide between leasing a warehouse for four years at the fixed annual rent of $24,098.04, payable in advance; and buying it outright for $100,000. If Campbell buys the warehouse, it can borrow the money at 12% interest for four years. The company will depreciate the warehouse on the straight-line basis for 10 years, but actually sell it for $30,000 after 4 years. The marginal tax rate of the company is 35%. Assume that the tax benefits of leasing are not available until the end of that year. Should Campbell buy or lease the warehouse?

The after-tax cost of debt, the discount rate, is 0.12(1 − 0.35) = 0.078.

The depreciation of the warehouse is $10,000 annually, and therefore its tax benefit is

.35(10,000) = $3500. The book value of the warehouse after 4 years is $60,000. The company plans to sell it for $30,000, which will create a loss of $30,000. But this loss also has a tax benefit of 30,000(0.35) = $10,500. The final sale brings in $30,000 in cash and $10,500 in tax benefits for a total of $40,500.

The NPV of the purchase decision is thus

4 3500

40‚500

NPV(buy) = −100,000 + 1.078i + 1.0784 = − $58,365.52

i=1

Assume that the company makes the lease payments at the beginning of each year, but their tax benefits are not available until the end of that year, the NPV of leasing is,

3 24‚098.04

4 24‚098.04(.35)

NPV(lease) = − 24,098.04 − 

i=1

1.078i + 

i=1

1.078i

= − 24,098.04 − 62,327.79 + 28,060.33 = − $58,365.50

Since the two costs are exactly alike, the company is indifferent between leasing and buying the warehouse. ♥

You can use WolframAlpha to verify the answer as follows:

-100000+Sum[3500/1.078^i,{i,1,4}]+40500/1.078^4

-Sum[24098.04/1.078^i,{i,0,3}]+Sum[.35*24098.04/1.078^i,{i,1,4}]

To do the problem in Excel, type in the following instructions:

	A	B
1	Buying
2	Initial cost	100000
3	Depreciation period	10
4	Holding period	4
5	Tax rate	.35
6	Cost of debt	.12
7	Discount rate	=(1-B5)*B6
8	PV of tax benefit of depreciation	=B2/B3*B5*(1-1/(1+B7)^B4)/B7
9	Book value	=B2-B4*B2/B3
10	Sales price	30000
11	PV of sales price	=(B10-(B10-B9)*B5)/(1+B7)^B4
12	NPV of buying	=-B2+B8+B11
13	Leasing
14	Lease payments, given	24098.04
15	PV of lease payments	=-B14-B14*(1-1/(1+B7)^(B4-1))/B7
16	PV of their tax benefits	=B14*B5*(1-1/(1+B7)^B4)/B7
17	NPV of leasing	=B15+B16

13.3. Altair Corporation is considering the alternatives of buying or leasing a new machine. It may buy the machine for $1 million, use it for five years, and then sell it for

$100,000. Altair will depreciate the machine completely over this period. Altair's after- tax cost of debt is 12% and its tax rate 30%. The lease will be for five years, with equal payments made in advance each year, with the tax benefits available after one year. What lease payment will make the cost of leasing equal to the cost of buying?

The after-tax cost of debt for the company is 12%, which is the proper discount rate. Next, consider the net cost of buying the machine. The cost of the machine is $1 million and the annual depreciation is $200,000. The annual tax benefit from the depreciation will be .3(200,000) = $60,000. The final sale price is $100,000, but after taxes it becomes 100,000(1 − .3) = $70,000. Combining these numbers, we get

5 60‚000

70‚000

NPV(buy) = − 1,000,000 + 

i=1

1.12i +

1.125 = − $743,993.55

Suppose the lease payments are x dollars annually, payable in advance, and their tax benefits, .3x, are available at the end of each year. Thus

4 x

5 .3x

NPV(lease) = − x − 1.12i + 1.12i

i=1

i=1

( [ ) ( −4 )= x − 1 − 1 − 1.12

.12

.3(1 − 1.12−5

+ .12

)] = −2.955916486x

Equate the two costs, as This gives

− 743,993.55 = − 2.955916486x x = $251,696.40 ♥

Thus, the annual lease payments are $251,696, in advance, which will equate the cost of leasing to the cost of buying this machine.

You can use WolframAlpha to verify the answer as follows:

-1000000+Sum[.3*1000000/5/1.12^i,{i,1,5}]+100000*(1-

.3)/1.12^5=-Sum[x/1.12^i,{i,0,4}]+Sum[.3*x/1.12^i,{i,1,5}]

13.4. Dallas Corporation has to decide between buying and leasing a computer. If Dallas buys the computer, it will cost $200,000, but an investment tax credit of 10% is also available. Dallas will depreciate the computer using a 5 year ACRS, with depreciation of 18%, 33%, 25%, 16% and 8% for the five years respectively. After 5 years, Dallas expects to sell the computer for $20,000. The tax rate of the company is 40% and its after-tax cost of debt is 11%. King Leasing Co will lease the same computer to Dallas for 5 years, but will charge the annual lease payments in advance every year. Dallas will get

the tax benefits immediately. Find the lease payment, which will make Dallas indifferent towards leasing or buying.

Let us first find the net cost of buying the computer. The purchase price is $200,000, the investment tax credit is $20,000, and so the net initial investment is $180,000.

The PV of tax benefits due to depreciation

0.18



0.33

0.25

0.16

0.08 

5= + $55,128.13

= 0.4 (180,000) 1.11 + 1.112 + 1.113 + 1.114 + 1.11 

The PV of after-tax proceeds of the sale =

20‚000 (1 −0.4)

1.115 = + $7,121.42

Adding these, we get the NPV(buy) = −180,000 + 55,128.13 + 7,121.42 = − $117,750.45

Suppose the lease payments are X dollars, payable in advance, and the tax benefits are also available at the same time. This gives

4 (1−.4)X

.6X(1 − 1.11−4)

NPV(lease) = − (1 − .4)X − 

i=1

1.11i = − .6X −

.11 = − 2.4614674 X

Equating the two costs, we get

− 2.4614674 X = − 117,750.45

X = the annual lease payment = $47,838 ♥

You can use WolframAlpha to verify the answer as follows:

.4*180000*(.18/1.11+.33/1.11^2+.25/1.11^3+.16/1.11^4+.08/1.11^5)+ 20000*(1-.4)/1.11^5-200000*(1-.1)=-Sum[(1-.4)*x/1.11^i,{i,0,4}]

13.5. Titan Trucking Company needs 10 new trucks. Each truck costs $45,000, lasts on the average 6 years, with no residual value. Titan depreciates the trucks on a straight-line basis. A 10% investment tax credit is also available. If it buys the trucks, Titan has to pay

$1,000 annually per truck for maintenance. Titan has tax rate of 40%. Hyperion Leasing Company will lease these trucks to Titan at an annual cost of $10,000 per truck, payable in advance. Titan can buy the trucks by borrowing money at the rate of 15% per annum. What is your recommendation to Titan whether to buy or to lease the trucks?

Let us calculate the result for one truck. Because of the 10% investment tax credit, the net cost of each truck is .9(45,000) = $40,500. The proper discount rate in this case is the after-tax cost of debt = (1 − 0.4)(0.15) = 0.09. We find the NPV as follows:

6 1000(1 − 0.4)

6 (40,500/6)(.4)

NPV(buy) = − 40,500 − 

i=1

1.09i + 

i=1

1.09i = − $31,079.57

Investment in the truck

PV of after-tax cost of maintenance

PV of tax benefits of depreciation

Assuming that the lease payments are payable in advance each year and the tax benefits are available at the end of the year, we get

5 10‚000

6 .4(10,000)

NPV(lease) = − 10,000 − 

i=1

1.09i + 

i=1

1.09i = − $30.952.80

First lease payment

PV of remaining five payments

PV of tax benefits of lease payments

Comparing the net cost of leasing and buying, we see that it is better to lease the trucks. ♥

You can use WolframAlpha to verify the answer as follows:

-40500-Sum[1000*.6/1.09^i,{i,1,6}]+Sum[40500/6*.4/1.09^i,{i,1,6}]

-Sum[10000/1.09^i,{i,0,5}]+Sum[.4*10000/1.09^i,{i,1,6}]

13.6. Dalton Company needs a new computer, which it may buy for $140,000. It will depreciate it on a straight-line basis over 7 years to zero value, and then sell for $10,000. Alternately, Dalton may lease the same computer for 7 years with annual lease payments of $25,000 payable in advance. Dalton can take the tax credit of lease payments immediately. The tax rate of Dalton is 35%, and its pre-tax cost of debt 10%. Should it buy or lease?

Using 6.5% as the after-tax cost of debt, set up the problem as follows:

7 20‚000(.35)

10‚000(1 −0.35)

NPV(buy) = − 140,000 + 

i=1

1.065i +

1.0657 = − $97,425.57

Initial investment

PV of tax benefit of depreciation

PV of after-tax resale value

6 25‚000(1−.35)

NPV(lease) = − 25,000(1 − .35) − 

i=1

1.065i = − $94,916.47

PV of first lease

payment, after tax

PV of remaining six

payments, after taxes

Comparing the cost of buying and leasing, we see that leasing is the better alternative. ♥

13.7. James Corporation needs a corporate jet for the next four years. It can buy the jet for $15 million. The company will depreciate the jet on a straight-line basis over the next 15 years, but sell it for $11 million after four years. The cost of debt for the company is 12% and its income tax rate 25%. Calculate the annual lease payment, payable in advance

each year, which will make the cost of leasing equal to the cost of buying. Assume that the tax benefits of the lease payments are available immediately.

The proper discount rate in this problem is (1 − t)kd = (1 − .25)(.12) = .09. The annual depreciation is $1 million and its tax benefit tD is $.25 million. Finally, the jet is sold for its book value, $11 million, and there is no payment of taxes. With amounts in $million,

4 .25

11

NPV(buy) = − 15 + 1.09i + 1.094 = −$6.397392709 million

i=1

Suppose the lease payment is L and its tax benefit is available immediately, then after taxes, it becomes (1 − .25)L = .75L. Thus

3 .75L

NPV(lease) = − .75L − 1.09i = −$2.648470999L million

i=1

Equating the two costs,

−$2.648470999L = −$6.397392709

which gives L = 2.415504157 = $2.4155 million ♥

You can use WolframAlpha to verify the answer as follows:

-15+Sum[.25/1.09^i,{i,1,4}]+11/1.09^4=-Sum[.75*L/1.09^i,{i,0,3}]

To simplify the leasing problems, we use the words immediately or right away to indicate that if the annual lease payment L is made at time t = 0, its tax benefit tL is also calculated at t = 0. This is the case in Examples 13.4, 13.6, and 13.7.

Similarly, the terms a year later, or at the end of the year signify that if the annual lease payment L is made at time t = 0, its tax benefit tL is available at t = 1. This happens frequently, as in Examples 13.1, 13.2, 13.3, and 13.5.

It is also possible to do the calculation twice, once with immediate benefits and once with delayed benefits. The average of these values will give a more realistic result.

We do this to simply the calculations. In real life, the lease payments are on a monthly basis and their cumulative tax benefit is available after several months. For example, a company may lease a machine in November 2011, pay $2000 fee in November 2011, and

$1000 monthly lease payments in November and December 2011. The total lease-related payments are $4000 in 2011. Suppose the income tax rate of the company is 30%. The company can claim the tax benefit = .3(4000) = $1200 in April 2012 when it files its income tax return.

14. INVESTMENT ANALYSIS

Objective: After reading this chapter, you will be able to analyze investment opportunities, particularly the real-estate investments.

14.1 Real Estate Investments

In this chapter, we shall look at some of the investment opportunities that present themselves to the corporations and individuals alike. We analyze the situations with the help of a powerful tool, the NPV analysis. The desirability of an investment depends on whether its NPV is positive or not. These examples provide an overview of the investment process.

Investing in real estate is quite popular. Many people buy a house, live in it comfortably, and then sell it at an appreciated price. We will look at real estate as an investment opportunity. First, consider a simple example where an investor buys a house, rents it out for a while, and then sells it at a profit.

To examine this situation analytically, assume that the purchase price of the house is H. Its selling price after n years is Hn. The profit, Hn − H, is taxable income, and the tax due at the time of sale is (Hn − H)t, where t is the income tax rate. The cash flow at the time of sale is thus Hn − (Hn − H)t. = Hn(1 − t) + Ht. Suppose the risk-adjusted discount rate for such an investment is r. Then the NPV of the investment is

( 233 )

NPV = − H +

Hn (1−t) + Ht

(1 + r)n (14.1)

Equation (14.1) is incomplete because it does not consider depreciation, rental income, or maintenance expenses. We can make the problem more realistic by renting the house at the annual rent R, and include the annual maintenance costs M. Assume that R and M are calculated at the end of the year. The maintenance expenses may also cover the real estate taxes. The net rental income R − M is taxable income, and its annual after-tax value is (R − M)(1 − t). At the same time, the homeowner can use the depreciation of the house as a tax deduction and create an annual benefit of tD, where D is the annual depreciation. The annual cash flow, C is thus

C = (R − M)(1 − t) + tD.

When the investor sells the house, the capital gain on the sale is the selling price minus the book value of the house. The book value of the house is given by H − nD. The after- tax cash flow from the sale of the house is thus

Hn − (Hn − H + nD)t

Including rental income, maintenance, and depreciation, (14.1) becomes

( A n al y t i ca l T ech ni q u e s ) ( 1 4 . I n v e s t m e n t A n a l y s i s )

n (R −M)(1−t) + tD

Hn −(Hn −H + nD)t

NPV = − H + 

i=1

(1 + r)i +

(1 + r)n (14.2)

We have assumed that the depreciation is on a straight-line basis and the income tax rate for the ordinary income and the capital-gains income is the same. We can make the model more complete by assuming a different tax rate t for ordinary income and tg for capital gains. This will make (14.2) to be

n (R −M)(1−t) + tD

Hn −(Hn −H + nD)tg

NPV = − H + 

i=1

(1 + r)i +

(1 + r)n (14.3)

We should modify equation (14.2) if the annual rents are available in advance every year. It will then become

n−1 R

n (M + D −R)t −M

Hn −(Hn −H + nD)t

NPV = − H + (1 + r)i + 

(1 + r)i +

(1 + r)n (14.4)

i=0

i=1

Next, we consider two other important costs: the closing costs when we buy the house and the selling costs when we sell it. The closing costs include the transfer taxes, attorney's fees, title insurance, loan origination fees, "points," document preparation fees, and deed recording fees. The bank may charge a fee, called points, when it approves a loan. It is usually between 0% and 3% of the amount of loan. In many places, the buyer and the seller pay the transfer taxes to the local government. At present, the transfer tax rate in Scranton, PA, is 2%, both for the buyer and the seller.

Let us consider a comprehensive problem about real estate investments. It is instructive to follow the details of the problem and to modify it to solve simpler problems.

Purchase price of the house = $150,000 (land = $30,000, building = $120,000) Depreciation schedule = 25 years, on a straight line

Initial rent = $1200 at the end of each month, expected to increase by 5% annually Maintenance, including real estate taxes = $300 per month, expected to increase by 4% per year

Expected rate of appreciation of the value of property = 5% per year Plan to sell the property after 10 years

Borrow 80% of the value of the house and pay 2 points Fixed closing costs at the beginning of the project = $335 Transfer tax rate = 1.85%

Fixed costs at the time of selling of the house = $100 Realtor's commission = 6%

Ordinary income tax rate = 28% Capital gains tax rate = 14%

Risk-adjusted discount rate = 10%

Is it a worthwhile investment?

Divide the problem into the following simpler problems.

(1) What is the initial investment?

(2) What is the present value of the tax benefits of depreciation?

(3) What is the present value of rental income, including taxes and inflation?

(4) What is the present value of expenses, including taxes and inflation?

(5) What is the present value of final sales price, including taxes and inflation?

(1) To find the initial investment, consider the following expenses:

(a) urchase price of the house, $150,000

(b) Payment of transfer taxes, .0185(150,000) = $2775 (c) “Points” paid to the bank, .02(.8)(150,000) = $2400

(d) Fixed cost at the time of purchase of the house (title insurance, lawyers fee, deed recording fee, etc.) = $335

Adding these numbers, we get 150,000 + 2775 + 2400 + 335 = $155,510. The expenses associated with buying the house, $5510 are tax deductible. Their tax benefit, available at the end of the first year, is .28(5510) = $1542.80. Its present value = 1542.80/1.1 =

$1402.55. Subtracting it from the total expenses of buying the house, we get 155,510 −

1402.55 = $154,107.45, which is the initial investment. (Negative cash flow) ♦

Let us define: H = purchase price of the house

tt = transfer tax rate

p = points paid to the bank. For 2 points, p = .02

γ = loan-to-value ratio; the amount of mortgage loan divided by the property value

F1 = fixed closing costs at the time of buying the property

r = risk-adjusted discount rate used to discount all cash flows

t = income tax rate

Assume that the tax benefit of the expenses occurs at the end of the year. This gives the first cash flow C1 as

C1 = − [H(1 + tt + pγ) + F1] +

t [H(tt + pγ) + F1]

(1 + r) (14.5)

Using Maple, one can write it as

C1:=-(H*(1+tt+p*gamma)+F1)+t*(H*(tt+p*gamma)+F1)/(1+r); subs(H=150000,tt=.0185,p=.02,gamma=.8,F1=335,r=.1,t=.28,C1);

(2) Only the building can be depreciated for tax purposes, not the land. The depreciation per year is 120,000/25 = $4800. Its tax benefit is .28*4800 = $1344. The present value of this amount over the 10-year holding period, discounted at the rate 10%, is

10 1344

1.10i = $8258.30 (Positive cash flow) ♦

i=1

Using the following symbols,

L = value of the land

N = number of years for depreciation

n = holding period of the property

one can write C2 as

n t (H −L)/N

t(H − L)[1 − (1 + r)−n]

C2 = 

i=1

(1 + r)i =

rN (14.6)

In Maple notation, it becomes,

C2:=t*(H-L)*(1-1/(1+r)^n)/r/N; subs(t=.28,H=150000,L=30000,r=.1,n=10,N=25,C2);

(3) Consider only the first year’s rent. The monthly rent is $1200 and the monthly discount rate is .1/12. The present value of the first-year rent is thus

12 1200

(1 + .1/12)i = $13,649.41

i=1

The total rent for the first year is 12(1200) = $14,400. Since the taxes are paid at the end of the year, they are .28(14,400) = $4032. The present value of the taxes is 4032/1.1 =

$3665.45.

The present value of the first year rental income, after taxes, is 13,649.41 − 3665.45 =

$9,983.96.

The rental income will go up by 5% annually for several years, and it will be discounted by 10% annually. The present value of 10-year rental income, including taxes, and inflation, is thus

9,983.96 + 9,983.96(1.05/1.1) + 9,983.96(1.05/1.1)2 + ... 10 terms

This is a geometric series, with a = 9,983.96, x = 1.05/1.1, and n = 10. Do the summation by using (1.4).

9‚983.96[1 − (1.05/1.1)10]

1 − 1.05/1.1 = $81,706.63 (positive cash flow) ♦

Using the following symbols,

R = rent per month, collected at the end of the month

fr = annual rent inflation

we can write C3 as

12 R

(  ) (    ) (  )12tR 1−[(1 + fr)/(1 + r)] n

C3 = 

i=1

(1 + r/12)i −

1 + r

1 − (1 + fr)/(1 + r) 

(    )1 − 1/(1 + r/12)12

( n ) (  )t 1 − [(1 + fr)/(1 + r)] 

( 3 )Or, C = 12R 



r − 1 + r

1 − (1 + fr)/(1 + r) 

(14.7)

In Maple notation, it becomes,

12*R*((1-1/(1+r/12)^12)/r-t/(1+r));

C3:=%*(1-((1+fr)/(1+r))^n)/(1-(1+fr)/(1+r)); subs(R=1200,r=.1,t=.28,fr=.05,n=10,C3);

(4) The calculation of the expenses proceeds in a similar way. The present value of the first-year expenses is

12 300

(1 + .1/12)i = $3412.35

i=1

The annual tax benefit due to expenses is .28(3600) = $1008, and its present value is 1008/1.1 = $916.36. The present value of the first-year expenses, after taxes, is 3412.35 − 916.36 = $2495.99.

Use an inflation rate of 4%, and discount rate of 10% to find the present value of the expenses for ten years as

2495.99 + 2495.99(1.04/1.1) + 2495.99(1.04/1.1)2 + ... 10 terms

Use (1.4) again to find the sum as

2495.99[1 − (1.04/1.1)10]/(1 − 1.04/1.1) = $19,644.76 (Negative cash flow) ♦

Defining the quantities,

M = monthly expenses, at the end of each month

fm = annual inflation rate for expenses

one can write C4 as,

12 M

(  ) 12tM 1−[(1 + fm)/(1 + r)] n

C4 = − 

i −  

i=1

(1 + r/12)

(1 + r)1 − [(1 + fm)/(1 + r)] 

(    )1 − 1/(1 + r/12)12

( n ) (  )t 1 − [(1 + fm)/(1 + r)] 

( 4 )Or, C = − 12M 



r − (1 + r)

1 − (1 + fm)/(1 + r) 

(14.8)

In Maple notation, it is

-12*M*((1-1/(1+r/12)^12)/r-t/(1+r)); C4:=%*(1-((1+fm)/(1+r))^n)/(1-(1+fm)/(1+r)); subs(M=300,r=.1,t=.28,fm=.04,n=10,C4);

(5) Assume that the property values are increasing at the rate of 5% per year. After 10 years, the selling price of the house is 150,000(1.05)10 = $244,334.19. The realtor will take 6% of this amount, namely, .06(244,334.19) = $14,660.05. Since the transfer taxes are applied to both the buyer and the seller, you have to pay transfer tax again, which

amounts to .0185(244,334.19) = $4520.18. You pay another $100 as fixed costs at selling.

After paying the realtor, the county transfer taxes, and fixed cost, you get 244,334.19 − 14660.05 − 4520.18 − 100 = $225,053.96. Defining,

fp = rate of appreciation of property values

one could write it as

H(1 + fp)n(1 − c − tt) − F2

In Maple notation, it looks like this

H*(1+fp)^n*(1-c-tt)-F2; subs(H=150000,fp=.05,n=10,c=.06,tt=.0185,F2=100,%);

The total amount of depreciation for ten years is 10(4800) = $48,000. The book value of the house is thus 150,000 – 48,000 = $102,000. The capital gain on the house is 225,053.96 – 102,000 = 123,053.96. The tax on the capital gain is .14(123,053.96) =

$17,227.55.

−{H(1 + fp)n(1 − c − tt) − F2 – [H – n(H – L)/N]}tg

-(H*(1+fp)^n*(1-c-tt)-F2-(H-n*(H-L)/N))*tg; subs(H=150000,fp=.05,n=10,c=.06,tt=.0185,%); subs(F2=100,L=30000,N=25,tg=.14,%);

The selling expenses on the house (realtor’s fee, transfer taxes, and fixed amount) add up to 14660.05 + 4520.18 + 100 = $19,280.23. Using this expense as a deduction against ordinary income, its tax benefit is .28(19,280.23) = $5398.47.

[H(1 + fp)n(c + tt) + F2]t

(H*(1+fp)^n*(c+tt)+F2)*t; subs(H=150000,fp=.05,n=10,c=.06,tt=.0185,F2=100,t=.28,%);

After paying taxes, the cash from selling the house is 225,053.96 − 17,227.55 + 5398.47

= $213,224.87. The present value of this sum is 213,224.87/1.110 = $82,207.42 (positive cash flow) ♦

C5 = [H(1 + fp)n(1 − c − tt) − F2 −{H(1 + fp)n(1 − c − tt) − F2 – [H – n(H – L)/N]}tg

+ [H(1 + fp)n(c + tt) + F2]t]/(1 + r)n (14.9)

In Maple notation, it becomes

H*(1+fp)^n*(1-c-tt)-F2;

%-(%-(H-n*(H-L)/N))*tg;

%+(H*(1+fp)^n*(c+tt)+F2)*t;

C5:=%/(1+r)^n; subs(H=150000,fp=.05,c=.06,tt=.0185,F2=100,n=10,L=30000,%); subs(N=25,tg=.14,r=.1,t=.28,%);

Adding all five cash flows, marked by a red diamond ♦, we get

NPV = −154,107.45 + 8258.30 + 81,706.63 – 19,644.76 + 82,207.42 NPV = −$1,579.87

In Maple notation, it is

NPV:=C1+C2+C3+C4+C5;

Considering the NPV, it is not a profitable project.

Examples

14.1. Suppose your marginal income-tax rate is 25%. Assume that 60% of the capital gains and the first $200 in dividends are tax exempt. The interest paid on borrowed funds and the transaction costs are tax deductible. On November 1, 2007, you borrow $8,000 from the bank at 7.5% interest rate. You already have $10,000 of your own funds. With this $18,000, you buy 100 shares of IBM at 90, with a dividend of $1.20 per share. You invest the remaining funds in PP&L 8s2018 bonds at 90. On November 1, 2008, you liquidate your portfolio, IBM at 100, and PP&L bonds at 96, and pay off the bank loan. The total transaction costs are $20 for stock and $50 for bonds. Calculate the after-tax rate of return on your own funds.

Consider the stock investment first. Money invested in IBM stock = $9000. Capital gain on IBM, including transaction costs = 10,000 − 9000 − 20 = $980. Since 60% of the capital gain is tax-exempt, the tax is due on 40% of the gain. Thus tax due = .4(980)(.25) = $98

Dividends received on IBM stock = $120 (all tax exempt)

After-tax capital gain on stock, plus dividends = 980 + 120 − 98 = $1002 (A)

You bought the bonds at 90, that is, 90% of their face value. Thus, you buy $10,000 face value of bonds for $9000. You sell the bonds at 96, meaning, 96% of their face value. Capital gain on bonds, including transaction cost = 10,000(0.96 − 0.90) − 50 = $550.

The tax due = .4(550)(.25) = $55.

The after-tax capital gain on bonds = 550 – 55 = $495 (B)

Interest received on bonds = 0.08 (10,000) = $800 Interest paid to the bank = 0.075 (8,000) = $600 Net interest income = 800 − 600 = $200

Tax due on interest income = 200(0.25) = $50 After-tax interest income = 200 − 50 = $150 (C)

Adding (A), (B), and (C), the total dollar return = 1002 + 495 + 150 = $1647

Overall percentage rate of return on your own investment = 1647/10,000 = 16.47% ♥

14.2. You are looking into the possibility of operating a Laundromat. You will rent a storefront and buy the washing machines and dryers. You expect to generate $20,000 annually in revenue. The rent of the building is $3,000 annually, payable in advance for the next five years. You will depreciate the machines on a straight-line basis over 5 years, with no resale value. You are in a 30% tax bracket, and your after-tax required rate of return is 12%. Assuming that the cash inflows occur at the end of the year, how much should you pay for the equipment?

To simplify the problem, assume that other expenses such as electricity and water are zero. To break even, the NPV of this investment is zero, arranged as follows:

	Action	Present value	Equals
1	Buy the Laundromat	−x	−x
2	PV of after-tax cash from machines	5 20‚000(1−.3)  1.12i i=1	50,466.86683
3	PV of rent payments, in advance	4 3000 − 3000 − 1.12i i=1	−12,112.04804
4	PV of tax benefit of rent	5 3000(.3)  1.12i i=1	3,244.298582
5	PV of tax benefit of depreciation	5 (x/5)(.3)  1.12i i=1	.2162865721 x
6	NPV	Sum of the above	= 0

Write the numbers in the last column as an equation,

−x + 50466.86683 − 12112.04804 + 3244.298582 + .2162865721 x = 0

Simplify it to get

.7837134279 x = 41599.11737

This gives x = 53,080. You should pay $53,080 for the equipment. ♥

You may do the problem on WolframAlpha as follows.

-x+Sum[20000*(1-.3)/1.12^i,{i,1,5}]-

Sum[3000/1.12^i,{i,0,4}]+Sum[3000*.3/1.12^i,{i,1,5}]+Sum[x/

5*.3/1.12^i,{i,1,5}]=0

To do the problem using Maple, enter the following instructions:

-I0;

%+sum(20000*(1-.3)/1.12^i,i=1..5);

%-sum(3000/1.12^i,i=0..4);

%+sum(3000*.3/1.12^i,i=1..5);

%+sum(I0/5*.3/1.12^i,i=1..5)=0;

solve(%);

To do the problem using Excel, set up the spreadsheet as follows. Adjust the value of the initial investment in cell B2 until the NPV in cell B15 becomes close to zero.

	A	B	C	D	E	F	G
1	Time, years	0	1	2	3	4	5
2	Initial investment, $	53080
3	Project life, years	5
4	Cost of capital	0.12
5	Income tax rate, t	0.3	1 − t	=1-B5
6	Cash from machines, $		20000	=C6	=C6	=C6	=C6
7	After-tax cash from machines, $		=C6*D5	=D6*D5	=E6*D5	=F6*D5	=G6*D5
8	Annual rent, $	3000	=B8	=B8	=B8	=B8
9	Depreciation for year		=B2/B3	=B2/B3	=B2/B3	=B2/B3	=B2/B3
10	Tax benefit of depreciation		=B5*C9	=B5*C9	=B5*C9	=B5*C9	=B5*C9
11	Tax benefit of rent		=B5*B8	=B5*B8	=B5*B8	=B5*B8	=B5*B8
12	Total cash flow	=-B2-B8	=C7- C8+C10+C11	=D7- D8+D10+D11	=E7- E8+E10+E11	=F7- F8+F10+F11	=G7+G10+G11
13	Discount factor	1	=1/(1+B4)	=1/(1+B4)^2	=1/(1+B4)^3	=1/(1+B4)^4	=1/(1+B4)^5
14	PV of cash flows	=B12*B13	=C12*C13	=D12*D13	=E12*E13	=F12*F13	=G12*G13
15	NPV	=SUM(B14:G14)

Video 14.3 14.3. Elbridge Gerry is planning to buy a house and rent it out for the next 5 years, collecting an annual rent of $6,000 in advance each year. He is in the 22% tax bracket. He will depreciate the house on a straight-line basis for 25 years. Gerry plans to sell the house after 5 years at a price that will be 20% higher than the purchase price, taking the profit as a long-term capital gain. Assume 60% of such capital gains are tax exempt. The after tax cost of capital for Gerry is 8%. Ignore the maintenance expenses of the house and real estate taxes. How much should Gerry pay for the house to break even?

Suppose the purchase price of the house is H. We can find the NPV by considering these factors and their respective present values:

(1) The initial investment = − H

4 6000

(2) PV of rents, in advance annually = 6000 + 1.08i = $25,872.76

i=1

(3) Assume that the taxes are due at the end of each year at the rate of 22% of rents.

5 6000(.22)

PV of taxes paid on the rental income = − 

i=1

1.08i = − $5270.38

(4) The annual depreciation is H/25.

5 (H/25)(.22)

The PV of tax benefits of depreciation = 

i=1

1.08i = .03514 H

(5) The sale price of the house is 1.2H. Gerry has already taken 20% of the depreciation of the house leaving its book value to be .8H. The profit on the sale is (1.2H − .8H). Only 40% of it is taxable at the rate of 22%.

PV of after-tax sales price =

1.2H −(1.2H −.8H)(.4)(.22)

1.085 = .7927433078 H

NPV is the sum of all these figures, and to break even, it should be zero. Thus

− H + 25872.76104 − 5270.377249 + .03513584833 H + .7927433078 H = 0

Solving for H, we have H = $119697.2041, or approximately, $119,700 ♥

We can also do the problem by using (14.4)

n−1 R

n (M + D −R)t −M

Hn −(Hn −H + nD)t

NPV = − H + (1 + r)i + 

(1 + r)i +

(1 + r)n (14.4)

i=0

i=1

We have to modify the equation slightly to accommodate the following changes:

(1) There are no maintenance expenses, which makes M = 0.

(2) The tax benefit of depreciation is available at the end of year 1-5. He pays the taxes on rental income at the end of the year.

(3) Since 60% of capital gain is tax exempt, we should take 40% of the capital gain, and pay taxes on it at the rate of 22%.

Incorporating these changes, equation (14.3) becomes

n−1 R

n (D −R)t

Hn −(Hn −H + nD)(.4)t

NPV = − H + (1 + r)i + 

(1 + r)i +

(1 + r)n

i=0

i=1

Now, put n = 5 years, R = $6000, t = .22, r = .08, Hn = 1.2H, D = H/25, and NPV = 0. This gives

4 6000

5 .22(H/25 − 6000)

1.2H − (1.2H − H + 5H/25)(.4)(.22)

− H + 1.08i + 

1.08i +

1.085 = 0

i=0

i=1

The WolframAlpha instruction to solve the equation is as follows.

-H+Sum[6000/1.08^i,{i,0,4}]+Sum[.22*(H/25-

6000)/1.08^i,{i,1,5}]+(1.2*H-(.2H+H/5)*.4*.22)/1.08^5=0

The result is $119,697.

To do the problem on Excel, set up the spreadsheet as follows. Adjust the value of the house in cell B2 until the NPV in cell B18 is almost zero.

	A	B	C	D	E	F	G
1	Time, years	0	1	2	3	4	5
2	Purchase price of house, $	119700	Depreciable life, years	25	Price appreciation	0.2
3	Project life, years	5
4	Cost of capital	0.08
5	Income tax rate, t	0.22	1 − t	=1-B5
6	Rent, $	6000	=B6	=B6	=B6	=B6
7	Tax on rent, $		=B5*B6	=B5*B6	=B5*B6	=B5*B6	=B5*B6
8	After-tax rent, $	=B6	=C6-C7	=C6-C7	=C6-C7	=C6-C7	=-C7
9	Depreciation for year		=B2/D2	=B2/D2	=B2/D2	=B2/D2	=B2/D2
10	Tax benefit of depreciation		=B5*C9	=B5*C9	=B5*C9	=B5*C9	=B5*C9
11	Sale price of house, $						=(1+F2)*B2
12	Book value of house						=B2-B3*B2/25
13	Capital gain						=G11-G12
14	Tax on capital gain						=0.4*B5*G13
15	Total cash flow	=-B2+B8	=C8+C10	=C8+C10	=C8+C10	=C8+C10	=G8+G10+G11- G14
16	Discount factor	1	=1/(1+B4)	=1/(1+B4)^2	=1/(1+B4)^3	=1/(1+B4)^4	=1/(1+B4)^5
17	PV of cash flows	=B15*B16	=C15*C16	=D15*D16	=E15*E16	=F15*F16	=G15*G16
18	NPV	=SUM(B17:G17)

To do the problem using Maple, we let

-H+sum(6000/1.08^i,i=0..4);

%-sum(.22*6000/1.08^i,i=1..5);

%+sum(.22*H/25/1.08^i,i=1..5);

%+(1.2*H-(1.2*H-.8*H)*.4*.22)/1.08^5=0;

solve(%);

14.4. Ralph Boston plans to buy a house and rent it out for a period of 5 years collecting an annual rent of $6,000 in advance each year. He will sell the house after five years and he believes that the price of the house will appreciate at the compound rate of 6% per annum. The maintenance and real estate taxes, paid at the end of each year, will be

$1,000 annually. Boston's income tax rate is 30%, and his after-tax cost of capital is 12%. He will depreciate the house on a straight-line basis over a 20-year period. How much should he pay for the house so that the NPV of this project is $5,000?

Suppose the purchase price of the house is H.

Increasing at the rate of 6% annually, its value after 5 years will become 1.065H = 1.338225578H.

The accumulated depreciation for 5 years will be 5(H/20) = .25H. The book value of the house will be H − .25H = .75H.

The capital gain on the house will be (1.338 − .75)H = .5882H. The tax on this capital gain will be .3(.5882H) = .1765H.

The after-tax proceeds from the sale will be (1.338 − .1765)H = 1.162H, with present value = (1.162/1.125)H = .6592H. Write the cash flows in a table:

	Action	Present value of cash flow	Equal to
1	Buy the house	−H	−H
2	Tax benefit of depreciation	5 .3H/20 1.12i i=1	.05407H
3	Rent for 5 years, in advance	4 6000 6000 + 1.12i i=1	24,224
4	Tax on rental income	5 .3(6000) −  1.12i i=1	-6489
5	Maintenance expenses, after taxes	5 (1000)(1−.3) −  1.12i i=1	-2523
6	Sell the house	1.065H − [1.065H − (H − .25H)](.3) 1.125	.6592H
7	Required NPV	Sum of the above	5000

Adding (1) through (7) and equating it to (8), we get

− H + .05407H + 6000 + 18224 − 6489 − 2523 + .6592H = 5000

Solve for H, H = $35,617.70 ♥

The WolframAlpha instruction to solve the equation is as follows.

-H+Sum[6000/1.12^i,{i,0,4}]+ Sum[(.3H/20-.3*6000-1000*(1-

.3))/1.12^i,{i,1,5}]+(1.06^5-(1.06^5-(1-

.25))*.3)*H/1.12^5=5000

The Maple code to solve the problem is as follows.

-H+sum(.3*H/20/1.12^i,i=1..5)+6000+sum(6000/1.12^i,i=1..4);

%-sum(.3*6000/1.12^i,i=1..5)-sum(1000*(1-.3)/1.12^i,i=1..5);

%+(H*1.06^5-.3*(H*1.06^5-.75*H))/1.12^5=5000;

solve(%);

14.5. Hart Co intends to buy an office building for $2 million and depreciate it on a straight-line basis over a 20-year period. However, Hart plans to sell the building after 5 years for $2.5 million. The tax rate of the company is 30% and its WACC is 8%. What is the minimum acceptable rental income (after paying expenses) calculated at the end of each year? The capital gains are fully taxable.

Write the cash flows in $million. Include the initial investment, PV of tax benefit of depreciation, PV of after-tax rental income, PV of after-tax resale value of building. To break even, set the NPV equal to zero. Arrange the items in a table.

	Action	Present value of cash flow	Equal to
1	Buy the building	−2	−2
2	Tax benefit of depreciation	5 .3(2/20)  1.08i i=1	.1197813011
3	After-tax rental income	5 R (1−.3)  1.08i i=1	2.794897026 R
4	Sell the building	2.5−[2.5−(2−5*2/20)](.3) 1.085	1.497283033
5	NPV	Sum of the above	0

− 2 + .1197813011 + 2.794897026 R + 1.497283033 = 0

This gives R = .1370124418 = $137,012 annually ♥

The WolframAlpha instruction to solve the equation is as follows.

-2+Sum[(.3*2/20+R*(1-.3))/1.08^i,{i,1,5}]+(2.5-(2.5-(2- 5*2/20))*.3)/1.08^5=0

The Maple code for this problem is as follows:

-2+sum(2/20*.3/1.08^i,i=1..5);

%+sum(R*(1-.3)/1.08^i,i=1..5);

%+(2.5-(2.5-2+5*2/20)*.3)/1.08^5=0;

solve(%);

14.6. Binghamton Company is planning to buy ATM machines and install them nationwide inside supermarkets. Each machine costs $6,000 and it will be depreciated on a straight-line basis over 5 years, although the useful life of each machine is expected to be 10 years. Binghamton will charge the users $1 per withdrawal. The company expects the expenses to be as follows: annual rent to the supermarket $3,000, payable in advance each year; maintenance, insurance, and service $2000 per year; and 25 cents per transaction to the banks whose cards are used at the machine. The income tax rate of Binghamton is 30% and its cost of capital 12%. Find the minimum transactions per year to break even. Realistically, is it a good project?

Suppose x is the number of annual withdrawals per machine to break even. Write an equation with the following items in it:

(1) Initial investment

(2) PV of tax benefit of depreciation

(3) PV of after-tax transaction fees collected

(4) PV of after-tax rent paid, and

(5) PV of after-tax maintenance costs Set the NPV equal to zero. This gives us

5 .3(6000/5)

10 (1 −.25)(1 −.3)x

NPV = − 6000 + 

i=1

1.12i + 

i=1

1.12i

9 3000(1 −.3)

10 2000(1 −.3)

– 3000(1 − .3) − 

i=1

1.12i − 

i=1

1.12i = 0

NPV = − 6000 + 1297.719433 + 2.966367090 x – 13,289.32456 − 7910.312240 = 0

It gives x = 8732 withdrawals/year ♥

The WolframAlpha instruction to solve the equation is as follows.

-6000+Sum[.3*6000/5/1.12^i,{i,1,5}]- Sum[3000*(1-

.3)/1.12^i,{i,0,9}]+ Sum[((1-.25)(1-.3)*x-2000*(1-

.3))/1.12^i,{i,1,10}]=0

The Maple solution is as follows:

-6000;

%+sum(.3*6000/5/1.12^i,i=1..5);

%+sum((1-.25)*(1-.3)*x/1.12^i,i=1..10);

%-3000*(1-.3)-sum(3000*(1-.3)/1.12^i,i=1..9);

%-sum(2000*(1-.3)/1.12^i,i=1..10);

solve(%=0);

8732 withdrawals per year are equal to 8732/365 = 24 transactions per day. It may work out as a reasonable project. ♥

14.7. Ardmore Corporation wants to set up a car wash. It will buy the land for $50,000, build a building for $150,000, and buy the car-wash equipment for $50,000. The machine will last for 5 years with no resale value. The company uses straight-line depreciation. Ardmore will depreciate the building over a 20-year period, but will sell the building and land for $200,000 after 5 years. The WACC for Ardmore is 12% and its income tax rate 30%. The company plans to charge $4.00 per car. Ardmore estimates per car expenses as, electricity 25¢, water 25¢, detergent 25¢, labor 50¢. Assume that all revenues and

expenses are available at the end of each year. In order to break even, how many cars should Ardmore wash per year?

Including machinery, land, and building, the initial investment in the car-wash business is

$250,000. (1)

The depreciation for the building per year is 150,000/20 = $7500, and for equipment 50,000/5 = $10,000, with a total of $17,500 per year. You cannot depreciate land. The present value of the tax benefit of depreciation for 5 years is

5 .3(17,500)

= 

i=1

1.12i = 18,925.07506 (2)

Suppose the company washes x cars per year to break even. The profit per car is (4.00 −

.25 − .25 − .25 − .50) = $2.75. After taxes it becomes 2.75(1 − .3) = $1.925. The present

5 1.925x

value of after-tax income for 5 years = 

i=1

1.12i = 6.939194190x (3)

The original value of building and land was $200,000. The total amount of depreciation taken on building and land is 7500*5 = $37,500. Its book value is thus 200,000 − 37,500

= $162,500. The taxable profit on building and land is 200,000 − 162,500 = $37,500. The taxes due are .3(37,500) = $11,250. The company sells the building and land for

$200,000.

PV of after-tax sales =

200‚000−11‚250

1.125 = 107,101.8190 (4)

Combine the cash flows (1) through (4), and set the total equal to zero to break even. Thus

−250,000 + 18,925.07506 + 6.939194190 x + 107,101.8190 = 0

Solve for x, x = 17,866 ♥

The car-wash must wash 17,866 cars every year to break even. If it is open 300 days a year, it should have 17,866/300 = 60 customers every day. It is unlikely to have that kind of traffic.

To verify the answer at WolframAlpha, try this:

-250000+Sum[.3*(150000/20+50000/5)/1.12^i,{i,1,5}]+Sum[2.75*(1-

.3)*x/1.12^i,{i,1,5}]+(200000-.3*(200000-(200000- 5*7500)))/1.12^5=0