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Unformatted text preview: R. 1. R. 2. R. 3. R. 4. R. 5. R. 6. R. 7. CHAPTER 10 REVIEW QUESTIONS Mutually exclusive projects are those projects whose acceptance will affect
the accept—reject decision of other projects. Decisions on mutually exclusive
projects cannot be made in isolation but must be analyzed against all other
potentially mutually exclusive projects. Projects that share common resources
such as land, equipment, and managerial time are mutually exclusive. For the case of mutually exclusive projects that cannot be repeated, NPV requires that projects be rank ordered with the best project being the one
with the highest NPV. Ranking and choosing the highest NPV project from a set of mutually exclusive
projects is the correct procedure if the projects cannot be repeated once
finished. If projects can be repeated, and if the projects being considered
have different lives, then a shorter term project with a lower NPV might be
preferred to a longer term project with a higher NPV. The reason is that the
collective NPV from the series of short lived projects could be higher than
the NPV of the longer term project. The equivalent annual annuity allows unequally lived projects to be directly
compared with each other under the assumption that the projects can be
replicated when they are completed. This approach converts the NPV of each
project into equivalent annuities and then compares the magnitudes of the
annuities. Given equivalent required rates of return, the best project is the
one with the highest equivalent annuity. True. By definition, NPV represents the change in wealth from an investment. Therefore, accepting a $1 million NPV project has the same effect on shareholder
wealth as the firm receiving an immediate gift of $1 million. In analyzing projects of unequal lives, both the equivalent annual annuity
approach and the common ending point approach seek to incorporate into the
analysis the potential to replicate projects. The common ending point
approach requires each project to be replicated until a common ending point is
reached, and then to accept the project with the highest NPV. The common
ending point approach can be more tedious but will reach identical decisions
compared with the equivalent annual annuity approach. The Fisher Effect states that the market interest rate can be separated into a
real rate of interest, or the rate of interest earned after the effects of
inflation have been removed, and an expected inflation rate. The economist
Irving Fisher called the sum of the real rate and the expected inflation rate
the nominal rate of interest. 69 R. 8. R. 9. The real rate of interest is the rate of interest earned after the effects Of
inﬂation have been removed. The real interest rate reﬂects real purchasing
power rather than dollars. The nominal interest rate adds the expected rate
of inﬂation to the real interest rate and therefore adjusts to reﬂect inﬂationary
expectations. The real rate of interest is the price of purchasing power through time. Although prices change in response to supply and demand, some argue that the real rate
of interest is relatively stable and tends to range around 2% to 3%. R. 10.Using nominal dollars in capital budgeting factors expected inﬂation into the analysis. Through NPV, expected inﬂation will cause the estimated cash ﬂows
to increase over time as the prices underlying the cash ﬂows rise, but
inﬂation will also cause the required rate of return to increase through
time. Using real dollars in capital budgeting removes expected inﬂation from
both the cash ﬂows and the required rate of return in NPV. Either approach
is acceptable as long as inﬂation is consistently treated in both the numerator
(cash ﬂow estimation) and in the denominator (required rate of return). R. 1.1.Capital rationing is the concept that a firm limits the amount that it will invest in new projects. Capital rationing is often argued to limit
investment. Modernists look skeptically at capital rationing and believe that
firms reject positive NPV projects not because of a lack of funding but
because of hidden costs such as the inability of management to efficiently
oversee excessive growth. ‘ R. 12.In cases when capital rationing is imposed, an approach to solve the problem is to rely on the profitability index. The profitability index expresses the
NPV of a project as a proportion of its cost so that it is a tool to rank
projects when there is a need to ration capital. This technique requires that
projects be ranked from high to low according to the profitability index, and to keep accepting projects until the allowable capital has been fully
allocated. R. 13.Integer linear programming is a mathematical technique for solving the capital rationing problem. The technique utilizes computer software to search for the
precise set of projects that should be accepted. The model’s objective is to
choose the set of projects that maximizes NPV within certain constraints such
as a limit on the capital that can be allocated. R. 14.As part of the integer linear programming model, the main objective is specified through the model’s objective function of maximizing NPV. The set
of constraints or limits to be placed on the decision variables includes a
limit on available capital but can also include constraints on the allowable
annual cash ﬂows or on certain combinations of projects. 70 CHAPTER 10 PROBLEMS 1. $31,547.08
As illustrated in Figure 10.2, the equivalent annuity is found by computing the
annual payment or annuity that has a present value equal to the NPV. The math
is performed either with a financial calculator or with formulas as demonstrated
in the annuity section of Chapter 4. With a financial calculator, set PV = $100,000, N = 4 and %I = 10, and then compute
the annuity payment. With formulas, divide the NPV by the annuity formula: Equivalent Annuity: ($NPV /((1/r)*(1—(1/((1+r)An)))))) = $31,547 ‘ 2 a. $11,200 b. $14,792 c. $9,877
Solved same as in Problem #1 (immediately above) except with new figures. (1. Project Reagan should be selected because it has the greatest equivalent annuity.
Even though the NPV of a single Project Reagan is lower than the NPV of a
single Project Roosevelt, by adopting the two year project (Project Reagan) rather
than the four year project (Project Roosevelt), the White House Corporation will
be able to repeat the two year project and earn more wealth in the long run. 3. Enquirer $261,026 Star $294,441
Solved as in Problem #1. Select Project Star with the higher equivalent annuity. 4. Project Peg. This problem involves two steps: first, compute the NPV’s, and second,
convert the NPV’s into equivalent annuities — — selecting the higher project. Equiv. Ann. NPV Initial Cost PVCF1 PVCF2 PVCF3 PVCF4
Al $761.90 $1,322 ($10,000) $6,364 $4,959 $0 $0
Peg $3,287.22 $10,420 ($20,000) $7,273 $7,438 $7,513 $8,196 5. Project Croix. This problem involves two steps: first, compute the NPV’s, and then
convert the NPV’s into equivalent annuities — — selecting the higher project. Equiv. Ann. NPV Initial Cost PVCF1 PVCF2 PVCF3
Thorr $11,500.00 $20,508 ($200,000) $83,333 $137,174 $0
Croix $39,603.75 $102,063 ($200,000) $74,074 $77,160 $150,828 6. Project Air. This problem involves two steps: first, compute the NPV’s, and then
convert the NPV’s into equivalent annuities — — selecting the higher project. Equiv. Ann. NPV Initial Cost PVCF1 PVCF2 PVCF3 PVCF4
Grd $101,893.60 $309,486 ($800,000) $267,857 $239,158 $284,712 $317,759
Air $297,735.85 $503,189 ($2,000,000) $1,785,714 $717,474 $0 $0 71 7. Project A. This problem involves one step: compute the present value of the costs
and select the one with the lowest total costs. PV—Costs Initial Cost PVCFl PVCF2 PVCF3 PVCF4
Project A $129,137 $100,000 $8,772 $7,695 $6,750 $5,921
Project B $135,518 $120,000 $7,018 $4,617 $2,700 $1,184
Project C $156,548 $40,000 $35,088 $30,779 $26,999 $23,683 8. Contracting out. This problem involves one step: compute the present value of the
costs and select the one with the lowest total costs. PV—Costs Initial Cost PV 20 Year Annuity Contracted Out $8,429,702 $0 $8,429,702
Do — It — Yourself $ 12, 5 12,376 $9,000,000 $3, 512, 376 9. The low quality model. This problem involves one step: compute the present value
of the costs and select the one with the lowest total costs. PV—Costs Initial Cost PV 10 Year Annuity
High Quality $19,528 $12,000 $7,528
Low Quality $18,038 $8,000 $10,038 10.Project A. This problem involves two steps: compute the present value of the costs
then convert them to equivalent annuities and select the lower one. Equiv. Ann. PV—Costs Initial Cost PVCF1 PVCF2 PVCF3 PVCF4
Sht $67,619 $117,355 $100,000 $9,091 $8,264 $0 $0
Med $72,465 $180,210 $160,000 $9,091 $6,612 $4,508 $0
Lng $103,094 $326,795 $200,000 $36,364 $33,058 $30,053 $27,321 11.The high quality model. The equivalent annuity approach reﬂects the advantage of
the high quality model’s longer life. Equiv Ann PV—Costs Initial Cost PV Annuity
High Quality $3,552 $20,771 $12,000 $8,771
Low Quality $3,594 $18,038 $8,000 $10,038 12.The Senior. This problem involves two steps: compute the present value of the costs
then compute the equivalent annuities and select the lower annuity. Equiv. Ann. PV—Costs Initial Cost PVCFl PVCF2 PVCF3 PVCF4
Tran. $184,070 $299,244 $100,000 $78,261 $120,983 $0 $0
Sen. $163,180 $465,876 $50,000 $69,565 $90,737 $118,353 $137,221 13. Each inﬂation rate may be computed as a Chapter 4 type problem. The final price 72 is the future value or FV, the initial price is the present value, or PV, and the
number of years is the time, represented as N or t. The problems may be easily
solved using a financial calculator or by using the formula method. Name PV—Year Price #1 FV—Year Price #2 Num Yrs Inf. Rate CPI 1940 42.0 1950 72.1 10 5.6%
CPI 1950 72.1 1960 88.7 10 2.1%
CPI 1960 88.7 1970 116.3 10 2.7%
CPI 1970 116.3 1980 246.8 10 7.8%
CPI 1980 246.8 1990 385.0 10 4.5%
Milk 1965 $1.00 2000 $2.25 35 2.3%
Gasoline 1965 $0.35 2000 $1.25 35 3.7%
Beer 1975 $1.25 2000 $3.00 25 3.6%
House 1950 $5,000 2000 $125,000 50 6.6%
Computer 1985 $5,000 2000 $1,500 15 — 7.7%
Tuition 1975 $2,000 2000 $5,000 25 3.7% 14. Nominal Real Rate Exp. Inf. Rate a. 8% 3% 5% All solved using formula:
b. 17% 2% 15% Nominal = Real + E(Inflation)
c 10% 4% 6%
d 8% 2% 6%
15. Nominal Real Rate Exp. Inf. Rate
8.15% 3.00% 5.00% All solved using formula in
17.30% 2.00% 15.00% problem and algebra. 10.00% 4.00% 5.77%
8.00% 1.89% 6.00% 999‘?” 16. NPV Initial Cost PVCF1 PVCF2 PVCF3 PVCF4 PVCF5
($195,483) — 1000000 $208,696 $181,474 $157,804 $137,221 $119,322 17a. $290,400.00 Second CF found from First CF * (1.10)
b. $319,440.00 Third CF found from Second CF * (1.10)
c. $351,384.00 Fourth CF found from Third CF * (1.10)
d. $386,522.40 Fifth CF found from Fourth CF * (1.10)
e NPV Initial Cost PVCF1 PVCF2 PVCF3 PVCF4 PVCF5
$52,261 — 1000000 $229,565 $219,584 $210,037 $200,905 $192,170
The larger cash inflows in years 1—5 created a higher NPV
g. If indeed the expected inflation rate is 10% and the cash flows should actually rise
by 10% each year, then this problem performs the superior analysis.
18a. NPV Initial Cost PVCF1 PVCF2 . PVCF3 PVCF4 PVCF5
$39,074 — 1000000 $228,571 $217,687 $207,321 $197,449 $188,046
b. NPV Initial Cost PVCF1 PVCF2 PVCF3 PVCF4 PVCF5
$52,128 — 1000000 $229,555 $219,565 $210,010 $200,870 $192,128 1"} 73 c. Nominal
15.00% Real Rate Exp. Inf. Rate
4.55% 10.00% Found as in problem #15 d. Answers 18(b) and 17(e) agree (except for minor rounding errors) because even
though one is computed nominally (17e) and one in real terms (18b), they are
performed consistently. The answer to 16 ignores inﬂation, while the answer to 18a. introduces a non—~trivia1 error due to the use of the approximate real interest
rate rather than the exact one. 19a. PV of $8,000 per year for 8 years with 10% discount rate = $42,679 b. $5,300.00 First FC found from $5,000*(1.06)
$5,618.00 Second CF found from First CF * (1.06)
$5,955.08 Third CF found from Second CF * (1.06)
$6,312.38 Fourth CF found from Third CF * (1.06)
$6,691.13 Fifth CF found from Fourth CF * ( 1.06)
$7,092.60 Sixth CF found from Fifth CF * (1.06)
$7,518.15 Seventh CF found from Sixth CF * (1.06)
$7,969.24 Eigth CF found from Seventh CF * (1.06) c. Year Cash Flow PV of CF (1. OOQQMAWNI—k $5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00 $4,807.69
$4,622.78
$4,444.98
$4,274.02
$4,109.64
$3,951.57
$3,799.59
$3,653.45 Sum of present values = $33,663.72 Nominal 10.00% Real Rate Exp. Inf. Rate
6.00% Found as in problem #15 3.77% Year Cash Flow PV of CF 1 \IONUIPUJN 8 $5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00
$5,000.00 $4,818.35
$4,643.30
$4,474.60
$4,312.04
$4,155.38
$4,004.42
$3,858.93
$3,718.74 Sum of present values = $33,985.75 74 Note that with a starting pension of only $5,000 and a life expectancy of only eight
years, Gene is better off accepting the $8,000 non—indexed pension. The eighth
year of the indexed pension would still be less than the $8,000 alternative. 20.The profitability index method of rationing capital is to select the projects with the
highest PI. The procedure has difficulty when acceptance of a project exceeds the
available capital but capital remains (in thousands): a. Capital Available Project CumulativeCapital Available
Before Investment Selected NPV NPV After Investment
$500 #1 $50 $50 $300
$300 #6 $12 $62 $250
$250 #7 $11 $73 $200
$200 #2 $40 $ 11 3 $0 b. Projects #1, #6, #7 and #2 have total NPV of $113,000
c. The second best project was skipped because there was insufficient money given
that project #1 was accepted. Let’s try skipping project #1: Capital Available Project CumulativeCapital Available
Before Investment Selected NPV NPV After Investment
$500 #3 $99 $99 $100
$100 #6 $12 $111 $50
$50 #7 $ 11 $122 $0 Using project #3 (and projects #6 & #7) produces a total NPV of $122,000 d. The profitability index failed to show that acceptance of the best project in terms
of RI. would preclude acceptance of the biggest project with almost as great a PI. 21.Define X1 as one if project one is selected and zero if it is not selected, define X2 as one if project two is selected and zero if it is not selected, and so forth. All dollar
numbers omit thousands. Assume all X’s < = 1. a. Max $50*X1 + $40*X2 + $99*X3 + $17*X4 + $19*X5 + $12*X6 + $11*X7
b. $200*X1 + $200*X2 + $400*X3+ $100*X4 + $100*X5 + $50*X6 + $50*X7 c. $1*X1 + $5*X2 + $10*X3+ $2*X4 + $3*X5 + $20*X6 + $15*X7 >= $2: $500
d.X3+X4<=1 e. X4+X5>=1 f. X3+X4+X5+X6+X7<=3 22. The solution is as shown in Problem 20c. 75 CHAPTER 10 DISCUSSION QUESTIONS 1. There is no capital budgeting method that is without difficulties. The process of
capital budgeting is complex especially in determining cash flows, project side effects
and so forth. NPV in general provides the best combination of few difficulties and
high accuracy and power. 2. The whole purpose of using the equivalent annuity is to incorporate the advantage
that a short lived project offers by finishing quicker and enabling the firm to begin
another project with (hopefully) a positive NPV. The method assumes that the same
project (or at least a project of similar NPV and project life) will be available each
time the shorter project ends. The terminology for this assumption is project
replication. 3. Least costs problems are assumed to have large benefits that would create positive
NPV’s if included in the analysis. However, since all alternatives within a least
cost problem have the same benefits, they can be ignored for simplicity. The ranking
that a least cost problem will produce is identical to the ranking that an NPV analysis
would produce if all of the alternatives produce exactly the same benefits. 4. Normal linear programming (with decision variables constrained to be greater than
or equal to zero and less than or equal to one) will almost always produce a solution
with several of the decision variables somewhere between zero and one. In fact, the
number of decision variables between zero and one will usually be equal to the
number of project constraints. The problem with a decision variable between zero
and one (e.g., 0.50) is that it recommends partial acceptance of a project. In most cases, a project can not economically be partially accepted —— it must either be
fully accepted (X: 1.00) or rejected (X=0.00). 5. Since (1 + Nominal) = (1 + Real) * (1 + Inflation), then
Nominal = Real + Inflation + (Real*1nflation)
Using Nominal = Real ignores the cross—product: Real*Inflation = error 6. Using continuously compounded interest rates, the annual form of the Fisher effect
must look like this: exp(norninal) = exp(real) * exp(inflation), where exp refers
to the exponential function. However, if we take the logarithm of each side, we get: nominal = real + inflation Thus, when continuously compounded rates are used, the above formula is the exact
formula rather than an approximation. 76 ...
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