Solution manual engineering economic analysis 9th edition ch07a

Appendix 7A: Difficulties Solving for An Interest Rate7A-1 There are 2 sign changes in the cash flow indicating there may be 2, 1, or zero positive interest rates.. At i = 0% NPW= +$5,00

Appendix 7A: Difficulties Solving for An Interest Rate

7A-1

There are 2 sign changes in the cash flow indicating there may be 2, 1, or zero positive interest rates

At i = 0% NPW= +$5,000

At i = ∞% NPW =+$4,000

This suggests that the NPW plot may look like one of the following:

After making a number of calculations, one is forced to conclude that Figure B is the general form of the NPW plot, and there is no positive interest rate for the cash flow

NPW

Figure B

$5,000

NPW

Figure A

$5,000

There is external investment until the end of the tenth year If an external interest rate (we will call it e* in Chapter 18) is selected, we can proceed to solve for the interest rate i for the investment phase of the problem

For external interest rate = 6%

Future worth of $4,000 a year for 10 years (11 payments)

= $4,000 (F/A, 6%, 11) = $4,000 (14.972) = $59,888

At year 10 we have +$59,888 -$75,000 = -$15,112

The altered cash flow becomes:

Year Cash Flow

11-19 +$4,000

At the beginning of year 10:

PW of Cost = PW of Benefits

$15,112 = $4,000 (P/A, i%, 9)

(P/A, i%, 9) = $15,112/$4,000 = 3.78

By linear interpolation from interest tables, i = 22.1%

The internal interest rate is sensitive to the selected external interest rate:

For external

Interest rate

Computed internal Interest rate

7A-2

The problem statement may be translated into a cash flow table:

Year Cash Flow

0 +$80,000

1 -$85,000

2 -$70,000

4 +$80,000

There are two sign changes in the cash flow indicating there may be as many as two positive rates of return

To search for positive rates of return compute the NPW for the cash flow at several interest rates This is done on the next page by using single payment present worth factors to compute the PW for each item in the cash flow Then, their algebraic sum represents NPW

at the stated interest rate

Year Cash Flow PW at 0% PW at 8% PW at 9% PW at

25%

PW at 30%

0 +$80,000 +$80,000 +$80,000 +$80,000 +$80,000 +$80,000

1 -$85,000 -$85,000 -$78,700 -$77,980 -$68,000 -$65,380

2 -$70,000 -$70,000 -$60,010 -$58,920 -$44,800 -$41,420

4 +$80,000 +$80,000 +$58,800 +$56,670 +$32,770 +$28,010

The plow of NPW vs i shows two positive interest rates: i ≈ 8.2% and i ≈ 25%

Using an external interest rate of 6%, the Year 0 cash flow is invested and accumulates to +

$80,000 (1.06) = $84,800 at the end of Year 1 The revised cash flow becomes:

Year Cash Flow

1 -$200

2 -$70,000

4 +$80,000

With only one sign change we know there no longer is more than one positive interest rate

PW of Benefit = PW of Cost, or PW(Benefit) – PW(Cost) = 0

$80,000 (P/F, i%, 4) - $200 (P/F, i%, 1) - $70,000 (P/F, i%, 2) = 0

Try i = 7%

80,000 (0.7629) - $200 (0.9346) - $70,000 (0.8734) = -$293

Try i = 6%

$80,000 (0.7921) - $200 (0.9434) - $70,000 (0.8900) = +$879

By interpolation, i = 6.75%

+$5,000

-$2,000

0% 10% 20% 30% i

(a)

Quarter Quarterly

Cash Flow

PW at 0%

PW at 20%

PW at 40%

PW at 45%

By interpolation, i ≈ 43% per quarter The nominal rate of return = 4 (43%) = 172% per year

(b)

Quarter Quarterly Cash Flow Transformed Cash Flow

Let X = amount required at end of quarter 1 to produce $50 at the end of 2 quarters:

X (1.03) = $50

X = $50/1.03 = $48.54

Solve the Transformed Cash Flow for the rate of return:

Quarter Transformed

Cash Flow

PW at 35% PW at 40%

Rate of return = 35% + 5% (2.56/(2.56 – (-5.34)))

= 36.6% per quarter Nominal annual rate of return = 36.6% x 4 = 146%

NOTE: Although there are three sign changes in the cash flow, the accumulated

cash flow sign test, (described in Chapter 18) indicates there is only a single positive rate of return for the untransformed cash flow It is 43%

(c) In part (a) the required external investment in Quarter 1, for return in Quarter 2, is assumed to be at the internal rate of return (which we found is 43% per quarter)

In part (b) the required external investment is at 3% per quarter.

The “correct” answer is the one for the computation whose assumptions more closely fit the actual problem Even though there is only one rate of return, there still exists the required external investment in Quarter 1 for Quarter 2 On this basis the Part (b) solution appears to have more realistic assumptions than Part (a)

7A-4

Year Cash Flow

0 -$500

1 +$2,000

2 -$1,200

3 -$300

Sum 0

There are two sign changes in the cash flow indicating as many as two positive rates of return The required disbursement in Year 2 & 3 indicate that money must be accumulated

in an external investment to provide the necessary Year 2 & 3 disbursements

Before proceeding, we will check for multiple rates of return This, of course, is not

necessary here

Since the algebraic sum of the cash flow = $0, we know that NPW at 0% = 0, and 0% is a rate of return for the cash flow

Looking for the other (possible) rate of return:

Year Cash Flow PW at

5%

PW at 50%

PW at 200%

PW at 219%

PW at 250%

PW at

∞%

1 +$2,000 +$1,905 +$1,333 +$667 +$627 +$571 $0

Solution using an external interest rate e* = 6%

How much of the +$2,000 at Year 1 must be set aside in an external investment at 6% to provide for the Year 2 and Year 3 disbursements?

+$500

Amount to set aside = $1,200 (P/F, 6%, 1) + $300 (P/F, 6%, 2)

= $1,200 (0.94.4) + $300 (0.8900)

= $1,399.08 The altered cash flow becomes:

Year Cash Flow

0 -$500

1 +$2,000 -$1,399.08 = +

$600.92

Solve the altered cash flow for the unknown i:

$500 = $600.92 (P/F, i%, 1)

(P/F, i%, 1) = $500/$600.92 = 0.8321

From tables: 20.2%

7A-5

Cash Flow

PW at 18%

PW at 20%

1 +$2,000 $200(1.06) 0

The rate of return is 18% + (2%) (23/(23 + 6)) = 19.6%

7A-6

Year Cash Flow PW at

20%

PW at 35%

PW at 50%

There is a single positive rate of return at 35%

Cash Flow

PW at 12%

PW at 15%

+$72 +$6 -$5 Rate of return ≈ 13.6%

For further computations, see the solution to Problem 18-4

7A-7

Some money flowing out of the cash flow in Year 2 will be required for the Year 3 investment

of $100 At 10% external interest, $90.91 at Year 2 becomes the required $100 at Year 3

Cash Flow

NPW at 20%

NPW at 25%

2 +$300 -$90.91(1.10) +$209.09 +$145.19 +$133.82

+$12.41 -$48.49 The rate of return on the transformed cash flow is 21% (This is only slightly different from the 21.4% rate of return on the original cash flow because the external investment is small and of short duration.)

7A-8

Cash Flow

PW at 15%

2 -$40.0 $20(1.10) -$18.0 -$13.6

-$0.1 From the computations we see that the rate of return on the internal investment is 15%

7A-9

Cash Flow

2 +$6,000 X 1.12 = +$6,720 0

4 +$4,000 Y (1.12)2 = +$1,280 +$4,000

5 +$4,000 Y = $1,020 +$4,000

Year Transformed

Cash Flow PW at 10% PW at 12%

0 -$15,000 -$15,000 -$15,000

Rate of Return = 10% + (2%) (638/(638+144)) = 11.6%

7A-10

The compound interest tables are for positive interest rates and are not useful here (Tables could be produced, of course, for negative values.)

PW of Cost = PW of Benefits

$50 = $20 (1 + i)-1 + $20 (1 + i)-2

let x = (1+ i)-1 thus, $50 = $20x + $20x2 or x2 + x – 2.50 = 0

x = - 1 + (12 – 4(-2.50))1/2 = - 1+ (11)1/2 = +1.159, -2.158

Solving for i:

x = (1 + i)-1 = +1.159 1 + i = 1/1.159 = 0.863 i = -0.137 = -13.7%

x = (1 + i)-1 = -2.158 1 + i = 1/-2.158 = -0.463 i = -1.463 = -146%

7A-11

Cash Flow

Year Transformed

Cash Flow

PW at 35% PW at 40%

Rate of return = 35% + 5% (0.7/(0.7+1.2)) = 36.8%

7A-12

Cash Flow

PW at 25%

2 +$500 X (1.10) = +$300 +$227.27 +$145.5

3 -$300 X = $300/1.10 = $272.73 0 0

-0.6 From the Present Worth computation it is clear that the rate of return is very close to 25% (Calculator solution says 24.96%)

7A-13

30.00% root

2 sign changes => 2 roots possible

6% external financing rate

12% external investing rate

8.8% MIRR value is less than external investing rate => not attractive

7A-14

0 -610 0% -110 =$B$2+NPV(D2,$B$3:$B$12)

10 -1300

2 sign changes => 2 roots possible

6% external financing rate

12% external investing rate

9.5% MIRR value is less than external investing rate => not attractive

7A-15

0 -500 0% -80 =$B$2+NPV(D2,$B$3:$B$5)

60% -68 No roots exist

2 sign changes => 2 roots possible

6% external financing rate

12% external investing rate

7.5% MIRR value is less than external investing rate => not attractive

0 -100 0% 0.00 =$B$2+NPV(D2,$B$3:$B$5)

50% -0.44 20.00% root 60% -1.17 40.00% root

3 sign changes => 3 roots possible All PW values = 0 given significant digits of cash flows

6% external financing rate

12% external investing rate

8.8% MIRR value is less than external investing rate => not attractive

7A-17

Year

Cash

0 -1200 -45% -422 =$B$2+NPV(D2,$B$3:$B$8)

2 sign changes => 2 roots possible

6% external financing rate

12% external investing rate

9.5% MIRR value is less than external investing rate => not attractive

7A-18

0 -3570 0% 2260 =$B$2+NPV(D2,$B$3:$B$10)

3 sign changes => 3 roots possible

800 down payment

55 monthly payment 1 sign change => 1 root possible

40 # payment

2500 final receipt

-0.75% IRR monthly =RATE(A3,-A2,-A1,A4)

-8.62% effective annual rate

=(1+A6)^12-1

7A-20

0 -850 0% -450 =$B$2+NPV(D2,$B$3:$B$12)

10 -1800

2 sign changes => 2 roots possible

6% external financing rate

12% external investing rate

9.1% MIRR value is less than external investing rate => not attractive

0 -16000 0% 1950 =$B$2+NPV(D2,$B$3:$B$7)

3 sign changes => 3 roots possible

7A-22

0 -200 0% 176 =$B$2+NPV(D2,$B$3:$B$10)

8 -124.5

4 sign changes => 4 roots possible

0 -210000 0% 127000 =$B$2+NPV(D2,$B$3:$B$9)

3 sign changes => 3 roots possible

Year

Cash

0 -103000 0% 37400 =$B$2+NPV(D2,$B$3:$B$7)

5 sign changes => 3 roots possible

0 -200 0% 100 =$B$2+NPV(D2,$B$3:$B$4)

2 sign changes => 2 roots possible

6% external financing rate

12% external investing rate

24.5% MIRR value is more than external investing rate => attractive

138,00 0

-101,352 IRR 8.0% 11.0% 19.2%

2 sign changes => 2 roots possible

6% external financing rate

12% external investing rate

5.1%

MIRR for A-B value is less than external investing rate => not attractive

0 -1000 0% 520 =$B$2+NPV(D2,$B$3:$B$7)

3 sign changes => 3 roots possible