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newnansm7app - Appendix 7A: Difficulties Solving for An...

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Appendix 7A: Difficulties Solving for An Interest Rate 7A-1 Year Cash Flow 0 +$4,000 1-9 +$4,000 10 -$71,000 11-19 +$4,000 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 0 i Figure B $5,000 NPW 0 i Figure A $5,000
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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 0 0 1-9 0 10 -$15,112 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 0% 3.1% 6% 22.1% 8% 45.9% 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 3 0 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.
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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 3 0 0 0 0 0 0 4 +$80,000 +$80,000 +$58,800 +$56,670 +$32,770 +$28,010 +$5,000 +$90 -$230 -$30 +$1,210 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 0 0 1 -$200 2 -$70,000 3 0 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
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This note was uploaded on 08/13/2010 for the course IME IME 314 taught by Professor Freeman during the Spring '10 term at Cal Poly.

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newnansm7app - Appendix 7A: Difficulties Solving for An...

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