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chapter3A - Additional Solutions Chapter Three Cash Flow...

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Additional Solutions: Chapter Three: Cash Flow Analysis 3S.1 Mr Dashwood must find the least value of N for which 1 500 < 100 (P/A,5%,N) which is equivalent to finding N such that (P/A,5%,N) > 15 Consulting Appendix A, this first occurs when N = 29 years. If the interest rate is 10%, we come to the end of the table in Appendix A before finding a value of N for which (P/A,10%,N) > 15. We therefore go on to look at the capitalized value of the annuity, which is 100/0.1, or £1 000. Thus, if he can invest his money at 10%, Mr Dashwood can afford to support his widowed half-sister indefinitely for a cost equivalent to a single present payment of £1 000. *3S.2 There are several ways of tackling this. One way is to convert the bi-annuity to an equivalent annuity: A1 = A(A/F,i,2) where ¥ A1 is the annual payment equivalent to getting a final payment of ¥ A at the end of two years. Then we can convert ¥ A1 to its present worth using a formula we already know: P=A1(P/A,i,2N)=A(A/F,i,2)(P/A,i,2N) (Don’t forget that it’s now 2 N rather than N .) Another solution is to calculate the effective biennial interest rate, j , from the equation j = (1 + i ) 2 -1 = 2 i + i 2 and then to use a conversion factor based on j : P = A(P/A, j, N) A third alternative is to construct a formula based on the one at the back of the book, (P/A, j, N) = ((1 + j ) N -1)/( j (1 + j ) N ) and substitute in the value of j , the effective biennial interest rate, as calculated above. This is a correct solution, but it’s more work. Also, I would advise against using these algebraic formulas, except when you’re constructing a spreadsheet – evaluating them is more work than looking a number up in the tables, and there are more opportunities for making a slip.
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3S.3 Having drawn a cash-flow diagram, we write down the PW equivalent of each of the costs: PW = 32 000 -16 000 (P/F, 15%,8 ) + 20 000 (P/ A , 15%,8 ) + 600 + 550 (P/A, 15%,7 ) – 50 (P/G, 15%,7) = 32 000 -16 000(0.3269) + 20 000 (4.6587) + 600 + 550(4.1604) – 50(10.192) = 122 322. The P/ A denotes a continuous cash flow, continuously compounded – this is the kind that you look up in Appendix C. We represent the decreasing insurance costs as an annuity plus a negative arithmetic gradient. There isn’t a P/G column in Appendix A, but we can make one by multiplying the P/A entry by the A/G entry. We have to treat the first insurance payment of
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