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ARCH 449 Chapter 3

ARCH 449 Chapter 3. Financial Mathematics. Notation. i = interest rate (per time period) n = # of time periods P = money at present F = money in future After n time periods Equivalent to P now, at interest rate i A = Equal amount at end of each time period on series E.g., annual.

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ARCH 449 Chapter 3

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  1. ARCH 449 Chapter3 Financial Mathematics

  2. Notation • i = interest rate (per time period) • n = # of time periods • P = money at present • F = money in future • After n time periods • Equivalent to P now, at interest rate i • A = Equal amount at end of each time period on series • E.g., annual

  3. Assumptions 500 End of second year + 200 200 0 1 2 3 4 5 Time _ 50 100 500 Biggining of third year # on the cash flow means end of the period, and the starting of the next period

  4. Assumptions P 0 1 2 3 n-1 n A If P and A are involved the Present (P) of the given annuals is ONE YEAR BEFORE THE FİRST ANNUALS

  5. If F and A are involved the Future (F) of the given annuals is AT THE SAME TIME OF THE LAST ANNUAL : F ………….. 1 2 3 .. .. n-1 n 0 Assumptions 0 A

  6. F ………….. 1 2 3 .. .. n-1 n 0 Assumptions P 0 A

  7. Overview • Converting from P to F, and from F to P • Converting from A to P, and from P to A • Converting from F to A, and from A to F

  8. Present to Future, and Future to Present

  9. Converting from Present to Future Fn …………. n P0 Fn = P (F/P, i%, n) To find F given P:

  10. Derive by Recursion • Invest an amount P at rate i: • Amount at time 1 = P (1+i) • Amount at time 2 = P (1+i)2 • Amount at time n = P (1+i)n • So we know that F = P(1+i)n • (F/P, i%, n) = (1+i)n • Single payment compound amount factor Fn = P (1+i)n Fn = P (F/P, i%, n)

  11. Example—Present to Future F = ?? P = $1,000 i = 10%/year 0 1 2 3 F3 = $1,000 (F/P, 10%, 3) = $1,000 (1.10)3 = $1,000 (1.3310) = $1,331.00 Invest P=$1,000, n=3, i=10% What is the future value, F?

  12. Converting from Future to Present Fn …………. n P (P/F, i%, n) = 1/(1+i)n • To find P given F: • Discount back from the future

  13. Converting from Future to Present • Amount F at time n: • Amount at time n-1 = F/(1+i) • Amount at time n-2 = F/(1+i)2 • Amount at time 0 = F/(1+i)n • So we know that P = F/(1+i)n • (P/F, i%, n) = 1/(1+i)n • Single payment present worth factor

  14. Example—Future to Present F9 = $100,000 ………… 0 1 2 3 8 9 P= ?? i = 15%/yr P = $100,000 (P/F, 15%, 9) = $100,000 [1/(1.15)9] = $100,000 (0.1111) = $11,110 at time t = 0 Assume we want F = $100,000 in 9 years. How much do we need to invest now, if the interest rate i = 15%?

  15. Annual to Present, and Present to Annual

  16. Converting from Annual to Present Fixed annuity—constant cash flow P = ?? ………….. 1 2 3 .. .. n-1 n 0 $A per period

  17. Converting from Annual to Present We want an expression for the present worth P of a stream of equal, end-of-period cash flows A P = ?? 0 1 2 3 n-1 n A is given

  18. Converting from Annual to Present Write a present-worth expression for each year individually, and add them The term inside the brackets is a geometric progression. This sum has a closed-form expression!

  19. Converting from Annual to Present Write a present-worth expression for each year individually, and add them

  20. Converting from Annual to Present This expression will convert an annual cash flow to an equivalent present worth amount: (One period before the first annual cash flow) • The term in the brackets is (P/A, i%, n) • Uniform series present worth factor

  21. Converting from Present to Annual Given the P/A relationship: We can just solve for A in terms of P, yielding: Remember:The present is always one period before the first annual amount! • The term in the brackets is (A/P, i%, n) • Capital recovery factor

  22. Future to Annual, and Annual to Future

  23. Converting from Future to Annual Find the annual cash flow that is equivalent to a future amount F $F ………….. 1 2 3 .. .. n-1 n 0 0 The future amount $F is given! $A per period??

  24. Converting from Future to Annual Take advantage of what we know Recall that: and Substitute “P” and simplify!

  25. Converting from Future to Annual First convert future to present: Then convert the resulting P to annual Simplifying, we get: • The term in the brackets is (A/F, i%, n) • Sinking fund factor (from the year 1724!)

  26. Example How much money must you save each year (starting 1 year from now) at 5.5%/year: In order to have $6000 in 7 years?

  27. Example Solution: The cash flow diagram fits the A/F factor (future amount given, annual amount??) A= $6000 (A/F, 5.5%, 7) = 6000 (0.12096) = $725.76 per year The value 0.12096 can be computed (using the A/F formula), or looked up in a table

  28. Converting from Annual to Future Given Solve for F in terms of A: • The term in the brackets is (F/A, i%, n) • Uniform series compound amount factor

  29. Converting from Annual to Future Given an annual cash flow: $F ………….. 1 2 3 .. .. n-1 n 0 0 Find $F, given the $A amounts $A per period

  30. Summary

  31. (F/A, 6%, 8)(A/P, 6%, 15)

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