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Understanding Financial Mathematics: Key Concepts of Present and Future Value

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This chapter delves into fundamental financial mathematics concepts, specifically focusing on the notation and relationships between present value (P), future value (F), and annual amounts (A). It explains how to convert values from present to future and vice versa using formulas and examples. Key principles, such as the single payment compound amount factor and the uniform series present worth factor, are outlined to facilitate better understanding of cash flows over time. This resource is crucial for anyone looking to grasp the essentials of financial calculations.

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Understanding Financial Mathematics: Key Concepts of Present and Future Value

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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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