Fundamentals of Financial Instruments - Sunil K. Parameswaran

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alt="normal upper F period normal upper V equals 2 comma 500 times FVIFA left-parenthesis 8 comma 25 right-parenthesis equals 2 comma 500 times 73.1059 equals dollar-sign 182 comma 764.75"/>

An illustration of Timeline for an Annuity Due

      Present Value

StartLayout 1st Row normal upper P period normal upper V period equals upper A plus StartFraction upper A Over left-parenthesis 1 plus r right-parenthesis EndFraction plus StartFraction upper A Over left-parenthesis 1 plus r right-parenthesis squared EndFraction plus minus minus minus minus plus StartFraction upper A Over left-parenthesis 1 plus r right-parenthesis Superscript upper N minus 1 Baseline EndFraction EndLayout

      Therefore,

StartLayout 1st Row normal upper P period normal upper V left-parenthesis 1 plus r right-parenthesis equals upper A left-parenthesis 1 plus r right-parenthesis plus upper A plus StartFraction upper A Over left-parenthesis 1 plus r right-parenthesis EndFraction plus minus minus minus minus plus StartFraction upper A Over left-parenthesis 1 plus r right-parenthesis Superscript upper N minus 2 Baseline EndFraction 2nd Row right double arrow normal upper P period normal upper V left-bracket left-parenthesis 1 plus r right-parenthesis minus 1 right-bracket equals upper A left-parenthesis 1 plus r right-parenthesis minus StartFraction upper A Over left-parenthesis 1 plus r right-parenthesis Superscript upper N minus 1 Baseline EndFraction 3rd Row right double arrow normal upper P period normal upper V equals StartFraction upper A Over r EndFraction left-bracket 1 minus StartFraction 1 Over left-parenthesis 1 plus r right-parenthesis Superscript upper N Baseline EndFraction right-bracket left-parenthesis 1 plus r right-parenthesis 4th Row Hence PVIFA Subscript upper A upper D Baseline left-parenthesis r comma upper N right-parenthesis equals PVIFA left-parenthesis r comma upper N right-parenthesis times left-parenthesis 1 plus r right-parenthesis EndLayout

      The present value of an annuity due that makes N payments is obviously greater than that of a corresponding annuity that makes N payments, because in the case of the annuity due, each of the cash flows has to be discounted for one period less. Consequently, the present value factor for an N period annuity due is greater than that for an N period annuity by a factor of (1 + r).

      An obvious example of an annuity due is an insurance policy, because the first premium has to be paid as soon as the policy is purchased.

      EXAMPLE 2.18

      David Mathew has just bought an insurance policy from MetLife. The annual premium is $2,500, and he is required to make 25 payments. What is the present value of this annuity due if the discount rate is 8% per annum?

StartLayout 1st Row 1st Column Blank 2nd Column PVIFA left-parenthesis 8 comma 25 right-parenthesis equals 10.6748 2nd Row 1st Column Blank 2nd Column PVIFA Subscript upper A upper D Baseline left-parenthesis 8 comma 25 right-parenthesis equals 10.6748 times 1.08 equals 11.5288 EndLayout

      Thus the present value of the annuity due is:

2 comma 500 times 11.5288 equals dollar-sign 28 comma 822

      Future Value

StartLayout 1st Row normal upper F period normal upper V period equals upper A left-parenthesis 1 plus r right-parenthesis Superscript upper N Baseline plus upper A left-parenthesis 1 plus r right-parenthesis Superscript upper N minus 1 Baseline plus upper A left-parenthesis 1 plus r right-parenthesis Superscript upper N minus 2 Baseline plus minus minus minus minus plus upper A left-parenthesis 1 plus r right-parenthesis EndLayout

      Therefore,

StartLayout 1st Row normal upper F period normal upper V left-parenthesis 1 plus r right-parenthesis equals upper A left-parenthesis 1 plus r right-parenthesis Superscript upper N plus 1 Baseline plus upper A left-parenthesis 1 plus r right-parenthesis Superscript upper N Baseline plus upper A left-parenthesis 1 plus r right-parenthesis Superscript upper N minus 1 Baseline 2nd Row plus minus minus minus minus plus upper A left-parenthesis 1 plus r squared right-parenthesis 3rd Row right double arrow normal upper F period normal upper V left-bracket left-parenthesis 1 plus r right-parenthesis minus 1 right-bracket equals upper A left-parenthesis 1 plus r right-parenthesis Superscript upper N plus 1 Baseline minus upper A left-parenthesis 1 plus r right-parenthesis 4th Row right double arrow normal upper F period normal upper V equals StartFraction upper A Over r EndFraction left-bracket left-parenthesis 1 plus r right-parenthesis Superscript upper N Baseline minus 1 right-bracket left-parenthesis 1 plus r right-parenthesis EndLayout

      Hence

StartLayout 1st Row FVIFA Subscript upper A upper D Baseline left-parenthesis r comma upper N right-parenthesis equals FVIFA left-parenthesis r comma upper N right-parenthesis times left-parenthesis 1 plus r right-parenthesis EndLayout

      Note 6: It should be reiterated that the future value of an N period annuity due is greater than that of an N period annuity if both the values are computed at time N that is after N periods. The future value of an annuity due as computed at time N − 1 will be identical to that of an ordinary annuity as computed at time N.

      EXAMPLE 2.19

      In the case of Mathew's MetLife policy, the cash value at the end of 25 years can be calculated as follows.

StartLayout 1st Row 1st Column Blank 2nd Column FVIFA left-parenthesis 8 comma 25 right-parenthesis equals 73.1059 2nd Row 1st Column Blank 2nd Column FVIFA Subscript upper A upper D Baseline left-parenthesis 8 comma 25 right-parenthesis equals 73.1059 times 1.08 equals 78.9544 EndLayout

      Thus the cash value of the annuity due is:

2 comma 500 times 78.9544 equals dollar-sign 197 comma 386

      An annuity that pays forever is called a perpetuity. The future value of a perpetuity is obviously infinite. But it turns out that a perpetuity has a finite present value. The present value of an annuity that pays for N periods is

normal upper P period normal upper V equals StartFraction upper A Over r EndFraction left-bracket 1 minus StartFraction 1 Over left-parenthesis 1 plus r right-parenthesis Superscript upper N Baseline EndFraction right-bracket

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