Fundamentals of Financial Instruments. Sunil K. Parameswaran
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Название: Fundamentals of Financial Instruments

Автор: Sunil K. Parameswaran

Издательство: John Wiley & Sons Limited

Жанр: Ценные бумаги, инвестиции

Серия:

isbn: 9781119816638

isbn:

СКАЧАТЬ alt="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 StartLayout 1st Row 1st Column Blank 2nd Column upper N right-arrow proportional-to 2nd Row 1st Column Blank 2nd Column StartFraction 1 Over left-parenthesis 1 plus r right-parenthesis Superscript upper N Baseline EndFraction right-arrow 0 EndLayout

      Thus, the present value of a perpetuity is A/r.

      EXAMPLE 2.20

      Let us consider a financial instrument that promises to pay $2,500 per year for ever. If investors require a 10% rate of return, the maximum amount they would be prepared to pay may be computed as follows.

normal upper P period normal upper V period equals 2 comma 500 slash 0.10 equals dollar-sign 25 comma 000

      Thus, although the cash flows are infinite, the security has a finite value. This is because the contribution of additional cash flows to the present value becomes insignificant after a certain point in time.

      The amortization process refers to the process of repaying a loan by means of regular installment payments at periodic intervals. Each installment includes payment of interest on the principal outstanding at the start of the period and a partial repayment of the outstanding principal itself. In contrast, an ordinary loan entails the payment of interest at periodic intervals, and the repayment of principal in the form of a single lump-sum payment at maturity. In the case of an amortized loan, the installment payments form an annuity whose present value is equal to the original loan amount. An Amortization Schedule is a table that shows the division of each payment into a principal component and an interest component and displays the outstanding loan balance after each payment.

      Take the case of a loan which is repaid in N installments of $A each. We will denote the original loan amount by L, and the periodic interest rate by r. Thus this is an annuity with a present value of L, which is repaid in N installments.

upper L equals upper A times PVIFA left-parenthesis r comma upper N right-parenthesis 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

      The interest component of the first installment

StartLayout 1st Row 1st Column Blank 2nd Column equals r times 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 2nd Row 1st Column Blank 2nd Column equals upper A left-bracket 1 minus StartFraction 1 Over left-parenthesis 1 plus r right-parenthesis Superscript upper N Baseline EndFraction right-bracket EndLayout