Fundamentals of Financial Instruments. Sunil K. Parameswaran

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Fundamentals of Financial Instruments - Sunil K. Parameswaran


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

      The outstanding balance at the end of the first payment

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

      In general, the interest component of the ‘t’th installment is

upper A left-bracket 1 minus StartFraction 1 Over left-parenthesis 1 plus r right-parenthesis Superscript upper N minus t plus 1 Baseline EndFraction right-bracket

      The principal component of the ‘t’th installment is

StartFraction upper A Over left-parenthesis 1 plus r right-parenthesis Superscript upper N minus t plus 1 Baseline EndFraction

      and the outstanding balance at the end of the ‘t’th payment is

StartFraction upper A Over r EndFraction left-bracket 1 minus StartFraction 1 Over left-parenthesis 1 plus r right-parenthesis Superscript upper N minus t Baseline EndFraction right-bracket

      EXAMPLE 2.21

      Sylvie has borrowed $25,000 from First National Bank and has to pay it back in eight equal annual installments. If the interest rate is 8% per annum on the outstanding balance, what is the installment amount, and what will the amortization schedule look like?

      Let us denote the unknown installment amount by A. We know that

StartLayout 1st Row 25 comma 000 equals StartFraction upper A Over 0.08 EndFraction times left-bracket 1 minus StartFraction 1 Over left-parenthesis 1.08 right-parenthesis Superscript 8 Baseline EndFraction right-bracket 2nd Row right double arrow upper A equals dollar-sign 4 comma 350.37 EndLayout


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Year Payment Interest Principal Repayment Outstanding Principal
0 25,000
1 4,350.37 2,000.00 2,350.37 22,649.63