Fundamentals of Financial Instruments - Sunil K. Parameswaran

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annual rate is the same.

left-parenthesis 1 plus r slash m right-parenthesis Superscript m upper N

      In the limit as m right-arrow proportional-to

left-parenthesis 1 plus r slash m right-parenthesis Superscript m upper N Baseline right-arrow normal e Superscript r upper N

      where e = 2.71828. Known as the Euler number or Napier's constant, e is defined by the expression:

normal e equals upper L i m Subscript n right-arrow infinity Baseline left-parenthesis 1 plus 1 slash n right-parenthesis Superscript n

      This limiting case is referred to as continuous compounding. If r is the nominal annual rate, then the effective annual rate with continuous compounding is er − 1.

      EXAMPLE 2.9

      Nigel Roberts has deposited $25,000 with Continental Bank for a period of four years at 8% per annum compounded continuously. The terminal balance may be computed as:

25 comma 000 times normal e Superscript 0.08 times 4 Baseline equals 25 comma 000 times 1.3771 equals dollar-sign 34 comma 428.19

      Continuous compounding is the limit of the compounding process as we go from annual, to semiannual, on to quarterly, monthly, daily, and even shorter intervals. This can be illustrated with the help of an example.

      EXAMPLE 2.10

      Sheila Norton has deposited $100 with ING Bank for one year. Let us calculate the account balance at the end of the year for various compounding frequencies. We will assume that the quoted rate in all cases is 10% per annum.

Compounding at Various Frequencies
Compounding Interval Terminal Balance
Annual 110.0000
Semi-annual 110.2500
Quarterly 110.3813
Monthly 110.4713
Daily 110.5156
Continuously 110.5171

      For instance, if we were to invest $P for N periods at a periodic interest rate of r%, then the future value of the investment is given by

normal upper F period normal upper V period equals upper P left-parenthesis 1 plus r right-parenthesis Superscript upper N

      The expression (1 + r)N is the amount to which an investment of $1 will grow at the end of N periods, if it is invested at a rate r. It is called the FVIF (Future Value Interest Factor). It depends only on two variables, namely the periodic interest rate, and the number of periods. The advantage of knowing the FVIF is that we can find the future value of any principal amount, for given values of the interest rate and time period, by simply multiplying the principal by the factor. The process of finding the future value given an initial investment is called compounding.

      EXAMPLE 2.11

      Shelly Smith has deposited $25,000 for four years in an account that pays interest at the rate of 8% per annum compounded annually. What is the future value of her investment?

      The factor in this case is given by FVIF left-parenthesis 8 comma 4 right-parenthesis equals left-parenthesis 1.08 right-parenthesis Superscript 4 Baseline equals 1.3605

      Thus, the future value of the deposit is dollar-sign 25 comma 000 times 1.3605 equals dollar-sign 34 comma 012.50

      Note 3: Remember that the value of N corresponds to the total number of interest conversion periods, in case interest is being compounded more than once per measurement period. Consequently, the interest rate used should be the rate per interest conversion period. The following example will clarify this issue.

      EXAMPLE 2.12

      Simone Peters has deposited $25,000 for four years in an account that pays a nominal annual interest of 8% per annum with quarterly compounding. What is the future value of her investment?

      8% per annum for four years is equivalent to 2% per quarter for 16 quarterly periods. Thus the required factor is FVIF(2,16) and not FVIF(8,4).

FVIF left-parenthesis 2 comma 16 right-parenthesis equals left-parenthesis 1.02 right-parenthesis Superscript 16 Baseline equals 1.3728

      Thus the future value of dollar-sign 25 comma 000 is 25 comma 000 times 1.3728 equals dollar-sign 34 comma 320

      Note 4: The FVIF is given in the form of tables in most textbooks, for integer values of the interest rate and number of time periods. If, however, either the interest rate or the number of periods is not an integer, then we cannot use such tables and would have to rely on a scientific calculator or a spreadsheet.

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