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
isbn: 9781119816638
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We can also derive the equivalent nominal rate if the effective rate is given.
We have already seen how to convert a quoted rate to an effective rate. We will now demonstrate how the rate to be quoted can be derived based on the desired effective rate.
Assume that HSBC Bank wants to offer an effective annual rate of 12% per annum with quarterly compounding. The question is what nominal rate of interest should it quote?
In this case, i = 12%, and m = 4. We have to calculate the corresponding quoted rate r.
Thus, a quoted rate of 11.49% with quarterly compounding is tantamount to an effective annual rate of 12% per annum. Hence HSBC should quote 11.49% per annum.
PRINCIPLE OF EQUIVALENCY
Two nominal rates of interest compounded at different intervals of time are said to be equivalent if they yield the same effective interest rate for a specified measurement period.
Assume that ING Bank is offering 10% per annum with semiannual compounding. What should be the equivalent rate offered by a competitor, if it intends to compound interest on a quarterly basis?
The first step in comparing two rates that are compounded at different frequencies is to convert them to effective annual rates. The effective rate offered by ING is:
The question is, what is the quoted rate that will yield the same effective rate if quarterly compounding were to be used?
Hence 10% per annum with semiannual compounding is equivalent to 9.88% per annum with quarterly compounding, because in both cases the effective annual rate is the same.
CONTINUOUS COMPOUNDING
We know that if a dollar is invested for N periods at a quoted rate of r% per period and if interest is compounded m times per period, then the terminal value is given by the expression
In the limit as
where e = 2.71828. Known as the Euler number or Napier's constant, e is defined by the expression:
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:
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.
The answers are depicted in Table 2.2. As can be seen, by the time we reach daily compounding, we have almost reached the limiting value.
| 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 |
FUTURE VALUE
We have already encountered the concept of future value in our discussion thus far, although we have not invoked the term. What exactly is the meaning of the future value of an investment? When an amount is deposited for a certain time period at a given rate of interest, the amount that is accrued at the end of the designated period of time is called the future value of the original investment.
For instance, if we were to invest $P for N СКАЧАТЬ