© 2000 John Petroff
|
|
© 2000 John Petroff |
1)- compunding and discounting single amount
Compounding menambah bunga ke nilai pokok dari setiap periode untuk memberi nilai masa depan (FV) dari Jumlah A yang ditanam pada tarif bunga i atas t periode waktu
FVt = A(1+i)t
Another method to calculate future value (especially popular before the invention of hand calculators) is to use precalculated products (1+i)t which are known as future value factors (FVF). These future value factors are listed in tables found in all actuarial and many finance manuals. The formula becomes
FV = A.FVFi,t
|
For instance, an amount of $1,000 placed in a savings account earning an interest rate of 10% per year for a period of two years, would grow to $1,100 the first year ($1,000 of principals plus 10% of $1,000, or $100 in interest), and to $1,210 the second year ($1,100 of principal plus 10% of $1,100, or $110 in interest). Here the future value factor is (1+0.1)2, or 1.21, and the future value can be obtained with the formula FV = 1000 (1 + 0.1)2= 1000 x 1.21 = $1,210 |
In discounting, the present value is obtained by diminishing the principal by an interest which is equal to the principal times the discount rate, and which is charged in each period until amount A is paid or received. The amount A is divided (rather than multiplied as it is in compounding) by t times (1 + i)
PV = A/(1+i)t or PV = A(1+i)-t
Or one may use present value factors listed in present value tables.
PV = A.PVFi,t
|
For instance, an amount of $1,000 received 2 years from now and subject to a discounting of 10% each year, would have a present value, or would be worth, $826.45 today PV = 1000 / (1 + 0.1)2 = 1000 / 1.21 = $ 826.45 Using tables, one finds that the present value factor for an interest rate of 10% and two years, is .82645 PV = 1000 x .82645 = $826.45 |
| The problem is sometimes presented in a reversed fashion: what amount must be placed today at an interest rate of 10% to allow a payment of $1,000 two years from today? The answer is naturally $826.45. |
When an amount is held or owed for less than one entire year, the interest to be added or charged is for a fraction of a year.
|
For instance, a 180 day, $10,000 face value treasury bill is purchased with a market yield of 10% for a price of TBV = 10000 / (1 + (0.10(180)/(360) ) ) = 10000 / ( 1 + 0.05) = $9,523.81 |
Note that, in the pricing of U.S. securities purchased at a discount, as in the example above, the number of days in a year is rounded to 360, and the number of days in a month is rounded to 30, by convention.
Discounting, as well as compounding, are normally performed on an annual basis. In some cases the interest owed or earned is recorded for periods shorter than a year, not just one time as in the example above, but on a regular basis. When a shorter period of time than a full year is used for compounding or discounting, the annual interest rate is divided, and the number of periods is multiplied by the number of times the interest is paid or charged per year. For instance, some savings accounts used to be credited quarterly for the interest earned during the quarter; then the interest rate is one fourth of the annual interest rate, and the compounding and discounting occur four times more often.
|
In the earlier example, the amount that needs to be placed today to generate $1,000 two years from now, with quarterly compounding, becomes a present value of PV0 = 1000 / (1 + (0.1 / 4))2x4= 1000 / (1 + 0.025)8 = 1000 / 1.2184 = 1000 x .8207485 = $820.75 Thus, a little less than the previous $826.45 is required with quarterly compounding. |
The general formulas for compounding and discounting a single amount when interest is earned or charged m times a year, become
FVt = A(1+i/m)mt
and
PV0 = A/(1+i/m)mt
Many banks today use daily compounding for credit cards and other accounts. In that case, the intraannual compounding factor m is 365.
|
We continue with the example of $1,000 needed two years from now. With daily compounding present value is PV0 = 1000 / (1 + (0.1 / 365 ))2x365= 1000 / (1 + 0.000274)730 = 1000 / 1.22137 = 1000 x 0.81875 = $818.75 Thus, with daily compounding even less is needed than with quarterly compounding. |
In some cases, continuous compounding is used with the help of natural (or Naperian) logarithms, and the present value formula becomes
PV0 = A / e it
where e is the natural logarithm base 2.7182818.
|
We complete the comparison in our example of $1,000 needed two years from now with continuous compounding PV0 = 1000 / e0.1x2 = 1000 / 1.2214027 = 1000 x 0.81873 = $818.73 The calculation shows that daily compounding gives almost the same result as continuous compounding. |
In practice, tables of present value and future value factors which have been calculated by actuaries and recorded in books, are sometimes still used. Today, most operations can be accomplished with an ordinary hand calculator or on a computer. For complex repayment schedules of annuities some may find the use of specialized tables convenient and visually comfortable. But, even in this case, however, the use of a financial calculator can be faster.
See review questions Q-2C1.1 through Q-2C1.18.
| Previous: Discounting |
|
Next: 2-Annuities |