Single payment compound amount factor (SPCAF)
The single payment compound amount factor is used to compute the future worth (F) accumulated after “n” years from the known present worth (P) at a given interest rate ‘i’ per interest period. It is assumed that the interest period is in years and the interest is compounded once per interest period.
The known present worth (P), unknown future worth (F) and the total interest period ‘n’ years are shown in Fig. 1.7.
 |
Fig 1.7 Cash flow diagram for ‘known P’ and ‘unknown F’ |
The future worth (F1) accumulated at the end of year 1 i.e. 1st year is given by;
| F1 = P + P X i = P(1+i) | (1) |
The future worth accumulated at the end of year 2 i.e. F2 will be equal to the amount that was accumulated at the end of 1st year i.e. F1 plus the amount of interest accumulated from end of 1st year to the end of 2nd year on F1 and is given by;
| F2 = F1 + F1 X i = F1(1+i) | (2) |
Putting the value of F1 from equation (1) in equation (2), the value of F2 is given by;
| F2 = F1(1+i) = P(1+i)(1+i) = P(1+i)2 | (3) |
Similarly, the future worth accumulated at the end of year 3 i.e. F3 is equal to the amount that was accumulated at the end of 2nd year i.e. F2 plus the amount of interest accumulated from end of 2nd year to the end of 3rd year on F2 and is given by;
| F3 = F2 + F2 X i = F2(1+i) | (4) |
Putting the value of F2 from equation (3) in equation (4), the value of F3 is given by;
| F3 = F2(1+i) = P(1+i)2(1+i) = P(1+i)3 | (5) |
Similarly, the future worth accumulated at the end of year 4 i.e. F4 is given by;
| F4= F3(1+i) = P(1+i)3(1+i) = P(1+i)4 | (6) |
Thus the generalized formula for the future worth at the end of ‘n' years is given by;
| F = P(1+i)n | (7) |
The factor (1+i)n in equation (7) is known as the single payment compound amount factor (SPCAF) . Thus if the value of ‘P' is known, one can easily calculate the future worth F at the end of ‘n' years at interest rate of ‘i' (per year) by multiplying the present worth with the single payment compound amount factor.