Therefore, the value of the bond is the face or par of the bond plus the present worth of this annuity, i.e.:
| (6) V = | P + | (C - Pr)(Rⁿ - 1) |
| rRⁿ |
This formula when simplified reduces to
| Pr + C(Rⁿ - 1) |
| rRⁿ |
which is identical with formula (5) and, of course, values the bond at the same amount.
Third Method. Valuation by either of the methods above is rather complex, requiring for easy solution the use of logarithm or compound interest tables. A third method, requiring only the ordinary arithmetic processes is sometimes used, although very burdensome when the periods are numerous. This is sometimes called the periodic method because working backwards from par, the value at the next preceding period is determined, and from that the next is found, and so on till the value at the desired period is reached. Reference to the second amortization schedule shown on page 270 will help in understanding the process. Using the same terminology as above and this additional:
- P₁ = the first preceding value above par
- ($1,004.90 in the schedule)
- P₂ = the second preceding value above
- par ($1,009.71 in the schedule)
- Etc.
formulas 7 to 9 below may be developed.
It was noted that any particular value of the bond is found by subtracting from the next preceding value the difference between the nominal and effective incomes. The nominal income is always based on par, and the effective on the value at the last interest period. For example, in the schedule referred to: $1,004.90-($25-$20.10) = $1,000, the $20.10 being the effective income, 2% on $1,004.90. Using symbols the above equation may be generalized as follows:
| (7)P₁ - (Pc - P₁r) = P, whence P₁ = | P + Pc | and, |
| 1 + r | ||
| (8)P₂ - (Pc - P₃r) = P₁, whence P₂ = | P + Pc | and, generalized, |
| 1 + r | ||
| (9)Pₙ - (Pc - Pₙr) = Pₙ₋₁, whence Pₙ = | Pₙ₋₁ + Pc | |
| 1 + r |