Cost Accounting (14th Edition)

INTEREST TABLES 841
2. Substitute: S 3 = 1 + (1.08) 1 + (1.08) 2
3. Multiply (2) by (1 + r):
(1.08)S 3 = (1.08) 1 + (1.08) 2 + (1.08) 3
4. Subtract (2) from (3): Note that all 1.08S 3 - S 3 = (1.08) 3 - 1
terms on the right-hand side are
removed except (1.08) 3 in equation (3)
and 1 in equation (2).
5. Factor (4): S 3 (1.08 - 1) = (1.08) 3 - 1
6. Divide (5) by (1.08 - 1):
S 3 = (1.08)3 - 1
= (1.08)3 - 1
= 0.2597
1.08 - 1 .08 0.08
7. The general formula for the amount of
S
an ordinary annuity of $1 becomes:
n = (1 + r )n - 1 Compound interest
or
r
Rate
= 3.246
This formula is the basis for Table 3, page 844. Check the answer in the table.
Table 4—Present Value of an Ordinary Annuity of $1
Using the same example as for Table 3, we can show how the formula of P n
, the present
value of an ordinary annuity, is developed.
End of Year 0 1 2 3
1st payment 1,000
1
(1.08)
= $ 926.14 $1,000
2nd payment 1,000
2
(1.08)
= $ 857.52 $1,000
3rd payment 1,000
3
(1.08)
= $ƒƒ794.00 $1,000
Total present value $2,577.66
We can develop the general formula for P n
by using the preceding example as a basis
where n = 3 and r = 0.08:
1.
P 3 =
2. Substitute:
P 3 = 1
1
3. Multiply by
1.08 :
1
1 + r + 1
(1 + r ) + 1
2 (1 + r ) 3
1.08 + 1
(1.08) 2 + 1
(1.08) 3
P 3
1
1.08 = 1
(1.08) 2 + 1
(1.08) 3 + 1
(1.08) 4
1
4. Subtract (3) from (2):
P 3 - P 3
1.08 = 1
1.08 - 1
(1.08) 4
1
5. Factor (4):
P 3 a1 -
(1.08) b = 1
1.08 c1 - 1
(1.08) d 3
6. or
P 3 a .08
1.08 b = 1
1.08 c1 - 1
(1.08) d 3
1.08
7. Multiply by
P 3 = 1
.08 c1 - 1
(1.08) d = .2062 = 2.577
.08 : 3 .08
The general formula for the present value of an annuity of $1.00 is as follows:
P n = 1 r c1 - 1 Compound discount
d =
n
(1 + r ) Rate
The formula is the basis for Table 4, page 845. Check the answer in the table. The present
value tables, Tables 2 and 4, are used most frequently in capital budgeting.
The tables for annuities are not essential. With Tables 1 and 2, compound interest and
compound discount can readily be computed. It is simply a matter of dividing either of
these by the rate to get values equivalent to those shown in Tables 3 and 4.