Concept & How to calculate about Compound interest & Compound Discount.

1. MATHEMATICS OF FINANCE
Compound Interest & Compound Discount

2. Compound Interest
P = Principal at the beginning of the first interest period.
i = Interest rate per conversion period.
P P(1+i) P(1+i)2
P(1+i)3 P(1+i)4
0 1 2 3 4 5
Interest due = Pi
Accumulated value = P + Pi
= P(1+i)
Interest due = (P(1+i))i
Accumulated value = P(1+i) +
(P(1+i))i
= P(1+i)2
Interest due = (P(1+i)2)i
Accumulated value = P(1+i)2 +
(P(1+i)2)i
= P(1+i)3

3. Compound Interest
S = P(1+i)n
S = Accumulated value of P at the end of n interest periods.
(1+i)n = Accumulation factor
The accumulated value S of principal P at rate jm for t years is
S = P(1+i)n = P(1+jm/m)mt = P[(1+jm/m)m]t

4. Compound Interest
Find the compound interest on $1000 for 2 years at 12%
compound semiannually.
At the end of period Interest Accumulated value
1 1000(0.12/2) = 60 1060
2 1060(0.12/2) = 63.6 1123.6
3 1123.6(0.12/2) = 67.416 1191.016
4 1191.016(0.12/2) = 71.46096 1262.47696
The compound interest is 1262.47696 – 1000 = 262.47696
S = P[(1+jm/m)m]t = 1000[(1+0.12/2)2]2 = 1000[(1.06)2]2
= 1000(1.06)4 = 1262.47696

5. Compound Interest
Find the compound interest on $1000 for 2 years at 12%
compound semiannually.
S = P[(1+jm/m)m]t
P = 1000, jm = j2 = 0.12, m = 2, t = 2
= 1000[(1+0.12/2)2]2 = 1000[(1.06)2]2
= 1000(1.06)4 = 1262.47696
The compound interest is 1262.47696 – 1000 = 262.47696

6. Discounted value
P = S(1+i)-n
P = S[(1+jm/m)-m]t
P = Discounted value of S (Present value of S)
(1+i)-n , [(1+jm/m)-m]t = Discounted factor

7. Compound Interest
Find the present value of $8000 due in 5 years at 7%
compounded quaterly.
P = S[(1+jm/m)-m]t
S = 8000, jm = j4 = 0.07, m = 4, t = 5
= 8000[(1+0.07/4)-4]5 = 8000[(1.0175)-4]5
= 8000(1.0175)-20 = 5654.60