Compound Interest

Wilson Leoputra

1 Compound interest

Compound interest is a method to calculate future value (FV) based on a set of parameters: present value (PV), interest, and the duration on which the present value is accumulated. The formula for compound interest can be written as:

(1)

where P is the principal (present value), r is the interest (in %), n is the duration the principal is accumulated (it can be day or month or year), and A is the future value. As an example if a client deposit $25000 with compound interest of 10% each year, what is the amount of money he will receive after 5 years? Substituting P = 25000,r = 10%,n = 5 to Eq. 1, we get: A = 25000(1 + 0.1)⁵ = 40262.75. This can also be written as an iterative process as:

Iteration	Future value
1	25000.00 * (1.1) = 27500.00
2	27500.00 * (1.1) = 30250.00
3	30250.00 * (1.1) = 33275.00
4	33275.00 * (1.1) = 36602.50
5	36602.50 * (1.1) = 40262.75

2 Compound interest added a constant yearly

In some situations, we need a formula to compute the compound interest with a constant being accumulated each year. Let b be the constant parameter, e.g. b = 25000, we obtain the following value sequences:

Iteration	Future value
1	25000.00 * (1.1) = 27500.00
2	(27500.00 + 25000) * (1.1) = 57750.00
3	(57750.00 + 25000) * (1.1) = 91025.00
4	(91025.00 + 25000) * (1.1) = 127627.50
5	(127627.50 + 25000) * (1.1) = 167890.25

The question is: what is the formula to generate these sequences?
The step by step derivation is shown below.

C0 = P

C1 = P * (1 + r) = P + Pr

C2 = (C1 + b)(1 + r)	= C1 + C1r + b + br
	= (P + Pr) + (P + Pr)r + b + br
	= P + Pr + Pr + Pr2 + b + br
	= P + 2Pr + Pr2 + b + br
	= P(1 + 2r + r2) + b + br
	= P(1 + r)2 + (r + 1)b

C3 = (C2 + b)(1 + r)	= C2 + C2r + b + br
	= (P + 2Pr + Pr2 + b + br) + (Pr + 2Pr2 + Pr3 + br + br2) + b + br
	= (P + 3Pr + 3Pr2 + Pr3) + br2 + 2b + 3br
	= P(1 + r)3 + (r2 + 3r + 2)b

C4 = (C3 + b)(1 + r)	= C3 + C3r + b + br
	= (P + 3Pr + 3Pr2 + Pr3 + br2 + 2b + 3br)+
	(Pr + 3Pr2 + 3Pr3 + Pr4 + br3 + 2br + 3br2) + b + br
	= (P + 4Pr + 6Pr2 + 4Pr3 + Pr4) + 3b + 6br + 4br2 + br3
	= P(1 + r)4 + (r3 + 4r2 + 6r + 3)b

C5 = (C4 + b)(1 + r)	= C4 + C4r + b + br
	= P(1 + r)5 + (3b + 6br + 4br2 + br3) + (3br + 6br2 + 4br3 + br4) + b + br
	= P(1 + r)4 + (r4 + 5r3 + 10r2 + 10r + 4)b

From these five sequences, we can see that

C_i = P(1 + r)ⁱ + something × b

The “something” can be plotted as

ith	Something	r = 1	r = 0.1
C2	r + 1	2	1.1
C3	r2 + 3r + 2	6	2.31
C4	r3 + 4r2 + 6r + 3	14	3.64
C5	r4 + 5r3 + 10r2 + 10r + 4	30	5.11

We obtain 2,6,14,30,…, if r = 1. Now we borrow the Geometric Series formula to solve this:

S_n = 

where a = initial value and x = ratio. Table shows that the first sequence takes the value of r + 1, so a = r + 1. If r = 1, the first sequence that we get is 2. If r = 0.1, the first sequence that we get is 1.1 and so on. After trial and error, we found that we can set a = r + 1 and x = r + 1, n = n - 1 in order to obtain the correct output. Substituting these to S_n give us:

S_n =  = 

To proof if this formula is correct, we subsitute n = 2,3,4,… to S_n

S₂ = = 0

S₃ ===   =   = r+1

S₄ = = = = = r² + 3r + 2

S₅ = = = = r³ + 4r² + 6r + 3

Alternatively, the trick is to multiply (C_i ×) + -

C3	=(r2 + 3r + 2) ×
	= + -
	=
	= ((r + 1)3 - (r - 1))

C4	= (r3 + 4r2 + 6r + 3) ×
	=  + -
	=
	=  ((r + 1)4 - (r - 1))

The formula for compound interest with a constant being accumulated each year is: