# Compound Interests Sequence Prediction

## 🚨🚧🚧🚧 Work in progress 🚧🚧🚧🚨

Iterations Val. @@V_i \%@@ Invest. @@I_i@@ Com. @@C_i \%@@ Delay @@\delta_i@@ Rate @@R_i \%@@ Divis. @@D_i@@ Row @@k@@ Int. @@J_k@@ Del. @@J'_k@@ Tot. i. @@T_k@@ Am. @@A_k@@ Profit @@P_k@@
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## Complete results

@@k@@ Int. @@J_k@@ Del. @@J'_k@@ Tot. i. @@T_k@@ Am. @@A_k@@ Profit @@P_k@@

## Explanations

The table on the top of the page sets all parameters for the computation and also display short results. (Put mouse pointer on some elements to see corresponding tooltip.) The first row defines the initial investment @@I_0@@. Each next row defines a succession of a given number of iterations. For each iteration @@k@@, the previous amount @@A_{k-1}@@ is added by a constant new investment @@I_k@@ and by the compound percentage @@C_k@@ of the interest computed on a previous amount @@J_{k-\delta_k} = A_{k-1-\delta_k} \frac{R_{k-\delta_k}}{100 D_{k-\delta_k}}@@ (with an eventual delay of @@\delta_k@@ iterations).
• Amount @@ \ \begin{array}[t]{|l} A_0 = I_0\\ A_k = \underbrace{A_{k-1}}_{\begin{array}{c}\text{previous}\\\text{amount}\\\end{array}} + \underbrace{I_k}_{\begin{array}{c}\text{current new}\\\text{investment}\\\end{array}} + \underbrace{\frac{C_k}{100} A_{k-1-\delta_k} \frac{R_{k-\delta_k}}{100 D_{k-\delta_k}}}_{\begin{array}{c}\text{current part of}\\\text{compound interest}\\\end{array}} = A_{k-1} + I_k + \frac{C_k}{100} J_{k-\delta_k}\\ \end{array} @@

The current amount @@A_k@@ is the sum of the previous amount @@A_{k-1}@@, a current new investment @@I_k@@, and a compound percentage @@C_k@@ of a previous amount interest @@J_{k-\delta_k} = A_{k-1-\delta_k} \frac{R_{k-\delta_k}}{100 D_{k-\delta_k}}@@.

When @@C_k = 0@@, there is a simple interest: nothing of the interest is reinjected to the amount, i.e. all is really a profit.
When @@C_k = 100@@, there is a compound interest: all the interest is reinjected to the amount (with a delay of @@\delta_k@@ iterations), and so there is no real profit.
When @@0 < C_k < 100@@, only @@C_k@@ percents are used as a compound interest, the remain @@100 - C_k@@ percents are profit.

When @@\delta_k = 0@@, there is no delay, i.e. the interest is computed on the immediate previous amount @@A_{k-1}@@.
When @@\delta_k > 0@@, the interest is computed on older amount @@A_{k-1-\delta_k}@@. That models a delay of @@\delta_k@@ iterations before injection on the interest.

Divisor @@D_k@@ is use to facilitate an interest rate percentage @@R_k@@ given for a period different than the period corresponding to one iteration. For example, for a @@R_k@@ given for one year and an iteration period of one day, then set @@D_k = 365@@.

• Current all interest (regardless compound percentage) for one iteration @@ \ \begin{array}[t]{|l} J_0 = 0\\ J_k = A_{k-1} \frac{R_k}{100 D_k}\\ \end{array} @@
• Delayed interest @@ \ \begin{array}[t]{|l} J'_0 = 0\\ J'_k = \sum\limits_{i + \delta_i = k} J_i = \text{sum of all }J_i\text{ such that }i + \delta_i = k\\ \end{array} @@

When delay @@\delta_k = 0@@ for all @@k@@, then @@J'_k = J_k@@ for all @@k@@.

• Total invested @@ \ \begin{array}[t]{|l} T_0 = I_0\\ T_k = T_{k-1} + I_k = I_0 + I_1 + I_2 + \cdots + I_k = \sum\limits_{i=0}^k I_i\\ \end{array} @@
@@I_0@@ is the initial investment.
For each @@k@@, the current total invested @@T_k@@ is the sum of the previous total invested @@T_{k-1}@@ and the current new investment @@I_k@@.
• Profit @@ \ \begin{array}[t]{|l} P_0 = 0\\ P_k = \underbrace{P_{k-1}}_{\begin{array}{c}\text{previous}\\\text{profit}\\\end{array}} + \underbrace{\frac{V_k}{100} I_k}_{\begin{array}{c}\text{current part of}\\\text{investment as}\\\text{kept profit value}\\\end{array}} + \underbrace{\frac{100 - C_k}{100} A_{k-1} \frac{R_k}{100 D_k}}_{\begin{array}{c}\text{current part of}\\\textit{not}\text{ compound interest}\\\end{array}} = P_{k-1} + \frac{V_k}{100} I_k + \frac{100 - C_k}{100} J_k\\ \end{array} @@
The current profit @@P_k@@ is the sum of the previous profit @@P_{k-1}@@, the percentage @@V_k@@ of the current new investment @@I_k@@ that is considered as a kept profit value, and the part of the previous amount interest @@J_k = A_{k-1} \frac{R_k}{100 D_k}@@ not reinjected to the amount.
When @@V_k = 0@@, all the investment is considered lost.
When @@V_k = 100@@, all the investment is considered as kept profit value.

## Formulas to compute directly the result after @@k@@ iterations under some circumstances

For starting amount @@\alpha \in \mathbb{R}@@, rate @@\tau \in \mathbb{R}@@ and number of iterations @@k \in \mathbb{N}@@.
• Simple interest on constant amount:
Amount @@A_k = \alpha \Rightarrow@@ Profits @@P_k = k \tau \alpha@@
• Evolution of the amount with compound interest:
@@\left.\begin{array}{ll} A_0 & = \alpha\\ A_{k+1} & = A_k + \tau A_k = (1 + \tau) A_k\\ \end{array}\right\} \Rightarrow A_k = (1 + \tau)^k \alpha@@
• Evolution of the amount with compound interest and perpetual adding of starting amount:
@@\left.\begin{array}{ll} A_0 & = \alpha\\ A_{k+1} & = A_k + \tau A_k + \alpha = (1 + \tau) A_k + \alpha\\ \end{array}\right\} \Rightarrow A_k = \alpha \sum\limits_{i=0}^k (1 + \tau)^i = \frac{(1 + \tau)^{k+1}\,-\,1}{\tau} \alpha@@ (when @@\tau \ne 0@@)