- Retained earnings for each year follow the recursive formula \(RE_n = RE_{n-1} + NI_n - D_n\), starting from \(RE_0 = 0\).
- Yearly calculations result in Year 1: \(\$500\), Year 2: \(\$1,800\), Year 3: \(\$3,100\), Year 4: \(\$6,300\), and Year 5: \(\$11,000\).
- Final balances: Year 1 \(\boxed{\$500}\), Year 2 \(\boxed{\$1,800}\), Year 3 \(\boxed{\$3,100}\), Year 4 \(\boxed{\$6,300}\), Year 5 \(\boxed{\$11,000}\).
Explanation
- Analysis of the Retained Earnings Calculation
We analyze the retained earnings for Waterhouse Manufacturing over its first five years. The retained earnings at the end of any given year (\(RE_{end}\)) are determined by the retained earnings from the previous year (\(RE_{begin}\)), plus the net income for the current year (\(NI\)), minus the dividends paid during the year (\(D\)). The recurrence relation is:
\[RE_{end} = RE_{begin} + NI - D\]
The initial condition is \(RE = 0\) at the beginning of Year 1. We will calculate the cumulative balance iteratively for each year.
- Step-by-Step Calculation of Retained Earnings
- Year 1: Starting with \(\$0\), the net income is \(\$2,100\) and dividends are \(\$1,600\).
\(RE_1 = 0 + 2,100 - 1,600 = 500\).
- Year 2: Starting with the Year 1 balance of \(\$500\), net income is \(\$2,400\) and dividends are \(\$1,100\).
\(RE_2 = 500 + 2,400 - 1,100 = 1,800\).
- Year 3: Starting with the Year 2 balance of \(\$1,800\), net income is \(\$3,400\) and dividends are \(\$2,100\).
\(RE_3 = 1,800 + 3,400 - 2,100 = 3,100\).
- Year 4: Starting with the Year 3 balance of \(\$3,100\), net income is \(\$5,300\) and dividends are \(\$2,100\).
\(RE_4 = 3,100 + 5,300 - 2,100 = 6,300\).
- Year 5: Starting with the Year 4 balance of \(\$6,300\), net income is \(\$7,300\) and dividends are \(\$2,600\).
\(RE_5 = 6,300 + 7,300 - 2,600 = 11,000\).
- Final Results
Based on the calculations, the retained earnings for each year are as follows:
Year 1: \(\boxed{\$500}\)
Year 2: \(\boxed{\$1,800}\)
Year 3: \(\boxed{\$3,100}\)
Year 4: \(\boxed{\$6,300}\)
Year 5: \(\boxed{\$11,000}\)
Examples
In accounting, calculating retained earnings is essential for understanding how much of a company's profit is being reinvested into its own operations rather than being paid out as dividends to shareholders. For example, a startup might keep all its earnings to fund new product development, while a mature company might distribute a larger portion as dividends. By tracking these figures, businesses can make informed decisions about future investments, growth strategies, and overall financial health.