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Understanding how to calculate the Return on Investment (ROI) is essential for any investor wishing to measure the efficiency of an investment. ROI is a simple ratio that compares the gain from an investment relative to its cost. The basic calculation involves subtracting the initial value of the investment from the final value, then dividing this net return by the cost of the investment, and finally multiplying by 100 to get a percentage.

To illustrate, let's consider our textbook example where Investment G had an up-front cost of \(\$30,000\) and returned \(\$55,000\) after six years. The ROI for Investment G can be calculated as follows:

\( ROI_{G} = \left(\frac{55,000 - 30,000}{30,000}\right) \times 100 = 83.33\% \)

Similarly for Investment H, with the same up-front cost and a return of \(\$90,000\) in 11 years, the ROI is:

\( ROI_{H} = \left(\frac{90,000 - 30,000}{30,000}\right) \times 100 = 200\% \)

By comparing the ROI percentages, we can gauge which investment would be more profitable. However, it's important to remember that ROI does not take into account the time value of money, which is a crucial factor when comparing investments over different periods.

Financial analysis is a broad field that encompasses the assessment of financial statements and investment potential to make informed economic decisions. In the context of ROI, financial analysis becomes essential in understanding not just the raw return numbers, but also the quality and risks associated with an investment.

As part of the financial analysis, one should consider the liquidity of the investment, the risk profile, the time horizon, and potential alternative uses of the capital. For example, the exercise provided compares two investments solely on their raw returns without considering risk or the time value of money.

A more comprehensive financial analysis might involve using discounted cash flow methods or net present value calculations that could alter the attractiveness of each investment. In the case of our investments, G and H, the longer maturity of Investment H might require a risk premium or a discount rate applied to adjust for time preferences of money.

Capital budgeting is the process of evaluating and selecting long-term investments that are aligned with the strategic goals of a firm or individual's financial strategy. It involves comparing the expected returns of potential investments against their costs, incorporating risk, and the time value of money.

Key techniques used in capital budgeting include Net Present Value (NPV), Internal Rate of Return (IRR), and Payback Period, among others. These methods are designed to help investors determine the viability and profitability of potential investments over a set time horizon.

Applying capital budgeting to the two investments from our example, an investor would need to estimate future cash flows, determine an appropriate discount rate to calculate NPV, or calculate IRR to see if it exceeds their required rate of return. This analysis helps in understanding not just the return on investment from a numerical perspective, but also how it fits into the larger financial plan and time frame.