Larry J. Goldstein, David C. Lay, David I. Schneider
Chapter 9: Problem 1
URL copied to clipboard! Now share some education!Find the present value of a continuous stream of income over 5 years when the rate of income is constant at \(\$ 35,000\) per year and the interest rate is \(7 \%\).
The rate of income is given as a constant \( \$35,000 \) per year, the time period is 5 years, and the interest rate is \( 0.07 \).
The formula to find the present value (PV) of a continuous stream of income is \(\text
In this case, \(R = 35,000\), \(r = 0.07\), and \(t = 5\). Substitute these values into the formula: \(\text
Substitute the value of \(e^ <-0.35>\) back into the equation: \(\text = \frac \times (1 - 0.705)\). Simplify it: \(\text = 500,000 \times 0.295\). Thus, \(\text = 147,500\).
These are the key concepts you need to understand to accurately answer the question.
The concept of present value (PV) is crucial in finance, as it allows us to determine the value of a sum of money at the present time, given a specific rate of return. Essentially, PV tells us how much a future sum of money is worth today. This is particularly important when dealing with continuous income streams, such as regular payments over a set period.
The formula to calculate the present value of a continuous income stream is: \(\text = \frac \times (1 - e^)\). Here, \(R\) stands for the rate of income per year, \(r\) is the annual interest rate (expressed as a decimal), and \(t\) is the total time period in years. By using this formula, we can discount future income back to its value today, which helps in making informed financial decisions.
A continuous income stream represents a flow of money received steadily over time, rather than in discrete intervals. This is in contrast to lump sum payments or periodic payments like monthly salaries.
To calculate the present value of such an income stream, we need to use a specialized formula because the income is received continuously throughout the period. In our example, the income stream is \(\text35,000\) per year. Continuous income streams are common in scenarios like rent income, royalties, or certain types of investments where payments are received on an ongoing basis. These calculations help determine the value at the starting point of receiving these continuous payments.
Exponential decay plays a significant role when calculating the present value of continuous income streams. Exponential decay refers to the process where quantities decrease at a rate proportional to their current value, often described by the mathematical constant \(e\).
In the calculation for present value, the term \(e^\) represents the decay over time due to the interest rate. For example, in the given problem, \(e^ \approx 0.705\). This indicates how much the income to be received in the future is discounted back to its present value, affected by the interest rate over the specified period. Exponential decay helps us understand the diminishing value of future earnings when discounted to today’s terms.
Interest rates are a fundamental part of financial computations, as they represent the cost of borrowing or the gain from investing money. In present value calculations, the interest rate, often expressed as a decimal, helps discount future sums to their value today.
In our exercise, the interest rate \(r\) is \(0.07\), or 7%. This rate influences how much future income is worth today. By using the formula \(\text = \frac \times (1 - e^)\), we incorporate the effect of the interest rate over time \(t\). Correctly applying the interest rate in these calculations ensures accurate financial assessments and decisions.
