The continuously compounded interest rate is 9%.
You invest $1,140 at this rate. What is the investment worth after seven years?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
What is the PV of $6.4 million to be received in ten years?
Note: Do not round intermediate calculations. Round your answer to the nearest whole dollar amount. Enter your answer in dollars, not millions of dollars.
What is the PV of a continuous stream of cash flows, amounting to $2,140 per year, starting immediately and continuing for 17 years?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
In the realm of finance, understanding the concepts of future value, present value, and continuous cash flows is crucial for making informed investment decisions. In this essay, we will explore these concepts and solve specific financial problems using them. We’ll use the given continuously compounded interest rate of 9% to perform these calculations.
Future Value: The future value (FV) of an investment can be calculated using the formula:
��=�����
Where: FV = Future Value PV = Present Value r = Annual interest rate (as a decimal) t = Time period (in years) e = Euler’s number (approximately 2.71828)
In the first scenario, you invest $1,140 at a continuously compounded interest rate of 9% for seven years. Plugging the values into the formula:
��=1140×�0.09×7≈2245.17
So, the investment will be worth approximately $2,245.17 after seven years.
The present value (PV) of a future cash flow can be determined using the formula:
��=�����
In the second scenario, we need to find the present value of $6.4 million to be received in ten years:
��=6,400,000�0.09×10≈3,042,715.78
Rounded to the nearest whole dollar, the present value is approximately $3,042,716.
Calculating the present value of a continuous stream of cash flows involves integrating the formula for present value over time. In this case, we have an annual cash flow of $2,140, starting immediately and continuing for 17 years.
��=∫0172140�0.09���
Solving this integral yields:
��≈23,337.16
So, the present value of the continuous stream of cash flows is approximately $23,337.16.
Understanding the concepts of future value and present value, as well as being able to calculate them, is essential for making sound financial decisions. In the scenarios discussed above, we used these concepts to determine the future value of an investment, the present value of a future cash flow, and the present value of a continuous stream of cash flows. These calculations can help individuals and businesses make informed choices when it comes to investments, loans, and financial planning.
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