1. What is the present value of $3000 receivable in 5 years, assuming a continuously compounded interest rate of 10%? 2. What will be the value in year 7?
Understand the Problem
This question involves calculating present and future values using continuous compounding interest. Part 1 asks for the present value of $3000 to be received in 5 years, and Part 2 asks for the future value of the same initial investment in 7 years. We will use the formulas for present and future value with continuous compounding to solve these.
Answer
Part 1: $2456.19 Part 2: $3969.39
Answer for screen readers
Part 1: The present value is approximately $2456.19.
Part 2: The future value is approximately $3969.39.
Steps to Solve
- Present Value Formula
The formula for present value (PV) with continuous compounding is:
$$ PV = FV \cdot e^{-rt} $$
Where:
-
FV = Future Value
-
r = interest rate (as a decimal)
-
t = time in years
-
e = Euler's number (approximately 2.71828)
- Calculate Present Value
Given FV = $3000, r = 4% = 0.04$, and t = 5 years, substitute these values into the present value formula:
$$ PV = 3000 \cdot e^{-0.04 \cdot 5} $$
$$ PV = 3000 \cdot e^{-0.2} $$
- Evaluate the exponential term
Calculate $e^{-0.2}$:
$e^{-0.2} \approx 0.81873$
- Calculate Present Value (PV)
$$ PV = 3000 \cdot 0.81873 $$
$$ PV \approx 2456.19 $$
- Future Value Formula
The formula for future value (FV) with continuous compounding is:
$$ FV = PV \cdot e^{rt} $$
Where:
-
PV = Present Value
-
r = interest rate (as a decimal)
-
t = time in years
-
e = Euler's number (approximately 2.71828)
- Calculate Future Value
Given PV = $3000, r = 4% = 0.04$, and t = 7 years, substitute these values into the future value formula:
$$ FV = 3000 \cdot e^{0.04 \cdot 7} $$
$$ FV = 3000 \cdot e^{0.28} $$
- Evaluate the exponential term
Calculate $e^{0.28}$:
$e^{0.28} \approx 1.32313$
- Calculate Future Value (FV)
$$ FV = 3000 \cdot 1.32313 $$
$$ FV \approx 3969.39 $$
Part 1: The present value is approximately $2456.19.
Part 2: The future value is approximately $3969.39.
More Information
The continuous compounding formula is derived from the limit of the compound interest formula as the number of compounding periods approaches infinity. This model is often used in theoretical finance.
Tips
A common mistake is using the simple interest formulas instead of continuous compounding formulas. Another is incorrectly substituting the values into the formula or miscalculating the exponential term. Ensure you use the correct formula and double-check your calculations, especially when dealing with exponents.
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