To find the accumulated present value $PV$ of an investment with continuous money flow, the formula is:

$PV = \int_0^T P(t) e^{-rt} \, dt$

Where:

• $P(t)$: Continuous money flow (constant at $8,000 per year)
• $r$: Interest rate (0.6% = 0.006 as a decimal)
• $T$: Total time period (6 years)

For a constant flow $P(t) = 8000$, the formula simplifies to:

$PV = 8000 \int_0^6 e^{-0.006t} \, dt$

Solve the integral:

The integral of $e^{-rt}$ is:

$\int e^{-rt} \, dt = -\frac{1}{r} e^{-rt}$

Using this, we calculate:

$PV = 8000 \left[ -\frac{1}{0.006} e^{-0.006t} \right]_0^6$

Evaluate at bounds:

$PV = 8000 \left( -\frac{1}{0.006} e^{-0.006 \cdot 6} + \frac{1}{0.006} e^{-0.006 \cdot 0} \right)$

$PV = 8000 \left( -\frac{1}{0.006} e^{-0.036} + \frac{1}{0.006} \right)$

Calculate $e^{-0.036}$: $e^{-0.036} \approx 0.96485$

Substitute: $PV = 8000 \left( -\frac{1}{0.006} \cdot 0.96485 + \frac{1}{0.006} \right)$

$PV = 8000 \left( -160.8083 + 166.6667 \right)$

$PV = 8000 \cdot 5.8584$

$PV \approx 46,867.20$

Final Answer:

The accumulated present value is approximately $46,867.20.

Would you like further explanation or have any questions? Here are 5 related questions:

1. How does continuous compounding differ from other compounding methods?
2. Can this formula be adapted for non-constant money flows?
3. How would changes in the interest rate affect the present value?
4. What is the significance of the exponential decay factor in this context?
5. How would the result differ if the investment period were extended to 10 years?

Tip: When solving for continuously compounded values, always ensure that the exponential factor is correctly computed for accuracy.