Hi Mickenzi!
We can solve this problem using the formulas for the present value and accumulated amount of a continuous income stream. Let's work through both calculations!
Step 1: Identify the given information
We are given:
Rate of money flow: f(t) = 390 dollars per year
Time period: T = 3 years
Annual interest rate: r = 10% = 0.10
Compounding: Continuous
Since the money flows in continuously, we use integrals rather than the standard compound interest formula for a single deposit.
Step 2: Calculate the present value
The present value tells us how much the future income stream is worth today.
The formula is:
PV = ∫₀ᵀ f(t)e^(-rt) dt
Substitute the given values:
PV = ∫₀³ 390e^(-0.10t) dt
Factor out the constant:
PV = 390 ∫₀³ e^(-0.10t) dt
Integrate:
PV = 390[-10e^(-0.10t)]₀³
Evaluate at the limits:
PV = -3900[e^(-0.30) - 1]
Rewrite:
PV = 3900(1 - e^(-0.30))
Using a calculator:
e^(-0.30) ≈ 0.740818
Therefore:
PV = 3900(1 - 0.740818)
PV ≈ 3900(0.259182)
Present Value ≈ $1,010.81
Step 3: Calculate the accumulated amount
The accumulated amount represents the total value of the continuous income stream at the end of the three-year period, including interest earned.
The formula is:
A = ∫₀ᵀ f(t)e^[r(T-t)] dt
Substitute:
A = ∫₀³ 390e^[0.10(3-t)] dt
We can simplify this using the formula:
A = (390/0.10)(e^(0.10 × 3) - 1)
Therefore:
A = 3900(e^0.30 - 1)
Using a calculator:
e^0.30 ≈ 1.349859
Substitute:
A = 3900(1.349859 - 1)
A = 3900(0.349859)
Accumulated Amount ≈ $1,364.45
Step 4: Understand the difference
Without any interest, the total amount of money received over three years would be:
390 × 3 = $1,170
However, the timing of the money matters.
The present value is less than $1,170 because future payments are discounted to their value today.
The accumulated amount is greater than $1,170 because payments received earlier earn interest until the end of the three-year period.
Final Answers
1. Present Value = $1,010.81
2. Accumulated Amount = $1,364.45
Key Takeaway
For a constant continuous income stream, we can use these simplified formulas:
Present Value:
PV = (R/r)(1 - e^(-rT))
Accumulated Amount:
A = (R/r)(e^(rT) - 1)
Where:
R = Constant annual income rate
r = Annual interest rate
T = Number of years
The present value measures what the income stream is worth today, while the accumulated amount measures what it will be worth at the end of the investment period.
I hope this helps!
