OK I understand the concept present value but I don't understand how to solve for present value in this context:
Assume that you won a local contest for which you gain a continuous income stream at a rate R(t)=5000e^0.1t per year for 5 years and r=0.05. Find present value.
Can anyone help?
To solve for the present value in this scenario, we need to calculate the present value of the continuous income stream. The present value represents the current worth of a future cash flow, taking into account the time value of money.
In this case, we have an income stream with a continuously compounded growth rate of 0.1 per year, and the rate at which we discount the future cash flows is 0.05 per year. To calculate the present value, we can use the formula:
PV = ∫ [R(t) * e^(-r*t)] dt
where PV represents the present value, R(t) is the income stream at time t, r is the discount rate, and dt represents an infinitesimal change in time.
Let's break down the steps to find the present value:
Step 1: Express the continuous income stream function R(t) in terms of the given rate and time period.
R(t) = 5000 * e^(0.1t)
Step 2: Substitute this expression into the present value formula.
PV = ∫ [5000 * e^(0.1t) * e^(-0.05t)] dt
Step 3: Simplify the expression inside the integral.
PV = ∫ [5000 * e^(0.05t)] dt
Step 4: Integrate the expression with respect to t.
PV = 5000 * ∫ e^(0.05t) dt
To integrate this expression, we can apply the power rule for integration:
int(x^n) = (x^(n+1))/(n+1)
Using this rule, we integrate e^(0.05t) with respect to t:
PV = 5000 * (e^(0.05t))/(0.05)
Step 5: Evaluate the integral limits.
Since we are interested in finding the present value of the income stream for a specific time period, we need to evaluate the integral at the upper and lower limits of the time period.
PV = 5000 * [(e^(0.05 * upper limit))/(0.05) - (e^(0.05 * lower limit))/(0.05)]
In this case, the time period is 5 years, so the upper limit is 5 and the lower limit is 0.
PV = 5000 * [(e^(0.05 * 5))/(0.05) - (e^(0.05 * 0))/(0.05)]
PV = 5000 * [(e^(0.25))/(0.05) - (1)/(0.05)]
Simplifying further:
PV = 5000 * [(20 * e^(0.25)) - 20 ]
Finally, calculate the numerical value:
PV ≈ 5000 * (20 * e^(0.25) - 20)
PV ≈ 5000 * (20 * 1.2840 - 20)
PV ≈ 5000 * (25.68 - 20)
PV ≈ 5000 * 5.68
PV ≈ $28,400
Therefore, the present value of the continuous income stream is approximately $28,400.