Find the total value of the given income stream and also...

Find the total value of the given income stream and also find its future value (at the end of the given interval) using the given interest rate. HIMT [See Examples 4, 5.]
$$R(t)=30,000+1,000 t, 0 \leq t \leq 10$, at $7 \%$$

Key Concepts

Future Value Calculation

Future value is the amount that a series of cash flows will grow to at a specified future date, taking into account the accrual of interest. For continuous payments, this involves integrating the income function after growing each payment at the specified interest rate until the future time.

Present Value Calculation

Present value involves computing the current equivalent value of future income by discounting it at an appropriate interest rate. For a continuous income stream, this is achieved by integrating the income function multiplied by the discount factor over the time interval.

Time Value of Money

This concept reflects the idea that a dollar received today is worth more than a dollar received in the future due to its potential earning capacity. It underpins the methods of discounting and compound interest that are used to evaluate income streams over time.

Continuous Income Stream

In financial analysis, an income stream that is modeled as a continuous function over a period of time represents payments accruing continuously rather than at discrete intervals. This requires the use of integral calculus to calculate total amounts accrued or discounted over the interval.

Transcript

00:01 Problem 27 we have to find the total value tv and the future value fv for the given income stream and interest rate since we noted that total value for the given income stream tv is given by the definite integral a up to be r of td.

00:40 Now by putting the values we have here the value of a is 0 and b is 10 and the income stream r of t is 30 ,000 plus 1000 times p d now we are going to evaluate the given this definite integral as the integral of 30 ,000 with respect to t is 30 ,000 t plus integral of 1 ,000 times the integral of t with respect to t will be t squared divided by 2 and the limits of integration are 0 up to 10.

01:54 So this will be 30 ,000 times t plus 1 ,000 divided by 2 is equal to 500 t squared, 500 t squared and the limits of integration are 0 up to 10.

02:17 Now we are going to apply the limits of integration.

02:24 So by putting the upper limit we get 300 ,000 plus 50 ,000s and when we put the lower limit we get 0.

02:47 So this will be 350 ,000s.

02:56 So the total value for the given income stream is $350 ,000.

03:20 Now we are going to find the future value of the given income stream using the formula, the definite integral a up to b, r of t times e raised to the power, r of b minus t d t d t now by putting the values we have 0 up to 10 r of t is equal to 30 ,000 plus 1 ,000 times t of e r r r r r r r r r 0 % percent are 0 .07 and the upper t is the upper limit of the given interval is 10 minus t b so this will be equal to 30 ,000 times the definite integral 0 up to 10 of e -raised to the power 0 .7 minus 0 .07 times t d t plus plus 1000 times definite integral 0 up to 10, t times e -res to the power 0 .7 minus 0 .07 times t dp.

05:34 Now we are going to evaluate these integrals...

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