Money Flow The rate of a continuous money flow starts at 1000 and increases exponentially at 5%

Money Flow The rate of a continuous money flow starts at ...

Key Concepts

Accumulated (Future) Value of Continuous Cash Flows

The accumulated or future value of continuous cash flows is computed by integrating each contribution along with its growth due to interest up to a future time. With continuous compounding, each cash flow is accumulated using the factor e^(r(T-t)), where r is the interest rate and T is the target time. This approach aggregates the future values of a continuous stream of cash flows.

Present Value of Continuous Cash Flows

Calculating the present value of a continuous cash flow involves integrating the stream of payments after discounting them back to the present time using an appropriate discount factor. The discounting accounts for the time value of money by applying a continuously compounded interest rate, resulting in an integral of the form ?[0, T] f(t)e^(-rt) dt.

Exponential Growth in Cash Flows

Exponential growth models describe processes where a quantity increases at a rate proportional to its current value. In the context of cash flows, this means that the rate of flow itself increases exponentially over time. This is typically represented by a function of the form A*e^(at), where A is the initial amount and a is the growth rate.

Continuous Compounding

Continuous compounding refers to the mathematical modeling of interest that is assumed to be earned and added to the account balance at every possible instant. Unlike periodic compounding, which applies interest at set intervals, continuous compounding uses the natural exponential function, leading to formulas involving e^(rt) for accumulation and e^(-rt) for discounting.

Transcript

00:01 We have that the rate of continuous money flow starts at $1 ,000 per year, and it will increase exponentially at a rate of 5 % per year for four years.

00:19 And what we want to do is find the present value and the accumulated if interest earned is 3 .5%, which is going to be compounded continued.

00:29 So to find accumulated, we need to find present value.

00:34 So let's first go ahead and find our present value.

00:39 So recall that present value is equal to 0 to t, where capital t is going to be our time interval, with the function f of t, which tells us our rate of money, which will bring you in per year, multiplied by e to the negative r, little t, t, where this little r is supposed to be r.

01:16 Rate our interests that we're getting.

01:20 So we're told that the money flow, so f of t is described by this part here.

01:32 So it starts at $1 ,000 and then exponentially increases at 5 % per, year.

01:39 So, because this is telling us is that f of t is going to be, so it starts at 1000 and then it's going to be multiplied by e since it's exponentially increasing times 0 .05 t.

02:05 So this here will describe our value because when we plug in one into this it would be 1 ,000 times e to the 0 .0 5t which would be just that much more times a thousand and so we have that so we can go ahead and plug this in to our present value equation so we have zero time interval is going to be 4 so that's our capital t f of t like we said over there is going to be 1 ,000 e through 0 .0.

02:51 Times e to the negative so our r is this 3 .5 % here so this is our little r so this would be 0 .00 so 0 .035 d t and what we are going to need to do now oh i'm and i drop that t right there.

03:33 So i'm going to first factor out from 1 ,000.

03:35 Still have my integral from 0 to 4.

03:38 And then i'm going to combine these two exponentials, and that will give 0 .25 minus 0 .035.

03:50 So this will be e to the 0 .015t, dt.

04:01 And you might recall to enterrate this is going to be e to 0 .015t divided by its new power, which is 0 .015...

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