[MUSIC] We start a new topic series of cash flows, at first we are going to consider general case of arbitrary cash flows. And look at how to find the values at different time moments, till now we considered only single cash flows, and then observed how the value of those cash flows change through time. In real life, we can have inflows or outflows at any time moment. So here in the slide, we have cash flow $100 today, this is an inflow. And another inflow $100 1 time period from today, we can consider these as payments at the beginning of the first period and beginning of the second period. I hope you remember, that we only can sum up money at the same time moment, so we can find the value of our cash flow at all three time moments. The additional cash flows can be equal as the previous example, or unequal arbitrary like here, and they can even be negative, We can visualize it at the scheme of periodic cash flows. Just like here in the slide, so here we have $100 inflow at time moment 0, $10 inflow at the time moment 1, and no inflows and outflows at time moment 2. Now, we will consider how to find future value and present value of such cash flows, let's go to an example. We need to find the future value of some investment and a visual scheme here is quite helpful to understand the data. So let's look, we have a project that requires $200 of initial investment and promises us cash flows of $100 and $200 at the end of the next two years. So required interest rate is 5%, And you might notice that the cash flows have different signs here. This happens because the first one is outflow, And these two are inflows, That's why this was negative and these two are positive. So to find the future value, we need to compound the cash flow 0 two periods forward, so it's going to be times (1 plus interest rate) power of 2. Then we need to compound cash flow 1 once, Which is times 1 plus interest rate and Cash flow 2 should not be compounded because it has taken at time moment 2 already. So, let's construct a table to make the calculations, so here we have our cash flows and, The places for the compounding factors and the future value. So compounding factor here is times 1.05 squared, here, it's just times 1.05, and here it is nothing, so we don't need to compound this value. The values, the future value of this first initial outflow is minus, $220.5. Here it is, $105 and here it is $200 so, the total future value of this investment is going to be the sum of these three values, which is going to be $84.5. So now, let's find the present value of the same investment, So here we are with the scheme again, but now we need not to compound but discount. So we want to find the value of all the cash flows here at time moment 0, so this cash flow 1, $100 should be discounted once. So we need to divide it by (1 plus r), the cash flow 2, $200 should become discounted twice, so we will need to divide it (1 plus r) squared. And the cash flow of 0 minus $200 initial investment should not be discounted at all, because this value is taken at the time moment that we need for relative. So let's fill the table once again, now it is a discounting table, so we will put the discounting factors here. So no discounts in here, discounting once here, divided by 1.05, divide by 1.05 squared here to discount twice. And the present values here is the same minus $200 here it is $94 and, 24 cents approximately and here it is approximately $181.4. So, to find the present value of the whole investment, we need to sum up these values, and we will have $76.64. So we considered how to find the present value and future value of the same investment, and now I'm going to show you some mathematical magic. So, let's look at the scheme, if I have the present value here the $76., 64. The magic is that the future value of this amount, is the value that we already found, $84.5, so how does this happen? Let's check, this is double compounding so, it should be times (1 plus r) power 2, so if we would like to find the future value of this amount of money. So future value after two periods of compounding, present value, (1 plus r) squared would be, $76.64 times (1 plus 0.05 squared). So you can check it with your own calculator, but the future value, calculated from this present value is just, $84.5. So, this means that it is not necessary to find the future value, or present value by the table or the scheme. So, we can find the one we like to, we can find for example future value and then discount it two periods back, and then find present value. Or we can find the present value and then compound it two periods forward, and then we will have the future value, and no need to discount or compound each cash flow one by one. We can generalize results obtained in the example, if we have a series of cash flows, we can first discount into its present value, and then compound it that present value to the future. Or do it at the reverse order, first compound the whole cash flow to the future value, and then discount this future value back to time moment zero. The diagram shows that results are the same and we saw it in the example, [MUSIC]