Video Transcript: Calculating Present Worth of Multiple Purchases over Time

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We’ll still work on another problem that Bridges present worth and annual present worth because you can do it either way. But it does illustrate some strength in annual worth and some of the weakness in present worth. Present worth you’re going to find out is really great with problems that have delay in them, but they have difficulties with some kind of repeated reciprocal kind of investment while annual worth is better with some of the reciprocal stuff.

This problem actually stems from a project I was considering doing; in my backyard it was my backyard aqua culture idea, and the idea I had was going to create a tank that had fish in it, in my back yard. I was thinking about Tilapia, and the idea is I can take my kitchen ways throw it in there. The fish can eat the kitchen ways, and they could poop and I can stick that in my garden, and I can still eat the fish so it was adding that extra cycle in there. In order to get this to work I was how ever going to need a little bit of pumping. So the pumping part actually ended up being the investment choice. So I had two choices for my pumps. I had what I am going to call pump style “A”, its right here. Pump style “A” cost $100, and after 3 years would happen, is that I could go ahead and salvage this pump out, and get myself $50 and that salvage value.

Pump “B” will be something that cost me $250, Lasted 5 years, and I could actually sell this guy for $150 for the reminder. So these are my two pumping opinions that were there. So what was interesting about this was, Is because we are dealing with these repeated purchases, we have to make sure we are still providing the same amount of service in each case, and we also have to sort out a planning horizon on this is supposed to be.

One of the common choices for planning horizon is that you choose the least common multiple of these two assets lays right here say well is how long I’m going to keep things. In this case we are talking about 15 years. Now for me on the project 15 years is reasonable, it’s something you suppose I do it because yes I do like to garden, yes I do like to do all that fun stuff, so this is actually kind of reasonable. If these were numbers like 9 or 10 or something along those lines, a 90 planning horizon is not a good idea for me. In 90 years I should be dead several decades, so it’s not the perfect response right there. But we are going to follow that very general rule of just choosing the least common multiple. And what that’s going to do it make it so that we don’t have to worry about salvaging these items early.

So let’s give our 15 year planning horizon, and what we have to know is in order to figure out what the best option is we can’t just take the present worth of these assets and then say well this one has the present worth closest to positive infinity because well these provide assets there not providing us with the 15 years of service. And so we have to think about there is going to be this repeating cycle of purchases that we are going to make.

So to illustrate what the cash flow diagrams are going to look like, were looking at in this case purchasing for $250 one of these assets in time period zero, then what we’re doing in year 5 were salvaging it out; and at the same time purchasing another one for $250 then after 5 more years brings us to time period 10 salvaging the one we purchase in time period 5 out, and getting another one for $250. Then go ahead in buying another one and savaging out which in year 15. And so what we are doing is we’re making 3 purchases of these assets one after another, in order to provide us with the with our full 15 years of pumping service. So it’s this whole sequence right here is actually the investment we’re valuing not the single purchase over here, but the sequence of these three purchases over time.

And were doing something similarly up here with asset “A”; here it is we’re purchasing it for $100 in time period zero. In time period three we would salvage it, buying another one. Then in time period 6 salvaging it, we’re buying another one. Then in time period 9 were salvaging it and buying another one. Then in time period 12 salvaging it and buying another one; then in time period 15 were salvaging it out. So again the investment we are valuing is this whole stream of investment right here. Not this individual one. So let’s go ahead and pull this together in a present worth fashion, before we start looking at it in an annual worth fashion. And our first step is going to be, to calculate the present worth of just one of these assets right here. And choosing a kind of nice interest rate, the interest rate we are going to use is 8% .

So let’s replace these two parts here, with our calculations so the present worth of single asset “A” purchase, should be the initial $100 dollar cost. And then what we are going to do is get our salvage at $50 which is 3 year in the future, and where going to keep the interest rate of 8% on this, so its 1 plus .08 to the third. And that will give us the salvage value, Sorry the present worth of a single purchase sale cycle, so it just this chunk right here. And that’s actually going to equal $60.31. This part over here present worth purchase single cycle of asset “B” is negative $250 that’s our original purchase, and then we have this $150 salvage. Divide by 1 plus .08, 1 plus the interest rate raised to the power of 5. And that’s going to give us a present worth of the purchase of sales of a single one of these pumping stuff, and that negative $147.91.

So now we have the present worth of a single purchase, well what we have to do then, is to figure out how to take those values and connect them with these full investments that we are evaluating, each one of them providing 15 years of service. And that’s usually the key phrase that I will use when I am talking about these kinds of repeated purchases. So in order to keep things fairly simple, what we are going to do is use this calculation right there, noting that what this does is it takes all of this value that’s scattered over throw year 0 and year 5, and concentrates it in time period zero. So you can think about it as what it’s doing, saying take these time values and turn them into the present worth of “B”. And then it takes all the values right here, and it turns them into present worth of “B”. And takes all the values right here, and it turns them into present worth of “B”.

So in order to make a calculation on this full 15 years of service, all we have to do is take present worth of “B” and then say well we have this other thing of present worth of “B” but we are going to have this purchase in time period 5, and we have another present worth of “B”. We’re going to make this purchase in time period 10. So if you were to place each one of these present worth’s “B” which is with that $147.91. What we’re going to do is find out that the cost they’re providing 15 years’ worth of pumping service with assets of type “B” is $137.09. Now the upper problem is a little bit longer, but essentially all that were doing is taking present worth of “A” that first purchase, then doing again for the second purchase in time period 3. Then again for the third purchase in time period 6, then again for the purchase in time period 9, and again for the purchase in time period 12. What you are going to find out is that the cost in providing 15 years of service with this kind of pump is $210.37. So clearly if your doing the present worth criteria pump “A” is the way to go.

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