My Math Forum Financial mathematics FV and PV Question help

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 May 22nd, 2011, 02:45 AM #1 Newbie   Joined: May 2011 Posts: 1 Thanks: 0 Financial mathematics FV and PV Question help Hi. I am taking introductory accounting as a core module for my university degree and I have an assignment due soon, but I'm stuck on two financial mathematics questions. I understand the principles, but they seem to be trick questions and I'm not sure whether to make 'n' = 1-4 or 0-3 for the first question and 0-4 or 1-5 for the second question. ANy help would be much appreciated! 1) Harris Company knows that in four year's time it must replace one of it's existing machines with a new one. To ensure that some funds are available to replace the machine in four years, the company is depositing $25 000 in a savings account at the end of each of the next four years. The savings account will earn 6% interest compounded annually. How much will be in the savings account at the end of 4 years when the new machine is to be purchased? * My thinking is that because the money is deposited at THE END of each year and it asks how much money is in the account at THE END of 4 years, one would assume that the last deposit doesn't earn any interest, so you calculate future value with the formula FV=PV(1+i)n (to the power of n) by making 'n' equal 0,1,2 and 3 in the respective calculations. The other option would be to assume that each year's money earns interest, so 'n' would be 1,2,3 and 4. Clarification on this would be very helpful! 2) Your company has just signed an agreement to purchase some equipment on an installment payment basis. The annual installment is$6000 payable for the next five years at the end of each year. Assume that the cost of capital is 12%. What is the present value of the future installments? What is the total interest paid over the term? * I'm not so confident about this question. I know to work out total interest, you find the present value of each of the installments and subtract those values from the $6000, but I don't know what to make 'n' in the formula once again to work out all the present values. Would it be 1,2,3,4 and 5 or 0,1,2,3 and 4? Thank you so much!  May 22nd, 2011, 09:35 PM #2 Senior Member Joined: Apr 2011 From: USA Posts: 782 Thanks: 1 Re: Financial mathematics FV and PV Question help I'm going to get the first one answered, then return. I know the answer to the second, but I'm trying to think how to explain it. You've got the right idea on the first one - the last payment doesn't earn any interest, so you'll get the correct answer using the 0, 1, 2 & 3. But you're doing it the long way. If you're required to use this method I don't know that I should be interfering with that, but you don't have to use my method. Both of these are annuities. That means that you have equal payments at equal time periods. That is, the 25,000 payment is the same amount each time, at the same time interval of a year. When this is the case, there is an equation we can use that shortcuts doing all the individual equations as separate cash flows, and it also eliminates having to think about stuff like, do I use 0-3 or 1-4. Cause it's already built in, as long as the payments are at the end of each period. (If they aren't, there's a quickie fix for that.) If you were to take just the $\text{(1\ +\ i)^n}$ part of each of those equations, and add them all up, you'd have a number you could multiply just one time by the 25,000. But there's another equation we can use: $\text{FV=Pmt\,\left(\frac{(1+i)^n-1}i\right)}$ The i and n mean the same thing. If you do just the part in parenthesis, you'll find it equals the same thing as if you total your four parts in parenthesis. So how nice to just do one equation. The equation is bigger, yes, but the advantages are that you only have one equation, which comes in terribly handy if you're talking about a 30 year mortgage (imagine doing 360 payments individually ), and again, you don't have to decide if it's 4 or 0. n is 4 cause that's the total periods. Do take note that it only works when the payments are equal, at equal time intervals. The one possible disadvantage is that you don't really get a view into the inner workings of why it works like what you're having to do. If you want to more thoroughly understand all this, it certainly doesn't hurt to think thru it like you did. (Just for fun. if you do end up with payments that start at the beginning of the period, just use that same equation, then take your answer and multiply by 1+i, so like 1.06 in this case. That's called an annuity due.) May 23rd, 2011, 12:36 AM #3 Senior Member Joined: Apr 2011 From: USA Posts: 782 Thanks: 1 Re: Financial mathematics FV and PV Question help I just caught two things. This is an introductory accounting class??? WOW! Second, these aren't "trick" questions. They are fairly common and yes, this is just the way it works. Quote:  Originally Posted by chou10 2) Your company has just signed an agreement to purchase some equipment on an installment payment basis. The annual installment is$6000 payable for the next five years at the end of each year. Assume that the cost of capital is 12%. What is the present value of the future installments? What is the total interest paid over the term? * I'm not so confident about this question. I know to work out total interest, you find the present value of each of the installments and subtract those values from the $6000, but I don't know what to make 'n' in the formula once again to work out all the present values. Would it be 1,2,3,4 and 5 or 0,1,2,3 and 4? OK, let's just get down to bare basics first, and then I'll put some extra explanation at the end. The short answer is use the 1-5, not the 0-4. This doesn't work like a future value. Also, again, this is an annuity: there's$6000 payments going on, an equal amount, each once a year so at equal time intervals. Makes it an annuity. But it's a present value annuity. That equation is:

$\text{PV=Pmt\,\left(\frac{1-(1+i)^{-n}}i\right)}$

(Note the negative exponent, something often missed.)

Again, if you were to use your lump sum present value equation, adding them all up would be the same number as the thing in parenthesis here will give you. That always works. Again, this equation is an easy shortcut to what you're doing, if you're allowed to use it.

As for interest, no it's not the difference between your present value and $6000. You have to remember that the$6000 is happening 5 times. Interest is the difference between what goes into something and what comes back out. (Or the other way around.) And it isn't just one $6000 payment; it's 5. So$6000 x 5 is the total payments. Then subtract the present value.

Now, if you're really into understanding this and want to know, I'm going to attempt to explain it. If that is enough and you don't want to know more, just stop here. I can go on a bit.

With a future value, you're looking at something you put into an investment now, and then you earn interest on it. So you put $25,000 in at the end of the year, and over the next year it earns interest, then another payment and earning interest again over the next year, etc. And that last payment, being made at the end of the 4 years, wouldn't earn any interest. Mathematically there's not much point to that last payment. But from the standpoint that you can add another$25,000 to something at the end of the time period and use it to contribute towards something, it does count into the total available.

Now present values are a little weird for most people to get, until you get some experience at them. It's easy for most people to get the idea that if you stick $1000 into something today, and earn 10% per year, you'd have$1100 at the end of the year. But what if you wanted $1000 at the end of one year? How much would you have to put in, at 10%, today to get to the$1000? 1000/1.10 = 909. What if you need the $1000 at the end of two years? Well, you'd need$909 at the beginning of the second year. So now we have to do this one more time to back up one more year: 909/1.10 = $826. So you'd have to have$826 today in order to have $1000 at the end of two years. The more years, the less you have to have today, cause it'll sit there longer earning. Also, you have to remember that with a present value annuity, you're starting off with some amount already -- so it gets to earn interest (or be charged interest) immediately. You start off with some value and after one year you take$6000 out of it. So one payment earned interest for one year. Then next $6000 doesn't happen until the end of the 2nd year, so it sat there for 2 years (2 periods). So that fifth$6000 doesn't come out until the end, but it did indeed compound interest for the entire 5 years. Cause it started compounding interest today, not one year from now.

The difference there is that the money already exists and the payment at the end of the year, is coming off a present value, and has earned interest in the meantime.

So a future value annuity starts at zero, you add payments to it, and it grows into something over time, with interest compounding. But since a payment is added at the end of the time period, that last payment has no interest. Whereas with a present value annuity, you start out with some figure, payments come off, it goes down to nothing, but it's compounding interest right from the start. So there's the difference.

(This might be easier with some visuals, but while I can do the equations on here, I don't always have a good way to make pictures.)

Any questions or clarifications needed, don't hesitate to ask.

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