+598 An investment allows for end-of-year payments that grow by 3% per payment for 20 years, starting with $2000 at the end of the first year. If the payments earn 4% compounded annually, how much was invested at the beginning?
Any help would be greatly appreciated, thanks
+118667 Let M=2000, G=1.03 R=1.04
Let P be the present value
\(Original\; value = P\\ End\; of \;1\; year = RP-M\\ End\; of \;2nd\; year = R(RP-M) - GM\\ End\; of \;2nd\; year = R^2P- RM - GM\\ End\; of \;2nd\; year = R^2P-M(R+G) \\ End\; of \;3rd\; year = R(R^2P- M( R+G)) - G^2M\\ End\; of \;3rd\; year = R^3P- M( R^2+GR) - G^2M\\ End\; of \;3rd\; year = R^3P- M( R^2+GR+G^2) \\ \)
\(End\; of \;4th\; year = R(R^3P- M( R^2+GR+G^2))-G^3M \\ End\; of \;4th\; year = R^4P- M( R^3+GR^2+G^2R)-G^3M \\ End\; of \;4th\; year = R^4P- M( R^3+GR^2+G^2R+G^3) \\ End\; of \;nth\; year = R^nP- M( R^{n-1}+R^{n-2}G^1+.......R^1G^{n-2}+G^{n-1}) \\ End\; of \;nth\; year = R^nP- M*\displaystyle\sum_{k=0}^{n-1}\;R^{n-1-k}G^k\)
NOW for our question the amont at the end of the 20th year =0
\(End\; of \;20th\; year = 1.04^{20}P- 2000*\displaystyle\sum_{k=0}^{19}\;1.04^{19-k}*1.03^k\\ 0 = 1.04^{20}P- 2000*\displaystyle\sum_{k=0}^{19}\;1.04^{19-k}*1.03^k\\ 2000*\displaystyle\sum_{k=0}^{19}\;1.04^{19-k}*1.03^k = 1.04^{20}P\\ 1.04^{20}P=2000*\displaystyle\sum_{k=0}^{19}\;1.04^{19-k}*1.03^k \\ P=\left[2000*\displaystyle\sum_{k=0}^{19}\;1.04^{19-k}*1.03^k \right]\div[ 1.04^{20}]\\\)
Initial value was $35,142.88 
An investment allows for end-of-year payments that grow by 3% per payment for 20 years, starting with $2000 at the end of the first year. If the payments earn 4% compounded annually, how much was invested at the beginning
This is the the most complicated of all the problems seen so far. There are at least two different formulae used in this problem. They are relatively complicated, because you have to take into account the increase of 3% in the regular annual payment of $2,000 per year. It is essentially the same as taking inflation into account, in that your payment increases by 3% every year. Based on all that the PV of the initial investment comes to=$35,142.88. By the way, for your teacher to give you these problems, the formulae must in your book, properly explained. You may have to look them up.
I have the advantage of having financial software on my computer that can solve all these problems almost instantaneously.
+118667 Let M=2000, G=1.03 R=1.04
Let P be the present value
\(Original\; value = P\\ End\; of \;1\; year = RP-M\\ End\; of \;2nd\; year = R(RP-M) - GM\\ End\; of \;2nd\; year = R^2P- RM - GM\\ End\; of \;2nd\; year = R^2P-M(R+G) \\ End\; of \;3rd\; year = R(R^2P- M( R+G)) - G^2M\\ End\; of \;3rd\; year = R^3P- M( R^2+GR) - G^2M\\ End\; of \;3rd\; year = R^3P- M( R^2+GR+G^2) \\ \)
\(End\; of \;4th\; year = R(R^3P- M( R^2+GR+G^2))-G^3M \\ End\; of \;4th\; year = R^4P- M( R^3+GR^2+G^2R)-G^3M \\ End\; of \;4th\; year = R^4P- M( R^3+GR^2+G^2R+G^3) \\ End\; of \;nth\; year = R^nP- M( R^{n-1}+R^{n-2}G^1+.......R^1G^{n-2}+G^{n-1}) \\ End\; of \;nth\; year = R^nP- M*\displaystyle\sum_{k=0}^{n-1}\;R^{n-1-k}G^k\)
NOW for our question the amont at the end of the 20th year =0
\(End\; of \;20th\; year = 1.04^{20}P- 2000*\displaystyle\sum_{k=0}^{19}\;1.04^{19-k}*1.03^k\\ 0 = 1.04^{20}P- 2000*\displaystyle\sum_{k=0}^{19}\;1.04^{19-k}*1.03^k\\ 2000*\displaystyle\sum_{k=0}^{19}\;1.04^{19-k}*1.03^k = 1.04^{20}P\\ 1.04^{20}P=2000*\displaystyle\sum_{k=0}^{19}\;1.04^{19-k}*1.03^k \\ P=\left[2000*\displaystyle\sum_{k=0}^{19}\;1.04^{19-k}*1.03^k \right]\div[ 1.04^{20}]\\\)
Initial value was $35,142.88
Hi Melody: Thanks for going to all that trouble calculating this problem in detail. However, there is a much easier method of doing it, which you can make note of for future references. It goes something like this:
Rate of investment 4%
Rate of increase in payment, or inflation rate of 3%, therefore the net investment rate is:
1.04/1.03=0.9708737...%. Then we simply use the standard TVM formula of finding the PV of 20 payments of $2,000 each, which is:PV=P[((1 + r)^n-1) / ((1 + r)^n.r)], which gives the following result: PV=2,000[((1.009708737)^20-1) / (1.009708737)^20* .009708737)], which gives:
PV=2,000 X 18.0985842...=$36,197.17. But this amount already includes 1.03 increase in the first payment. Therefore the PV=$36,197.17 / 1.03=$35,142.88.
+598 Thanks for all the help guys!
Just out of curiousity Mr. Guest. What financial software do you use?
Supply
+118667 This is a mathematics site Mr Banker. Not a finanical site.
Why don't you show some ability and show the derivation of your formula.
I know the derivation has already been done by someone else so you should be able to find it easily enough on the net.
I have seen no evidence that you are capable of deriving it for yourself.
NeedSupply: Most of the financial software that I have in my computer is proprietary, written and programmed into my computer by myself, because I worked as an Investment Banker for some 25 years. In addition to that, I have a certain mathematical ability to fully understand the Math that is used in investments. In your case, because you are a student, I recommend that you buy youself a "Financial calculator". I would recommend that buy one that you can afford such as HP-10b11, on which you can do most, if not all, your problems that I have seen so far. I have not seen the price of it, but I believe you can buy one for as little as $30.00US. It has many memories as well as scientific functions, statistics, cash flow analysis, date and time calculations.....etc.
Good luck to you.
Melody: Did you read my note in answer #4?. As I mentioned there, the only formula you need is the STANDRAD TVM formula for PV of a stream of payments into the future!!. It's listed there!. Look at my note in #4 above. The derivation of most of the standard TVM formulae was derived by Sir Isaac Newton some 300-350 years ago!. There are some modern formulas that are used in special but rare situations, which I have not come across yet.
