how much money should the department deposit at the end of each quarter?

Question

The Math department purchased a copy machine for $ 12000. After 5 years, the machine will be worthless. How much money should the department deposit at the end of each quarter, if the money is worth 4% compounded quarterly, in order to save enough to buy a new copy machine at the end of 5 years?

Answer 1

There's actually several different ways to solve these, so it always makes it difficult when we don't know what kind of class this is for or what methods you're using.

 

There's a typical algebra equation that goes something like:

                             [ (1 + r/m)ⁿ - 1 ]

     FV   =   Pmt *   ------------------------

                                    (r/m)

 

(Sorry, not too easy to write on here.)

 

Different letters can be used, but FV is future value, i.e. your 12,000, which is the amount in the future you're trying to get to. 

r = annual rate and m is how often it compounds in a year.   So your problem would be (.04/4) - it's once per quarter, so that's 4 times a year.  You need the rate per compounding period.

n = total number of periods, which 4 times per year * 5 years = 20 periods

(See Steve's post on those last 2 - he's doing the same thing up at the top of hist post)

 

Hence,

                        [ (1 + .01)²⁰ - 1 ]

12,000 = Pmt     ---------------------

                                  .01

 

If you're using a similar equation, see if you can solve that.  (Steve's answer is the one that's correct.  Although depending on any rounding you are allowed to do, it doesn't have to be exact but very close.)

 

 

You can also do this in Excel using the Pmt function.  N is still 20, and FV is still 12,000.  The interest you need to adjust yourself and put in as 1%.  (Excel doesn't care how many years - it cares around periods and everything is adjusted to quarterly periods.)  There is no PV so you leave that blank, and you can also leave Type blank since this is what's called an ordinary annuity.

 

You can also use a financial calculator, which will be much like using Excel, except it will probably want the annual rate of interest, 4%, and then P/YR (or something similar) which is 4, i.e. 4 periods per year.

 

You also might have a table/chart to do this on, although a payment chart is a bit more unusual.

 

--------------------------

The things to understand here, regardless of method used, is that it's a future value annuity and it compounds quarterly.  The 12,000 is needed in the future - the quarterly deposits happen along a time line, leading up to a saved up amount of 12,000.  It's an annuity because you have a series of payments. ("end of every something" is one clue)  And you have to deal with the quarters because interest is re-calculated each quarter, so everything is done based on periods, not years.

 

If you're not understand something here, you need to ask specific questions.  Time value of money involves a lot of stuff and not something people usually understand overnight.

Answer 2

d = each quarterly deposit

r = (4% / year) * (1 year/4 quarters)

r = 1% / quarter

t = 5 years * 4 quarters / year

t = 20 quarters

Bn = account Balance at end of quarter n

B0 = 0

B1 = 0 + d

B2 = d(1+1%) + d = d(1.01) + d

B3 = (d(1.01)+d)(1.01) + d = d(1.01)^2+d(1.01) + d

B4 = d(1.01)^3 + d(1.01)^2 + d(1.01) + d
…
B20 = d(1.01)^19 + … + d(1.01) + d

(1.01)B20 = d(1.01)^20 + d(1.01)^19 + … + d(1.01)

(1.01)B20–B20 = d(1.01)^20 – d = d((1.01)^20 – 1)

(1.01–1)B20 = d((1.01)^20 – 1)

(0.01)12000 = d((1.01)^20 – 1)

d = 120/((1.01)^20 – 1)

d ≈ $544.9837786866084 ≈ $544.99 must be deposited each quarter

Answer 3

P ( 1 + 0.04/ 4) ^5(4) = 12000

 

p ( 1.01^20) =12000

 

p = 12000/ 1.22 = 9874.53

 

  9874.53/ 20 = 491.73