Question

1. You are currently saving for your child's college education. The current cost of college is $10,000 a year. You expect that college costs will continue to increase at a rate of 5 percent a year. Your child is scheduled to begin attending a four-year college 10 years from now (i.e., college payments will be made at t=10, t=11, t=12, and t=13). You currently have $25,000 in an account which earns 6 percent after taxes. You would like to have all of the necessary savings by the time your child enters college, and you would like to contribute a constant amount at the beginning of each of the next 10 years in order to provide the necessary amount. (You want to make 10 equal contributions starting in Year 0 and ending at Year 9.) How much should you contribute to the account each year in order to fully provide for your child's education?

Answer 1

lets find out the future value of college cost at year 10, 11,12,13

Future Value = Present Value X   (1+r)^t          ;

                 PV =    10000 ( college exp at t=0)   

   r =       5% or   0.05 ( growth rate of college exp )

               t    =      10     ( no. of years to college)

FV of college cost at 10th year   = 10000 X (1+.05)^{10 .} like wise we find the college fee for all 4 years.

the excel formulas are

so we know the amount to be spent each year.

Now, lets find out the balance required in the savings account at the beginning of the 10th year to be able to pay the above fees.

we will consider the fee paid in each year as Growing annuity (5% increase). so we need to find the PV of Growing annuity.

             Pmt = 1st year payment of college fee 16288.95

               i      = 6% or     0.06      ( interest rate)

              g       = 5% or    0.05       ( annuity increase rate-fee increase rate)

               n      = 4 years

substituting we get,

PV            =        16288.95 X (1+0.06) X (1-[(1+0.05)⁴ X (1+0.06)^-4])    /    (0.06-0.05)

               =                         17266.283   X     ( 1-[1.21550625 X0.792094])    /   0.01

               =                         17266.283   X     ( 1-[0.962794798])    /   0.01

                =                         17266.283   X     ( 0.0372052)    /   0.01

               =                       $ 64239.55

The savings Account must have a balanceof $64,239.55 at the beginning of year 10

now we need to compute the addtional savings required

the savings account has already 25000- which earns 6% interest for 10 years.

lets find the FV of this at the 10th year  

         FV   =    PV    X (1+r)ⁿ

                     =   25000 X (1+.06)¹⁰

                      =   $ 44, 771.19

therefore shortfall in savings account at 10th year

                    =    $64,239.55   - $ 44, 771.19

                    =     $ 19,468.26        

we need to find the annuity investment to be made each year from 0 to 9 for 10 years..at 6% to get the above sum of $ 19,468.26        

     we   already have r and n,, but we have only the future value of annuity i.e. 19468.26. we need to discount it to year 0 to find the PV of this annuity

             PV   =   FV   /   (1+r)ⁿ

                           = 19468.26/(1+.06)¹⁰

                           =10871.03

annuity due    =       

                           

                            

                            

                           

the amount to be contributed each year from Year 0 and ending at Year 9. is    $1,393.42