Scott and Linda have been saving to pay for their daughter Casie's college education.Casie just turned 10 at (t = 0),and she will be entering college 8 years from now (at t = 8).College tuition and expenses at State U.are currently $14,500 a year,but they are expected to increase at a rate of 3.5% a year.Ellen should graduate in 4 years-if she takes longer or wants to go to graduate school,she will be on her own.Tuition and other costs will be due at the beginning of each school year (at t = 8,9,10,and 11). So far,Scott and Linda have accumulated $15,000 in their college savings account (at t = 0).Their long-run financial plan is to add an additional $5,000 in each of the next 4 years (at t = 1,2,3,and 4).Then they plan to make 3 equal annual contributions in each of the following years,t = 5,6,and 7.They expect their investment account to earn 9%.How large must the annual payments at t = 5,6,and 7 be to cover Casie's anticipated college costs?
A) $1,965.21
B) $2,068.64
C) $2,177.51
D) $2,292.12
E) $2,412.76

Question

Quizplus · Accepted Answer

This is a very difficult problem.It should only be used as a take-home assignment. Curtent college costyear&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp$14,500
College cost inflation&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp3.5%
Retun an investrent accuunt&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp9.0%
Payments at $t = 1,2,3$, and &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp$5,000
Account balance at  t =0  &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp $15.000 
1.Determine the cost of each year during college and its PV at t = 8,discounted at the return on investment. $\begin{array}{lllrl}
&&&\text { \underline{Cost} }  & \text { \underline{PV at t=8} }  \
\text { Year } 1(t=8) & = & \text { Current cost } \times(1+\text { infl })^{s}= & -19,093.73 & {-19,093.73} \
\text { Year } 2(t=9) & = & \text { Prior year } \times(1+\text { infl })= & -19,762.01 & -18,130.29 \
\text { Year } 3(t=10) & = & \text { Prior year } \times(1+\text { infl })= & -20,453.68 & -17,215.45 \
\text { Year } 4(t=11) & = & \text { Prior year } \times(1+\text { infl })= & -21,169.56 & \underline{-16,346.79} \
\text { Find PV }(\text { at } t=8) \text { of } & =\text { amount needed at } t=8: & & &-70,786.26\
\text { all college costs } & & &
\end{array}$ 
2.Create a time line with those cash flows,plus the known initial CFs,as shown below.Put X in for the unknown values for t = 5-7.We show the time line on two sets of rows.Ours now has the solution value,but it didn't originally. 11ea42d0_c071_b6dc_9abf_2f80d33b94c0_TB2090_00 
 3. We found the $PV$ of the callege costs $( t =8- 11 )$ at $t = 8$ abave.
Their sum is shown to the right.$- 70.786 .26$ 
 $\begin{array}{llrrr}
\text { 4. Find the FV of } t=0 \& 4 \text { positive CFs at } t=8 & 0 & \$ 15,000.00 & \$ 29,888.44 \
& 1 & \$ 5,000.00 & \$ 9,140.20 \
& 2 & \$ 5,000.00 & \$ 8,385.50 \
& 3 & \$ 5,000.00 & \$ 7,693.12 \
& 4 & \$ 5,000.00 & \underline{\$ 7,057.91}\
&&& \$ \underline{62,165.16}
\end{array}$ 5. Find the difference between the positve and negative $t = 8$ values:
$- \mathbf {\$ 8 , 6 2 1 . 0 9 }$
6. Find PMT for a 3-year annuity due whose $F V$ is equual to this difference:
$\$2,412.76$

Scott and Linda Have Been Saving to Pay for Their