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 yearsσif 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).σσSo 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.
 $\begin{array}{lr}
\text { Current college cost/year } & \$ 14,500 \
\text { College cost inflation } & 3.5 \% \
\text { Return on investment account } & 9.0 \% \
\text { Payments at } t=1,2,3, \text { and } 4 & \$ 5,000 \
\text { Account balance at } t=0 & \$ 15,000
\end{array}$ 
 Determine the cost of each year during college and its PV at $ t=8 $, discounted at the return on investment. 
 $\begin{array}{ll}
\end{array}$
$\begin{array}{lll}
\begin{array} { l } 
\
\text { Year } 1(t=8) & =&\text { Current cost } \times(1+\text { infl })^{8}= \
\text { Year } 2(t=9) & =&\text { Prior year } \times(1+\text { infl })= \
\text { Year } 3(t=10) & =&\text { Prior year } \times(1+\text { infl })= \
\text { Year } 4(t=11) & =&\text { Prior year } \times(1+\text { infl })= \
\text { Find PV }(\text { at } t=8) \text { of  all college costs} & =&\text { amount needed at } t=8
\end{array}
\begin{array} { l } 
\text {Cost}&\text { PV at t=8  }\
-19,093.73 & -19,093.73 \
-19,762.01 & -18,130.29 \
-20,453.68 & -17,215.45 \
-21,169.56 & -16,346.79\
&-70,786.26
\end{array}
\end{array}$
 
 Create a time line with those cash flows, plus the known initial CFs, as shown below.
2. Put $ X $ in for the unlknown 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. 
 $\begin{array}{lcccc}
&0&1&2&3&4&5\
\hline \text { Known values;  }&\
 \text {X for unknown  }&\$ 15,000.00 &\$ 5,000.00 & \$ 5,000.00 & \$ 5,000.00 & \$ 5,000.00 & X \
 \text { Solution value  }&& & & &\$ 2,412.76 \
 \text { for X:  }&\
\hline   \text { Cash flows, }&\$ 15,000.00& \$ 5,000.00 & \$ 5,000.00 & \$ 5,000.00 & \$ 5,000.00 & \$ 2,412.76 \
\hline
\end{array}$
$\begin{array}{l}
&6&7&8&9&10&11\
\hline  &X &X  &-\$ 19,093.73&-\$ 19,762.01&-\$ 20,453.68&-\$ 21,169.56\
&\$ 2,412.76 &\$ 2,412.76\
\hline \text { Cash flows, }&\
\text { continued: }&\$ 2,412.76 &\$ 2,412.76&-\$ 19,093.73&-\$ 19,762.01&-\$ 20,453.68&-\$ 21,169.56\
\hline 
\end{array}$ 
 3. We found the $ P V $ of the college costs $ (t=8-11) $ at $ t=8 $ above.
Their sum is shown to the right.$ \quad $  $ \quad $ $ \quad $ $ \quad $ $ \quad $ $ \quad $ $ \quad $ $ \quad $ $ \quad $ $ \quad $ $ \quad $ $ \quad $ $ \quad $ $ -70,786.26 $ 
 $\begin{array}{llrrr}
\text { 4. Find the FV of } t=0 \& 4 \text { positive CFs at } t=8 & 0 & \$ 15,000.00 & \$ 29,888.44 \
& 1 & \$ 5,000.00 & \$ 9,140.20 \
& 2 & \$ 5,000.00 & \$ 8,385.50 \
& 3 & \$ 5,000.00 & \$ 7,693.12 \
& 4 & \$ 5,000.00 & \$ 7,057.91 \
& & & \$ 62,165.16
\end{array}$ 
 $\begin{array}{lccc}
\text { 5. Find the difference between the positive and negative =8  values:} & -\$ 8,621.09  \
 \text {6. Find PMT for a 3-year annuity due whose FV is equal to this difference: } & \$ 2,412.76
\end{array}$

Scott and Linda Have Been Saving to Pay for Their