1) Suppose that a young couple has just had their first baby and they wish to ensure that enough
money will be available to pay for their child’s college education. Currently, college tuition, books,
fees, and other cost $20,000 per year. On average, tuition and other costs have historically
increased at a rate of 6% per year. Assume the first college payment is made at the beginning year
19 (i.e. immediately after the child’s 18th birthday).
a) Assuming that college costs continue to increase an average of 6% per year and that all her
college savings are invested in an account paying 8% interest, then what is the amount of money
she will need to have available at age 18 to pay for all four years of her undergraduate
education?
b) How much does the couple need to save every year until their child’s 18th birthday to achieve
their goal, assuming they make their first savings payment on their child’s first birthday, the last
one on her 18th birthday? Assume they save the same amount every year.
a) METHOD 1
Step 1: Determine the cost of the first year of college.
𝑔𝑔 = 5%
𝑟𝑟 = 9%
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶18 = 𝑃𝑃𝑃𝑃 × (1 + 𝑔𝑔)𝑁𝑁 = $20,000 × (1 + .06)18 = $57,086.78
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶19 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶18 × (1 + 𝑔𝑔) = $57,086.78 × (1 + .06) = $60,511.99
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶20 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶19 × (1 + 𝑔𝑔) = $60,511.99 × (1 + .06) = $64,142.71
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶21 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶20 × (1 + 𝑔𝑔) = $64,142.71 × (1 + .06) = $67,991.27
Step 2: Figure out the value for four years of college.
𝐶𝐶1 1+𝑔𝑔 𝑁𝑁 $57,086.78 1+.06 4
𝑃𝑃𝑃𝑃17 = × �1 − � � �= × �1 − � � � = $205,631.58
𝑟𝑟−𝑔𝑔 1+𝑟𝑟 .08−.06 1+.08
𝑃𝑃𝑃𝑃18 = 𝑃𝑃𝑃𝑃17 × (1 + 𝑟𝑟) = $205,631.58 × (1 + .08) = $𝟐𝟐𝟐𝟐𝟐𝟐, 𝟎𝟎𝟎𝟎𝟎𝟎. 𝟏𝟏𝟏𝟏
METHOD 2
Step 1: Same as method 1
Step 2: Call the college payments 𝐶𝐶18 , 𝐶𝐶19 , 𝐶𝐶20 𝑎𝑎𝑎𝑎𝑎𝑎 𝐶𝐶21 , where the subscript denote the timing
(ie. 𝐶𝐶18 is the payment immediately after the child’s 18th birthday)
𝐶𝐶19 1 + 𝑔𝑔 𝑁𝑁 $60,511.99 1 + .06 3
𝑃𝑃𝑃𝑃18 = 𝐶𝐶18 + × �1 − � � � = $57,086.78 + × �1 − � � �
𝑟𝑟 − 𝑔𝑔 1 + 𝑟𝑟 . 08 − .06 1 + .08
= $57,086.78 + $164,995.32
𝑃𝑃𝑃𝑃18 = $𝟐𝟐𝟐𝟐𝟐𝟐, 𝟎𝟎𝟎𝟎𝟎𝟎. 𝟏𝟏𝟏𝟏
METHOD 3
Step 1: Same as above
Step 2: We can discount each college payment individually
19 𝐶𝐶 20 𝐶𝐶 21 𝐶𝐶 $60,511.99 $64,142.71 $67,991.27
𝑃𝑃𝑃𝑃18 = 𝐶𝐶18 + (1+𝑟𝑟) + (1+𝑟𝑟)2 + (1+𝑟𝑟)3 = $57,086.78 + (1+.08)
+ (1+.08)2
+ (1+.08)3
= $57,086.78 + $56,029.62 + $54,992.03 + $53,973.66 = $𝟐𝟐𝟐𝟐𝟐𝟐, 𝟎𝟎𝟖𝟖𝟖𝟖. 𝟏𝟏𝟏𝟏
, b) The couple needs to have $222,082.10 saved up by their child’s18thbirthday. So this is the
future value of their savings between now and their child’s18th birthday. If they save C
dollars every year, then:
𝐶𝐶
$222,082.10 = ∗ ((1 + .08)18 − 1)
. 08
Solving for C gives:
𝑪𝑪 = $𝟓𝟓, 𝟗𝟗𝟗𝟗𝟗𝟗. 𝟎𝟎𝟎𝟎
2) As a future graduate of the University of Minnesota’s prestigious Carlson School of Management,
someday you would like to endow a scholarship (meaning give the university money in your name)
to pay for tuition expenses for future CSOM students. Assume you just graduated
(congratulations!). You plan to work for fifteen years after graduation before endowing this
scholarship (at the end of the fifteenth year AFTER graduation). Annual tuition at UMN is $10,000
today, and is expected to grow at the long term average rate of inflation of 3% per year forever.
Savings is expected to earn a return of 7% per year forever.
a) If the first tuition payment is due one year after the scholarship is endowed, and you would like
the scholarship to pay all tuition for one student per year for the twenty years following the
creation of the endowment, how much money do you need to endow the scholarship?
b) If you would like the scholarship to pay for tuition for one student per year forever, how much
money do you need to endow the scholarship?
c) You plan to start saving for the endowment starting your first year after graduation (meaning
first savings is one year after graduation). You plan to increase the amount you save each year
by 5%, because you expect to have more income per year as time goes on. How much money
do you need to save in the first year, so that you will have enough to endow the scholarship
from part b (that pays tuition forever)?
d) HOW would your answer to part c. change if your savings earned 10% per year instead of 7% per
year.
a) We want the endowment to pay for tuition starting 16 years from today, however the
question is asking for how much we need to endow the sholarship15 years from now. The
first payment will be what tuition costs 16 years from today. If the $10,000 tuition today
grows at the inflation rate (3% per year) forever, the cost of school 16 years from today is
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇16 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇0 ∗ (1 + .03)16 = $16,047.06
This is the first payment in a growing annuity of tuition payments that will last 20 years. We
need to have the present value of this annuity saved 15 years from now to pay for it. That is
we need to have saved:
𝐶𝐶1 1 + 𝑔𝑔 𝑁𝑁 16,047.06 1 + .03 20
𝑃𝑃𝑃𝑃(𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎) = ∗ �1 − � � �= ∗ �1 − � � �
𝑟𝑟 − 𝑔𝑔 1 + 𝑟𝑟 . 07 − .03 1 + .07
= $𝟐𝟐𝟐𝟐𝟐𝟐, 𝟗𝟗𝟗𝟗𝟗𝟗. 𝟎𝟎𝟎𝟎
b) You are being asked the same question as in part (a), but now we want the scholarship to
last forever. Consequently, we are expecting to pay a growing perpetuity of tuition payments
money will be available to pay for their child’s college education. Currently, college tuition, books,
fees, and other cost $20,000 per year. On average, tuition and other costs have historically
increased at a rate of 6% per year. Assume the first college payment is made at the beginning year
19 (i.e. immediately after the child’s 18th birthday).
a) Assuming that college costs continue to increase an average of 6% per year and that all her
college savings are invested in an account paying 8% interest, then what is the amount of money
she will need to have available at age 18 to pay for all four years of her undergraduate
education?
b) How much does the couple need to save every year until their child’s 18th birthday to achieve
their goal, assuming they make their first savings payment on their child’s first birthday, the last
one on her 18th birthday? Assume they save the same amount every year.
a) METHOD 1
Step 1: Determine the cost of the first year of college.
𝑔𝑔 = 5%
𝑟𝑟 = 9%
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶18 = 𝑃𝑃𝑃𝑃 × (1 + 𝑔𝑔)𝑁𝑁 = $20,000 × (1 + .06)18 = $57,086.78
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶19 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶18 × (1 + 𝑔𝑔) = $57,086.78 × (1 + .06) = $60,511.99
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶20 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶19 × (1 + 𝑔𝑔) = $60,511.99 × (1 + .06) = $64,142.71
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶21 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶20 × (1 + 𝑔𝑔) = $64,142.71 × (1 + .06) = $67,991.27
Step 2: Figure out the value for four years of college.
𝐶𝐶1 1+𝑔𝑔 𝑁𝑁 $57,086.78 1+.06 4
𝑃𝑃𝑃𝑃17 = × �1 − � � �= × �1 − � � � = $205,631.58
𝑟𝑟−𝑔𝑔 1+𝑟𝑟 .08−.06 1+.08
𝑃𝑃𝑃𝑃18 = 𝑃𝑃𝑃𝑃17 × (1 + 𝑟𝑟) = $205,631.58 × (1 + .08) = $𝟐𝟐𝟐𝟐𝟐𝟐, 𝟎𝟎𝟎𝟎𝟎𝟎. 𝟏𝟏𝟏𝟏
METHOD 2
Step 1: Same as method 1
Step 2: Call the college payments 𝐶𝐶18 , 𝐶𝐶19 , 𝐶𝐶20 𝑎𝑎𝑎𝑎𝑎𝑎 𝐶𝐶21 , where the subscript denote the timing
(ie. 𝐶𝐶18 is the payment immediately after the child’s 18th birthday)
𝐶𝐶19 1 + 𝑔𝑔 𝑁𝑁 $60,511.99 1 + .06 3
𝑃𝑃𝑃𝑃18 = 𝐶𝐶18 + × �1 − � � � = $57,086.78 + × �1 − � � �
𝑟𝑟 − 𝑔𝑔 1 + 𝑟𝑟 . 08 − .06 1 + .08
= $57,086.78 + $164,995.32
𝑃𝑃𝑃𝑃18 = $𝟐𝟐𝟐𝟐𝟐𝟐, 𝟎𝟎𝟎𝟎𝟎𝟎. 𝟏𝟏𝟏𝟏
METHOD 3
Step 1: Same as above
Step 2: We can discount each college payment individually
19 𝐶𝐶 20 𝐶𝐶 21 𝐶𝐶 $60,511.99 $64,142.71 $67,991.27
𝑃𝑃𝑃𝑃18 = 𝐶𝐶18 + (1+𝑟𝑟) + (1+𝑟𝑟)2 + (1+𝑟𝑟)3 = $57,086.78 + (1+.08)
+ (1+.08)2
+ (1+.08)3
= $57,086.78 + $56,029.62 + $54,992.03 + $53,973.66 = $𝟐𝟐𝟐𝟐𝟐𝟐, 𝟎𝟎𝟖𝟖𝟖𝟖. 𝟏𝟏𝟏𝟏
, b) The couple needs to have $222,082.10 saved up by their child’s18thbirthday. So this is the
future value of their savings between now and their child’s18th birthday. If they save C
dollars every year, then:
𝐶𝐶
$222,082.10 = ∗ ((1 + .08)18 − 1)
. 08
Solving for C gives:
𝑪𝑪 = $𝟓𝟓, 𝟗𝟗𝟗𝟗𝟗𝟗. 𝟎𝟎𝟎𝟎
2) As a future graduate of the University of Minnesota’s prestigious Carlson School of Management,
someday you would like to endow a scholarship (meaning give the university money in your name)
to pay for tuition expenses for future CSOM students. Assume you just graduated
(congratulations!). You plan to work for fifteen years after graduation before endowing this
scholarship (at the end of the fifteenth year AFTER graduation). Annual tuition at UMN is $10,000
today, and is expected to grow at the long term average rate of inflation of 3% per year forever.
Savings is expected to earn a return of 7% per year forever.
a) If the first tuition payment is due one year after the scholarship is endowed, and you would like
the scholarship to pay all tuition for one student per year for the twenty years following the
creation of the endowment, how much money do you need to endow the scholarship?
b) If you would like the scholarship to pay for tuition for one student per year forever, how much
money do you need to endow the scholarship?
c) You plan to start saving for the endowment starting your first year after graduation (meaning
first savings is one year after graduation). You plan to increase the amount you save each year
by 5%, because you expect to have more income per year as time goes on. How much money
do you need to save in the first year, so that you will have enough to endow the scholarship
from part b (that pays tuition forever)?
d) HOW would your answer to part c. change if your savings earned 10% per year instead of 7% per
year.
a) We want the endowment to pay for tuition starting 16 years from today, however the
question is asking for how much we need to endow the sholarship15 years from now. The
first payment will be what tuition costs 16 years from today. If the $10,000 tuition today
grows at the inflation rate (3% per year) forever, the cost of school 16 years from today is
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇16 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇0 ∗ (1 + .03)16 = $16,047.06
This is the first payment in a growing annuity of tuition payments that will last 20 years. We
need to have the present value of this annuity saved 15 years from now to pay for it. That is
we need to have saved:
𝐶𝐶1 1 + 𝑔𝑔 𝑁𝑁 16,047.06 1 + .03 20
𝑃𝑃𝑃𝑃(𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎) = ∗ �1 − � � �= ∗ �1 − � � �
𝑟𝑟 − 𝑔𝑔 1 + 𝑟𝑟 . 07 − .03 1 + .07
= $𝟐𝟐𝟐𝟐𝟐𝟐, 𝟗𝟗𝟗𝟗𝟗𝟗. 𝟎𝟎𝟎𝟎
b) You are being asked the same question as in part (a), but now we want the scholarship to
last forever. Consequently, we are expecting to pay a growing perpetuity of tuition payments
