CH 4 Practice Problems

Question 1

Your goal is to establish a trust fund to provide scholarships for students. If you aim for the fund to
distribute $10,000 annually, starting next year and continuing in perpetuity, and you anticipate a 6%
rate of return on the investment, how much should you deposit now?

What would be the required initial contribution now if the scholarship is set to commence 3 years
from today?

Answer 1

PV0 = 𝐶1/𝑟
= 10000/0.06
= $166,667

Step 1:
PV2 = 𝐶3/𝑟
= 10000/0.06
= $166,667

Step 2:
𝑃𝑉0 = 𝑃𝑉2/(1+𝑟)^2
= 166667/(1+0.06)^2
= $148,333

Question 2

After becoming a renowned novelist, you decide to fund an annual literary event at the UH. You aim for the event to cost $20,000 each year. If the UH gets a 7% annual return on its investments and plans to host the first event in one year, how much would you need to contribute now to fund the event
indefinitely?

How much would you need to contribute now if you want to start the annual events 2 years from
today?

Answer 2

PV0 = 𝐶1/𝑟
= 20000/0.07
= $285,714

Step 1:
PV1 = 𝐶2/𝑟
= 20000/0.07
= $285,714

Step 2:
𝑃𝑉0 = 𝑃𝑉1/(1+𝑟)^1
=285714/(1+0.07)^1
= $267,023

Question 3

As a successful business owner, you wish to establish an annual winter music festival in the town
square. You estimate that organizing the festival will cost $25,000 every year. Considering the town’s
trust fund achieves a consistent 4% annual return on its investments, and you aim for the first festival to
be held next winter, how much would you need to donate now to guarantee the music festival is
celebrated every year without fail?

How much would you need to contribute now if you want to start the annual events 10 years from
today?

Answer 3

PV0 = 𝐶1/𝑟
= 25000/0.04
= $625,000

Step 1:
PV9 = 𝐶10/𝑟
= 25000/0.04
= $625,000

Step 2:
𝑃𝑉0 = 𝑃𝑉9/(1+𝑟)^9
=625000/(1+0.04)^9
= $439,117

Question 4

You are considering acquiring shares in AdvanceBio, a company known for its annual dividends. If the
dividend for next year is forecasted at $2 and is expected to grow by 3% annually thereafter, and given an 8% discount rate, what would be the estimated value of the stock?

Answer 4

PV0 = 𝐶1/(𝑟 − 𝑔)
= 2/(0.08 − 0.03)
= 2/0.05
= $40

Question 4

You are considering buying a unique investment security that offers indefinite annual interest payments. If the interest for the upcoming year is projected at $25 and anticipated to increase by 1.5% each subsequent year, and given an 10% discount rate, what is the estimated value of this security?

Answer 4

PV0 = 𝐶1/(𝑟 − 𝑔)
= 25/(0.10 − 0.015)
= 25/0.085
= $294.12

Question 5

You’re thinking about buying a rental property. You project the first year’s rent revenue to be $30,000, and you anticipate this amount to grow by 3.5% every year. Given that this rent revenue is projected to persist indefinitely and the discount rate stands at 9%, what is the estimated value of the
property?

What would be the value of the property today if you can start collecting rents 3 years from now?

Answer 5

PV0 = 𝐶1/(𝑟 − 𝑔)
= 30000/(0.09 − 0.035)
= 30000/0.055
= $545,455

Step 1:
PV2 = 𝐶3/(𝑟−𝑔)
= 30000/(0.09−0.035)
= $545,455

Step 2:
𝑃𝑉0 = 𝑃𝑉2/(1+𝑟)^2
= 545455/(1+0.09)^2
= $459,099

Question 6

Jane, at age 30, plans to start saving for her retirement. She decides to contribute $2000 at the end of every year to her retirement account. The account offers an annual interest rate of 6%. If Jane
continues this pattern of saving until she is 65, how much will she have in her retirement account?

Answer 6

N: 65 - 30 = 35
I/Y: 6%
PV: 0
PMT: -2,000
FV: 222,870 (?)

Question 7

Kai wishes to buy a house in 10 years and wants to save up for a down payment. She plans to deposit
$10,000 every year into a savings account that offers an annual interest rate of 6%. How much will Kai
have saved up at the end of 10 years for her down payment?

Answer 7

N: 10
I/Y: 6%
PV: 0
PMT: -10,000
FV: 131,808 (?)

Question 8

Kekoa wants to buy a new car and has found a lender who offers a 0.5% monthly interest rate. If
Kekoa can afford to make monthly repayments of $400 for the next 4 years, what is the maximum loan
amount (present value) he can borrow?

Answer 8

N: 4 * 12 = 48
I/Y: 0.5%
PV: 17,032 (?)
PMT: -400
FV: 0

Question 9

Emma wants to save up to pay the down payment for a house. She estimates she will need a total of
$100,000. If she deposits a certain amount at the end of every year into an account that offers an annual
interest rate of 5%, how much should her annual deposit be if she plans to make these deposits for 10
years in order to reach her goal?

Answer 9

N: 10
I/Y: 5%
PV: 0
PMT: -7,950 (?)
FV: 100,000

Question 10

Sophia is considering leasing an apartment in the city center. The landlord requires monthly
payments at the beginning of each month. The monthly rent is $1,500 and the lease term is for 3 years.
If Sophia uses a discount rate of 0.8% monthly, what is the present value of her lease payments?

Answer 10

N: 3 * 12 = 36
I/Y: 0.8%
PV: 47,132 (?)
PMT: -1,500
FV: 0

Question 11

Ava is given the option to pay her rent upfront for her new apartment. The rent is $1,200 per month,
payable at the beginning of the month, and the lease is for 2 years. If Ava’s personal discount rate is 1%
monthly, what lump sum should she be willing to pay now for the entire lease term?

Answer 11

N: 2 * 12 = 24
I/Y: 1%
PV: 25,746 (?)
PMT: -1,200
FV: 0

Question 12

Alex wants to save money for his child’s college education. He decides to deposit $5,000 at the end
of this year into a special savings account. He plans to increase the deposit by 3% every subsequent year
for the next 10 years. If the account offers an interest rate of 6% annually, how much will he have saved
up in the account at the end of the 10 years?

Answer 12

FV𝑡 = 𝐶1/(𝑟 − 𝑔)((1 + 𝑟)^𝑇 − (1 + 𝑔)^𝑇)

FV10 =5000/(0.06 − 0.03)((1 + 0.06)^10 − (1 + 0.03)^10)

Prepare each component.
= 5000/(0.06 − 0.03)
= 5000/0.03
= 166667

= (1 + 0.06)^10
= 1.06^10
= 1.7908

= (1 + 0.03)^10
= 1.03^10
= 1.3439

Therefore
FV10
= 166667 × (1.7908 − 1.3439)
= $74,483

Question 13

Emily contributes $12,000 to her retirement fund at the end of every year. She expects to increase
her contribution by 2% each year. If the fund offers an average return of 7% annually, how much will
Emily have in her fund at the end of 20 years?

Answer 13

FV𝑡 = 𝐶1/(𝑟 − 𝑔)((1 + 𝑟)^𝑇 − (1 + 𝑔)^𝑇)

FV20 = 12000/(0.07 − 0.02)((1 + 0.07)^20 − (1 + 0.02)^20)

Prepare each component.
= 12000/(0.07 − 0.02)
= 12000/0.05
= 240000

= (1 + 0.07)^20
= 1.07^20
= 3.8697

= (1 + 0.02)^20
= 1.02^20 = 1.4859

Therefore
FV20
= 240000 × (3.8697 − 1.4859)
= $572,112

Question 14

Your child’s college tuition for the freshman year is estimated to be $20,000, due next year. Over the
following three years, the tuition is expected to increase by 5% annually. If you can earn an 8% return on your investments, how much should you set aside today?

If the college tuition for the freshman year is due 5 years from now instead, how much should you
set aside today?

Answer 14

PV𝑡 = 𝐶𝑡+1/(𝑟 − 𝑔)(1 − (1 + 𝑔/1 + 𝑟)^𝑇)

PV0 = 20000/(0.08 − 0.05)(1 − (1 + 0.05/1 + 0.08)^4)

Prepare each component.
= 20000/(0.08 − 0.05)
= 20000/0.03
= 666667

= (1+0.05/1+0.08)^4
= (1.05/1.08)^4
= (0.9722)^4
= 0.8934

Therefore
PV0
= 666667 × (1 − 0.8934)
= $71,066

Step 1:
PV4 = $71,066

Step 2:
𝑃𝑉0 =𝑃𝑉4/(1+𝑟)^4
= 71066(1+0.08)^4
= $52,236

N: 4
I/Y: 8%
PV: -52,236 (?)
PMT: 0
FV: 71,066