Question

Erich and Mallory are 22, newly married, and ready to embark on the journey of life.   They both plan to retire 45 years from today. Because their budget seems tight right now, they had been thinking that they would wait at least 10 years and then start investing $1800 per year to prepare for retirement.   Mallory just told Erich, though, that she had heard that they would actually have more money the day they retire if they put $1800 per year away for the next 10 years - and then simply let that money sit for the next 35 years without any additional payments – then they would have MORE when they retired than if they waited 10 years to start investing for retirement and then made yearly payments for 35 years (as they originally planned to do).   Please help Erich and Mallory make an informed decision:   

Assume that all payments are made at the END a year (or month), and that the rate of return on all yearly investments will be 7.5% annually.  

(Please do NOT ROUND when entering “Rates” for any of the questions below)

1. How much money will Erich and Mallory have in 45 years if they do nothing for the next 10 years, then put $1800 per year away for the remaining 35 years?

2. How much money will Erich and Mallory have in 10 years if they put $1800 per year away for the next 10 years?

3. How much will the amount you just computed grow to if it remains invested for the remaining 35 years, but without any additional yearly deposits being made?
   
4. How much money will Erich and Mallory have in 45 years if they put $1800 per year away for each of the next 45 years?

5. How much money will Erich and Mallory have in 45 years if they put away $150 per MONTH at the end of each month for the next 45 years? (Remember to adjust 7.5% annual rate to a Rate per month!)

6. If Erich and Mallory wait 25 years (after the kids are raised!) before they put anything away for retirement, how much will they have to put away at the end of each year for 20 years in order to have $700,000 saved up on the first day of their retirement 45 years from today?

Which set of Cash Flows is worth more now?

Assume that your grandmother wants to give you generous gift. She wants you to choose which one of the following sets of cash flows you would like to receive:

Option A: Receive a one-time gift of $ 10,000 today.   

Option B: Receive a $1400 gift each year for the next 10 years. The first $1400 would be received 1 year from today.    Option C: Receive a one-time gift of $17,000 10 years from today.

Compute the Present Value of each of these options if you expect the interest rate to be 3% annually for the next 10 years.    Which of these options does financial theory suggest you should choose?

1. Option A would be worth $__________ today.
2. Option B would be worth $__________ today.
3. Option C would be worth $__________ today.
4. Financial theory supports choosing Option _______

       

Compute the Present Value of each of these options if you expect the interest rate to be 6% annually for the next 10 years. Which of these options does financial theory suggest you should choose?

5. Option A would be worth $__________ today.
6. Option B would be worth $__________ today.
7. Option C would be worth $__________ today.
8. Financial theory supports choosing Option _______

Compute the Present Value of each of these options if you expect to be able to earn 9% annually for the next 10 years. Which of these options does financial theory suggest you should choose?

9. Option A would be worth $__________ today.
10. Option B would be worth $__________ today.
11. Option C would be worth $__________ today.
12. Financial theory supports choosing Option _______

Answer 1

Erich & Mallory:

a) Start investing after 10 years

• Annual payment = $1,800
• Invest for years = 35
• earned interest = 7.50% p.a

This problem to be solved in 3 steps:

Step 1: First calculate Present Value in 10th year [i.e; PV(10)] for annual investments made from 10th year on wards till 45 years. (i.e invested for 35 years)

• Investments from year-10 to year-45 (total 35 years)
• Annual payment = 1800 for n=35 years at rate = 7.5%
• Present Value at 10th year, PV(10) = 1800/(1+0.075)^1 + 1800/(1+0.075)^2 + ...+ 1800/(1+0.075)^10
• PV(10) = $12,355.35

Step 2: Then from PV(10) find Current PV(0) by discounting this lump sum investment for 10 years.

• PV(0) = PV(10) / (1+r)^10
• PV(0) = $12,355.35/ (1+0.075)^10
• PV(0) = $5,994.74

Step 3: Then calculate the Future Value of PV(0) after 45 years.

• Future Value after 45 years, FV(45) = PV(0) * [ (1+r) ^ 45]
• FV(45) = $5,994.74 * [ 1+0.075]^45
• FV(45) = $155,292.74

Answer:(a) Value after 45 years = $155,292.74

b) Invest for 10 years, annual amount = $1,800 for next 10 years.

• Future value after 10 years = 1800*(1+0.075)^9 + 1800*(1+0.075)^8 + ... + 1800*(1+0.075)^2 + 1800*(1+0.075)^1 + 1800
• FV = $25,464.76

c)Invest for 10 years and after that no additional annual payments

• Future value of $25,464.76 if stayed invested for next 35 years compounded annually at 7.5% with out any additional annual payments:
• FV = $25,464.76 * (1+0.075)^35 = $320,063.24

d) Start investing from next year on wards for next 45 years

• Annual payment = $1,800
• Invest for years = 45
• earned interest = 7.50% p.a

Future value at retirement = 1800*(1+0.075)^44 + 1800*(1+0.075)^43 + ... + 1800*(1+0.075)^2 + 1800*(1+0.075)^1 + 1800

FV = $597,716.13

e)Monthly investment of $150 for next 45 years

• Monthly amount = $150, for periods = 540 (i.e; 45 * 12 months) at rate = 0.625% (i.e; 7.5% / 12 )
• Future value = 150 * (1+0.00625)^539 + 150 * (1+0.00625)^538 + ...+ 150 * (1+0.00625)^2 + 150 * (1+0.00625)^1 + 150
• FV = $670,054.64

f)Savings per year if started investing after 25 years.

This is solved in 2 steps:

Step 1: First need to calculate present value at 25th year of Retirement amount of $700,000

• PV(25) = $700,000 / (1+0.075)^20 = $164,789.20

Step2:

• With this amount $164,789.20 as present value, annuity payments need to be calculated for 20 years at rate 7.5% as:
• $164,789.20 = C/(1+0.075)^1 + C/(1+0.075)^2 + ...+ C/(1+0.075)^19 + C/(1+0.075)^20
• Solving for C, we get C = $16,164.53

Grandma gift:

Option A: One-time gift of $10,000 today

Option B: $1,400 end of year payment for next 10 years at 3%

• Present value = 1400/(1+0.03)^1 + 1400/(1+0.03)^2 + ...+ 1400/(1+0.03)^9 + 1400/(1+0.03)^10
• PV = $11,942.28

Option C: one-time gift of $17,000 after 10 years

• Current Value = $17,000 / (1+0.03)^10
• Value = $12,649.60

Option A is worth more than B, C. So Option A to be chosen.

At 6% interest rate:

Option A: $10,000

Option B:

• Present value = 1400/(1+0.06)^1 + 1400/(1+0.06)^2 + ...+ 1400/(1+0.06)^9 + 1400/(1+0.06)^10
• PV = $10,304.12

Option C:

• Current Value = $17,000 / (1+0.06)^10
• Value = $9,492.71

With 6% rate, Option B is chosen as its value is higher.

At 9% rate,

Option A: $10,000

Option B:

• Present value = 1400/(1+0.09)^1 + 1400/(1+0.09)^2 + ...+ 1400/(1+0.09)^9 + 1400/(1+0.09)^10
• PV = $8,984.72

Option C:

• Current Value = $17,000 / (1+0.09)^10
• Value = $7,180.98

With 9% rate, Option A is chosen as its value is higher.

Summary:

	3%	6%	9%
Option A	$10,000	$10,000	$10,000
Option B	$11,942.28	$10,304.12	$8,984.72
Option C	$12,649.60	$9,492.71	$7,180.98
Choose Option	C	B	A