Question

BUSI 320 Comprehensive Problem 3 FALL 2020

Use what you have learned about the time value of money to analyze each of the following decisions:

Decision #1:   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?

       Option A would be worth $__________ today.

      Option B would be worth $__________ today.

       Option C would be worth $__________ today.

       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?

       Option A would be worth $__________ today.

       Option B would be worth $__________ today.

       Option C would be worth $__________ today.

      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?

       Option A would be worth $__________ today.

       Option B would be worth $__________ today.

       Option C would be worth $__________ today.

       Financial theory supports choosing Option _______

Decision #2 begins at the top of page 2!

Decision #2: Planning for Retirement

Evan and Gina 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 $2700 per year to prepare for retirement. Gina just told Evan, though, that she had heard that they would actually have more money the day they retire if they put $2700 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 Evan and Gina 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 7.2% annually.  

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

1. How much money will Evan and Gina have in 45 years if they do nothing for the next 10 years, then put $2700 per year away for the remaining 35 years?

2. How much money will Evan and Gina have in 10 years if they put $2700 per year away for the next 10 years?

b2) 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?

3. How much money will Evan and Gina have in 45 years if they put $2700 per year away for each of the next 45 years?

How much money will Evan and Gina have in 45 years if they put away $225

4. per MONTH at the end of each month for the next 45 years? (Remember to adjust 7.2% annual rate to a Rate per month!)

e.If Evan and Gina 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 $1,000,000 saved up on the first day of their retirement 45

Answer 1

Decision 1

Computing present value of three options where interest rate is 3% annually

Option A - $ 10,000 today

Present value shall be $ 10,000 only.

Option B - $ 1,400 at the end of each year for next 10 years

Present value = Annual Cash flow * Present value annuity factor of 3% for 10 years

= 1,400 * 8.5302

= $ 11,942.28

Option C - $ 17,000 at the end of 10^th year

Present value = Cash flow * Present value factor of 3% at the end of 10^th year

= 17,000 * 0.7441

= $ 12,649.70

Hence,

- Option A would be worth $ 10,000 today.

- Option B would be worth $ 11,942.28 today.

- Option C would be worth $ 12,649.70 today.

Financial theory supports choosing Option C

Computing present value of three options where interest rate is 6% annually

Option A - $ 10,000 today

Present value shall be $ 10,000 only.

Option B - $ 1,400 at the end of each year for next 10 years

Present value = Annual Cash flow * Present value annuity factor of 6% for 10 years

= 1,400 * 7.3601

= $ 10,304.14

Option C - $ 17,000 at the end of 10^th year

Present value = Cash flow * Present value factor of 6% at the end of 10^th year

= 17,000 * 0.5584

= $ 9,492.80

Hence,

- Option A would be worth $ 10,000 today.

- Option B would be worth $ 10,304.14 today.

- Option C would be worth $ 9,492.80 today.

Financial theory supports choosing Option B

Computing present value of three options where interest rate is 9% annually

Option A - $ 10,000 today

Present value shall be $ 10,000 only.

Option B - $ 1,400 at the end of each year for next 10 years

Present value = Annual Cash flow * Present value annuity factor of 9% for 10 years

= 1,400 * 6.4177

= $ 8,984.78

Option C - $ 17,000 at the end of 10^th year

Present value = Cash flow * Present value factor of 9% at the end of 10^th year

= 17,000 * 0.4224

= $ 7,180.80

Hence,

- Option A would be worth $ 10,000 today.

- Option B would be worth $ 8,984.78 today.

- Option C would be worth $ 7,180.80 today.

Financial theory supports choosing Option A

It may be observed that as the interest rate increases, the present value of the amount decreases. This is because the person is earning interest at higher rate and hence he would expect a higher sum today in lieu of the future cash flow.

Present value annuity factor is derived from the annuity table. It can also be computed by using formula [1-(1+r)^-n/r], where ‘r’ is rate of interest and ‘n’ is period. Present value annuity factor of 9% for 10 years is 6.4177. It means if one person earns interest rate of 9% per annum, then, he would be indifferent if he receives $ 6.4177 now or he receives $1 at the end of each year for 10 years.

Similarly, present value factor is derived from present value table. It can also be computed by using formula (1/(1+r)ⁿ), where ‘r’ is rate of interest and ‘n’ is period. It implies the present value of $1 received after ‘n’ number of years at the rate of interest of ‘r’.

Decision 2

A. Investing $2,700 per year for next 35 years

Future value = C x [(1+r)ⁿ-1/r],

C = Annual cash flow at the end of each period; $ 2,700 in the instant case

R = Rate of interest; 7.2% in the instant case

n = Number of payment; 35 in the instant case.

Applying the above formula in the instant case,

Future value shall be

= 2,700 [11.39771-1/0.072)

= $ 3,89,914

B. Investing $2,700 per year for next 10 years

Future value = C x [(1+r)ⁿ-1/r],

C = Annual cash flow at the end of each period; $ 2,700 in the instant case

R = Rate of interest; 7.2% in the instant case

n = Number of payment; 10 in the instant case.

Applying the above formula in the instant case,

Future value shall be

= 2,700 [2.004231-1/0.072)

= $ 37,658.68

C. Future Value of above amount at the end of next 35 years

Future Value = P(1+r)^t

P = Principal value; $ 37,658.68 in the instant case

R = Rate of interest; 7.2% in the instant case

t = time period; 35 in the instant case

Future value shall be

= 37,658.68 (1+0.072)³⁵

= $ 429,222.60

D. Investing $2,700 per year for next 45 years

Future value = C x [(1+r)ⁿ-1/r],

C = Annual cash flow at the end of each period; $ 2,700 in the instant case

R = Rate of interest; 7.2% in the instant case

n = Number of payment; 45 in the instant case.

Applying the above formula in the instant case,

Future value shall be

= 2,700 [2.004231-1/0.072)

= $ 819,136.60

It may be seen that this is actually the aggregate of the amount computed in Para A and Para C above.

E. Investing $225 every month for next 45 years

Future value = C x [(1+r)ⁿ-1/r],

C = Monthly cash flow at the end of each month; $ 225 in the instant case

R = Rate of interest for each month; 0.6% (7.2/12) in the instant case

n = Number of payment; 540 (45*12) in the instant case.

Applying the above formula in the instant case,

Future value shall be

= 225 [25.287715-1/0.006)

= $ 910,789.31

F. $1,000,000 after 20 years

Future value = C x [(1+r)ⁿ-1/r],

Future value is given as $ 1,000,000

C = Annual cash flow at the end of each period to be computed

R = Rate of interest; 7.2% in the instant case

n = Number of payment; 20 in the instant case.

Applying the above formula in the instant case,

1,000,000 = C x [4.01694-1/0.072)

C = $ 23,865.21