MBA/MFM 253 TVM Practice Problems 2 Fall 2011

1. You are considering buying a new car. The current price of the car is $25,000. The dealer has offered you a special nominal interest rate of 3% each year for the next 3 years if you finance through the dealership.
a) What is your monthly car payment?
PV = 25,000 I = 0.25 N = 36 FV = 0 PMT =? = $727.03
25,000 = PMT (1-(1/(1.0025)36))/.0025
b) You are considering putting a $5,000 down payment on the car, what would your payment be if you did this assuming you could still get the special interest rate? $581.62
c) In either case what is the effective rate of interest you paid on the car?
1.002512=1.030416 3.0416%

2. A given rate is quoted at an effective annual rate of 12.55%. If you know that the interest compounds quarterly, what is the nominal annual rate?
12% Need to find the rate that when compounded four periods will equal .1255 solve for r in the following equation (1+(r/4))4=1.125
3. What is the PV of an annuity that pays 10,000 a year at the end of each of the next 15 years assuming a 5% return?
$103,796.58PMT = 10,000 N=15 I = 5 FV = ) PV = ?
What is the PV of an annuity that pays 10,000 a year at the beginning of each of the next 12 years assuming a rate of 7%? $84,986.743

3. PV and FV problems that are a little harder ( you need to figure out the pieces, the final answers are there – but you need to figure out how it set it up – the set up is usually the trickier part and the part people make mistakes on—this forces you to think about how to set it up as opposed to just looking at the answer)
Your rich uncle has decided to help you save for your daughter’s college education he thinks the first year of tuition for her will be $40,000. Your daughter enters college in 9 years and you can earn 7% on a special college savings account.
a) Your uncle would like to make a one time deposit today into an account to cover the cost of the first year of tuition. How much should he deposit into the account? $21,757.35
b) Your uncle has decided that he wants to make a deposit into the account each year at the end of the year (9 equal deposits). How much should each deposit be? $3,339.46 (you need to solve for Payment)
c) Your uncle decides that he wants to make a deposit today, followed by 8 more deposits each one being a year apart (an annuity due starting today) How much should each deposit be? $3,120.99
d) Your uncle wants to make a deposit today that will cover $40,000 each year for four years. The first $40,000 would be paid at the end of the 9th year from today. How much would he need to have in the bank when your daughter starts college at the end of the 9th year to generate the four annuity payments of $40,000? (Draw it on a time line if it helps --- you need the PV of a 4 year annuity due with $40,000 each year) $144,972.64
e) Using your answer in part d) how much would your uncle need to deposit today into the account to be able to make the 4 $40,000 payments? $78,855.51
f) How much would heed to deposit at the end of each of the next 9 years to be able to provide $40,000 each year in tuition?
$12,103.25

4. You have $500 that you would like to invest you have two options, Saving Account A pays 8% compounded quarterly, Saving Account B pays 8.25% compounded yearly, which account would you prefer? Explain your choice. You would prefer 8.25% annually WHY?

5. To celebrate the completion of your graduate degree you have decided to take a trip around the world. You have 18 months until you complete the program. You will need to pay $15,000 for the trip at the time of your graduation.
a. If you can earn an annual rate of 6% each year (.5% each month) how much will you need to deposit at the end of each of the next 18 months to have enough saved for the trip? $798.47
b. You currently have $5,000 saved for the trip, if you deposit it today in your account how much would you need to deposit at the end of each of the next 18 months to have enough for the trip? $507.32 6. Your sister would like your help planning for her new daughters college education. She has estimated that the cost of 1 year of college (tuition, room, and board) is currently $60,000 a year. The cost of college is estimated to increase at an annual rate of 5%. She would like you to figure out how much she would need to deposit at the end of each month for the next 18 years to cover her daughter’s tuition assuming her deposits will earn 12% each year (1 % a month) (Note your account will earn interest a 1% a month even if you are not making monthly payments after you stop paying in at the end of the 18th year).
a) What is the monthly payment $687.67
b) What if your parents want to make a $20,000 contribution to the account today? How much would each monthly payment need to be? $461.28

7. Your parents are saving for the college education of your younger sister who just turned 10. Your parents are planning on making eight payments at the end of each year of $1,500 each. Starting at the beginning of the ninth year (the end of the eighth year), your sister will then be allowed to make four equal withdraws from the account to finance her college education each year. How much will she have for books and tuition each year? Use 10% as your required return. PMT = $4,919.57

8. Your rich Aunt Oprah has recently decided that she wants to invest in your future. She has decided that for your 25th birthday she is going to open a trust account in your name. She is going to make a one time deposit into the trust account and you will have total access to the money in 15 years on your 40th birthday. The trust officer has presented her with two options, depositing $500,000 today and earning an 8% return each year on the deposit, or depositing $450,000 in one year and earning 9.99% each year on the deposit (you would not receive interest until the end of the second year in this case).
a. Represent the two choices on a time line.
b. Your aunt has asked you to help her decide which option to take. Use excel to find the FV of each possibility at the end of each of the 15 years. Use excel to illustrate the value of the two different possibilities in a line graph that compares the values of the account (one line for the value of each account with years on the x axis and account value on the y axis). Write a quick explanation of your results for your aunt, reference your graph in your explanation and use the idea of compounding interest to explain the difference between the value of each account over the next 15 years to your aunt.
c. Would your choice in b) be the same if you could take the money out of the account on your 30th birthday? Explain in detail why or why not and support your answer using future values.
d. Is there a point in time when you would be indifferent between the two options? Explain

a and b)

Age 25 26 27 28 36 40
Year 0 1 2 3 11 15

Option 1 500,000 540,000 583,200 629,856 1,165,819.50 1,568,084.56
Option 2 0 450,000 494,955 544,401 1,166,123.46 1,706,700.60

The FV of the deposit for each option need to be found. By the time you can access the money when you are 40 the lower contribution of $450,000 will be worth $450,000(1.0999)14=$1,706,700.60 which is greater than the value of option 1 which would be worth $500,000(1.08)15=$1,568,084.56

c) No the choice is not the same. If you can withdraw the money when you are 30 the value of option 1 will be higher. At age 30 option 1 is worth 500,000(1.08)5=734,664.04 compared to 450,000(1.0999)4=658,605.45 for option 2. Since you could then invest either sum however you chose you would prefer to have the option worth more at that point in time.

d) At approximately year 11 (age 36) you are almost indifferent. At this point in time option 2 is worth a little over $300 more than option 1. Option 2 is worth more from this point on.

6. Following graduation from the MBA program, you have decided to start saving and return to Law school at an ivy league institution in 5 years. You plan to work for 5 years prior to work on your Law degree which will take three years of full time study. Currently it would cost you $50,000 each year for tuition and books for the program, however you expect the price of tuition and books to increase at a rate of 5% each year. You have been offered a special graduate education saving account that earns a nominal annual return of 12%, however your interest will compound monthly. Assume that you decide to start saving for your education exactly 60 months (5 years) before your targeted starting date. You plan to make a deposit at the beginning of the first month (at time 0) then at the end of each of the next 60 months (5 years) to finance your graduate education. At the end of the 60 months you should have enough money in your account to finance your education for both years (the tuition for the second year will stay the account until you need to pay it out at the beginning of the second year). How much will you need to deposit each month in order to finance your graduate education?

You first need to calculate the value of tuition in each of the three years.

Tuition for year 5 = 50,000(1.05)5 = 63,814.08
Tuition for year 6 = 50,000(1.05)6 = 67,004.78
Tuition for year 7 = 50,000(1.05)7 = 70,355.02

To pay for tuition you need to find the PV of each of the tuition payments at year 5. This is the amount you would need in the bank to pay for all three years of law school. Since the account pays interest compounded monthly you need to either use 1% for each month or the effective annual rate of (1+(.12/12)12-1 = .126825

Value at year 5 of tuition 5 = 63,814.08 = 63,814.08
Value at year 5 of tuition 6 = 67,004.78/1.126825 = 59,463.34
Value at year 5 of tuition 7 = 70,355.02/(1.126825)2 = 55,409.23

The sum of these three numbers is the amount you need to have saved prior to starting law school.
This is equal to $178,686.65

There are 5 years or 60 months until you need the money. You plan to make a payment today and one at the end of each of the next 60 months. Since you are making 61 payments the value of each payment needs to be calculated not as a regular 60 period annuity or annuity due. The FV of the payments will have the same value as a 61 period annuity (make sure to know why for the test). So you can solve for the payment that would make the FV of a 61 period annuity equal to $123,277.42

PMT (FVIFA61,1%) = $178,686.65

The payment is equal to $2,140.31

7. You have decided to purchase a new home for $235,000. You decide to finance it using a 30 year fixed rate mortgage. Your banker says you can currently borrow for 30 years at 4.5% interest per year, however you plan to make monthly payments. How much will your monthly payment be?

You first need to find the payment. This can be found from an ordinary annuity. The PV is 235,000 the interest rates is .045/12 = .00375 per month and there are 360 months.

235,000= PMT (PVIFA360,.00375) PMT = 1190.71

8. You younger brother is almost completed his undergraduate education. Following graduation, he wants to go to Graduate School to get your MBA. He plans to work for 4 years and save for tuition then attend an Ivy League MBA program full-time for two years. Currently it would cost him $55,000 each year for tuition and books for the MBA program, however he expects the price of tuition and books to increase at a rate of 7% each year. He has been offered a special graduate education saving account that earns a nominal annual return of 12%, however interest will compound monthly. Assume that he decides to start saving for his education exactly 48 months (4 years) before the targeted starting date. He plans to make a deposit at the beginning of the first month (at time 0) then at the end of each of the next 48 months (4 years) to finance his graduate education for a total of 49 payments. At the end of the 48 months he should have enough money in his account to finance his education for both years (the tuition for the second year will stay in the account until it is needed at the beginning of the second year). How much will he need to deposit each month in order to finance his graduate education?

You first need to calculate the value of tuition in each of the two years.
Tuition for year 4 = 55,000(1.07)4 = 72,093.78
Tuition for year 5 = 55,000(1.07)5 = 77140.35

To pay for tuition you need to find the PV of each of the tuition payments at year 4. This is the amount you would need in the bank to pay for both years of school. Since the account pays interest compounded monthly you need to either use 1% for each month or the effective annual rate of (1+(.12/12)12-1 = .126825

Value at year 5 of tuition 5 = 72,093.78 = 70,093.78
Value at year 4 of tuition 5 = 77,140.35/1.126825 = 68,458.15

The sum of these two numbers is the amount you need to have saved prior to starting law school.
This is equal to $140,551.93

There are 4 years or 48 months until you need the money. You plan to make a payment today and one at the end of each of the next 48 months. Since you are making 49 payments the value of each payment needs to be calculated not as a regular 48 period annuity or annuity due. You need to find the V of all payments. This can be thought of as a payment at time 0 plus a 48 period annuity or:

PMT(1.01)48 + PMT(FVIFA1%,48) = 140,551.93
PMT[(1.01)48 + (FVIFA1%,48)]= 140,551.93
PMT = 2236.85

Another way to calculate this would be to consider the fact that the FV of the payments will have the same value as a 49 period annuity (make sure to know why for the test). So you can solve for the payment that would make the FV of a 49 period annuity equal to $140,551.93

PMT (FVIFA49,1%) = $140,551.93

The payment is equal to $2,236.85