SM132 Calculator Study Guide
Three years ago, you deposited $1000 in a savings account with a 4% APR. The bank pays interest monthly with no compounding. How much interest did you earn last year?
$40 (no compounding, its just 4% of 1000 every time)
1. Convert 7% EAR to an APR with monthly compounding.
((1 + 0.7)^ (1/12) - 1) x 12 = 0.0678 = 6.78%
1. Convert 0.23% monthly periodic rate to an EAR
(1 + 0.23)^12 - 1 = 0.028 = 2.80%
What is the value today of 20 monthly payments of $600 beginning 1 year from today? Assume a monthly interest rate of 0.75%.
1. N = 20 PMT = 600 I/YR = 0.75 FV = 0 PV = ? = 11104.81 2. FV = 11104.81 I/YR = 0.75 N = 11 (not 12) PV = 10228
1. You receive $1000 per year for 10 years beginning 1 year from today and you receive an additional $4000 10 years from today. If the interest rate is 6%, what is the value today of this stream of cash flows?
1. PMT: 1000 N: 10 I/YR: 6 FV: ??? = 13180.8 2. FV = 13180.8 + 4000 = 17180.8 I/YR: 6 N: 10 PV:??? = 9593.7
1. You have raised $700,000 to fund research into harbor seals and want to make an annual award to a graduate student every year forever beginning 6 years from today. Assuming an interest rate of 8% EAR, what amount will the research award be?
1. PV = 700,000 N = 5 I/YR = 8 FV = 1028529.653 2. 102859.653 x 0.08 = ?
1. Charles River Kayakers (CRK) is renting space at the Museum of Science to store the kayaks that it uses for Boston Harbor expeditions. CRK has agreed to pay $1000 per month for 5 months beginning on May 1. What is the value of the contract on January 1st of the same year? Assume a monthly interest rate of 1.5%.
1. n = 5 I/YR = 1.5 PMT = 1000 PV = ? = 4782.65 2. n = 3 I/YR = 1.5 FV = 4782.645 PV = ? = 4574
You want to endow a scholarship that will pay $10,000 per year forever, starting one year from now. 1. If the school's endowment discount rate is 6%, what amount must you donate to endow the scholarship? 2. How would your answer change if you endow it now, but it makes the first award to a student 10 years from today?
1. 10000/0.06 = 166666.67 2. N = 9 I/YR = 6 FV = 166666.67 PV = ?
You are thinking of purchasing a house. The house costs $250,000.You have $36,000 in cash that you can use as a down payment on the house, but you need to borrow the rest of the purchase price. The bank is offering a 30-year mortgage that requires annual payments and has an interest rate of 6% per year. What will be your annual payment if you sign thismortgage?
1. 250000 - 36000 = 214000 2. N = 30 PV = 214000 FV = 0 I/YR = 6 PMT = ??
1. You expect to receive monthly payments for 5 years beginning 1 month from today. If the value of the annuity today is $2,000, what is the monthly payment if the interest rate is 5% APR with monthly compounding?
1. 5%/12 = .41667% (1 + 0.0041667)^12 - 1 = 5.115% I/YR = 5.115/12 = 0.426 N = 5 x 12 = 60 PV = 2000 PMT = ?
Assuming interest rates are 5% APR, what is the value at T0 of each of the following 4 year annuities:
1. Find FV 2. Find PV with the FV
You are considering a car loan with a stated APR of 6.14% based on monthly compounding. What is the effective annual rate of this loan?
1. Monthly Interest Rate = APR/12 = 6.14/12 = 0.512% 2. Effective Annual Rate = (1+ Monthly interest rate)^12 - 1 = (1.00512^12)-1 = 0.0632 = 6.32%
You receive $1000 per year for 10 years beginning 1 year from today and you receive an additional $4000 10 years from today. If the interest rate is 6%, what is the value today of this stream of cash flows?
1. PMT = 1000 N = 10 I/YR = 6 FV = ? (13180.795) -> 2. FV + 4000 = 17180.795 -> 3. FV = 17180.795 I/YR = 6 N = 10 PV = ?
Consider the following perpetuity: the first payment of $1000 will be one year from today and the payments will increase by 3% each year. If interest rates are 8%, what is the value of this perpetuity today?
1000 / (0.08 - 0.03) - 20,000
An annuity pays $300 per year for 8 years beginning one year from today. What is the remaining value of the annuity immediately before the fourth $300 payment? Assume an interest rate of 2%.
1142 + 300
Your roommate bought you a lottery ticket before you went home for Thanksgiving. You come back to school and receive an email telling you that you have a winning ticket. You are given three choices - the choices are 1) a lump sum payment today of $120,000, 2) $240,000 twelve years from now, or 3) twelve annual payments of $14,000 beginning today. If the interest rate is 6%, which would option is the most valuable?
3.
1. How much will you have in the bank 3 years from today if you deposit $100 in a savings account which pays 4.5% APR with monthly compounding?
4.5%/12 = 0.375% 1.00375^12 - 1 = 0.0459 I/YR = 4.59% PV = -100 N = 3 FV = ? = 114.42
You plan to fund a scholarship that will pay $4000 per year forever beginning one year from today. If the interest rate is 5%, what would you have to invest today to fund the scholarship?
4000 / .05 = 80000
You have decided to buy a perpetual bond. The bond makes one payment at the end of every year forever and has an interest rate of 6%. If the bond initially costs $4,000, what is the payment every year?
4000 x 0.06 = 240
Partners in Health (PIH) has decided to set up a scholarship for Haitian college graduates to attend Medical School in Boston. The scholarship will award $60,000 each year forever and the first scholarship award will be 6 years from today. If the interest rate is 8%, how much will it cost PIH to fully fund this donation today?
60000/0.08 = Value at 5 years = 750,000 N = 5 FV = 750000 I/YR = 8 PV = ?
1. You plan to borrow money to buy a snowmobile. You find the following three banks that offer the same APR and the same total number of years, but different payment intervals. Which bank would you prefer to borrow from? A) Bank A which compounds quarterly B) Bank B which compounds monthly C) Bank C which compounds daily
A
1. Convert 7% APR to an EAR with weekly compounding
[1 + (0.07/52)]^52 - 1 = 0.07245 = 7.25%
You are considering two ways of financing a spring break vacation. You could put it on your credit card, at 18% APR, compounded monthly, or borrow the money from your parents, who want an interest payment of 10% every six months. Which is the lower rate? a) EAR of your credit card = ? b) EAR for the loan from your parents = ?
a) 1. 18%/12 = 1.5% = 0.015 2. ((1 + 0.015)^12) - 1 = 0.1956 = 19.56% b) 1. 10% + 10% = 20% APR (6 months + 6 months) 2. 20% / 2 (periods in a year) = 10% = periodic Rate 3. (1.10^2)-1 = .2100 = 21.00%
You have just taken out a $18,000 car loan with a 7% APR, compounded monthly. The loan is for five years. When you make your first payment... A) ________ will go toward the principal of the loan and B) _________ will go toward the interest.
B) 18000 x (.07 / 12) = 105 (interest payment) A) N = 60 I/YR = 7/12 PV = 18000 FV = 0 PMT = ? = 356.42 Principle Payment = 356.42 - 105 = 251.42
1. You plan to open a savings account to save for a trip after you graduate. You find the following three banks that offer the same APR, but different compounding options. Which bank should you choose if you want to have the most money for your trip when you graduate? A) Bank A which compounds monthly B) Bank B which compounds weekly C) Bank C which compounds daily D) You would need to know the APR in order to determine which would be the best choice
Bank C which compounds daily
A $1000 investment today will pay you $300 three months from now, $500 six months from now and $800 in one year. If the interest rate is 8% APR (compounded quarterly), what is the present value of these cash flows?
CFJ0 = -1000 CFJ1 = 300 CFJ2 = 500 CFJ3 = 0 CFJ4 = 800 I/YR = 8/4 = 2 NPV = ?
1. You plan to do a snorkeling trip to the Great Barrier Reef when you graduate. The trip will cost $15,000. You already have $7,000 in the bank and you plan to make a deposit every year beginning one year from today. If your bank pays 6% interest, how much does the yearly deposit need to be if you are to have $15,000 4 years from today?
FV: -15000 PV: 7000 I/YR: 6 N: 4 PMT: ? = 1409
Majid has $12,000 in his savings account and can save an additional $3600 at the end of each year. If interest rates are 12%, how long will it take his savings to grow to $47,000 (round to the nearest year)?
FV: -47000 PV: 12000 PMT: 3600 I/YR: 12 N= ? = 5.3
1. A $600 investment today will pay you $689 in 3 years. What is the annual interest rate that you will have earned on your investment?
FV: 689 PV: -600 N: 3 I/YR: ?
1. A bank is negotiating a loan. The loan can either be paid off as a lump sum of $80,000 at the end of four years, or as equal annual payments at the end of each of the next four years. If the interest rate on the loan is 6%, what annual payments should be made so that both forms of payment are equivalent?
FV: 80000 PV: 0 I/YR: 6 N: 4 PMT: ? (18287.3)
1. A perpetuity will pay $4000 every three months beginning three months from today. If the interest rate is 12% EAR, what is the PV of this perpetuity?
Find APR APR = 11.4949% 0.114949 / 4 = 0.0287373 4000 / 0.0287373 = 139191.9
1. A group of concerned students has raised money to help victims of Hurricane Matthew in Haiti. They plan to invest the money to provide monthly payments to help a small town on the north coast rebuild homes. The town will receive the first payment in a month and the payment will be $3000. Each month after that, the town will receive a payment that is 0.5% larger than the last payment. This pattern of payments will go on forever. Assume that the interest rate is 12% EAR. What is the value of the gift immediately after the first payment is made?
Find APR/12 = 0.9488% perpetuity grew by 15 3015/(0.009488 - 0.005) = ?
You are offered the right to receive $4,000 per year forever, starting in one year. If your discount rate is 3%, what is this offer worth to you?
Guide: ($4000) / 0.04 = $133,333
Your grandmother has been putting $5,000 into a savings account on every birthday since your first (that is, when you turned one). The account pays an interest rate of 10%. How much money will be in the account immediately after your grandmother makes the deposit on your 18th birthday
Guide: N = 18 PMT = 5000 I/YR = 10 FV = ? (find)
You expect to receive $100 per year for 40 years beginning next year. If you receive 8% on your investments, what is the present value of this annuity?
Guide: N = 40 I/YR= 8 PMT = 100 FV = 0 PV = ?
You receive $300 per year for 20 years beginning one year from today. Assuming an interest rate of 4%, what is the value of this stream of payments today?
Guide: PMT = 300 N = 20 I/YR = 4 PV = ?
1. You are considering investing in a new gold mine in South Africa. Gold in South Africa is buried very deep, so the mine will require an initial investment of $250 million. Once this investment is made, the mine is expected to produce revenues of $40 million per year for the next 20 years. It will cost $10 million per year to operate the mine. What is the IRR of this investment? Assuming a discount rate of 8%, what is the NPV of this investment?
IRR: PV = -250 PMT = (40-10) = 30 N = 20 I/YR = 10.3% NPV: *PV of the annuity - 250mil initial investment* 1. N = 20 PMT = 30 I/YR = 8 PV = ? (294.54mil) 2. 294.54 - 250 = 44.54 mil
You have decided to refinance your mortgage. You plan to borrow whatever is outstanding on your current mortgage. The current monthly payment is $1,850 and you have made every payment on time. The original term of the mortgage was 30 years, and the mortgage is exactly four years and eight months old. You have just made your monthly payment. The mortgage interest rate is 5.750% (APR). How much do you owe on the mortgage today?
N = (30 x 12) - ((4x12) + 8) = 304 I/YR = 5.75/12 = 0.479167 PMT = 1850 FV = 0 PV = ?
1. You deposited $100 in a savings account 11 months ago. You currently have $107.68. What was the EAR if the bank paid interest with monthly compounding?
N = 11 PV = -100 FV = 107.68 I/YR = 0.67493% = periodic rate 1.006749^12 - 1 = 0.08406 = 8.41%
You are thinking about buying a savings bond. The bond costs $49 today and will mature in 11 years with a value of $98. What annual interest rate will the bond earn?
N = 11 PV = -49 FV = 98 I/YR = ? (6.5)
1. You receive $100 every 3 months beginning 3 months from today for 5 years and an additional $2000 5 years from today. If the interest rate is 3% APR with quarterly compounding, what is the value today of this stream of cash flows?
N = 20 PMT = 100 I/YR = .75 PV = ? = 1850.80 FV = 2000 I/YR = 0.75 N = 20 PV = ? = 1722.379 1722 + 1850.8 = 3573
An annuity pays $300 per year for 8 years beginning one year from today. What is the remaining value of the annuity immediately after the fourth $300 payment? Assume an interest rate of 2%.
N = 4 I/YR = 2 PMT = 300 PV = ? = 1142
Suppose Capital One is advertising a 60-month, 5.78% APR motorcycle loan. If you need to borrow $8,400 to purchase your dream Harley-Davidson, what will be your monthly payment?
N = 60 I/YR = (5.78/12) = 0.4817 PV = 8400 FV = 0 PMT = ?
Car Price is 100,000 I/YR: 2.44 N: 60 months Loan: 100,000 how much do you pay each month?
N = 60 I/YR = 2.44/12 = 0.2033 PV = 100000 FV = 0 PMT = ?
1. The Fall 2021 TAs love helping BU finance students so much that they have formed a company to tutor students. They negotiate a contract with Questrom to be paid $20,000 per year beginning one year from today for the next 10 years. What is the value of this contract of the contract today if you assume an interest rate of 7%?
PMT = 20000 N = 10 I/YR = 7 PV = ?
1. You plan to put $2500 in the bank every year for 30 years beginning one year from today. It interest rates are 2.5%, how much will you have in the bank 30 years from today right after you make your last deposit?
PMT = 2500 N = 30 I/YR = 2.5 FV = solve Answer: 109757
You receive a $3,000 check from your grandparents for graduation. You decide to save it toward a down payment on a house. You invest it earning 12% per year and you think you will need to have $6,000 saved for the down payment. How long will it be before the $3,000 has grown to $6,000?
PV = -3000 FV = 6000 I/YR = 12 N = ?
Suppose you currently have $5,500 in your savings account, and your bank pays interest at a rate of 0.48% per month. If you make no further deposits or withdrawals, how much will you have in the account in 6 years?
PV = 5500 I/YR = 0.48 N = 6 x 12 = 72 FV = ?
A) If you deposit $1 into a bank account that pays 5.5% per year for three years: The amount you will receive after three years is ___________ B) If you deposit $1 into an account that pays 2.3% every six months for three years: The amount you will receive after three years is ________ ? C) An account that pays 6.9% every 18 months for 3 years? If you deposit $1 into a bank account that pays 6.9% every 18 months for three years: The amount you will receive after three years is ________ ? D) An account that pays 0.56% per month for three years? If you deposit $1 into a bank account that pays 0.56% per month for three years. The amount you will receive after three years is ________?
A) N = 3 I/YR = 5.5 PV = 1 FV = ? = 1.17424 B) 2.3% = periodic rate, m = 2 (1.023^2)-1 = 0.014618 I/YR = 1.4618% N = 3 PV = 1 FV = ? C) 6.9% = periodic rate. m = 12 months /18 months = 0.6667 (1.069^0.6667) - 1 = 0.0454889 = 4.54889% = I/YR N = 3 PV = 1 FV = ? D) 0.56% = periodic Rate m = 12 (1 + 0.0056)^12 - 1 = 0.0693089 = I/YR = 6.93089% N = 3 PV = 1 FV = ?
Your bank account pays interest with an EAR of 8%. A) The APR quote for the account with semiannual compounding is ________% B) The APR quote for the account with monthly compounding is ___________ %
A) (1 + 0.08)^(1/2) - 1 = 0.03923 0.03923 x 2 = 0.07846 = 7.846% B) (1 + 0.08)^(1/12) - 1 =0.00643403 0.00643403 x 12 = 0.0772084 = 7.721%
An online bank is offering to pay 0.30% interest per month on deposits. Your local bank offers to pay 0.80% interest quarterly (every 3 months). Which is the higher interest rate? A) The EAR for the online bank is ________ B) The EAR for the local Bank is ________ C) The offer of ___________ is higher.
A) 0.30% = periodic rate, m = 12 ((1 + 0.003)^12) - 1 =0.0366 = 3.66% B) 0.80% = periodic rate, m = 4 (quarterly) ((1 + .008)^4 - 1) = 0.032386 = 3.2386% C) 0.30% per month
1. You have a small business breeding Australian sheepdog puppies and have found a buyer for your last puppy. The puppy costs $4000 and the buyer has offered you three payment options. If the interest rate is 6%, which of the following would you prefer to receive? A) One payment of $4000 today B) Two payments of $2450 each. The first payment is 3 years from today and the second payment is 4 years from today C) Three annual payments of $1497 beginning one year from today
A) 4000 B) CFJ0 = 0 CFJ1 = 0 CFJ2 = 0 CFJ3 = 2450 CFJ4 = 2450 I/YR = 6 NPV = *Find* C) CFJ0 = 0 CFJ1 = 1497 CFJ2 = 1497 CFJ3 = 1497 NPV = *Find* Answer = C
You have found three investment choices for a one-year deposit: 9.2%APR compounded monthly, 9.2%APR compounded annually, and 8.5%APR compounded daily. Compute the EAR for each investment choice. (Assume that there are 365 days in the year.) A) The EAR for the first investment choice is _____ ? B) The EAR for the second investment choice is _____ ? C) The EAR for the third investment choice is _____ ?
A) 9.2/12 = 0.766667% (1.00766667 ^12) - 1 = 0.09598 = 9.598% B) =9.2% C) 8.5/365 = 0.02329% 1.0002329^365 - 1 = 8.87064%
The Boston chapter of Habitat for Humanity wants to raise money to rebuild homes destroyed by Hurricane Michael. Because they realize that hurricanes will continue to strike the southeastern states, they plan to fund the construction of one house per month forever. They want the funding to increase by 0.5% per month to cover increases in material costs. If they decided to have the funding grow at 1% instead of 0.5%, the amount required to fund the endowment will: A) Increase B) Decrease C) Remain unchanged D) Cannot be determined without knowing the interest rate
A) higher growth rate means the value of the perpetuity is going to be larger -> Increase
You are trying to decide how much to save for retirement. Assume you plan to save $6,000 per year with the first investment made one year from now. You think you can earn 9.5% per year on your investments and you plan to retire in 38 years, immediately after making your last $6,000 investment. a. How much will you have in your retirement account on the day you retire? b. If, instead of investing $6,000 per year, you wanted to make one lump-sum investment today for your retirement that will result in the same retirement saving, how much would that lump sum need to be? c. If you hope to live for 28 years in retirement, how much can you withdraw every year in retirement (starting one year after retirement) so that you will just exhaust your savings with the 28th withdrawal (assume your savings will continue to earn 9.5% in retirement)? d. If, instead, you decide to withdraw $385,000 per year in retirement (again with the first withdrawal one year afterretiring), how many years will it take until you exhaust your savings? (Use trial-and-error, a financial calculator: solve for "N", or Excel: function NPER) e. Assuming the most you can afford to save is $1,200 per year, but you want to retire with $1,000,000 in your investment account, how high of a return do you need to earn on your investments? (Use trial-and-error, a financial
A. N = 38 PMT = 6000 I/YR = 9.5 FV = ? B. FV = 1923687.96 PMT = 0 N = 38 I/YR = 9.5 PV = ? C. N = 28 PV = 1923687.96 I/YR = 9.5 PMT = 0 FV = ? = 24419117.45 ... FV = 24419117.45 I/YR = 9.5 N = 28 PMT = ? D. PMT = -385000 FV = 0 PV = 1923687.96 I/YR = 9.5 N = ? E. PMT = -1200 FV = 1,000,000 PV = 0 N = 38 I/YR = ?
A rich relative has bequeathed you a growing perpetuity. The first payment will occur in a year and will be $5,000. Each year after that, you will receive a payment on the anniversary of the last payment that is 8% larger than the last payment. This pattern of payments will go on forever. Assume that the interest rate is 12% per year. a. What is today's value of the bequest? b. What is the value of the bequest immediately after the first payment is made?
A. 5000 / (0.12 - 0.08) = $125000 B. 1000 x 1.04 = 1040 (1040)/(.16 - .04) = 8667 *grows by 4%, then divide by difference of interests*
Assume that your parents wanted to have $160,000 saved for college by your 18th birthday and they started saving on your first birthday. They saved the same amount each year on your birthday and earned 8.0% per year on their investments. a. How much would they have to save each year to reach their goal? b. If they think you will take five years instead of four to graduate and decide to have $200,000 saved just in case, how much would they have to save each year to reach their new goal?
A: N = 18 FV = 160000 I/YR = 8 PMT = ? (find) B: N = 18 FV = 200000 I/YR = 8 PMT = ?
You make monthly payments on your car loan. It has a quoted APR of 5.6% (monthly compounding). What percentage of the outstanding principal do you pay in interest each month?
APR/12 = 5.6/12 = 0.466667%
When Alfred Nobel died, he left the majority of his estate to fund five prizes, each to be awarded annually in perpetuity starting one year after he died (the sixth one, in economics, was added later). a. If he wanted the cash award of each of the five prizes to be $40,000 and his estate could earn 9% per year, how much would he need to fund his prizes? b. If he wanted the value of each prize to grow by 2% per year (perhaps to keep up with inflation), how much would he need to leave? Assume that the first amount was still $40,000. c. His heirs were surprised by his will and fought it. If they had been able to keep the amount of money you calculated in (b), and had invested it at 9% per year, how much would they have in 2014, 118 years after his death?
a. (5 x 40,000) / (0.09) = ? b. (5 x 40,000) / (0.09 - 0.02) = ? *200,000/0.07* C. N = 118 I/YR = 9 PV = 2857142.86 PMT = 0 FV = ?
The British government has a consol bond outstanding paying £100 per year forever. Assume the current interest rate is 4% per year. a. What is the value of the bond immediately after a payment is made? b. What is the value of the bond immediately before a payment is made?
a. 100/.04 = 2500 b. 2500 + 100 (because when the payment is made it loses that 100 and goes back to 2500)
You are looking to buy a car and you have been offered a loan with an APR of 6.1%, compounded monthly. a. What is the true monthly rate of interest? b. What is the EAR?
a. 6.1%/12 = 0.5083% = 0.005083 b. [(1 + 005083)^12]-1 = 0.06273 = 6.2730%