Financial Management Topic 5 (Time Value of Money)
FV
future value FV = PV (1 + r)^n
PV
present value PV = FV / (1 + r)^n
(True/False) The effective rate is always lower than or equal to the stated rate. TrueFalse
False The answer is False. The effective rate is always higher than or equal to the stated rate since the effective rate considers the impact of compounding.
What is the present value of an ordinary annuity with the payment of $300 for 5 years if the discount rate is 10%? $1,137.24 $1,228.53 $1,831.53 $1,530.30
$1,137.24 PVAordinary = PMT [(1-{1/(1+r)^n})/r] = 300 [(1-{1/(1.10)^5})/0.10] = 1,137.24 [Calculator: 300 PMT, 10 I/Y, 5 N, compute PV]
What is the present value of an ordinary annuity with the payments of $500 for 3 years if the discount rate is 10%? $1,655.00 $1,367.77 $1,243.43 $1,820.55
$1,243.43 PVAordinary = PMT [(1-{1/(1+r)^n})/r] = 500 [(1-{1/(1.10)^3})/0.10] = $1,243.43 [Calculator: 500 PMT, 10 I/Y, 3 N, compute PV].
Your friend plans to contribute $315 a month at the end of each month to his retirement account. His employer will match the contribution on a dollar for dollar basis. He expects an annual return of 6%. How much will he have in 40 years? $1,170,000 $1,254,639 $1,240,200 $1,260,912
$1,254,639 PV: 0FV: $1,254,639.16 PMT: -630 N: 480 I: 0.50% Mode: 0 (End)
What is the future value of an ordinary annuity with payments of $300 for 5 years if the discount rate is 10%? $1,831.53 $1,530.30 $1,137.24 $1,228.53
$1,831.53 FVAordinary = PMT [({1+r}^n - 1)/r] = 300 [({1.10}^5 - 1)/0.10] = 1831.53 [Calculator: 300 PMT, 5 N, 10 I/Y, compute FV]
Suppose you turned 25 today and got married to your dream spouse (what a day!). You receive a wedding gift of $5,000. In consultation with your new spouse, you decide to put the money in an account that is earning 8% so that you can travel around the world when you retire at the age of 65. How much will you have in your account in 40 years? (allow $1 rounding) $108,623 $100,576 $117,312 $21,000 FEEDBACK
$108,623 FV = PV(1+r)t = 5000(1+0.08)40 = $108,622.61 Found in the following section(s) of the text: 5.3: Single Sum Calculations
You calculated the present value of an ordinary annuity to be $10,000. The discount rate is 10%. Which of the following is the most likely solution for the present value of an annuity due with the same cash flows? $11,000 $9,500 $9,000 $10,000
$11,000 If the stream of cash flows are the same between ordinary annuities and annuities due, then the present value of an annuity due must be higher. (The annuity due method shifts payments closer to time 0 which means the cash flows are discounted less).
Your grandparents decided to give you $3,500 a year at the beginning of each year for 4 years while you are in college. If instead they were to give you an equivalent single sum today, how much would they have to give you today? Assume that the discount rate is 6%. $16,229.83 $12,127.87 $15,311.16 $12,855.54
$12,855.54 PV: ($12,855.54) FV: 0 PMT: 3500 N: 4 I: 6% Mode: 1 (Begin)
During all 4 years of your college life, you save $3,000 a year at the end of each year. If you can earn a 6% annual return on your investment, how much will you have when you graduate in 4 years? $10,395 $13,124 $13,911 $12,000
$13,124 PV: 0 FV: $13,123.85 PMT: -3000 N: 4 I: 6% Mode: 0(End)
You and your spouse welcomed a new baby boy today. Your parents gave you $5,000 for the child's education and you invest it in an account that earns 7% annual return, so that he can use the money to go to a college in 18 years. How much will he have when he starts going to college? $15,794 $16,900 $19,348 $18,083
$16,900 FV = PV(1+r)t = 5000(1+0.07)^18 = $16,900
Your grandparents set-up a family education fund that pays $5,000 every quarter in perpetuity to fund family educational pursuits. The fund earns 12%. How much did your grandparents set aside to establish the fund? Assume that the fund started making distributions 3 months after establishment. $41,667 $46,667 $166,667 $125,000
$166,667 Since payments are made every 3 months, you need to divide the rate by 4 (quarterly rate of 12%/4 = 3%). PVperpetuity = PMT/Rate = 5000/0.03 = $166,667 Found in the following section(s) of the text: 5.7: Other Cash Flow Patterns
Today is your 30th birthday. As luck would have it, you also welcome a new baby into your home as your birthday present. You desire to provide some financial support for your baby in 18 years and a lump sum of money when you retire at the age of 60. Four years of college tuition costs a total of $21,657 in today's dollars. Additionally, you desire to have purchasing power of $1,000,000 (at today's value) when you retire. If the average inflation rate is 2% and you can earn 8% on your account, how much do you need in your account today to cover your baby's future college tuition and your retirement goal? (allow $5 rounding error) $104,797 $326,213 $187,749 $181,698
$187,749 · Step 1: $21657 in 18 years = FV = PV(1+r)^t = 21657(1+0.02)^18 = $30931.53 · Step 2: $1,000,000 in 30 years = FV = PV(1+r)^t = 1000000(1+0.02)^30 = $1,811,362 · Step 3: Discount tuition for 18 years = FV = PV(1+r)^t = 30931.53/(1+0.08)^18 = $7741 · Step 4: Discount retirement for 30 years = PV = FV/(1+r)^t = 1811362/(1+0.08)^30 = $180,008 Step 5: Add PVs together; Total amount needed today = $7741 + 180008 = $187,749.
Starting on your child's 10th birthday, you plan to make annual contributions of $1,994.70 to an account that earns a 5% annual return. Your final contribution will be on your child's 17th birthday. How much will your child have in the account when they start college on their 18th birthday? $23,094 $19,048 $21,995 $20,000
$20,000 PV: 0 FV: ($20,000) PMT: 1,994.70 N: 8 I: 5% Mode: 1 (Begin)
You are considering buying a security that will pay $1,000 every year forever. If you buy the asset, you will start receiving the first $1,000 payment immediately (today). If the discount rate is 5%, what is the most you should be willing to pay for this security? $19,000 $22,000 $20,000 $21,000
$21,000 Remember that the perpetuity formula assumes the payments come at the end of the period (i.e., the first payment is one year from today). If the payment starts immediately, use the perpetuity formula to get the value of payments at time 1 through infinity and add one additional payment of $1000. So PV = (1000/0.05) + 1000 = $21,000. Found in the following section(s) of the text: 5.7: Other Cash Flow Patterns
You just turned 25 and graduated from a university. You are already thinking about your retirement. You want to have $500,000 in today's dollar value when you retire. You have a lot of savings today and you were wondering if the savings may be enough if you invest wisely. You are anticipating an average inflation rate of 2% and you can earn 10% return on average. If you retire 40 years from now, how much savings do you need to have today to reach your goal? $23,015 $11,047 $10,248,724 $24,393
$24,393 First, you need to find the future value of $500,000 with the inflation rate. Then you need to find how much you need today if you can earn 10% a year for 40 years. Step 1: FV = PV(1+r)^t = 500000(1+0.02)^40 = $1,104,020 [calculator: 500000 PV, 2 I/Y, 40 N, compute FV] Step 2: PV = FV/(1+r)^t = 1104020/(1+0.10)^40 = $24,393 [calculator: 1104020 FV, 10 I/Y, 40 N, compute PV]
You decided to invest $2,000 a year for next ten years starting one year from today. If you make the annual contribution into an account paying 5%, how much will you have in 10 years? (round to nearest $1) $26,414 $25,156 $27,156 $25,500
$25,156 PV FV PMT N I Mode - ($25,155.79) 2,000.00 10 5.00% 0 (End) Found in the following section(s) of the text: 5.3: Single Sum Calculations
4 years ago you received a student loan of $20,000 with the annual interest rate of 8% compounded monthly. Because it is a student loan, you did not make any payments during the time you were in school (interest was still accruing). What is the current balance of the student loan? $27,557.09 $20,000.00 $20,538.69 $27,209.78
$27,557.09 PV: 20000 FV: $27,557.09 PMT: 0 N: 48 I: 0.67% Mode: 0 (End)
You have a 5-year-old son. You want to provide financial help when he goes to college in 13 years. You are planning to give him $20,000 a year at the beginning of each year. You will be make annual contribution at the end of each year to an account which will earn 5%. If you have $5,000 in the account already, how much will your annual deposit be? $3,471.51 $3,496.85 $4,203.98 $3,671.70
$3,671.70 PV: ($74,464.96)FV: 0PMT: 20000N: 4I: 5%Mode: 1 (Begin) PV: ($5,000.00)FV: $74,464.96 PMT: $(3,671.70)N: 13I: 5.00%Mode: 0 (End)
Your firm is considering a new project that will initially cost $25 million and will generate cash flows of $5 million in year 1, $10 million in year 2, $12.5 million in year 3, $7.5 million in year 4 and $2.5 million in year 5. If the required rate is 10%, what is the total present value of cash flows? (Be sure to subtract out the initial cost of $25 million.) $37.50 million $12.50 million $3.88 million $28.88 million
$3.88 million Step 1: find the present value of the future cash inflows: Year 1 2 3 4 5 FV 5 10 12.5 7.5 2.5 PV $4.55 $8.26 $9.39 $5.12 $1.55 PV of CFs $28.88 Step 2: subtract the initial cost from the present value of the inflows calculated in step 1: PV = $28.88 - $25 = 3.88 million. Use:PV = FV / (1 + r)^n = 150 / (1.07)^8 = $87.30
4 years ago, you received a student loan of $20,000 with the annual interest rate of 8% compounded monthly. Because it is a student loan, you did not make any payments while you were in school (although interest was still accruing). You just graduated and your payments start at the end of each month (assume that today is 1st of the month). If you plan to pay back the loan over 10 years, what is the amount of your monthly payment? $331.60 $330.13 $334.93 $327.94
$334.93 Step 1: Find the amount of the loan at the beginning of the current month. PV: 20000FV: $27,557.09PMT: 0N: 48I: 0.67%Mode: 0 Step 2: Calculate payment for a 10 year amortization of the loan PV: $27,557.09FV: 0PMT: 334.93N: 120I: 0.67%Mode: 0 (End)
You just bought a new car. You will make a down payment of $4,000 now, and take a loan for the remaining amount. The loan is for 5 years as with an APR of 4.8%. If you are going to make monthly payments of $563.39 at the end of each month, what is the price of the car you purchased? $33,268 $30,120 $30,000 $34,000
$34,000 PV: 30000FV: 0PMT: -563.39N: 60 I: 0.4%Mode: 0
You have a 5 year old son. You want to provide financial help when he goes to college in 13 years. You are planning to give him $20,000 a year at the beginning of each year during his 4 years of college. How much do you have to set aside today if you can earn annual interest of 5% in an account you deposit the money? $39,490 $42,426 $37,610 $24,904
$39,490 Step 1: PV: ($74,464.96) FV: 0 PMT 20000 N: 4 I: 5% Mode: 1 (Begin) Step 2: PV: ($39,490.36) FV: $74,464.96 PMT: 0 N: 13 I: 5% Mode: 0
You are starting a four-year educational program today. Since you did not save enough money, you plan to take a loan of $10,000 each year for four years. The loan has an interest rate of 4.5% and you are taking the first $10,000 installment today. You plan to finish your education in exactly four years. What will be the balance owing on your loan when you graduate? (Round to the nearest $1) $41,800 $44,707 $42,782 $40,000
$44,707 PV- FV PMT N I Mode - ($44,707) 10,000.00 4 4.50% 1 (Begin) Found in the following section(s) of the text: 5.3: Single Sum Calculations
Your friend will retire a year from today and she is hoping to have enough in her retirement account for her to withdraw $50,000 each year for 15 years with the first withdrawal coming on the day she retires (i.e., one year from today). She does not plan to contribute any money during her final year before retirement. She has been earning 7% return on her retirement account and is assuming that will continue in the future. How much should she have today in her account? $487,273 $467,883 $455,396 $437,273
$455,396 PV: ($455,395.70) FV: 0 PMT: 50000 N: 15 I: 7% Mode: 0
You just graduated from a university and have some student loan debt. The interest rate on your loan is 4%. If you make monthly payments of $504.54 for 10 years at the beginning of each month (starting today) your debt will be completely repaid. The amount you owe today is closest to which of the following? (Round to the nearest $1) $50,000 $49,834 $60,545 $49,107
$50,000 PV FV PMT N I Mode ($50,000) $0.00 504.54 10 x 12= 120 4/12 = 0.3333% 1 (Begin) Found in the following section(s) of the text: 5.7: Other Cash Flow Patterns
You just turned 20 years old today and started thinking about retirement savings. If you deposit $2,000 at the end of each year into an account earning annual interest of 8%, how much will you have if you retire in 40 years? $427,219 $518,113 $399,270 $559,562
$518,113 PV: 0 FV: ($518,113.04) PMT: 2000 N: 40 I: 8% Mode: 0
You are planning for retirement and need $4,000,000 when you retire in exactly 40 years. Your employer will match all contributions to your retirement account on 1-for-1 basis and you can earn 8% on all invested funds. Currently, you have $10,000 in your retirement account. To reach your goal, you plan to make monthly contributions starting today. How much do you have to contribute to your retirement account each month? (Use at least four decimal places of accuracy for the interest rate and round to the nearest $1) $538.13 $1,069.14 $1,076.27 $534.57
$534.57 PV($10,000.00) FV$4,000,000.00 PMT($1,069.14) N40 x 12 = 480 I 8 / 12 = 0.67%1 (Begin) This table represents TVM buttons on your financial calculator. Step 1: Solve for the total monthly contribution (PMT). The highlighted cell is the solution. (Put calculator in Begin mode, enter PV, FV, N, I, and solve for PMT.) Step 2: Divide the total monthly payment by two to account for the 1-for-1 matching contribution. 1069.14/2 = $534.57 Found in the following section(s) of the text: 5.9: Solving for Any Variable
Your Mom is asking you for help with her retirement planning. She and your Dad are hoping to start withdrawing from their retirement accounting beginning in one year from today, and they want to make sure that they have enough in the account to make annual withdrawals of $60,000 for 20 years. Their retirement account pays 8% annually. How much do they have to have now in the account? (Round to the nearest $1) $600,000 $622,313 $589,089 $636,216
$589,089 PV FV PMT N I Mode ($589,089) $0.00 60,000.00 20 8.00% 0 (End) Found in the following section(s) of the text: 5.7: Other Cash Flow Patterns
Your professor just talked about going on a trip when he retires in 10 years. He anticipates the cost of the trip to be $10,000. He said he will deposit some money today so that he will have $10,000 in 10 years. If he can earn an annual rate of 5% on his account, how much does he have to set aside today? $6,139 $5,847 $19,672 $21,049
$6,139 PV = FV/(1+r)^t = 10000/(1+0.05)^10 = $6,139
If you were to set up a perpetual educational fund for your family which will pay $5,000 a year forever, how much do you have to set a side today? Assume the appropriate discount rate is 8%. $65,000 $62,500 $67,500 $70,000
$62,500 The correct answer is $62,500. PVperpetuity = PMT/I = 5000/0.08 = $62,500
Your firm is considering a new project. The project will cost the firm $50 million initially and will generate cash flows of $10 million in year 1, $20 million in year 2, $25 million in year 3, $15 million in year 4 and $5 million in year 5. If the required rate is 10%, what is the total present value of cash flows? (Be sure to subtract out the initial cost of $50 million.) $75.00 million $57.75 million $25.00 million $7.75 million
$7.75 million Step 1: Present values of future cash flows: Year 1 FV: $10 PV: $9.09 Year 2 FV: $20 PV: $16.53 Year 3 FV: $25 PV: $18.78 Year 4 FV: $15 PV: $10.25 Year 5 FV: $5 PV: $3.10 PV - Inflow: $57.75 Step 2: PV of CFs - Initial Cost = $57.75 million - $50 million = 7.75 million Use: PV = FV / (1 + r)^n = 150 / (1.07)^8 = $87.30
You have a son who just turned 8 years old. You want him to go to a university and would like to provide financial support. Your goal is to give him a gift of $15,000 when he enters college in exactly 10 years. Assuming you can earn 5%, how much do you have to set aside today? (round to the nearest $1) $9,209 $24,433 $7,500 $15,000
$9,209 PV = FV/(1+r)^t = 15000/(1+0.05)^10 = $9,208.70 Found in the following section(s) of the text: 5.3: Single Sum Calculations
int
(I/YR) Interest Rate (consistent with number of periods)
Multi-step TVM Problems: When the number of periods or payments change in a problem, you must use multiple steps, like how? Break apart the problem going from the future to the present. Example: Savings for 15 years for college and then withdrawing $20k for 4 years using 6%. What do you have to deposit per year in savings?
*Find PV of 20k for 4 years at 6% interest (-0-FV) *Use PV as FV to find PMT for 15 years at 6% interest (-0-PV) Solved: 6 I/Y, 4N, 20,000 PMT, CPT PV =-73,460.23 FV73,460.23, N15, I/Y 6, PV 0, CPT PMT=-2,977.4
Basic TVM Problem: A couple wants to build up their savings to equal $2,500,000 when they retire in forty years. They currently have $50,000 in the bank and feel they will earn 5.5% interest. What do they have to save each year to reach their goal? PV -50,000 N 40 I/Yr 5.5 FV 2,500,000 What is the Payment (PMT)? (Pd at EOY)
-$15,184.84
What is the discount rate of a stream of cash flows of 50,000 that have a present value of 450,000? .12 .10 .11 .75
.11 50,000 / 450,000 = 11.1%
Inflation Problem If you live for 20 years in retirement, how much will inflation of 3% impact the purchasing power of your retirement income? (hint: choose an income level and calculate the equivalent amount after 20 years)
1 [PV], 20 [N], 3 [I/YR], solve for [FV] = $1.806 Point: You will need $1.81 in the 20th year of retirement to buy as much as $1 when you initially retired. The purchasing power of your income is cut almost in half!
A woman has just found out that a rich great-aunt has bequeathed a trust fund that pays $50,000 to her and to her descendants forever. If the trust fund earns 3.5% interest, what is the amount of the trust fund? 1,782,425 1,428,571 5,000,000 2,529,123
1,428,571 50,000 / .035 = PV or 1,428,571
What is the future value of an ordinary annuity with payments of $500 for 3 years if the discount rate is 10%? $1,243.43 $1,655.00 $1,820.55 $1,367.77
1,655.00 FVAordinary = PMT [({1+r}n - 1)/r] = 500 [({1.10}3 - 1)/0.10] = 1,655.00 Found in the following section(s) of the text: 5.4: Ordinary Annuities
What is the present value of a stream of cash flows of $125,000 at a discount rate of 7%? 1,552,667 875,000 1,785,714 1,250,000
1,785,714 125,000 / .07 = 1,785,714
You just purchased a new car that cost $20,000 with all taxes and fees. You are making a down payment of $5,000 and can afford $200 payments each month starting one month from today. If the APR of the loan is 3%, how many months will it take to repay the loan? (round to nearest month) 75 months 78 months 83 months 84 months
83 months PV FV PMT N I Mode PV$15,000.00 $0.00 -200 83.2 0.25% 0 (End) Found in the following section(s) of the text: 5.8: Non-Annual Compounding
Your friend is trying to finance a home purchase and asks for your help. The home she wishes to buy will cost $300,000. She has enough cash to make a down payment of 20% of the price of the house and can afford monthly mortgage payments of $2,500 for principal and interest (ignore taxes and insurance). The mortgage contract calls for monthly payments due at the beginning of each month with the first payment due at the signing of the loan agreement. If the loan will be completely repaid in 20 years, the annual interest rate is closest to: (Note: use four decimal places of accuracy) 11.14% 0.94% 11.28% 8.33%
11.28% Step 1: Calculate the monthly rate. PV$240,000.00 FV$0.00 PMT(2,500) N240 I0.9402% MODE 1 (due) Step 2: Annualize the monthly rate (multiply by 12). Using monthly data in Step 1 gives us a monthly rate. To annualize, multiply by 12: .9402 x 12 =11.28%APR. Note the first payment is due at contract signing; hence, you must use the annuity due mode (i.e., Begin mode) on your calculator. If you erroneously used the end mode, you would get answer C (11.14%).
A couple has $25,000 in their retirement savings today. How many years do they have to save at 6%, putting in $1,000 at the beginning of each year, to reach $80,000? 20.0 22.2 34.8 14.2
14.2
A couple wants to save up for a down payment on a house. They think they need to save $100,000 in five years. If the interest rate is 4% and they start at the end of the year when they both get bonuses from their employers, how much do they have to put aside annually? 18,462.71 17,752.61 22,096.37 15,962.84
18,462.71 FV = 100,000, N = 5, I/Y = 4, PV = 0, Solve for PMT This is an END problem = 18,462.71
Which of the following gives the largest effective rate? 18.6% APR compounded daily 18.7% APR compounded monthly 19.0% APR compounded quarterly 20.4% APR compounded annually
18.6% APR compounded daily (Note: the solution below uses your calculator to find the highest FV and associated effective rate. You could also use the formula: Eff Rate = (1 + i/m)^m - 1; Where, i is the stated rate and m is the number of compounding periods per year.) PV FV PMT N I APR Effective Yield PV($100) $120.40 0 1 20.4000% 20.4% 20.40% PV($100) $120.40 0 4 4.7500% 19.0% 20.40% PV($100) $120.39 0 12 1.5583% 18.7% 20.39% PV($100) $120.44 0 365 0.0510% 18.6% 20.44% Found in the following section(s) of the text: 5.10: The Effective Rate
Which of the following gives the smallest effective yield? 18.7% APR compounded monthly 20.4% APR compounded annually 18.6% APR compounded daily 19.0% APR compounded quarterly
18.7% APR compounded monthly Recall the formula: EY = (1 + [stated/m])m - 1. Use the formula to calculate the effective yield for each option: 20.4% compounded annually: EY = (1 + [.204/1])1 - 1 = 20.40% (with m = 1, stated and EY are identical) 19% compounded quarterly: EY = (1 + [.19/4])4 - 1 = 20.40% 18.7% compounded monthly: EY = (1 + [.187/12])12 - 1 = 20.39% (smallest effective yield) 18.6% compounded daily: EY = (1 + [.186/365])365 - 1 = 20.44%
Set Payment Beginning Payments at Begin or End of each compounding period-set proper mode
2nd BGN 2nd ENTER (BGN/END only relate to payments in a problem). Make sure for BGN it says BGN at topic, END is default. or 2nd BGN(PMT key)-2nd-SET(enter key) Toggle between BEG and END by hitting 2nd - Set Press C/CE to get back to working register
Clear TVM registers
2nd CLR TVM (ALL xP/Y, P/Y, AMORT,BGN) or 2nd-CLR TVM(FV key)
Set Decimal to 4th place:
2nd Format then number 1 then ENTER
Set payment period to 1: P/Yr should always be set to 1.
2nd P/Y then 1 then ENTER or 2nd-P/Y (I/Y key)-1-Enter
A mother wants to contribute to her child's higher education fund. She wishes to have $15,000 available each year for six years. Her child starts college in 15 years and she can save 6% before school starts if she puts her end-of-year bonus into a trust fund and figures that the fund will earn 4% after her child begins her college education. How much does she have to put aside annually if the money is withdrawn for college at the beginning of each year attending college? 3,346.19 4,159.87 5,802.74 3,513.38
3,513.38 Step 1: PMT = 15,000, I/Y = 4, N = 6, FV = 0, Solve for PV Begin mode PV = 81,777.34 Step 2: Take PV in Step 1 and use it as FV in Step 2. N = 15, I/Y = 6, PV = 0, Solve for PMT End problem = 3,513.38
A man has just inherited $250,000. If he invests the money at 4.5%, how much can he expect to have at the end of 15 years when he retires? 120,254.27 519,732.04 477,862.41 483,820.61
483,820.61 PV = 250,000, I/Y = 4.5, N = 15, PMT = 0, Solve for FV No PMT, so it does not matter what mode. = 483,820.61
What is the cash flow stream for a present value 1,000,000 at 5% paid in equal installments in the future? 35,000 500,000 50,000 20,000
50,000 1,000,000 * .05 = 50,000
You just turned 20 years old today and you were thinking that it would be nice to have $4,000,000 when you retire. If you deposit $14,759 at the end of each year into an account which earns annual interest of 10%, how old will you be when you can retire with $4,000,000? 54 years old 35 years old 50 years old 55 years old
55 years old PV: 0FV: $4,000,000.00 PMT: -14,759 N: 35 I: 10% Mode: 0 Current age (20 years) + How long it takes (35 years) = 55 years old.
A person wants to put aside $500 at the beginning of each month for 10 years. If she estimates an interest rate of 5.5%, how much will she have in her savings account at the end of the 10 years? 80,118.33 86,437.68 70,154.99 76,905.66
80,118.33 PMT = 500, N = 120, I/Y = 0.4583 (5.5/12), PV = 0, Solve for FV This is a BEGIN problem 80,118.33
Which one of the following is NOT a characteristic of annuities due? Payments are of equal amounts. All are characteristics of annuities due. Payments are made at the beginning of each period. Payments are equally spaced.
All are characteristics of annuities due. Found in the following section(s) of the text: 5.4: Ordinary Annuities
Regulatory Compliance
All firms face regulation. When evaluating the regulatory structure of potential new markets or new products, or when faced with regulatory changes, TVM allows managers to evaluate compliance alternatives. Alternatives such as outsourcing, capital investment, and even whether to participate in a particular industry are evaluated with TVM tools.
Which of the following best describes the difference between an annuity due and an ordinary annuity? An ordinary annuity has a limited life, but an annuity due goes on forever. An ordinary annuity pays at the end of the period, but an annuity due pays at the beginning. An ordinary annuity goes on forever, but an annuity due has a limited life. An ordinary annuity pays at the beginning of the period, but an annuity due pays at the end.
An ordinary annuity pays at the end of the period, but an annuity due pays at the beginning.
m=compounds per year: pertaining to non annual compounding
Calculate monthly payments directly by stating r and n in monthly terms Process: 1) Set-up calculator, 2) enter 333,583 [PV], 6 / 12 = .5 [I/YR], and 30 × 12 = 360 [N], solve for [PMT] = $2,000 (some rounding error) Instead of a 30-year, 6% loan, calculate payments for a 360-month loan at .5% per month loan! The process is the same for quarterly loans (m = 4; N = years × 4, I/YR = APR / 4), semi-annual loans, etc. To deal with compounding problems, divide r and multiply n by the number of compounds per year.
Capital Budgeting:
Capital budgeting is the process of deciding in which potential projects a firm will invest. This requires an exacting calculation (covered in Topic 11) of cash flows over time. Of course, TVM allows us to understand the benefits and costs of accepting a particular project. The creation of value within the firm requires careful analysis of these complex long-lived cash flows.
Ordinary Annuity?
Cash flows at the end of the period
Social Responsibility:
Corporate citizenship can be an important part of firm operations. To the extent that alternatives involve long-term decisions, TVM will be a valuable tool in optimizing decisions.
Which of the following is another name for the effective yield? AFY APY ARR APR
Correct Answer: APY In finance, the Annual Percentage Yield (APY) is frequently called the effective yield.
Entrepreneurial Finance:
Early stage companies frequently need to attract capital to develop products and markets. The providers of this capital, known as venture capitalists, require compensation for investing in these risky ventures. Venture capitalists use TVM extensively to evaluate the benefits and related costs of investing.
Effective Yield Equation
Effective yield=[1+(i/m)]^m−1
Future Value FV: If you invest $100 today in an account that pays 10% interest, how much will you have one year from now? It is probably obvious that you would have $110 in one year. The equation that describes this relationship is
FV = PV (1 + r)^n = 100 (1.1)^1 = 110 or two years investment FV = PV (1 + r)^n = 100 (1.1)^2 = 121
PV entered as a negative number
FV must have a different sign than PV. if not the ERROR 5
FV
Future Value- The value of the principal at a point in the future
I/Yr
Interest rate per period.
A compounding period is what?
Is where the interest rate is applied to the principal. *A semi-annual bond has two compounding periods. Two payments of interest per year. *A monthly loan has twelve compounding periods. Twelve payments of interest per year.
The payments do not have to by annuities, why?
It all depends on the timing of the payments and when interest is earned.
You are planning to go on a great vacation when you retire in 30 years. You calculated the travel expense for several potential trips. To fund the travel, you decided to save $100 a year (starting at the end of the current year) for 30 years in an account with an annual interest rate of 7.5%. You want to use all of the savings for the trip. Which of the following four countries is the choice that is both feasible and will come closest to exhausting your savings? Japan - cost of $10,000. France - cost of $11,000. Australia - cost of $12,000. Brazil - cost of $9,000.
Japan - cost of $10,000. PV- FV($10,339.94) PMT100.00 N30 I 7.50% Mode 0 (End) Calculator: PV: 0. PMT: 100, N: 30, I: 7.5%, Mode: 0, solve for FV: $10,339.94.
You just graduated with a bachelor's degree in mechanical engineering and are starting a new job tomorrow. You have a goal to start an MBA program in exactly three years and need to have $30,000 saved at that time. To fund your MBA, you are saving $690 a month at the end of each month starting a month from now in an account paying 12%. Will you have $30,000 when you start your MBA? No; after three years, I only have $27,940 in my account. Yes; after three years, I will have $30,020 in my account. Yes; after three years, I will have $31,293 in my account. No; after three years, I only have $29,723 in my account.
No; after three years, I only have $29,723 in my account. PV0 FV($29,723.05) PMT690 N3 x 12 =36 I 12/12 = 1% Mode 0 (End)
n
Number of periods (i.e. years, months)
Solve Present Value, what buttons do you push?
PMT,N,I/yr,FV CPT & PV The calculator will solve for PV
Present Value (PV): Suppose you anticipate receiving $150 in eight years. What is the present value if we discount at 7%? Present value calculations for single sums use the same formula as future values (solved for PV):
PV = FV / (1 + r)^n = 150 / (1.07)^8 = $87.30
Basic Loan Calculations on a calculator: You ender every data point except for the one you want to solve for: Solve for interest rate, what buttons do you push?
PV,PMT,N,FV then CPT & INT The calculator will solve for INT
What if we use an annuity due approach? Calculating value as an annuity due simply shifts the timing of the PVA and FVA one period further into the future. The annuity due formulas, applied to our example, are as follows:
PVAdue=pmt(1−1/(1+r)^n/r)(1+r) 1000(1−1/(1.18)^5/.18)(1.18)=$3,690.061000(1−1(1.18)5.18)(1.18)=$3,690.06 FVAdue=pmt((1+r)^n−1/r)(1+r) 1000((1.18)^5−1/.18)(1.18)=$8,441.97
Present Value: In analogous fashion, we can calculate the present value of the same ordinary annuity (PVA) as
PVAordinary=pmt 1− 1/(1+r)^n/r 100(1−1(1.13)3.13)=$236.12100(1−1(1.13)3.13)=$236.12 Amazingly Important Interpretation: Be sure you understand what this $236.12 represents. Specifically, if you deposit $236.12 in an account paying 13% today (i.e., time 0), you can take out $100 at time 1, time 2, and time 3; the third withdrawal will completely exhaust your account. The important thing to notice is that with this formula/shortcut, we get a present value one period before the first payment . This will be an important distinction when we get to annuities due in a little while. Note that the calculation yielding a value one period before the first payment is a result of the assumption that payments are received at the end of each period with ordinary annuities. We'll change this to beginning of period payments in a later section (see Annuities Due)
Perpetuities are different from ordinary annuities in that perpetuities: Do not have equally spaced cash flows. Pay an infinite number of payments. Pay at the beginning of each period. Do not have equal dollar amount payments.
Pay an infinite number of payments. Perpetuities are annuities with unending payments. Found in the following section(s) of the text: 5.7: Other Cash Flow Patterns
PMT
Payment
PMT
Payment-the amount of any periodic payment
Which one of the following is NOT a characteristic of ordinary annuities? Payments are equally spaced. Payments are made at the beginning of each period. Payments are of equal amounts. All are characteristics of ordinary annuities.
Payments are made at the beginning of each period. With ordinary annuities, payments come at the END of the period. Found in the following section(s) of the text: 5.4: Ordinary Annuities
PV
Present Value- The Value of the principal today
An ordinary annuity is best defined as: Series of equally spaced cash flows of the same magnitude paid at the end of each period. Series of equally-spaced cash flows. Series of cash flows paid at the end of each period. Series of cash flows of the same magnitude.
Series of equally spaced cash flows of the same magnitude paid at the end of each period.
Suppose you are 30 years old today and desire a retirement income of $10,000 per month in today's dollars. If you believe inflation will average 3% over the next 40 years, how much income will you need in 40 years to have the same purchasing power as $10,000 today? How much would you need in your savings account today to fund one month of retirement income 40 years from now? Assume you can earn 9% on all invested funds.
Step 1 - Income in 40 years: FV = 10,000 (1.03)^40= 32,620.38 [Interpretation: If inflation is 3% per year, you will need $32,620.38 in 40 years to have the same purchasing power at $10,000 today] Step 2 - Current savings: PV = 32,620.38/(1.09)^40 = $1,038.55 [Interpretation: if you invest $1,038.55 at 9% it will grow to $32,620.38 in 40 years]
For over 90% of TVM calculations, there are four variables and one equation. Hence, we must know three of the variables and solve for the fourth. This is sometimes referred to as a "four-find-three" game. The basic steps for TVM calculations are(List the 3 steps)
Step 1- Set-up your calculator: This means clearing the memory registers and/or telling your calculator that you are solving a TVM problem. For the HP10B and the TI BAII+, the keystrokes are HP10B - press [2nd] and [clear all] TI BAII+ - press [2nd] and [clr TVM] Step 2 - Enter the three variables you know: There are five TVM buttons on your calculator: [FV], [PV], [PMT], [ N], and [ I/Yr]. If you have cleared your calculator as indicated in Step 1, all contain a value of zero. Simply enter the three "knowns" in any order. Step 3 - Solve for the unknown variable: Given Steps 1 and 2, simply solve for the fourth variable. For example, if calculating future value, HP10B - press [FV] TI BAII+ - press [cpt] [FV]
To perform annuity due calculations on your financial calculator, simply change to the "due mode" as follows
TI BAII+ 1) press [2nd] [BGN] (Which is the [ PMT] button) 2) press [2nd] [SET] (Which is the [ENTER] button)
Risk Assessment:
TVM plays a critical role in risk assessment. Frequently, this takes one of two forms: Risk buckets: Projects are grouped into "risk buckets" with different return requirements. For instance, high-risk projects are assessed in one bucket with high return requirements. TVM is used to assess the expected return of each project for comparison to the appropriate required return. Sensitivity analysis: TVM is used to assess the present value of projects under different input assumptions. For example, if you are evaluating a potential merger, sensitivity analysis allows you to find the value of the target firm under different revenue and/or cost projections. The valuations inform managers about the risk of the merger at various prices.
Future Value?
The future value of an ordinary annuity is a function of three variables: 1) payment size, 2) number of payments, and 3) the discount rate. Assuming the discount rate is 13%, the future value (FVA) of our three-payment, $100-per-payment annuity is FVAordinary=pmt[(1+r)^n−1/r] =100 (1+0.13)^3−1/0.13=$340.69 Interpretation: It is vital that you understand exactly what this means. Generally, this is the value of n consecutive equal deposits growing at a rate r. The value is stated at the time of the last payment. Thus, $340.69 is the future value of three annual payments of $100 growing at 13% stated at the time we make the last payment. The last $100 payment is included in the $340.69, but has not earned any return.
Define Interest Rate or Discount Rate
The rate at which a dollar chages in value due to the passage of time. cost of capital, hurdle rate.
Extended Example: Consider a single cash flow of $1000 at time 6 (i.e., six years from today). Assuming the discount rate is 12%, find the time-adjusted value at time 0, time 3, time 6, and time 20.
Time 0 - Discount for 6 years: PV = 1000 / (1.12)^6 = $506.63 Time 3 - Discount for 3 years: PV = 1000 / (1.12)^3 = $711.78 Time 6 - Discount for 0 years: PV = 1000 / (1.12)^0 = $1000 Time 20 - Compound for 14 years: FV = 1000 (1.12)^14 = $4,887.11 Interpretation: If you put $506.63 in an account earning 12%, it will grow to $711.78 in three years, $1000 in six years, and $4,887.78 in twenty years!
Annuity due
To Calculate the PC at the time of the first payment.
An Annuity is due when?
When the payment occurs at the beginning of the period (BEGIN) e.g Loans where payment is upfront
Ordinary Annuity is?
When the payment occurs at the end of the period (END) e.g. bonds where interest payments area after interest has been earned.
N
Years
Annuity
a stream of equal payments
Annuity is what?
an equally spaced series of cash flows all of the same magnitude. That is, payments of $100 at time 1, time 2, and time 3 would be a three-period, $100-per-payment annuity. However, there is no requirement that the first payment should come at time 0 or time 1. A series of $100 payments at time 17, time 18, and time 19 would also be a three-period, $100 annuity. Generally, we define an annuity as a series of equally-spaced cash flows of the same size.
Interest goes in at what
whole number Input '4' for 4% rather than '.04'
EOY
end of the year
PV always entered as a negative and always comes up as a what?
negative result
Interest rate formula
r-Real Risk-Free Rate + Inflation + Risk Premium where *r = Nominal discount rate (note: nominal means inflation is included) *Real risk-free rate = The rate earned on riskless investments with 0% inflation *Inflation = The annual decay in the purchasing power of money *Risk premium = Compensation for bearing the risk of a particular investment 1) amount of the cash flows, 2) the timing of the cash flows, and 3) the rate at which the value of the cash flows changes due to the passage of time.
The Effective rate
the lender is required to make a number of disclosures including APR and Annual Percentage Yield (APY) . In finance, APY is frequently called the effective yield. The difference between APR and effective yield is a result of compounding frequency. Recall, that m is the number of compounds per year. If compounding frequency is greater than once per year (m > 1), then the effective yield will be greater than the APR.
