1.1.4 The Future Value and Present Value of a Series of Equal Cash Flows (Ordinary Annuities, Annuity Dues, and Perpetuities) Flashcards
(35 cards)
Annuity
Annuity is a finite set of sequential cash flows, all with the same value.
Ordinary annuity
Ordinary annuity has a first cash flow that occurs one period from now (indexed at t = 1). In other words, the payments occur at the end of each period.
What is the formula of Future value of a ordinary annuity?
A = annuity amount
N = number of regular annuity payments
r = interest rate per period
FV = [A*(1+r)^(n−1) ] / r
What is the formula of Present value of a ordinary annuity?
PV = A * [ (1 - (1/(1+r)^n) ] / r
Annuity due has a first cash flow that is paid immediately (indexed at t = 0). In other words, the payments occur at the beginning of each period.
What is the formula of Future value of an annuity due?
FV = A * [ (1+r)^(n−1) / r ] * (1+r)
This consists of two parts: the future value of one annuity payment now, and the future value of a regular annuity of (N -1) period. Calculate the two parts and add them together. Alternatively, you can use this formula.
Note that, all other factors being equal, the future value of an annuity due is equal to the future value of an ordinary annuity multiplied by (1 + r).
What is the formula of Present value of an annuity due?
PV = A * [ 1 - (1/(1+r)^n / r ] * (1+r)
This consists of two parts: an annuity payment now and the present value of a regular annuity of (N - 1) period. Use the above formula to calculate the second part and add the two parts together. This process can also be simplified to a formula:
Note that, all other factors being equal, the present value of an annuity due is equal to the present value of an ordinary annuity multiplied by (1 + r).
Perpetuity ?
An example of a perpetuity is a stock paying constant dividends.
A perpetuity is a perpetual annuity: an ordinary annuity that extends indefinitely.
In other words, it is an infinite set of sequential cash flows that have the same value, with the first cash flow occurring one period from now.
Example: Future value of a regular annuity
An analyst decides to set aside $10,000 per year in a conservative portfolio projected to earn 8% per annum. If the first payment he makes is one year from now, calculate the accumulated amount at the end of 10 years.
Method 1: Using a formula
Identify the given variables: A = 10,000, r = 0.08, N = 10
Identify the appropriate formula: FV = A x {[(1 + r)N - 1] / r}
Solve for the unknown: FV = 10,000 {[(1 + 0.08)10 - 1] / 0.08} = $144,865
- If you owed $200 at the end of each year for the next three years, the present value of the obligation would be ______.
A. less than it would be if you owed all $600 at the end of three years
B. the same as it would be if you had to pay $300 today and $300 at the end of three years
C. less than it would be if you had to pay $300 today and $300 at the end of this year
Correct Answer: C
2. What is the present value of the following annuity due? Payment amount = $100 Payment frequency = annual, at the beginning of each year Number of payments = 20 Interest rate = 8% per year A. $981.81 B. $1,060.36 C. $1,145.19
Correct Answer: B
3. What is the present value of the following regular (ordinary, deferred) annuity? Payment amount = $100 Payment frequency = annual, at the end of each year Number of payments = 20 interest rate = 8% per year A. $981.81 B. $1,840.00 C. $2,000.00
Correct Answer: A
4. What is the future value of the following annuity due? Payment amount = $100 Payment frequency = annual, at the beginning of each year Number of payments = 20 Interest rate = 8% per year A. $2,000.00 B. $4,576.20 C. $4,942.29
Correct Answer: C FV = 100(1.08)20 + 100(1.08)19 + 100(1.08)18 + … + 100(1.08)2 + 100(1.08)1 = $4,942.29 (Or use the formula to calculate.)
- You have invested in an annuity that pays you $1,500 per year. Payments are always made at the end of the year. If each payment is invested at the rate of 11% per year, what is the total amount you will have accumulated (payments plus interest) by the end of 12 years?
A. $19,980.00
B. $30,981.87
C. $34,069.78
Correct Answer: C FV = 1,500.00(1.11)11 + 1,500.00(1.11)10 + 1,500.00(1.11)9 + … + 1,500.00(1.11)1 + 1,500.00(1.11)0 = $34,069.78 (Or use the formula to calculate.)
- You have invested in an annuity that pays you $1,500 per year. Payments are always made at the end of the year. If each payment is invested at the rate of 11% per year, what is the total amount you will have accumulated (payments plus interest) by the end of 12 years?
A. $19,980.00
B. $30,981.87
C. $34,069.78
Correct Answer: C FV = 1,500.00(1.11)11 + 1,500.00(1.11)10 + 1,500.00(1.11)9 + … + 1,500.00(1.11)1 + 1,500.00(1.11)0 = $34,069.78 (Or use the formula to calculate.)
6. What is the future value of the following regular (ordinary, deferred) annuity? Payment amount = $255 Payment frequency = annual, at the end of each year Number of payments = 7 Interest rate = 4% per year A. $1,856.40 B. $2,014.07 C. $2,590.80
Correct Answer: B FV = 255(1.04)6 + 255(1.04)5 + 255(1.04)4 + … + 255(1.04)1 + 255(1.04)0 = $2,014.07 (Or use the formula to calculate.)
- What is the present value of 15 payments of $100 each received every 18 months at a discount rate of 9%?
A. $951.28
B. $1209.10
C. $620.43
Correct Answer: C (1+i)18/12 = 1.137993 or 13.8% CALC: N=15 FV=0 PV=? I=13.7993 PMT=100; PV = 620.43
- What is the present value of 15 payments of $100 each received every 18 months at a discount rate of 9%?
A. $951.28
B. $1209.10
C. $620.43
Correct Answer: C (1+i)18/12 = 1.137993 or 13.8% CALC: N=15 FV=0 PV=? I=13.7993 PMT=100; PV = 620.43
- An annuity is defined as ______.
A. equal cash flows at equal intervals of time forever
B. equal cash flows at equal intervals of time for a specific period
C. unequal cash flows at equal intervals of time forever
Correct Answer: B
- Which of the following amounts is closest to the end value of investing $80,000 for 3 years compounded continuously at a rate of 12%?
A. $112,750
B. $113,550
C. $114,667
Correct Answer: C EAR = (er - 1) = e.12 - 1 = 0.1275 (12.75%) End value after 3 years = 80,000 (1 + 0.1275)3 = $114,667.31
- You are choosing between investments offered by two different banks. One promises a return of 10% for three years in simple interest while the other offers a return of 10% for three years in compound interest. You should ______.
A. choose the simple interest option because both have the same basic interest rate
B. choose the compound interest option because it provides a higher return than the simple interest option
C. choose the compound interest option only if the compounding is for monthly periods
D. choose the simple interest option only if compounding occurs more than once a year
E. choose the compound interest option only if you are investing less than $5,000
Correct Answer: B
- Your rich aunt has offered to give you $150 at the end of each of the next 30 months. You plan to put the money into your savings account, which pays an interest rate of 5.5% per year compounded monthly. How much do you expect to have at the end of the 30 months?
A. $4,500.00
B. $4,747.50
C. $4,812.26
Correct Answer: C Use the time value of money functions of your calculator: n = 30 i = 5.5/12 = 0.45833 PV = 0 PMT = 150 CPT FV => FV = $4,812.26
- You expect to receive annual payments of $425 per year forever. If these payments can be invested at the rate of 11% per year, what is the present value of this perpetuity?
Correct Answer: $3,863.64 PV = 425/0.11
- A perpetuity is ______.
A. an annuity with 40 periods to maturity
B. a growing stream of payments for 100 years
C. a lump sum payment received in 100 years
D. an annuity with no maturity
E. one single cash payment
Correct Answer: D
- You are examining two perpetuities which are identical in every way except that perpetuity A will begin making annual payments of $P to you two years from today while the first $P payment of perpetuity B will occur one year from today. It must be true that ______
A. the current value of perpetuity A is greater than that of B by $P.
B. the current value of perpetuity B is greater than that of A by $P.
C. the current value of perpetuity B is equal to that of perpetuity A.
D. the current value of perpetuity A exceeds that of B by the PV of $P for one year.
E. the current value of perpetuity B exceeds that of A by the PV of $P for one year.
Correct Answer: E
