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Chapter 1 > 1.1.4 The Future Value and Present Value of a Series of Equal Cash Flows (Ordinary Annuities, Annuity Dues, and Perpetuities) > Flashcards

1.1.4 The Future Value and Present Value of a Series of Equal Cash Flows (Ordinary Annuities, Annuity Dues, and Perpetuities) Flashcards

1
Q

Annuity

A

Annuity is a finite set of sequential cash flows, all with the same value.

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2
Q

Ordinary annuity

A

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.

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3
Q

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

A

FV = [A*(1+r)^(n−1) ] / r

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4
Q

What is the formula of Present value of a ordinary annuity?

A

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.

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5
Q

What is the formula of Future value of an annuity due?

A

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).

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6
Q

What is the formula of Present value of an annuity due?

A

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).

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7
Q

Perpetuity ?

An example of a perpetuity is a stock paying constant dividends.

A

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.

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8
Q

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.

A

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

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9
Q
  1. 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
A

Correct Answer: C

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10
Q 
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
A

Correct Answer: B

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11
Q 
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
A

Correct Answer: A

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12
Q 
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
A

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.)

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13
Q
  1. 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
A

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.)

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14
Q
  1. 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
A

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.)

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15
Q 
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
A

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.)

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16
Q
  1. 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
A

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

17
Q
  1. 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
A

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

18
Q
  1. 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
A

Correct Answer: B

19
Q
  1. 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
A

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

20
Q
  1. 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
A

Correct Answer: B

21
Q
  1. 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
A

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

22
Q
  1. 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?
A

Correct Answer: $3,863.64 PV = 425/0.11

23
Q
  1. 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
A

Correct Answer: D

24
Q
  1. 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.
A

Correct Answer: E

25
Q
  1. You wish to accumulate $500,000 by making annual deposits of $9,869.82 into a money market account. All deposits will be made at the beginning of each year. The account pays an interest rate of 16%. How many deposits will be necessary?
    A. 13
    B. 14
    C. 50
A

Correct Answer: B BGN; FV = $500,000; PMT = -$9,869.82; i = 16%; CPT n = 14

26
Q
  1. At the end of 15 years, you wish to have accumulated $500,000. You plan to accumulate this money by making annual deposits of $10,003.93 into a money market account. All deposits will be made at the beginning of each year. What interest rate must the account pay each year?
    A. 8%
    B. 12%
    C. 14%
A

Correct Answer: C BGN; FV = $500,000; PMT = -$10,003.93; n = 15; CPT i = 14%

27
Q
  1. You are trying to accumulate $5,000 in a savings account. The savings account pays 4% per year and you plan to make yearly deposits of $416.45 at the end of each year. How many yearly deposits must you make?
    A. 10
    B. 12
    C. 15
A

Correct Answer: A By calculator: PMT = -$416.45; i = 4%; FV = $5,000.00; CPT n = 10

28
Q
  1. At the end of 15 years, you wish to have accumulated $500,000. You plan to accumulate this money by making annual deposits into a money market account paying 8% per year. All deposits will be made at the beginning of each year. How much must be deposited each year?
    A. $17,050.72
    B. $18,414.77
    C. $15,787.70
A

Correct Answer: A BGN, FV = $500,000; n = 15; i = 8%; CPT PMT = -$17,050.72

29
Q
  1. You are trying to accumulate $5,000 in a savings account at the end of 4 years. The savings account pays 4% per year and you plan to make yearly deposits at the end of each year. How large must the deposits be?
    A. $1,177.45
    B. $1,250.00
    C. $1,377.45
A

Correct Answer: A

30
Q
  1. You are trying to accumulate $5,000 in a savings account at the end of 4 years. You plan to make yearly deposits of $1,061.63 at the end of each year. What interest rate must the savings account pay in order for you to accumulate $5,000?
    A. 10%
    B. 11%
    C. 12%
A

Correct Answer: B By calculator: PMT = -$1,061.63; n = 4; FV = $5,000.00; CPT i = 11%

31
Q
  1. You are to receive $1500 every year forever from the federal government as the winner of the national fiscal prudency and awareness contest. The government has also provided you with the option of choosing $1700 per year over the next 30 years. Payments are to be received semi-annually. If the market rate of interest is 8% annually, what is the value of the two options?
    A. $18,382; $10,323
    B. $18,750; $19,230
    C. $18,750; $19,138
A

Correct Answer: B (1500/2)/0.04 = 18,750; N=60, I=4%, PMT=850,FV=0, PV=? = 19,229.97

32
Q
  1. You are to receive $1500 every year forever from the federal government as the winner of the national fiscal prudency and awareness contest. The government has also provided you with the option of choosing $1700 per year over the next 30 years. Payments are to be received semi-annually. If the market rate of interest is 8% annually, what is the value of the two options?
    A. $18,382; $10,323
    B. $18,750; $19,230
    C. $18,750; $19,138
A

Correct Answer: B (1500/2)/0.04 = 18,750; N=60, I=4%, PMT=850,FV=0, PV=? = 19,229.97

33
Q
  1. A share of preferred stock pays a specific dividend on a specific schedule for as long as the issuing company exists. Assume that a share of preferred stock pays an annual per-share dividend at the end of each year. The present value of this share of preferred stock is $75.62. Assume that the company paying the dividends will exist forever. If the dividends can be invested at 4% per year, what is the amount of each dividend?
    A. $3.02
    B. $78.64
    C. infinite
A

Correct Answer: A A = (PV)(r) = (75.62)(0.04) = 3.02

34
Q
  1. You expect to receive a series of annual payments of $6572.89 forever. The present value of this series of payments is $45,000. At what rate of interest can these payments be invested?
    A. 1.461%
    B. 6.85%
    C. 14.61%
A

Correct Answer: C r = A/PV

35
Q
  1. You expect to receive a series of annual payments forever. The present value of this series of payments is $10,000. If these payments can be invested at a rate of 15% per year, what is the amount of each cash flow?
    A. $666.67
    B. $1,500.00
    C. infinite
A

Correct Answer: B A = (PV)(r) = (10,000.00)(.15) = 1,500.00

Chapter 1 (5 decks)

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