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MULTIPLE CHOICE

 

  1. Suppose someone offered you your choice of two equally risky annuities, each paying $5,000 per year for 5 years. One is an annuity due, while the other is a regular (or deferred) annuity. If you are a rational wealth maximizing investor which annuity would you choose?

a. The annuity due.

b. The deferred annuity.

c. Either one, because as the problem is set up, they have the same present value.

d. Without information about the appropriate interest rate, we cannot find the values of the two annuities, hence we cannot tell which is better.

e. The annuity due; however, if the payments on both were doubled to $10,000, the deferred annuity would be preferred.

 

  2. Suppose someone offered you the choice of two equally risky annuities, each paying $10,000 per year for five years. One is an ordinary (or deferred) annuity, the other is an annuity due. Which of the following statements is most correct?

a. The present value of the ordinary annuity must exceed the present value of the annuity due, but the future value of an ordinary annuity may be less than the future value of the annuity due.

b. The present value of the annuity due exceeds the present value of the ordinary annuity, while the future value of the annuity due is less than the future value of the ordinary annuity.

c. The present value of the annuity due exceeds the present value of the ordinary annuity, and the future value of the annuity due also exceeds the future value of the ordinary annuity.

d. If interest rates increase, the difference between the present value of the ordinary annuity and the present value of the annuity due remains the same.

e. Answers a and d are correct.

 

  3. Which of the following statements is most correct?

a. Other things held constant, an increase in the number of discounting periods per year increases the present value of a given annual annuity.

b. Other things held constant, an increase in the number of discounting periods per year increases the present value of a lump sum to be received in the future.

c. The payment made each period under an amortized loan is constant, and it consists of some interest and some principal. The later we are in the loan's life, the smaller the interest portion of the payment.

d. There is an inverse relationship between the present value interest factor of an annuity and the future value interest factor of an annuity, (i.e., one is the reciprocal of the other).

e. Each of the statements above is true.

 

  4. Which of the following is most correct?

a. The present value of a 5-year annuity due will exceed the present value of a 5-year ordinary annuity. (Assume that both annuities pay $100 per period and there is no chance of default.)

b. If a loan has a nominal rate of 10 percent, then the effective rate can never be less than 10 percent.

c. If there is annual compounding, then the effective, periodic, and nominal rates of interest are all the same.

d. Answers a and c are correct.

e. All of the answers above are correct.

 

  5. Which of the following statements is most correct?

a. An investment which compounds interest semiannually, and has a nominal rate of 10 percent, will have an effective rate less than 10 percent.

b. The present value of a three-year $100 annuity due is less than the present value of a three-year $100 ordinary annuity.

c. The proportion of the payment of a fully amortized loan which goes toward interest declines over time.

d. Statements a and c are correct.

e. None of the answers above is correct.

 

  6. Which of the following statements is most correct?

a. If an investment has interest compounded annually, its nominal rate must always equal its effective rate.

b. The present value of a 5-year ordinary annuity will be greater than the present value of a 5-year annuity due. (Assume that both annuities pay $100 per period, and that there is no chance of default).

c. In an amortized loan with monthly payments, the proportion of the payment that goes toward repayment of principal falls steadily over time.

d. Answers a and b are correct.

e. Answers a and c are correct.

 

  7. Which of the following statements is most correct?

a. The present value of a 5-year ordinary annuity paying $100 a year will be greater than the present value of a 5-year annuity due paying $100 a year.

b. If interest is paid more than once a year, then an investment's effective rate must exceed its nominal rate.

c. On an amortized loan, the percentage of each payment which goes toward the repayment of principal falls steadily over time.

d. None of the answers above is correct.

e. Answers b and c are correct.

 

  8. Frank Lewis has a 30-year, $100,000 mortgage with a nominal interest rate of 10 percent and monthly compounding. Which of the following statements regarding his mortgage is most correct?

a. The monthly payments will decline over time.

b. The proportion of the monthly payment which represents interest will be lower for the last payment than for the first payment on the loan.

c. The total dollar amount of principal being paid off each month gets larger as the loan approaches maturity.

d. Statements a and c are correct.

e. Statements b and c are correct.

 

  9. Your subscription to Jogger's World Monthly is about to run out and you have the choice of renewing it by sending in the $10 a year regular rate or of getting a lifetime subscription to the magazine by paying $100. Your cost of capital is 7 percent. How many years would you have to live to make the lifetime subscription the better buy? Payments for the regular subscription are made at the beginning of each year. (Round up if necessary to obtain a whole number of years.)

a. 15 years

b. 10 years

c. 18 years

d.  7 years

e.  8 years

 

 10. Assume you are to receive a 20-year annuity with annual payments of $50. The first payment will be received at the end of Year 1, and the last payment will be received at the end of Year 20. You will invest each payment in an account that pays 10 percent. What will be the value in your account at the end of Year 30?

a. $6,354.81

b. $7,427.83

c. $7,922.33

d. $8,591.00

e. $6,752.46

 

 11. You are contributing money to an investment account so that you can purchase a house in five years. You plan to contribute six payments of $3,000 a year--the first payment will be made today (t = 0), and the final payment will be made five years from now (t = 5). If you earn 11 percent in your investment account, how much money will you have in the account five years from now (at t = 5)?

a. $19,412

b. $20,856

c. $21,683

d. $23,739

e. $26,350

 

 12. Your uncle has agreed to deposit $3,000 in your brokerage account at the beginning of each of the next five years (t = 0, t = 1, t = 2, t = 3 and t = 4). You estimate that you can earn 9 percent a year on your investments. How much will you have in your account four years from now (at t = 4)? (Assume that no money is withdrawn from the account until t = 4.)

a. $13,719.39

b. $17,954.13

c. $19,570.00

d. $21,430.45

e. $22,436.12

 

 13. You just put $1,000 in a bank account which pays 6 percent nominal annual interest, compounded monthly. How much will you have in your account after 3 years?

a. $1,006.00

b. $1,056.45

c. $1,180.32

d. $1,191.00

e. $1,196.68

 

 14. Assume that you can invest to earn a stated annual rate of return of 12 percent, but where interest is compounded semiannually. If you make 20 consecutive semiannual deposits of $500 each, with the first deposit being made today, what will your balance be at the end of Year 20?

a. $52,821.19

b. $57,900.83

c. $58,988.19

d. $62,527.47

e. $64,131.50

 

 15. You have $2,000 invested in a bank account that pays a 4 percent nominal annual interest with daily compounding. How much money will you have in the account at the end of July (i.e., in 132 days)? (Assume there are 365 days in each year.)

a. $2,029.14

b. $2,028.93

c. $2,040.00

d. $2,023.44

e. $2,023.99

 

 16. You are interested in saving money for your first house. Your plan is to make regular deposits into a brokerage account which will earn 14 percent. Your first deposit of $5,000 will be made today. You also plan to make four additional deposits at the beginning of each of the next four years. Your plan is to increase your deposits by 10 percent a year. (That is you plan to deposit $5,500 at t = 1, and $6,050 at t = 2, etc.) How much money will be in your account after five years?

a. $24,697.40

b. $30,525.00

c. $32,485.98

d. $39,362.57

e. $44,873.90

 

 17. Assume that your required rate of return is 12 percent and you are given the following stream of cash flows:

                 Year        Cash Flow

                   0          $10,000

                   1           15,000

                   2           15,000

                   3           15,000

                   4           15,000

                   5           20,000

 

If payments are made at the end of each period, what is the present value of the cash flow stream?

a. $66,909

b. $57,323

c. $61,815

d. $52,345

e. $62,029

 

 18. You are given the following cash flows. What is the present value (t = 0) if the discount rate is 12 percent?

                       0  12% 1      2            3           4       5      6         Periods

                            0      1    2,000  2,000  2,000    0   -2,000

a. $3,277

b. $4,804

c. $5,302

d. $4,289

e. $2,804

 

 19. You are given the following cash flow information. The appropriate discount rate is 12 percent for Years 1-5 and 10 percent for Years 6-10. Payments are received at the end of the year.

                       Year      Amount

                       1-5      $20,000

                       6-10     $25,000

 

What should you be willing to pay right now to receive the income stream above?

a. $166,866

b. $158,791

c. $225,000

d. $125,870

e. $198,433

 

 20. A project with a 3-year life has the following probability distributions for possible end-of-year cash flows in each of the next three years:

                         Year 1                  Year 2                    Year 3

         Prob  Cash Flow    Prob  Cash Flow    Prob  Cash Flow

         0.30    $300            0.15    $100             0.25    $200

         0.40     500             0.35     200               0.75     800

         0.30     700            0.35     600                0.15     900

 

Using an interest rate of 8 percent, find the expected present value of these uncertain cash flows. (Hint: Find the expected cash flow in each year, then evaluate those cash flows.)

 a. $1,204.95

b. $  835.42

c. $1,519.21

d. $1,580.00

e. $1,347.61

 

 21. You just graduated, and you plan to work for 10 years and then to leave for the Australian "Outback" bush country. You figure you can save $1,000 a year for the first 5 years and $2,000 a year for the next 5 years. These savings cash flows will start one year from now. In addition, your family has just given you a $5,000 graduation gift. If you put the gift now, and your future savings when they start, into an account which pays 8 percent compounded annually, what will your financial "stake" be when you leave for Australia 10 years from now?

a. $21,432

b. $28,393

c. $16,651

d. $31,148

e. $20,000

 

 22. Foster Industries has a project which has the following cash flows:

 

                            t     Cash Flow

                            0     -$300.00

                            1       100.00

                            2       125.43

                            3        90.12

                            4          ?

 

What cash flow will the project have to generate in the fourth year in order for the project to have an internal rate of return = 15%?

a. $ 15.55

b. $ 58.95

c. $100.25

d. $103.10

e. $150.75

 

 23. You recently purchased a 20-year investment which pays you $100 at t = 1, $500 at t = 2, $750 at t = 3, and some fixed cash flow, X, at the end of each of the remaining 17 years. The investment cost you $5,544.87. Alternative investments of equal risk have a required return of 9 percent. What is the annual cash flow received at the end of each of the final 17 years, that is, what is X?

 

a. $600

b. $625

c. $650

d. $675

e. $700

 

 24. John Keene recently invested $2,566.70 in a project that is promising to return 12 percent per year. The cash flows are expected to be as follows:

 

                             End of       Cash

                              Year        Flow

                                1         $325

                                2          400

                                3          550

                                4           ?

                                5          750

                                6          800

 

What is the cash flow at the end of the 4th year?

a. $1,187

b. $  600

c. $1,157

d. $  655

e. $1,267

 

 25. An investment costs $3,000 today and provides cash flows at the end of each year for 20 years. The relevant cost of capital is 10 percent. The projected cash flows for years 1, 2, and 3 are $100, $200, and $300 respectively. What is the annual cash flow received for each of the years 4 through 20 (17 years)? (Assume the same payment for each of these years.)

 

a. $285.41

b. $313.96

c. $379.89

d. $417.87

e. $459.66

 

 

 26. If you buy a factory for $250,000 and the terms are 20 percent down, the balance to be paid off over 30 years at a 12 percent rate of interest on the unpaid balance, what are the 30 equal annual payments?

a. $20,593

b. $31,036

c. $24,829

d. $50,212

e. $ 6,667

 

 27. Drexel Corporation has been enjoying a phenomenal rate of growth since its inception one year ago. Currently, its assets total $100,000. If growth continues at the current rate of 12 percent compounded quarterly, what will total assets be at the end of 10 quarters?

a. $142,571

b. $126,678

c. $148,016

d. $136,855

e. $134,392

 

 28. If it were evaluated with an interest rate of 0 percent, a 10-year regular annuity would have a present value of $3,755.50. If the future

(compounded) value of this annuity, evaluated at Year 10, is $5,440.22, what effective annual interest rate must the analyst be using to find the future value?

a.  7%

b.  8%

c.  9%

d. 10%

e. 11%

 

 29. On January 1, 1993, a graduate student developed a 5-year financial plan which would provide enough money at the end of her graduate work (January 1, 1998) to open a business of her own. Her plan was to deposit $8,000 per year for 5 years, starting immediately, into an account paying 10 percent compounded annually. Her activities proceeded according to plan except that at the end of her third year (1/1/96) she withdrew $5,000 to take a Caribbean cruise, at the end of the fourth year (1/1/97) she withdrew $5,000 to buy a used Prelude, and at the end of the fifth year (1/1/98) she had to withdraw $5,000 to pay to have her dissertation typed. Her account, at the end of the fifth year, was less than the amount she had originally planned on by how much?

a. $15,373

b. $16,550

c. $32,290

d. $38,352

e. $13,975

 

 30. Suppose you put $100 into a savings account today, the account pays a nominal annual interest rate of 6 percent, but compounded semiannually, and you withdraw $100 after 6 months. What would your ending balance be 20 years after the initial $100 deposit was made?

a. $226.20

b. $115.35

c. $ 62.91

d. $  9.50

e. $  3.00

 

31. You have just taken out a 30-year, $120,000 mortgage on your new home. This mortgage is to be repaid in 360 equal end-of-month installments. If each of the monthly installments is $1,500, what is the effective annual interest rate on this mortgage?

a. 15.87%

b. 14.75%

c. 13.38%

d. 16.25%

e. 16.49%

 

 32. You have just borrowed $20,000 to buy a new car. The loan agreement calls for 60 monthly payments of $444.89 each to begin one month from today. If the interest is compounded monthly, then what is the effective annual rate on this loan?

a. 12.68%

b. 14.12%

c. 12.00%

d. 13.25%

e. 15.08%

 

 33. You have just taken out a 30-year mortgage on your new home for $120,000. This mortgage is to be repaid in 360 equal monthly installments. If the stated (nominal) annual interest rate is 14.75 percent, what is the amount of each of the monthly installments?

a. $1,515.00

b. $1,472.38

c. $1,493.41

d. $1,522.85

e. $1,440.92

 

 34. A baseball player is offered a 5-year contract which pays him the following amounts:

       Year 1: $1.2 million

       Year 2:  1.6 million

       Year 3:  2.0 million

       Year 4:  2.4 million

       Year 5:  2.8 million

 

Under the terms of the agreement all payments are made at the end of each year.

Instead of accepting the contract, the baseball player asks his agent to negotiate a contract which has a present value of $1 million more than that which has been offered. Moreover, the player wants to receive his payments in the form of a 5-year annuity due. All cash flows are discounted at 10 percent. If the team were to agree to the player's terms, what would be the player's annual salary (in millions of dollars)?

a. $1.500

b. $1.659

c. $1.989

d. $2.343

e. $2.500

 

 35. Your company must make payments of $100,000 each year for 10 years, with the first payment to be made 10 years from today. To prepare for these payments, your company must make 10 equal annual deposits into an account which pays a nominal interest rate of 7 percent, daily compounding (360-day year). Funds will remain in the account during both the accumulation period (the first 10 years) and the distribution period (the last 10 years), and the same interest rate will be earned throughout the entire 20 years. The first deposit will be made immediately. How large must each deposit be?

a. $47,821.11

b. $49,661.86

c. $51,234.67

d. $52,497.33

e. $53,262.39

 

 36. Your lease calls for payments of $500 at the end of each month for the next 12 months. Now your landlord offers you a new 1-year lease which calls for zero rent for 3 months, then rental payments of $700 at the end of each month for the next 9 months. You keep your money in a bank time deposit that pays a nominal annual rate of 5 percent. By what amount would your net worth change if you accept the new lease? (Hint: Your return per month is 5%/12 = 0.4166667%.)

a. -$509.81

b. -$253.62

c. +$125.30

d. +$253.62

e. +$509.81

 

 37. Josh and John (2 brothers) are each trying to save enough money to buy their own cars. Josh is planning to save $100 from every paycheck (he is paid every 2 weeks). John plans to put aside $150 each month but has already saved $1,500. Interest rates are currently quoted at 10 percent. Josh's bank compounds interest every two weeks while John's bank compounds interest monthly. At the end of 2 years they will each spend all their savings on a car (each brother buys a car). What is the price of the most expensive car purchased?

a. $5,744.29

b. $5,807.48

c. $5,703.02

d. $5,797.63

e. None of the answers above is correct.

 

 38. An investment pays $100 every six months (semiannually) over the next 2.5  years. Interest, however, is compounded quarterly, at a nominal rate of 8 percent. What is the future value of the investment after 2.5 years?

a. $520.61

b. $541.63

c. $542.07

d. $543.98

e. $547.49

 

 39. You have just bought a security which pays $500 every six months. The security lasts for ten years. Another security of equal risk also has a maturity of ten years, and pays 10 percent compounded monthly (that is, the nominal rate is 10 percent). What should be the price of the security that you just purchased?

a. $6,108.46

b. $6,175.82

c. $6,231.11

d. $6,566.21

e. $7,314.86

 

 40. You have been offered an investment that pays $500 at the end of every 6 months for the next 3 years. The nominal interest rate is 12 percent; however, interest is compounded quarterly. What is the present value of the investment?

a. $2,458.66

b. $2,444.67

c. $2,451.73

d. $2,463.33

e. $2,437.56

 

 41. A ten-year security generates cash flows of $2,000 a year at the end of each of the next three years (t = 1, 2, 3). After three years, the security pays some constant cash flow at the end of each of the next six years. (t = 4, 5, 6, 7, 8, 9). Ten years from now (t = 10) the security will mature and pay $10,000. The security sells for $24,307.85, and has a yield to maturity of 7.3 percent. What annual cash flow does the security pay for years 4 through 9?

a. $2,995

b. $3,568

c. $3,700

d. $3,970

e. $4,296

 

 42. Your company is planning to borrow $2,500,000 on a 10-year, 9 percent, annual payment, fully amortized term loan. What fraction of the payment made at the end of the third year will represent repayment of principal?

a. 29.83%

b. 50.19%

c. 35.02%

d. 64.45%

e. 72.36%

 

 43. You have just bought a house and have a $125,000, 25-year mortgage with a fixed interest rate of 8.5 percent with monthly payments. Over the next five years, what percentage of your mortgage payments will go toward the repayment of principal?

a.  8.50%

b. 10.67%

c. 12.88%

d. 14.93%

e. 17.55%

 

 44. You have just taken out an installment loan for $100,000. Assume that the loan will be repaid in 12 equal monthly installments of $9,456 and that the first payment will be due one month from today. How much of your third monthly payment will go toward the repayment of principal?

a. $7,757.16

b. $6,359.12

c. $7,212.50

d. $7,925.88

e. $8,333.33

 

 45. The Desai Company just borrowed $1,000,000 for 3 years at a quoted rate of 8 percent, quarterly compounding. The loan is to be amortized in end-of-quarter payments over its 3-year life. How much interest (in dollars) will your company have to pay during the second quarter?

a. $15,675.19

b. $18,508.81

c. $21,205.33

d. $24,678.89

e. $28,111.66

 

 46. You have a 30-year mortgage with a nominal annual interest rate of 8.5 percent. The monthly payment is $1,000. What percentage of your total payments over the first three years goes toward the repayment of principal?

a. 1.50%

b. 3.42%

c. 5.23%

d. 6.75%

e. 8.94%

 

 47. You have a $175,000, 30-year mortgage with a 9 percent nominal rate. You make payments every month. What will be the remaining balance on your mortgage after 5 years?

a. $ 90,514.62

b. $153,680.43

c. $167,790.15

d. $173,804.41

e. $174,514.83

 

 48. You just bought a house and have a $150,000 mortgage. The mortgage is for 30 years and has a nominal rate of 8 percent (compounded monthly). After 36 payments (3 years) what will be the remaining balance on your mortgage?

a. $110,376.71

b. $124,565.82

c. $144,953.86

d. $145,920.12

e. $148,746.95

 

 49. Your family purchased a house three years ago. When you bought the house you financed it with a $160,000 mortgage with an 8.5 percent nominal interest rate (compounded monthly). The mortgage was for 15 years (180 months). What is the remaining balance on your mortgage today?

a. $ 95,649

b. $103,300

c. $125,745

d. $141,937

e. $159,998

 

 50. You have just taken out a 10-year, $12,000 loan to purchase a new car. This loan is to be repaid in 120 equal end-of-month installments. If each of the monthly installments is $150, what is the effective annual interest rate on this car loan?

a. 6.5431%

b. 7.8942%

c. 8.6892%

d. 8.8869%

e. 9.0438%

 

ANSWER KEY FOR TEST - UNTITLED

 

  1. a. The annuity due.

Annuities

  2. c. The present value of the annuity due exceeds the present value of the ordinary annuity, and the future value of the annuity due also exceeds the future value of the ordinary annuity.

Annuities

 

By definition, an annuity due is received at the beginning of the year while an ordinary annuity is received at the end of the year. Because the payments are received earlier, both the present the future values of the annuity due are greater than those of the ordinary annuity

3. c. The payment made each period under an amortized loan is constant, and it consists of some interest and some principal. The later we are 

in the loan's life, the smaller the interest portion of the payment.

  4. e. All of the answers above are correct.

Time value concepts

  5. c. The proportion of the payment of a fully amortized loan which goes towards interest declines over time.

Time value concepts

Statement c is correct; the other statements are false. The effective rate of the investment in statement a is 10.25%. The present value of the annuity due is greater than the present value of the ordinary annuity.

  6. a. If an investment has interest compounded annually, its nominal rate must always equal its effective rate.

Miscellaneous concepts  

Statement a is correct; the other statements are false. The annuity due's cash flows are received sooner than those of the ordinary annuity. 

  7. b. If interest is paid more than once a year, then an investment's effective rate must exceed its nominal rate.

Miscellaneous concepts

 Statement b is correct; the other statements are false. The present value of a 5-year annuity due is greater than the present value of a 5-year ordinary annuity, other things equal. The percentage of each payment applied to principal rises over time.

  8. e. Statements b and c are correct.

Amortization

 Statements b and c are correct; therefore, statement e is the correct choice. Monthly payments will remain the same over the life of the loan. 

  9. a. 15 years

PV of an annuity

Financial calculator solution:

Inputs: I = 7; PV = -90; PMT = 10; FV = 0. Output: N = 14.695 15 years.

10. b. $7,427.83

FV of an annuity

Financial calculator solution:

Calculate FV at Year 20, then take that lump sum forward 10 years to year 30 at 10%.

Inputs: N = 20; I = 10; PV = 0; PMT = -50. OutputYear Ž™: FV = $2,863.75.

At Year 30

Inputs: N = 10; I = 10; PV = -2,863.75; PMT = 0.

OutputYear ™: FV = $7,427.83.                              

 11. d. $23,739

FV of annuity due

 

There are a few ways to do this. One way is shown below.

To get the value at t = 5 of the first 5 payments:

BEGIN mode

N = 5

I = 11

PV = 0

PMT = -3,000

FV = $20,738.58

 

Now add on to this the last payment that occurs at t = 5.

$20,728.58 + $3,000 = $23,738.58 $23,739.

 12. b. $17,954.13

FV of annuity due

 One of the several ways of doing this is to treat this as a 4-year annuity due plus a payment in year 4.

BEGIN Mode

N = 4

I = 9

PV = 0

PMT = -3,000

FV = $14,954.13.

 Plus the $3,000 at the end of Year 4 = $14,954.13 + $3,000 = $17,954.13. 

 13. e. $1,196.68

FV under monthly compounding

 

N = 3 x 12 = 36

I = 6/12 = 0.5

PV = -1,000

PMT = 0

Solve for FV = $1,196.68.                  

 14. d. $62,527.47

FV under semiannual compounding

 Financial calculator solution:

Calculate the FV as of Year 10

BEGIN mode, Inputs: N = 20; I = 6; PV = 0; PMT = -500.

Output: FV = $19,496.36.

Calculate the FV as of Year 20 using FVas the PV

END mode, Inputs: N = 20; I = 6; PV = -19,496.36; PMT = 0.

Output: FV = $62,527.47. 

15. a. $2,029.14

FV under daily compounding

 The answer is a. Solve for FV as N = 132, I = 4/365 = 0.0110, PV = -2,000, PMT = 0, and solve for FV = ? = $2,029.14.

 16. e. $44,873.90

FV of lump sum and annuity

 

First, calculate the payment amounts:

PMT™ = $5,000, PMT = $5,500, PMTŽ = $6,050, PMT = $6,655, PMT“ = $7,320.50.

Then, find the future value of each payment at t = 5:

For PMT™, N = 5, I = 14, PV = -5,000, PMT = 0; thus, FV = $9,627.0729.

Similarly, for PMT, FV = $9,289.2809, for PMTŽ, FV = $8,963.3412, for PMT, FV = $8,648.8380, and for PMT“, FV = $8,345.3700.

Finally, summing the future values of the respective payments will give the balance in the account at t = 5 or $44,873.90.

17. a. $66,909

PV of an uneven CF stream

Financial calculator solution:

Using cash flows

Inputs: CF™ = 10,000; CF = 15,000; Nj = 4 times; CFŽ = 20,000;

        I = 12.

Output: NPV = $66,908.78 $66,909.

 18. a. $3,277

PV of an uneven CF stream

 inancial calculator solution:

Inputs: CF™ = 0; CF = 1; CFŽ = 2,000; Nj = 3 times; CF = 0;

        CF“ = -2,000; I = 12.

Output: NPV = $3,276.615 $3,277.

 19. d. $125,870

PV of an uneven CF stream

Financial calculator solution:

Solve for PV at time = 0 of $20,000 annuity

Inputs: CF™ = 0; CF = 20,000; Nj = 5 times; I = 12.

Output: NPV™ = $72,095.524.

Solve for PV at time = 5 of $25,000 annuity using its value at t = 5

Inputs: CF™ = 0; CF = 25,000; Nj = 5 times; I = 10.

Output: NPV” = 94,769.669.

Solve for PV at time = 0 of $25,000 annuity

Inputs: N = 5; I = 12; PMT = 0; FV = -94,769.669.

Output: PV = $53,774,855.

Add the two PVs together

PVBoth annuities = $72,095.524 + $53,774.855 = $125,870.38 $125,870. 

 20. e. $1,347.61

PV of uncertain cash flows

Financial calculator solution:

Using cash flows,

Inputs: CF™ = 0; CF = 500; CFŽ = 430; CF = 650; I = 8.

Output: NPV = $1,347.61.

21. d. $31,148

FV of an uneven CF stream

Financial calculator solution:

Solution using NFV: (Note: Some calculators do not have net future value function. Cash flows can be grouped and carried forward or PV can be used; see alternative solution below.)

Inputs: CF™ = 5,000; CF = 1,000; Nj = 5; CFŽ = 2,000; Nj = 5; I = 8.

Output: NFV = $31,147.79 $31,148.

 

Alternative solution: Calculate PV of the cash flows, then bring them forward to FV using the interest rate.

Inputs: CF™ = 5,000; CF = 1,000; Nj = 5; CFŽ = 2,000; Nj = 5; I = 8.

Output: PV = $14,427.45.

Inputs: N = 10; I = 8; PV = -14,427.45.

Output: FV = $31,147.79 $31,148.

22. d. $103.10

Value of missing cash flow

Enter the first 4 cash flows, enter I = 15, and solve for NPV = $58.945. The future value of $58.945 will be the required cash flow.

PV = -58.945; N = 4; I/YR = 15; PMT = 0; solve for FV = $103.10.

23. d. $675

Value of missing payments

 

Find the FV of the price and the first three cash flows at t = 3.

To do this first find the present value of them.

CF™ = -5,544.87

CF = 100

CFŽ = 500

CF = 750

I = 9; solve for NPV = -$4,453.15.

 

N = 3

I = 9

PV = -4,453.15

PMT = 0

FV = $5,766.96.

 

Now solve for X.

N = 17

I = 9

PV = -5,766.96

FV = 0

Solve for PMT = $675.

24. c. $1,157

Value of missing payment

 

Find the present value of each of the cash flows:

PV of CF = $325/1.12 = $290.18. PV of CFŽ = $400/(1.12)‚ = $318.88.

PV of CF = $550/(1.12)„ = $391.48. PV of CF” = $750/(1.12)‡ = $425.57.

PV of CF• = $800/(1.12)ˆ = $405.30. Summing these values you obtain $1,831.41. The present value of CF“ must then be $2,566.70 - $1,831.41 = $735.29. The value of CF“ is ($735.29)(1.12)… = $1,157.

25. d. $417.87

Value of missing payment     

 

The project's cost should be the PV of the future cash flows. Use the cash flow key to find the PV of the first 3 years of cash flows.

 

CF™ = 0, CF = 100; CFŽ = 200, CF = 300, I/YR = 10, NPV = $481.59.

 

The PV of the cash flows for years 4 - 20 must be:

$3,000 - $481.59 = $2,518.41.

 

Take this amount forward in time 3 years:

N = 3, I/YR = 10, PV = -2,518.41, PMT = 0, solve for FV = $3,352.00.

This amount is also the present value of the 17-year annuity.

N = 17, I/YR = 10, PV = -3,352, FV = 0, solve for PMT = $417.87.

26. c. $24,829

Amortization         

Financial calculator solution:

Inputs: N = 30; I = 12; PV = -200,000; FV = 0.

Output: PMT = $24,828.73 $24,829.

27. e. $134,392

Non-annual compounding       

Financial calculator solution:

Inputs: N = 10; I = 3; PV = -100,000; PMT = 0.

Output: FV = $134,391.64 $134,392.

28. b.  8%

Effective annual rate       

Financial calculator solution:

Calculate the PMT of the annuity

Inputs: N = 10; I = 0; PV = -3,755.50; FV = 0. Output: PMT = $375.55.

Calculate the effective annual interest rate

Inputs: N = 10; PV = 0; PMT = -375.55; FV = 5,440.22.

Output: I = 7.999 8.0%.

29. b. $16,550

Annuity value         

Financial calculator solution:

Calculate the FV of the withdrawals which is how much her actual account fell short of her plan.

END mode  Inputs: N = 3; I = 10; PV = 0; PMT = -5,000.

Output: FV = $16,550.

 

Alternative solution: Calculate FV of original plan.

BEGIN mode  Inputs: N = 5; I = 10; PV = 0; PMT = -8,000.

Output: FV = $53,724.88.

Calculate FV of actual deposits less withdrawals, take the difference.

Inputs: CF™ = 8,000; CF = 8,000; Nj = 2; CFŽ = 3,000; Nj = 2;

        CF = -5,000; I = 10.

Output: NFV = $37,174.80.

Difference: $53,724.88 - $37,174.80 = $16,550.08.

30. d. $  9.50

FV of a sum         

 Financial calculator solution: (Step 2 only)

Inputs: N = 39; I = 3; PV = -3.00; PMT = 0. Output: FV = $9.50.

 31. a. 15.87%

Effective annual rate       

Financial calculator solution:

Calculate periodic rate

Inputs: N = 360; PV = -120,000; PMT = 1,500; FV = 0.

Output: I = 1.235% per period.

Use interest rate conversion feature

Inputs: NOM% = 1.235 x 12 = 14.82; P/YR = 12.

Output: EFF% = 15.868% 15.87%.

32. a. 12.68%

Effective annual rate       

 Financial calculator solution:

Calculate periodic rate and nominal rate

Inputs: N = 60; PV = -20,000; PMT = 444.89; FV = 0.

Output: I = 1.0. NOM% = 1.0% x 12 = 12.00%.

Use interest rate conversion feature

Inputs: P/YR = 12; NOM% = 12.0. Output: EFF% = EAR = 12.68%.

 33. c. $1,493.41

Required annuity payments      

Financial calculator solution:

Inputs: N = 360; I = 14.75/12 1.2292; PV = -120,000; FV = 0.

Output: PMT = $1,493.409 $1,493.41.

34. c. $1.989

Required annuity payments     

 

Enter CFs:

CF™ = 0

CF = 1.2

CFŽ = 1.6

CF = 2.0

CF“ = 2.4

CF” = 2.8

I = 10%; NPV = $7.2937 million.

$1 + $7.2937 = $8.2937.

 

Now, calculate the annual payments. BEGIN mode

N = 5; I/YR = 10; PV = -8.2937; FV = 0; PMT = ? + $1.989 million.

35. b. $49,661.86

Annuities and daily compounding    

The FV of the DEP annuity at T = 10 must be sufficient to make the 10 payments of $100,000 each.

Step 1  Find the PV of the $100,000 payments at the end of Year 10.

        This is a 10-year annuity due. What rat do we use? 7% is not

        correct, and if we use the periodic rate, that won't work

        either in an annuity setup. We want a rate that's consistent

        with an annual annuity. That means we must use the EAR.

             Use the interest conversion feature on your financial

        calculator to find EAR = 7.2501%.

        P/YR = 360

        NOM% =   7

        Solve for EFF% = 7.2501%.

 

        Now find the PV of the annuity:

        BEGIN

         10     7.2501             100,000    0

        N     I     PV     PMT    FV

                       -744,647,80

 

Step 2  Determine the amount of the annuity due by using the present

        value of the $100,000 payments at Year 10 as the future value

        of the annuity due.

        BEGIN

         10     7.2501      0             744,647.80

        N     I     PV     PMT    FV

                                    -49,661.86

 

        Deposits of $49,661.86 will provide the needed funds

 36. b. -$253.62

 37. d. $5,797.63

FV under non-annual compounding    

 First, find the FV of Josh's savings as: N = 2 x 26 = 52, I = 10/26 = 0.3846, PV = 0, PMT = -100, and FV = ? = $5,744.29. John's savings will have two components, a lump sum contribution of $1,500 and his monthly contributions. The FV of his regular savings is: N = 2 x 12 = 24, I = 10/12 = 0.8333, PV = 0, PMT = -150, and FV = ? = $3,967.04. The FV of his previous savings is: N = 24, I = 0.8333, PV = -1,500, PMT = 0, and FV = ? = $1,830.59. Summing the components of John's savings yields $5,797.63 which is greater than Josh's total savings. Thus, the most expensive car purchased costs $5,797.63.

 38. c. $542.07

FV under quarterly compounding    

 The effective rate is given by:

NOM% = 8

P/YR = 4

Solve for EFF% = 8.2432%.

 The nominal rate on a semiannual basis is given by:

EFF% = 8.2432

P/YR = 2

Solve for NOM% = 8.08%.

 The future value is given by:

N = 2.5 x 2 = 5

I = 8.08/2 = 4.04

PV = 0

PMT = -100

PV = 0

Solve for FV = $542.07.

 39. b. $6,175.82

PV under monthly compounding    

 

Start by calculating the effective rate on the second security:

P/YR = 12

NOM% = 10

Solve for EFF% = 10.4713%.

Then, convert this effective rate to a semiannual rate:

EFF% = 10.4713

P/YR = 2

NOM% = 10.2107%.

Now, calculate the value of the first security as follows:

N = 10 x 2 = 20, I = 10.2107/2 = 5.1054, PMT = 500, FV = 0, thus, PV = -$6,175.82.

40. c. $2,451.73

PV under non-annual compounding    

The answer is c. First, find the effective annual rate for a nominal rate of 12% with quarterly compounding: P/YR = 4, NOM% = 12, and EFF% = ? = 12.55%. In order to discount the cash flows properly, it is necessary to find the nominal rate with semiannual compounding that corresponds to the effective rate calculated above. Convert the effective rate to a semiannual nominal rate as P/YR = 2, EFF% = 12.55, and NOM% = ? = 12.18%. Finally, find the PV as N = 2 x 3 = 6, I = 12.18/2 = 6.09, PMT = 500, FV = 0, and PV = ? = -$2,451.73.

41. c. $3,700

Value of missing payments     

There are several different ways of doing this. One way is:

Find the future value of the first three years of the investment at Year 3.

N = 3

I = 7.3

PV = -24,307.85

PMT = 2,000

FV = $23,580.68.

Find the value of the final $10,000 at Year 3.

N = 7

I = 7.3

PMT = 0

FV = 10,000

PV = -$6,106.63.

Add the two Year 3 values (remember to keep the signs right).

$23,580.68 + -$6,106.63 = $17,474.05.

Now solve for the PMTs over years 4 through 9 (6 years) that have a PV of $17,474.05.

N = 6

I = 7.3

PV = -17,474.05

FV = 0

PMT = $3,700.00.

 42. b. 50.19%

Amortization         

 Financial calculator solution:

Calculate the principal portion of PMT using amortization function:(Note: The steps below are specific to the Hewlett-Packard 17B II but the basic steps generalize to a variety of calculators.)

Inputs: N = 10; I = 9; PV = -2,500,000. Output: PMT = $389,550.22.

Inputs: [AMORT], #P = 3, [NEXT] or [AMORT].

Output: [=] or [PRIN] = 195,502.12.

Principal fraction = $195,502.12/$389,550.22 = 0.5019 = 50.19%.

43.  d. 14.93%

 Amortization        

 

   N =   25 x 12

   I =    8.5/12

  PV =  -125,000

  FV =         0

 PMT = $1,006.53

 

 Do amortization:

 Enter:      1 INPUT 60 AMORT

 Int = $51,375.85

 Prin = $9,015.95

 Bal = $115,984.05

 

 Total payments = 5 x 12 x $1,006.53

                = $60,391.80.

 

 % Repayment     $9,015.95 of principal   $60,391.80 =  0.1493 = 14.93%.

 44. a. $7,757.16

Amortization        

 

Given: Loan Value =      $100,000   Repayment Period = 12 months

       Monthly Payment =   $9,456

 

N = 12

PV = -100,000

PMT = 9,456

FV = 0

Solve for I/YR = 2.00% x 12 = 24.005.

 

To find the amount of principal paid in the third month (or period), use the calculator's amortization feature. Enter:

3 INPUT 3  (to indicate the starting and ending period you're

           concerned with)

AMORT (to activate the amortization function)

Interest = $1,698.84

Principal = $7,757.16

Balance = $77,181.86

 45.  b. $18,508.81

 Amortization         

 46.  e. 8.94%

 Amortization         

 

 Enter information into the calculator to use its amortization

 feature:

 N = 360

 I/YR = 8.5/12 = 0.7083

 PMT = 1,000

 FV = 0

 Solve for PV = -$130,053.64 = Original value of mortgage.

 

 Enter: 1 INPUT 36 AMORT

 Int 1-36  = $32,782.14

 Prin 1-36 = $3,217.86

 

 Total payments 1-36 = $36,000.

                                                  

 Percentage of total payments ($3,217.86/$36,000) which is principal =  8.94%.

 47. c. $167,790.15

Remaining balance on mortgage    

 

Solve for the monthly payment as follows:

N = 30 x 12 = 360

I = 9/12 = 0.75

PV = -175,000

FV = 0

Solve for PMT = $1,408.09/month.

 

Use the calculator's amortization feature to find the remaining principal balance:

5 yrs = 5 x 12 = 60 payments.

1 INPUT 60 AMORT

Interest   $77,275.55

Principal  $ 7,209.85

Balance   $167,790.15

 48. d. $145,920.12

Remaining balance on mortgage    

 

Solve for the monthly payment as follows:

N = 30 x 12 = 360

I = 8/12 = 0.667

PV = -150,000

FV = 0

Solve for PMT = $1,100.65/month.

 

Use the calculator's amortization feature to find the remaining principal balance:

3 x 12 = 36 payments

1 INPUT  36 AMORT

Interest  $ 35,543.52

Principal $  4,079.88

Balance   $145,920.12

 49. d. $141,937

Remaining balance on mortgage    

 

Solve for the monthly payment as follows:

N = 12 x 15 = 180

I = 8.5/12 = 0.7083

PV = -160,000

FV = 0

PMT = $1,575.58.

 

Use the calculator's amortization feature to find the remaining principal balance:

1 INPUT  36 AMORT

Interest  $ 38,658.34

Principal $ 18,062.54

Balance   $141,937.46                                   

 50. e. 9.0438%

Effective annual rate      

 

Given: Loan Value =   $12,000  Loan Term = 10 years (120 months)

       Monthly Payment = $150

 

N = 120

PV = -12,000

PMT = 150

FV = 0

Solve for I/YR = 0.7241 x 12 = 8.6892%. However, this is a nominal rate. To find the effective rate, enter the following:

NOM% = 8.6892

P/YR = 12

Solve for EFF% = 9.0438%.