Reading 5: The Time Value of Money-LOS f习题精选 2010, university, recently, college, savings

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Session 2: Quantitative Methods: Basic Concepts
Reading 5: The Time Value of Money

LOS f: Draw a time line and solve time value of money applications (e.g., mortgages and savings for college tuition or retirement).

 

 

 

Natalie Brunswick, neurosurgeon at a large U.S. university, was recently granted permission to take an 18-month sabbatical that will begin one year from today. During the sabbatical, Brunswick will need $2,500 at the beginning of each month for living expenses that month. Her financial planner estimates that she will earn an annual rate of 9% over the next year on any money she saves. The annual rate of return during her sabbatical term will likely increase to 10%. At the end of each month during the year before the sabbatical, Brunswick should save approximately:

A) $3,505.

B) $3,330.

C) $3,356.

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This is a two-step problem. First, we need to calculate the present value of the amount she needs over her sabbatical. (This amount will be in the form of an annuity due since she requires the payment at the beginning of the month.) Then, we will use future value formulas to determine how much she needs to save each month (ordinary annuity).

Step 1: Calculate present value of amount required during the sabbatical

Using a financial calculator: Set to BEGIN Mode, then N = 12 × 1.5 = 18; I/Y = 10 / 12 = 0.8333; PMT = 2,500; FV = 0; CPT → PV = 41,974

Step 2: Calculate amount to save each month

Make sure the calculator is set to END mode, then N = 12; I/Y = 9 / 12 = 0.75; PV = 0; FV = 41,974; CPT → PMT = -3,356

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John is getting a $25,000 loan, with an 8% annual interest rate to be paid in 48 equal monthly installments. If the first payment is due at the end of the first month, the principal and interest values for the first payment are closest to:

       Principal        Interest

A)  $410.32 $200.00

B)  $443.65   $166.67

C)  $443.65   $200.00

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Calculate the payment first:

N = 48; I/Y = 8/12 = 0.667; PV = 25,000; FV = 0; CPT PMT = 610.32.

Interest = 0.006667 × 25,000 = $166.67; Principal = 610.32 – 166.67 = $443.65 .

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An individual borrows $200,000 to buy a house with a 30-year mortgage requiring payments to be made at the end of each month. The interest rate is 8%, compounded monthly. What is the monthly mortgage payment?

A) $2,142.39.

B) $1,467.53.

C) $1,480.46.

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With PV = 200,000; N = 30 × 12 = 360; I/Y = 8/12; CPT → PMT = $1,467.53.

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It will cost $20,000 a year for four years when an 8-year old child is ready for college. How much should be invested today if the child will make the first of four annual withdrawals 10-years from today? The expected rate of return is 8%.

A) $66,243.

B) $30,683.

C) $33,138.

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First, find the present value of the college costs as of the end of year 9. (Remember that the PV of an ordinary annuity is as of time = 0. If the first payment is in year 10, then the present value of the annuity is indexed to the end of year 9). N = 4; I/Y = 8; PMT = 20,000; CPT → PV = $66,242.54. Second, find the present value of this single sum: N = 9; I/Y = 8; FV = 66,242.54; PMT = 0; CPT → PV = 33,137.76.

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An investor who requires an annual return of 12% has the choice of receiving one of the following:

1. 10 annual payments of $1,225.00 to begin at the end of one year.
2. 10 annual payments of $1,097.96 beginning immediately.

Which option has the highest present value (PV) and approximately how much greater is it than the other option?

A) Option B's PV is $27 greater than option A's.

B) Option A's PV is $42 greater than option B's.

C) Option B's PV is $114 greater than option A's.

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Option A: N = 10, PMT = -$1,225, I = 12%, FV = 0, Compute PV = $6,921.52.
Option B: N = 9, PMT = -$1,097.96, I = 12%, FV = 0, Compute PV → $5,850.51 + 1,097.96 = 6,948.17 or put calculator in Begin mode N = 10, PMT = $1,097.96, I = 12%, FV = 0, Compute PV → $6,948.17. Difference between the 2 options = $6,921.52 ? $6,948.17 = -$26.65.

Option B's PV is approximately $27 higher than option A's PV.

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A recent ad for a Roth IRA includes the statement that if a person invests $500 at the beginning of each month for 35 years, they could have $1,000,000 for retirement. Assuming monthly compounding, what annual interest rate is implied in this statement?

A) 7.411%.

B) 7.625%.

C) 6.988%.

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Solve for an annuity due with a future value of $1,000,000, a number of periods equal to (35 × 12) = 420, payments = -500, and present value = 0. Solve for i. i = 0.61761 × 12 = 7.411% stated annually. Don’t forget to set your calculator for payments at the beginning of the periods. If you don’t, you’ll get 7.437%.

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Which of the following statements about compounding and interest rates is least accurate?

A) Present values and discount rates move in opposite directions.

B) All else equal, the longer the term of a loan, the lower will be the total interest you pay.

C) On monthly compounded loans, the effective annual rate (EAR) will exceed the annual percentage rate (APR).

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Since the proportion of each payment going toward the principal decreases as the original loan maturity increases, the total dollars interest paid over the life of the loan also increases.

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Nikki Ali and Donald Ankard borrowed $15,000 to help finance their wedding and reception. The annual payment loan carries a term of seven years and an 11% interest rate. Respectively, the amount of the first payment that is interest and the amount of the second payment that is principal are approximately:

A) $1,468; $1,702.

B) $1,650; $1,468.

C) $1,650; $1,702.

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Step 1: Calculate the annual payment.

Using a financial calculator (remember to clear your registers): PV = 15,000; FV = 0; I/Y = 11; N = 7; PMT = $3,183

Step 2: Calculate the portion of the first payment that is interest.

Interest₁ = Principal × Interest rate = (15,000 × 0.11) = 1,650

Step 3: Calculate the portion of the second payment that is principal.

Principal₁ = Payment ? Interest₁ = 3,183 ? 1,650 = 1,533 (interest calculation is from Step 2)

Interest₂ = Principal remaining × Interest rate = [(15,000 ? 1.533) × 0.11] = 1,481

Principal₂ = Payment ? Interest₁ = 3,183 ? 1,481 = 1,702

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How much should an investor have in a retirement account on his 65^th birthday if he wishes to withdraw $40,000 on that birthday and each of the following 14 birthdays, assuming his retirement account is expected to earn 14.5%?

A) $234,422.

B) $272,977.

C) $274,422.

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This is an annuity due so set your calculator to the BGN mode. N = 15; I/Y = 14.5; PMT = –40,000; FV = 0; CPT → PV = 274,422.50. Switch back to END mode.

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Sarah Parker is buying a new $25,000 car. Her trade-in is worth $5,000 so she needs to borrow $20,000. The loan will be paid in 48 monthly installments and the annual interest rate on the loan is 7.5%. If the first payment is due at the end of the first month, what is Sarah’s monthly car payment?

A) $483.58.

B) $416.67.

C) $480.57.

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N = 48; I/Y = 7.5 / 12 = 0.625; PV = 20,000; FV = 0; CPT → PMT = 483.58.