Problems1_Time Value of Money

5/21/2018 Problems1_Time Value of Money

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Lesson 1: The Time Value of Money

Study Questions

1. Your mother, who gave you life (and therefore everything), has encouraged you to borrow$65,000 in student loans. The interest rate is a record-low 2.8% APR, and the repayment

calls for 10 years of monthly payments. The first payment is at the end of the first month.How big will your payments be?

Get in 12 payments per year.

N = 120 = 10 years 12 payments per year

I/YR = 2.8%

PV = $65,000

FV = 0

Solve for PMT = $621.66

2. How many months will it take to pay off a $9,000 loan with monthly payments of $225? TheAPR is 18%.

Get in 12 payments per year.

I/YR = 18%

PV = $9,000

PMT = $225;

FV = 0

Solve for N = 61.54

It will take 62 months to repay the loan.

By the way, if youre an A+ student, you should be able to see that your last payment

would be only $122.60. One way to see that is to solve for FV with N = 61; I/YR = 18%; PV =

$9,000; PMT = $225. Solve for FV = $120.79. $120.79 is the loan balance at the end of

month 61. The payment at the end of month 62 that gets you out of debt is $122.60:

N = 1; I/YR = 18%; PV = $120.79; PMT = $0. Solve for FV = $122.60.

Alternatively,

N = 1; I/YR = 18%; PV = $120.79; FV = $0. Solve for PMT = $122.60.

3. You have just won $5,000 playing the lottery. You are going to save this for your retirementin 30 years. If your investment yields 12%, how much will you have saved for your

retirement?

Get in 1 payment per year. N = 30; I/YR = 12%; PV = $5,000; PMT = $0.

Solve for FV = $149,799.61

The business about the lottery is just distracter information.


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4. You have just inked the deal on a 15-year fixed rate mortgage at 6.5%, financing $100,000and obligating yourself to 180 monthly payments of $871.11. If you pay an extra $100 permonth, in how many months will your house be paid for?

Start with the base case: 12 payments per year; PV = $100,000; N = 180; I/YR = 6.5;

PMT = -871.11. Now set PMT = -971.11 and solve for N = 151.

5. You have just graduated with $90,000 of student loans. The terms of the loan call formonthly repayment over 10 years. The interest rate is 5%. What is the amount of yourmonthly payment?

N= 120 = 10 years 12 payments per year

I/Y = 5

PV = $90,000

PMT = $954.59

FV = 0

6. You are considering loaning your brother-in-law $12,000. He has agreed to a monthlyrepayment schedule lasting 5 years, with the first payment due in one month. If your interestrate is 9%, what is the amount of his monthly payment?

N = 60

I/Y = 9

PV = 12,000

PMT = 249.10

7. You have just signed a lease for a car. You are required to pay $500 per month for 60months. Each payment is due at the START of the month. The first payment is due today. Ifthe correct interest rate is 6% APR, what is the correct value of this car?

CF0 = $500

CF1 = $500

F01 = 59

INT = .5 = 6/12

NPV = $25,992.09

Get in BEGIN mode

N = 60

I/Y = 6

PMT = $500

Solve for PV = $25,992.09

8. You have saved $20,000 as a down payment on a $100,000 house. Mortgage rates at 5.5%APR with monthly compounding. If you finance for 15 years, what will be the amount ofyour monthly payment?

N = 180

I/Y = 5.5


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PV = 80,000

PMT = 653.67

9. You have incurred $45,000 in student loans, to be repaid monthly for 10 years at 5% APR.Your intention is to pay an extra $50 per month each month. How many months will it take

until you are out of debt?

N = 120

I/Y = 5

PV = $45,000

PMT = $477.29

PMT = $527.29

I/Y = 5

PV = $45,000

N = 105.68

106 months or 14 months early

10. What rate of return is offered by an investment that costs $5,000 today, pays $2,000 per yearfor three years and then requires shut-down expense of $500 in the fourth year?

Use the cash flow menu

CF0 = $5,000

CF1 = $2,000

F01 = 3

CF2 = = $500

Compute IRR = 5.41%

You could also use the time value of money keys, but its a

N = 4

PV = $5,000

PMT = $2,000

FV = $2,500

Solve for I/YR = 5.41%

11. What rate of return is offered by an investment that costs $5,000 today and pays $2,000 peryear at the end of each year, for three years?

Use the cash flow menu

CF0 = $5,000

CF1 = $2,000

F01 = 3

Compute IRR = 9.70%

You could also use the time value of money keys:

N = 3

PV = $5,000

PMT = $2,000

FV = 0

Solve for I/YR = 9.70%

12. What rate of return is offered by a retirement plan where you invest $2,000 per year for thenext 10 years (the first payment is due today) and receive $250,000 in one payment 30 years

from today?

Use the cash flow menu.

CF0 = $2,000

CF1 = $2,000

F01 = 9

CF2 = 0


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F02 = 20

CF3 = $250,000

Compute IRR = 10.24%

13. What is the effective annual interest rate of a credit card that has a nominal rate of 24% and

monthly compounding?

Enter the ICONV menu; set C/Y = 12; NOM = 24% and compute EFF = 26.82%

14. What is the effective annual interest rate of a credit card that has a nominal rate of 12.5% andmonthly compounding?

Enter the ICONV menu; set C/Y = 12; NOM = 12.5% and compute EFF = 13.24%

15. What is the effectiveannual interest rate on the following offer: When you buy $1,000 worthof lumber at Boone County Lumber, the terms are 310 net thirty, which means that you canhave a 3% discount for paying within 10 days, but no matter what, the bill must be settledwithin 30 days.

3-10 net thirty represents an offer of a 20-day loan of $970. To see this, draw a

timeline.

Day 0 Day 10 Day 20 Day 30

Get lumber;

walk outwithout paying.

Pay $970 (3%

discount if paidby day 10).

or Pay $1,000.

Pay by day 30no matter

what.

Get in 365 payments per year. N = 20; PV = 970; FV = 1,000; compute I/Y = 55.63%.

Enter the ICONV menu; set C/Y = 365; NOM = 55.63% and compute EFF = 74.35%

Notice that the problem asked for an effective rate, not an APR.

16. Three years ago, you bought a Mini Cooper, financing the $30,000 purchase price for 5 years

with a 7% APR loan and monthly payments. You have just made you 36th payment and areconsidering the purchase of a Chrysler PT Cruiser. How much do you owe on your loan?Round your answer to the nearest dollar.

N = 60

I/Y = 7

PV = 30,000

PMT = 594.04

AMORT

P1 = 1

P2 = 36

BAL = 13,267.70


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17. You are considering the purchase of a Mini Cooper. The dealer offers you a 36-month 7%

APR loan with NO payments for 6 months. The first payment is due 6 months from today.THE LAST PAYMENT IS AT THE END OF THE MONTH 36 MONTHS FROM TODAY.Thus there are 30 payments. Payments are calculated on a $28,000 principal. The car is worth

$28,000. How big should the payment be?

Step one

N = 5

I/Y = 7

PV = 28,000

PMT = 0

CPT FV = 28,826.25

Step two

N = 30

I/Y = 7

PV = 28,826.25

PMT = 1,050.20

Check:

CF0 = 0

CF01 = 0; F01 = 5

CF02 = 1,050.20; F02 =

I = 7/12 CPT NPV = 28

18. You have $50,000 in student loans: $20,000 is financed at 5% APR with monthly payments

over 10 years; $30,000 is financed at 24% APR with monthly payments over 10 years. Whatis the interest rate on your portfolio of debt?

Take the sum of the payments on the two loans: $873.57 = $212.13 + $661.44

With PV = $50,000 solve for I/Y = 17.14%. A common mistake would be$20,000 $30,000

16.4% 5% 24%$50,000 $50,000

= +

19. You are considering refinancing your house. Exactly 10 years ago, you borrowed $150,000 at9% APR, agreeing to make 360 monthly payments. You have just made your 120th payment.Interest rates have dropped from 9% to 6%. You want to refinance with a 20-year loan at 6%.

By how much would your monthly payment decline?

First, we need to figure the balance on the old loan:

N = 360

I/Y = 9

PV = $150,000

PMT = $1,206.93

AMORT

P1 = 1

P2 = 120

BAL = $134,145.30Next, we need to figure the payment on the new loan:

N = 240 = 20 years 12 payments per year

I/Y = 6

PV = $134,145.30

PMT = $961.06

Monthly savings = $245.87 = $1,206.93 $961.06


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Hard problem. Hard like a rock.

20. Suppose you graduated last May and began repayment on $25,000 of student debt financedover 10 years at 3% APR with a payment of $241.40 on the last day of May. When you doyour taxes next April, how much interest will you be able to deduct?

N = 120

I/Y = 3

PV = $25,000

PMT = $241.40

AMORT

P1 = 1

P2 = 8 (When you pay taxes in April, it will be on income and expenses of the prior

year.)

INT = $487.40

Problems1_Time Value of Money

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