CH 5: Time Value of Money

Suppose you bought a condo and took out a 30-year, $100,000 amortized loan at a nominal annual rate of 8% with annual end-of-year payments. How large would your annual payments be?

Loan Amorization Schedule, $100,000 at 10% for 5 Years Year Beginning Amount Payment % Interest % Repayment of Principle Ending Balance 1 $100,000.00 $8,882.74 90.06% $8,000.00 9.94% $882.74 $99,117.26 2 $99,117.26 $8,882.74 89.27% $7,929.38 10.73% $953.36 $98,163.89 3 $98,163.89 $8,882.74 88.41% $7,853.11 11.59% $1,029.63 $97,134.26 4 $97,134.26 $8,882.74 87.48% $7,770.74 12.52% $1,112.00 $96,022.26 5 $96,022.26 $8,882.74 86.48% $7,681.78 13.52% $1,200.96 $- Answer - 8882.74

The U.S. government offers to sell you a bond for $362.45. No payments will be made until the bond matures 15 years from now, at which time it will be redeemed for $1,000. What annual interest rate would you earn if you bought this bond for $362.45?

N = 115 PV = 362.45 PMT = 0 FV = 1000 Answer = 7%

Amortized Loan

A loan that is repaid in equal payments over its life.

Amorizatoin Schedule

A table showing precisely how a loan will be repaid. It gives the required payment on each payment date and a breakdown of the payment, showing how much is interest and how much is repayment of principal.

What annual payment must you receive in order to earn a 6.5% rate of return on a perpetuity that has a cost of $1,250?

$81.25

A dollar today is worth less than a dollar tomorrow.

False

If we are given a periodic interest rate, say a monthly rate, we can find the nominal annual rate by dividing the periodic rate by the number of periods per year.

False

Assume that you have $10,000 which you plan to invest for 3 years. How much more would you have if you invested the funds at 10% with annual compound interest versus 10% with simple interest? No funds will be paid until the end of the 3rd year.

$310

Assume that you just received an ordinary annuity with 5 annual payments of $1,000 each. You plan to invest the payments at a 6% annual interest rate. What will the value of the first payment be at the end of the 5th year?

$1262 Interest 6% Year Amount PV FV 1 $1,000 $943.40 $1,262.48 2 $1,000 $890.00 3 $1,000 $839.62 4 $1,000 $792.09 5 $1,000 $747.26 Total Present Value $4,212.36

Suppose your rich uncle gave you $50,000, which you plan to use for graduate school. You will make the investment now, you expect to earn an annual return of 6%, and you will make 4 equal annual withdrawals, beginning 1 year from today. Under these conditions, how large would each withdrawal be so there would be no funds remaining in the account after the 4th withdrawal?

$14,429.57 SOLUTION: N = 4 PV = $50,000 (input -$50,000) I/YR = 6% FV = $0 PMT = 14429.57

Your father is now planning to retire, and his employer has promised him a guaranteed, but fixed, income of $50,000 per year for the rest of his life. If the rate of inflation is 5% per year, how much in current dollars will the payment at the end of 20th year be worth?

$18,844.47

company's sales today are $50 million. If sales grow at 7% annually, what will they be (in millions of dollars) 20 years later?

$193.48

Suppose a U.S. government bond promises to pay $1,000 three years from now, with no payments until the end of the 3rd year. If the going annual interest rate on such bonds is 6%, how much is the bond worth today?

$839.62

Your uncle is about to retire, and he wants to buy an annuity that will provide him with $75,000 of income a year for 20 years, with the first payment coming immediately. The going rate on such annuities is 5.25%. How much would it cost him to buy the annuity today?

$963,213

Assume that you just received an ordinary annuity with 8 annual payments of $1,000 each. You plan to invest the payments at a 6% annual interest rate. How much would you have, in total, at the end of the 8th year?

. $ 9,897.47 Interest 6% Year Amount PV 1 $1,000 $943.40 2 $1,000 $890.00 3 $1,000 $839.62 4 $1,000 $792.09 5 $1,000 $747.26 6 $1,000 $704.96 7 $1,000 $665.06 8 $1,000 $627.41 Total Present Value $6,209.79 NPV present value $6,209.79 total future value $9,897.47

Assuming that all interest payments are reinvested, how long would it take $5,000 to double if it were invested in a bank that pays 5% per year?

14.21 years

A bank states that it charges a 21% APR (or annual percentage rate) on credit card balances where the cardholder has been late making a payment. However, the bank compounds monthly. What EFF% is the bank charging?

23.14% Use Effect excel function to solve

Your younger sister is just starting high school, and 4 years from today she should be entering college. Your father plans to start a college fund for her, beginning today. He will invest $6,000 per year in a mutual fund, beginning today, and he expects to earn an annual return of 9%. What is the expected value of the college fund when your sister enters college?

29,980 Interest 9% Year Amount PV 1 $6,000 $5,660.38 $6,540.00 2 $6,000 $5,339.98 $7,128.60 3 $6,000 $5,037.72 $7,770.17 4 $6,000 $4,752.56 $8,469.49 Total Present Value $20,790.63 $29,908.26

Longstreet Corporation's sales today are $10 million. If sales grow at a rate of 10% per year, how long will it take Longstreet's sales to hit $15 million?

4.25 years

PV and loan eligibility You have saved $3,000 for a down payment on a new car. The largest monthly payment you can afford is $300. The loan will have a 11% APR based on end-of-month payments. What is the most expensive car you could afford if you finance it for 48 months? Round your answer to the nearest cent. $ 14606.55 What is the most expensive car you could afford if you finance it for 60 months? Round your answer to the nearest cent. $ 16796.64

48 Months Interst rate /12 = .917 N = Number of months, in this case 48 PV ? PMT = Max you can pay so $300 FV = $0 solve = 11606.55 + 3000 down payment = 14606.55 60 months N = 60 I = .917 PV ? PMT $300 FV = $0 solve = 13796.64 + 3000 down payment = $16796.64

The "cost" referred to in the concept of opportunity cost is always monetary.

False

If you deposit money today in an account that pays 8.5% annual interest, how long will it take to double your money? Round your answer to two decimal places.

Answer: 8.5 years if the interest rate was 10% it would take 10 years.

TRUE STATEMENTS:

Cash flows do not have to occur for every period shown on the time line. A lump sum cash flow can be depicted as easily as annuities on a time line. Time lines can show cash flows that occur over years, quarters, or any other periods. Time lines put verbal information into a diagram that is useful for seeing the cash flows that occur, when they occur, and the interest rate used in the analysis. Time lines typically show dollars below the line and years above the line.

Which of the following statements is TRUE?

Compound interest means that interest in future periods is earned on the interest earned in the past, whereas under simple interest, interest is earned only on the original investment.

If the discount (or interest) rate is positive, the future value of an expected series of payments will always exceed the present value of the same series.

False

you have saved $4000 for a down payment on a new car. the largest monthly payment you can afford is $350. the loan would have a 12% APR based on end of the month payments. what is the most expensive car you could afford if you finance it for 48 months? for 60 months? For 48 months:

For 48 months: N = 48 I = 1 PMT = 350 FV = 0 PV = ? = $13,290.89 Value of car = $13,290.89 + $4000 = $17,290.89 For 60 months: N = 60 I = 1 PMT = 350 FV = 0 PV = ? = $15,734.26 Value of car = $15,734.26 + $4000 = $19,734.26

You are considering investing in a bank account that pays a nominal annual rate of 7%, compounded monthly. If you invest $3,000 at the end of each month, how many months will it take for your account to grow to $150,000?

I/YR 7.0% I/MO 0.583333% Monthly annuity, so interest must be calculated on monthly basis PV $0 PMT $3,000 FV $150,000 N 44.0021 = $44.0021

What's the present value of a 4-year ordinary annuity of $1,000 plus an additional $2,000 at the end of Year 4 if the annual interest rate is 10%?

Interest 10% Year Amount PV 1 $1,000 $909.09 2 $1,000 $826.45 3 $1,000 $751.31 4 $3,000 $2,049.04 Total Present Value $4,535.89 Answer = $4535.89

What is the present value of the following uneven cash flow stream: $0 at t = 0, $200 at t = 1, $300 at t = 2, and $400 at t = 3 if the appropriate annual interest rate is 10%?

Interest 10% Year Amount PV 1 $200 $181.82 2 $300 $247.93 3 $400 $300.53 Total Present Value $730.28 Answer = $730.28

What's the future value of this cash flow stream: $1,200 at the end of Year 1, $0 at the end of Year 2, and $500 at the end of Year 3? The appropriate annual interest rate is 12%.

Interest 12% Year Amount PV 1 $1,200 $1,071.43 2 $0 $0.00 3 $500 $355.89 Total Present Value $1,427.32 Future value $2,005.28 Answer $ 2,005.28

Your uncle will sell you his bicycle shop for $250,000, with "seller financing," at a 6.0% nominal annual rate. The terms of the loan would require you to make 12 equal end-of-month payments per year for 4 years, and then make an additional final (balloon) payment of $50,000 at the end of the last month. What would your equal monthly payments be?

Monthly annuity, so interest must be calculated on a monthly basis. Years 4 Payments/year 12 N 48 Nominal rate 6.0% PV $250,000 I/period 0.5% FV $50,000 PMT $4,947.01 = $4947.01

Your child's orthodontist offers you two alternative payment plans. The first plan requires a $4,000 immediate up-front payment. The second plan requires you to make monthly payments of $137.41, payable at the end of each month for 3 years. What nominal annual interest rate is built into the monthly payment plan?

N 36 PV $4,000 PMT $137.41 FV $0 I/MO 1.20% Monthly annuity, so interest must be calculated on monthly basis Answer: 14.36%

Longstreet Corporation's sales today are $10 million. Sales are expected to grow by $1 million in each of the next 5 years, so that sales in 5 years are expected to be $15 million. What average rate of annual growth does Longstreet expect over the next five years?

N = 5 PV = -10,000,000 PMT = 0 FV = 15,000,000 Answer = 8%

What is discounting?

Reducing the future value of cash flows

Your uncle the banker offers to lend you $25,000 to start a new business. You will have to make a payment of $7,000 at the end of each of the next 3 years plus a final payment of $10,000 at the end of Year 4. What annual interest rate is built into this loan?

Set up: -10,000 (press CFj on calculator) 7,000 (press CFj on calculator) 7,000 (press CFj on calculator) 7,000 (press CFj on calculator) 10,000 (press CFj on calculator) then press Shift (red) IRR/YR =8.66%

Suppose you won a $75,000 after-tax cash prize in the lottery. You want to start a new business that you think will lose money for a while, after which it will be up and running and bringing in big bucks. You plan to invest the funds immediately in securities that are expected to earn 10% per year. Suppose you would need only $20,000 per year during the start-up period. How long could you operate before you would require cash from the new business, i.e., how long could you receive payments of $20,000 per year? The first withdrawal will be made a year from today, and your answer will contain a fraction of a year.

Solve for N PV = -$75,000 I/YR = 10% PMT = $20,000 FV = $0 = 4.93 years

Is this statement true or false? A rational person should choose to receive cash flows from an annuity due of $1,000 per year rather than from a similar ordinary annuity.

TRUE

Which of the following statements is FALSE?

The FV as shown on a time line is always the cash flow at Time = 1.

A $50,000 loan is to be amortized over 7 years, with annual end-of-year payments. Which of these statements is CORRECT?

The proportion of each payment that represents interest as opposed to repayment of principal would be lower if the interest rate were lower.

The concept of opportunity cost is described in which of the following statements?

The value of what certain resources could have produced had they been used in the best alternative way & The cost of choosing between two alternatives

A lender should prefer to lend at a rate of 10% with semiannual compounding, but a borrower would prefer a loan with a rate of 10%, annual compounding. True or false?

True

As a result of compounding, the effective annual rate on a bank deposit (or a loan) is always equal to or greater than the nominal rate on the deposit (or loan).

True

Is this statement true or false? A rational person should choose to receive cash flows from an annuity due of $1,000 per year rather than from a similar ordinary annuity.

True

Is this statement true or false? If one set up time lines for an ordinary annuity of $1,000 per year for 3 years and a 3-year, $1,000 annuity due, the primary difference between the two time lines is that the $1,000 payments for the annuity due would begin at t = 0 and end at t = 2, whereas the ordinary annuity's payments would begin at t = 1 and end at t = 3.

True

Is this statement true or false? If you calculated the value of an ordinary annuity, you could find the value of the corresponding annuity due by multiplying the FV of the ordinary annuity by (1 + I), because this would take into account that each annuity due payment occurs one year earlier.

True

Time lines can be constructed in situations where some of the cash flows occur annually but others occur quarterly.

True

Under an amortized loan, the periodic payments are all equal. However, the fraction of the payment that represents interest declines over time. True or false?

True

What is the PV of an ordinary annuity with 5 annual payments of $10,000 each if the appropriate annual interest rate is 6%? a. $42,966.11 b. $42,123.64 c. $44,701.94 d. $45,595.98 e. $43,825.44

b. $42,123.64

Suppose you deposited $1,000 in a credit union that pays a nominal rate of 7% with daily compounding and a 365-day year. How much could you withdraw after nine months, assuming this is three-fourths of a year? a. $1,073.56 b. $ 980.69 c. $1,011.02 d. $ 951.27 e. $1,053.90

e. $1,053.90 I Per = (0.07)/365 = .000191781 (9/12)(365) = 274 EFF% = (1+0.07/365)^365 - 1 = .072500983 Or $1000(1.000191781)^274 = 1053.9478 = $1053.90