Homework answers / question archive / CHAPTER 5 – Time Value of Money 2: ANALYZING ANNUITY CASH FLOWS     Questions   LG1     5-1 How can you add a cash flow in year two and a cash flow in year four  in year seven?   LG2     5-2 People can become millionaires in their retirement years quite easily if they start saving early in employer 401(k) or 403(b) programs (or even if their employers don’t offer such programs)

CHAPTER 5 – Time Value of Money 2: ANALYZING ANNUITY CASH FLOWS     Questions   LG1     5-1 How can you add a cash flow in year two and a cash flow in year four  in year seven?   LG2     5-2 People can become millionaires in their retirement years quite easily if they start saving early in employer 401(k) or 403(b) programs (or even if their employers don’t offer such programs)

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CHAPTER 5 – Time Value of Money 2: ANALYZING ANNUITY CASH FLOWS

 

 

Questions

 

LG1     5-1 How can you add a cash flow in year two and a cash flow in year four  in year seven?

 

LG2     5-2 People can become millionaires in their retirement years quite easily if they start saving early in employer 401(k) or 403(b) programs (or even if their employers don’t offer such programs).  Demonstrate the growth of a $250 monthly contribution for 40 years earning 9 percent APR.

 

 

LG3     5-3 When you discount multiple cash flows, how does the future period that a cash flow is paid affect its present value and its contribution to the value of all the cash flows?

 

 

LG4     5-4 How can you use the present value of an annuity concept to determine the price of a house you can afford?

 

 

LG5     5-5 Since perpetuity payments continue for ever, how can a present value be computed? Why isn’t the present value infinite?

 

LG6     5-6 Explain why you use the same adjustment factor, (1 + i), when you adjust annuity due payments for both future value and present value.

 

 

LG7     5-7 Use the idea of compound interest to explain why EAR is larger than APR.

 

LG8     5-8 Would you rather pay $10,000 for a five year $2,500 annuity or a ten-year $1,250 annuity?  Why?

 

 

LG9     5-9 The interest on your home mortgage is tax deductible. Why are the early years of the mortgage more helpful in reducing taxes than in the later years?

 

 

LG10   5-10 How can you use the concepts illustrated in computing the number of payments in an annuity to figure how to pay off a credit card balance?  How does the magnitude of the payment impact the number of months?

 

Problems

Basic

Problems      5-1 Future Value Compute the future value in year 8 of a $1,000 deposit in year 1 and

 LG1             another $1,500 deposit at the end of year 3 using a 10% interest rate.

 

LG1     5-2 Future Value Compute the future value in year 7 of a $2,000 deposit in year 1 and another $2,500 deposit at the end of year 4 using a 8% interest rate..

 

LG2     5-3 Future Value of an Annuity What is the future value of a $500 annuity payment over 5 years if interest rates are 9 percent?

 

 

LG2     5-4 Future Value of an Annuity What is the future value of a $700 annuity payment over 4 years if interest rates are 10 percent?

 

 

LG3     5-5 Present Value Compute the present value of a $1,000 deposit in year 1 and another $1,500 deposit at the end of year 3 if interest rates are 10 percent.

 

LG3     5-6 Present Value Compute the present value of a $2,000 deposit in year 1 and another $2,500 deposit at the end of year 4 using an 8% interest rate.

 

LG4     5-7 Present Value of an Annuity What’s the present value of a $500 annuity payment over 5 years if interest rates are 9 percent?

 

LG4     5-8 Present Value of an Annuity What’s the present value of a $700 annuity payment over 4 years if interest rates are 10 percent?

 

LG5     5-9 Present Value of a Perpetuity What’s the present value, when interest rates are 7.5 percent, of a $50 payment made every year forever?

 

 

LG5     5-10 Present Value of a Perpetuity What’s the present value, when interest rates are 8.5 percent, of a $75 payment made every year forever?

 

 

LG6     5-11 Present Value of an Annuity Due If the present value of an ordinary, 7-year annuity is $6,500 and interest rates are 8.5 percent, what’s the present value of the same annuity due?

 

LG6     5-12 Present Value of an Annuity Due If the present value of an ordinary, 6-year annuity is $8,500 and interest rates are 9.5 percent, what’s the present value of the same annuity due?

 

LG6     5-13 Future Value of an Annuity Due If the future value of an ordinary, 7-year annuity is $6,500 and interest rates are 8.5 percent, what is the future value of the same annuity due?

 

LG6     5-14 Future Value of an Annuity Due If the future value of an ordinary, 6-year annuity is $8,500 and interest

rates are 9.5 percent, what’s the future value of the same annuity due?

 

LG7     5-15 Effective Annual Rate A loan is offered with monthly payments and an 11 percent APR.  What’s the loan’s effective annual rate (EAR)?

 

 

LG7     5-16 Effective Annual Rate A loan is offered with monthly payments and a 12 percent APR.  What’s the loan’s effective annual rate (EAR)?

 

ems               5-17 Future Value Given a 6 percent interest rate, compute the year 6 future value of deposits made in years 1, 2, 3, and 4 of $1,000, $1,200, $1,200, and $1,500. 

LG1    

 

LG1     5-18 Future Value Given a 7 percent interest rate, compute the year 6 future value of deposits made in years 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400. 

 

LG2     5-19 Future Value of Multiple Annuities Assume that you contribute $200 per month to a retirement plan for 20 years.  Then you are able to increase the contribution to $400 per month for another 20 years.  Given a 7 percent interest rate, what is the value of your retirement plan after the 40 years? 

 

LG2     5-20 Future Value of Multiple Annuities Assume that you contribute $150 per month to a retirement plan for 15 years.  Then you are able to increase the contribution to $350 per month for the next 25 years.  Given an 8 percent interest rate, what is the value of your retirement plan after the 40 years?

 

LG3     5-21 Present Value Given a 6 percent interest rate, compute the present value of payments made in years 1, 2, 3, and 4 of $1,000, $1,200, $1,200, and $1,500. 

 

LG3     5-22 Present Value Given a 7 percent interest rate, compute the present value of payments made in years 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400. 

 

LG2     5-23 Present Value of Multiple Annuities A small business owner visits his bank to ask for a loan.  The owner states that he can repay a loan at $1,000 per month for the next three years and then $2,000 per month for two years after that.  If the bank is charging customers 7.5 percent APR, how much would it be willing to lend the business owner?   

 

LG2     5-24 Present Value of Multiple Annuities A small business owner visits his bank to ask for a loan.  The owner states that she can repay a loan at $1,500 per month for the next three years and then $500 per month for two years after that.  If the bank is charging customers 8.5 percent APR, how much would it be willing to lend the business owner?

 

 

LG5     5-25 Present Value of a Perpetuity A perpetuity pays $100 per year and interest rates are 7.5 percent.  How much would its value change if interest rates increased to 8.5 percent?  Did the value increase or decrease?

 

 

 

 

LG5     5-26 Present Value of a Perpetuity A perpetuity pays $50 per year and interest rates are 9 percent.  How much would its value change if interest rates decreased to 8 percent?  Did the value increase or decrease?

 

 

 

LG6     5-27  Future and Present Value of an Annuity Due  If you start making $50 monthly contributions today and continue them for 5 years, what’s their future value if the compounding rate is 10 percent APR?  What is the present value of this annuity?

 

 

LG6     5-28  Future and Present Value of an Annuity Due  If you start making $75 monthly contributions today and continue them for 4 years, what is their future value if the compounding rate is 12 percent APR? What is the present value of this annuity?

 

 

                         

 

LG7     5-29 Compound Frequency Payday loans are very short-term loans that charge very high interest rates. You can borrow $250 today and repay $300 in two weeks. What is the compounded annual rate implied by this 20 percent rate charged for only two weeks? 

 

 

LG7     5-30 Compound Frequency Payday loans are very short-term loans that charge very high interest rates. You can borrow $500 today and repay $575 in two weeks. What is the compounded annual rate implied by this 15 percent rate charged for only two weeks? 

 

 

LG8     5-31 Annuity Interest Rate  What’s the interest rate of a 5-year, annual $5,000 annuity with present value of $20,000? 

 

 

LG8     5-32 Annuity Interest Rate What’s the interest rate of a 7-year, annual $4,000 annuity with present value of $20,000? 

 

 

LG8     5-33 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a $1,000 per month contribution to grow to $75,000 in 5 years? 

 

LG8     5-34 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a $500 per month contribution to grow to $45,000 in 6 years? 

 

LG9     5-35 Loan Payments You wish to buy a $25,000 car.  The dealer offers you a 4-year loan with a 10 percent APR.  What are the monthly payments? How would the payment differ if you paid interest only?  What would the consequences of such a decision be?

 

LG9     5-36 Loan Payments You wish to buy a $10,000 dining room set.  The furniture store offers you a 3-year loan with a 11 percent APR.  What are the monthly payments?  How would the payment differ if you paid interest only?  What would the consequences of such a decision be?

 

 

LG10   5-37 Number of Annuity Payments Joey realizes that he has charged too much on his credit card and has racked up $5,000 in debt.  If he can pay $150 each month and the card charges 17 percent APR (compounded monthly), how long will it take him to pay off the debt?

LG10   5-38 Number of Annuity Payments Phoebe realizes that she has charged too much on her credit card and has racked up $6,000 in debt.  If she can pay $200 each month and the card charges 18 percent APR (compounded monthly), how long will it take her to pay off the debt?

 

ems               5-39 Future Value Given a 8 percent interest rate, compute the year 7 future value if deposits of $1,000 and

LG1     $2,000 are made in years 1, and 3, respectively, and a withdrawal of $500 is made in year 4.

LG1     5-40 Future Value  Given a 9 percent interest rate, compute the year 6 future value if deposits of $1,500 and $2,500 are made in years 2, and 3, respectively, and a withdrawal of $700 is made in year 5. 

 

LG4     5-41 Low Financing or Cash Back?  A car company is offering a choice of deals.  You

LG9     can receive $500 cash back on the purchase, or a 3 percent APR, 4-year loan. The price of the car is $15,000 and you could obtain a 4-year loan from your credit union, at 7 percent APR. Which deal is cheaper?

 

 

LG4     5-42 Low Financing or Cash Back?  A car company is offering a choice of deals.  You

LG9     can receive $1,000 cash back on the purchase, or a 2 percent APR, 5-year loan. The price of the car is $20,000 and you could obtain a 5-year loan from your credit union, at 7 percent APR.  Which deal is cheaper?

 

 

LG9     5-43 Amortization Schedule Create the amortization schedule for a loan of $15,000, paid monthly over 3 years using a 9 percent APR. 

 

 

 

LG9     5-44 Amortization Schedule Create the amortization schedule for a loan of $5,000, paid monthly over 2 years using a 8 percent APR. 

 

LG4     5-45 Investing for Retirement Monica has decided that she wants to build enough       

LG9     retirement wealth that, if invested at 8 percent per year, will provide her with $3,500 of monthly income for 30 years. To date, she has saved nothing, but she still has 25 years until she retires.  How much money does she need to contribute per month to reach her goal?

 

 

LG4     5-46 Investing for Retirement Ross has decided that he wants to build enough retirement

LG9     wealth that, if invested at 7 percent per year, will provide him with $4,000 of monthly income for 30 years. To date, he has saved nothing, but he still has 20 years until he retires.  How much money does he need to contribute per month to reach his goal?

 

LG9     5-47 Loan Balance Rachel purchased a $15,000 car two years ago using an 8 percent, 4-year loan.  He has decided that he would sell the car now, if he could get a price that would pay off the balance of his loan. What is the minimum price Rachel would need to receive for his car?

 

LG9     5-48 Loan Balance Hank purchased a $20,000 car two years ago using an 9 percent, 5-year loan.  He has decided that he would sell the car now, if he could get a price that would pay off the balance of his loan. What’s the minimum price Hank would need to receive for his car?

 

 

LG9     5-49 Teaser Rate Mortgage  A mortgage broker is offering a $183,900 30-year mortgage with a teaser rate.  In the first two years of the mortgage, the borrower makes monthly payments on only a 4 percent APR interest rate.  After the second year, the mortgage interest rate charged increases to 7 percent APR. What are the monthly payments in the first two years? What are the monthly payments after the second year?

 

 

 

LG9     5-50 Teaser Rate Mortgage  A mortgage broker is offering a $279,000 30-year mortgage with a teaser rate.  In the first two years of the mortgage, the borrower makes monthly payments on only a 4.5 percent APR interest rate.  After the second year, the mortgage interest rate charged increases to 7.5 percent APR. What are the monthly payments in the first two years? What are the monthly payments after the second year?

 

5-51 Excel Problem Consider a person who begins contributing to a retirement plan at age 25 and contributes for 40 years until retirement at age 65.  For the first ten years, she contributes $3,000 per year.  She increases the contribution rate to $5,000 per year in years 11 through 20.  This is followed by increases to $10,000 per year in years 21 through 31 and to $15,000 per year for the last ten years.  This money earns a 9 percent return, first compute the value of the retirement plan when she turns age 65.  Then compute the annual payment she would receive over the next 40 years if the wealth was converted to an annuity payment at 8 percent.

 

 

Research It!

Retirement Income Calculators

The Internet provides some excellent retirement income calculators.  You can find one by Googling “retirement income calculator.”  Many of the calculators allow you to determine your predicted annual income from a retirement nest egg under different assumptions.  For example, you can spend only the investment income generated from the nest egg.  Most retirees try not to touch the principal.  Or, you can spend both the income and the nest egg itself.  These calculators let you input the size of the retirement wealth and the investment return to be earned. They then make time value computations to determine the annual income the nest egg will provide.

 

Go to a retirement income calculator like the one at MSN Money.  Use the calculator to create a retirement scenario.  Use the TVM equations or a financial calculator to check the Internet results.

 

(http://moneycentral.msn.com/investor/calcs/n_retire/main.asp)

 

 

Integrated Mini Case:  Paying on your Stafford loan

 

Consider Gunther, a new freshman who has just received a Stafford student loan and started college.  He plans to obtain the maximum loan from Stafford at the beginning of each year.  Although Gavin does not have to make any payments while he is in school, the 6.8 percent interest owed (compounded monthly) accrues and is added to the balance of the loan.

 

Stafford loan limits:

Freshman         $2,625

Sophomore      $3,500

Junior              $5,500

Senior              $5,500

 

After graduation, Gavin gets a six month grace period.  This means that monthly payments are still not required, but interest is still accruing. After the grace period, the standard repayment plan is to amortize the debt using monthly payments for 10 years.

 

a. Show a time line of when the loans will be taken.

b. What will be the loan balance when Gavin graduates after his fourth year of school?

c. What is the loan balance six months after graduation?

d. Using the standard repayment plan and a 6.8 percent APR interest rate, compute the monthly payments Gavin owes after the grace period.

 

Combined Chapter 4 and Chapter 5 Problems

 

4&5-1 Future Value Consider that you are 35 years old and have just changed to a new job. You have $80,000 in the retirement plan from your former employer.  You can roll that money into the retirement plan of the new employer.  You will also contribute $5,000 each year into your new employer’s plan.  If the rolled-over money and the new contributions both earn a 7 percent return, how much should you expect to have when you retire in 30 years?

 

4&5-2 Future Value Consider that you are 45 years old and have just changed to a new job. You have $150,000 in the retirement plan from your former employer.  You can roll that money into the retirement plan of the new employer.  You will also contribute $8,000 each year into your new employer’s plan.  If the rolled-over money and the new contributions both earn an 8 percent return, how much should you expect to have when you retire in 20 years?

 

 

4&5-3  Future Value and Number of Annuity Payments Your client has been given a trust fund valued at $1 million.  He cannot access the money until he turns 65 years old, which is in 25 years.  At that time, he can withdrawal $25,000 per month.  If the trust fund is invested at a 5.5 percent rate, how many months will it last your client once he starts to withdraw the money?

 

 

4&5-4 Future Value and Number of Annuity Payments Your client has been given a trust fund valued at $1.5 million.  She cannot access the money until she turns 65 years old, which is in 15 years.  At that time, she can withdraw $20,000 per month.  If the trust fund is invested at a 5 percent rate, how many months will it last your client once she starts to withdraw the money?

 

4&5-5  Present Value and Annuity Payments A local furniture store is advertising a deal in which you buy a $3,000 dining room set and do not need to pay for two years (no interest cost is incurred).  How much money would you have to deposit now in a savings account earning 5 percent APR, compounded monthly, to pay the $3,000 bill in two years? Alternatively, how much would you have to deposit in the savings account each month to be able to pay the bill?

 

4&5-6 Present Value and Annuity Payments A local furniture store is advertising a deal in which you buy a $5,000 living room set with 3 years before you need to make any payments (no interest cost is incurred).  How much money would you have to deposit now in a savings account earning 4 percent APR, compounded monthly, to pay the $5,000 bill in three years? Alternatively, how much would you have to deposit in the savings account each month to be able to pay the bill?

 

4&5-7 House Appreciation and Mortgage Payments Say that you purchase a house for $200,000 by getting a mortgage for $180,000 and paying a $20,000 down payment.  If you get a 30-year mortgage with a 7 percent interest rate, what are the monthly payments?  What would the loan balance be in 10 years?  If the house appreciates at 3 percent per year, what will be the value of the house in 10 years? How much of this value is your equity?

 

                       

4&5-8 House Appreciation and Mortgage Payments Say that you purchase a house for $150,000 by getting a mortgage for $135,000 and paying a $15,000 down payment.  If you get a 15-year mortgage with a 7 percent interest rate, what are the monthly payments?  What would the loan balance be in 5 years?  If the house appreciates at 4 percent per year, what will be the value of the house in 5 years? How much of this value is your equity?

 

4&5-9 Construction Loan You have secured a loan from your bank for 2 years to build your home.  The terms of the loan are that you will borrow $100,000 now and an additional $100,000 in one year. Interest of 10 percent APR will be charged on the balance monthly.  Since no payments will be made during the two-year loan, the balance will grow at the 10% compounded rate.  At the end of the two years, the balance will be converted to a traditional 30-year mortgage at a 6 percent interest rate.  What will you paying monthly mortgage payments (Principal and Interest only)?

 

4&5-10 Construction Loan You have secured a loan from your bank for 2 years to build your home.  The terms of the loan are that you will borrow $100,000 now and an additional $50,000 in one year. Interest of 9 percent APR will be charged on the balance monthly.  Since no payments will be made during the two-year loan, the balance will grow.  At the end of the two years, the balance will be converted to a traditional 30-year mortgage at a 7 percent interest rate.  What will you pay as monthly mortgage payments (Principal and Interest only)?