Chapter 2: The Time Value of Money PDF

ANSWERS TO CHAPTER QUESTIONS Chapter 2 The Time Value of Money 1) Compounding is interest paid on principal and interest accumulated. It is important because normal compounding over many years can result in a more accurate and...

ANSWERS TO CHAPTER QUESTIONS Chapter 2 The Time Value of Money 1) Compounding is interest paid on principal and interest accumulated. It is important because normal compounding over many years can result in a more accurate and greater accumulated sum at the end of the period than what may have been anticipated. On the other hand, returns on accumulated sums can be appreciably higher under compounding than calculated through simple return methods. 2) It is important to assess the value of a sum of money at different points in time. Among other things, it leads to incorporation of the required return on monies invested in forming decisions. These decisions may be too complex to determine through simple guesstimates and could lead to wrong conclusions. 3) The present value is the value today of sums to be paid in the future. The value is established by taking future cash flows and discounting them back to the present at an appropriate rate of return. The future value is the accumulated sum at the end of the period. It is calculated by taking cash flows prior to that time frame and compounding them by the appropriate rate of return. 4) The rate of return that could be received on marketable investments having the same level of risk. 5) When a discount rate is raised, the present value of a future sum is reduced. Alternative investments are now providing a higher return which makes the future sum to be received on the investment being considered less valuable. 6) The lump sum today. The reason is the lump sum today has more compounding periods. Assuming a similar market established rate of return for both, a sum invested Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.  in the future will have a lower present value than one that exists today and a lump sum invested today will have a greater future value as well. 7) A regular annuity is a series of payments made or received at the end of the period. An annuity due indicates payments made or received at the beginning of the period. Annuity dues have higher values because they have one full period more of compounding. An example of an annuity due is annual payments made on January 1 each year as contributions toward retirement. Annual payments received on December 31 are an example of a regular annuity. 8) The rate of return is the sum you receive expressed as compensation to you for making an investment. An inflation-adjusted return adjusts for a rise in the cost of living. Making that adjustment allows returns to be expressed in purchasing power terms. Doing so is particularly important in personal financial planning which uses investments to fund future expenditures with these future costs often rising with inflation. 9) When payments are due at the end of the period they are called a regular annuity. When payments are due at the beginning of the period they are called an annuity due. 10) The Rule of 72 gives a quick estimate on when your investment return will double based on the investment return percentage. 11) Future value is the value that a set amount of money will be worth using today’s dollars and discounted by the rate of inflation. a) Future value = Cash Flow x (1+interest rate) number of periods 12) The consequence of not accounting for inflation means not accounting for the decrease in the purchasing power of the dollar. That same dollar that could have Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.  bought you a candy bar today may only be able to purchase half a candy bar 10 years from now. 13) The internal rate of return takes into account the time valuation of money, and cash inflows and outflows. The IRR is often used to determine the profitability of a capital expenditure. Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.  ANSWERS TO CHAPTER PROBLEMS Chapter 2 Time Value of Money 1) What is the present value of a $20,000 sum to be given 6 years from now if the discount rate is 8 percent? Excel Solution A B C D 6 Inputs 7 Future Cash Flow $20,000 8 Discount Rate 8% 9 Number of Years 6 10 11 Solution =PV(B8,B9,0,B7) 12 Present Value ($12,603) Calculator Solution Inputs 6 8 20,000 N I/Y PV PMT FV Solution -12,603 2) What is the future value of an investment of $18,000 that will earn interest at 6 percent and fall due in 7 years? Excel Solution A B C D 6 Inputs 7 Present Cash Flow $18,000 8 Interest Rate 6% 9 Number of Years 7 10 11 Solution =FV(B8,B9,0,-B7) 12 Future Value $27,065 Calculator Solution Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Inputs 7 6 -18,000 N I/Y PV PMT FV Solution 27,065 3) Jason was promised $48,000 in 10 years if he would deposit $14,000 today. What would his compounded annual return be? Excel Solution A B C D 6 Inputs 7 Present Cash Flow $14,000 8 Future Cash Flow $48,000 9 Number of Years 10 10 11 Solution =RATE(B9,0,-B7,B8) 12 Annual Return 13% Calculator Solution Inputs 10 -14,000 48,000 N I/Y PV PMT FV Solution 13 4) How many years would it take for a dollar to triple in value if it earns a 6 percent rate of return? Excel Solution Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.  A B C D 6 Inputs 7 Present Value $1 8 Future Value $3 9 Interest Rate 6% 10 11 Solution =NPER(B9,0,-B7,B8) 12 Number of Years 19 Calculator Solution Inputs 6 -1 3 N I/Y PV PMT FV Solution 19 5) Marcy placed $3,000 a year into an investment returning 9 percent a year for her daughter’s college education. She started when her daughter was 2. How much did she accumulate by her daughters 18th birthday? Excel Solution A B C D 7 Inputs 8 Payment $3,000 9 Interest Rate 9% 10 Number of Years 16 11 12 Solution =FV(B9,B10,-B8,0) 13 Future Value $99,010 Calculator Solution Inputs 16 9 -3,000 Copyright © 2017 NMcGraw-Hill Education. I/Y All rights reserved. PV No reproduction PMT or distribution FV without the prior written consent of McGraw-Hill Education. Solution 99,010 6) Todd was asked what he would pay for an investment that offered $1,500 a year for the next 40 years. He required an 11 percent return to make that investment. What should he bid? Excel Solution A B C D 7 Inputs 8 Payment $1,500 9 Interest Rate 11% 10 Number of Years 40 11 12 Solution =PV(B9,B10,B8,0) 13 Present Value ($13,427) Calculator Solution Inputs 40 11 1,500 N I/Y PV PMT FV Solution -13,427 7) Ann was offered an annuity of $20,000 a year for the rest of her life. She was 55 at the time and her life expectancy was 84. The investment would cost her $180,000. What would the return on her investment be? Excel Solution Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.  A B C D 7 Inputs 8 Payment $20,000 9 Present Value $180,000 10 Number of Years 29 11 12 Solution =RATE(B10,B8,-B9,0) 13 Rate of Return 10.5% Calculator Solution Inputs 29 -180,000 20,000 N I/Y PV PMT FV Solution 10.5 8) How many years would it take for $2,000 a year in savings earning interest at 6 percent to amount to $60,000? Excel Solution A B C D 6 Inputs 7 Payment $2,000 8 Future Value $60,000 9 Interest Rate 6% 10 11 Solution =NPER(B9,-B7,0,B8) 12 Number of Years 18 Calculator Solution Inputs 6 -2,000 60,000 Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior N written consent ofI/Y McGraw-Hill Education. PV PMT FV Solution 18 9) Aaron has $50,000 in debt outstanding with interest payable at 12 percent annual. If Aaron intends to pay off the loan through 4 years of interest and principal payment, how much should he pay annually? Excel Solution A B C D 7 Inputs 8 Present Value of the Loan $50,000 9 Interest Rate 12% 10 Number of Years 4 11 12 Solution =PMT(B9,B10,B8,0) 13 Payment ($16,462) Calculator Solution Inputs 4 12 50,000 N I/Y PV PMT FV Solution -16,462 10) What is the difference in amount accumulated between a $10,000 sum with 12 percent interest compounded annually and one compounded monthly over a one-year period? Excel Solution Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.  A B C D E F 6 Inputs 7 Present Value $10,000 8 Annual Interest Rate 12% 9 10 11 Solution 12 Comparison of Accumulated Amounts 13 14 Frequency Periods per Year FV =FV($B$8/B15,B15,0,-$B$7) 15 Annual 1 $11,200.00 =-FV($B$8/B16,B16,0,$B$7) 16 Monthly 12 $11,268.25 17 18 Difference in Amounts $68.25 Calculator Solution Annual Compounding: Inputs 1 12 -10,000 N I/Y PV PMT FV Solution 11,200 Monthly Compounding: Inputs 12 1 -10,000 N I/Y PV PMT FV Solution 11,268.25 Difference in Amounts = 11,268.25 - 11,200 = 68.25 11) What is the difference in future value between savings in which $3,000 is deposited each year at the beginning of the period and the same amount deposited at the end of the period? Assume an interest rate of 8 percent and that both are due at the end of 19 years. Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Excel Solution A B C D E 7 Inputs 8 Payment $3,000 9 Interest Rate 8% 10 Number of Years 19 11 12 Solution 13 1) Deposit at the beginning of the period =FV(B9,B10,-B8,0,1) 14 Future Value $134,286 15 16 2) Deposit at the end of the period =FV(B9,B10,-B8,0,0) 17 Future Value $124,339 18 19 Difference in Amounts $9,947 Calculator Solution Deposit at the beginning of the period: Set the calculator in the BEGIN mode Inputs 19 8 -3,000 N I/Y PV PMT FV Solution 134,286 Deposit at the end of the period: Set the calculator back to the END mode Inputs 19 8 -3,000 N I/Y PV PMT FV Solution 124,339 Difference in Amounts = 134,286 – 124,339 = 9,947 Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 12) Kenneth made a $20,000 investment in year 1, received a $5,000 return in year 2, made $8,000 cash payment in year 3, and received his $20,000 back in year 4. If his required rate of return is 8 percent, what was the net present value of his investment? Excel Solution A B C D 7 Inputs 8 Cash Flow Year 1 ($20,000) 9 Cash Flow Year 2 $5,000 10 Cash Flow Year 3 ($8,000) 11 Cash Flow Year 4 $20,000 12 Discount Rate 8% 13 14 Solution =NPV(B12,B8:B11) 15 Net Present Value ($5,882) 13) John had $50,000 in salary this year. If this salary is growing 4 percent annually and inflation is projected to rise 3 percent per year, calculate the amount of return he will receive in nominal and real dollars in the fifth year. Excel Solution A B C D E F G 7 Inputs 8 Present Value of Salary $50,000 9 Growth Rate 4% 10 Inflation Rate 3% 11 Number of Years 5 12 13 Solution 14 1) Calculate Real Rate of Return =(1+B9)/(1+B10)-1 15 Real Return 1% 16 17 2) Calculate the amount of reaturn in nominal and real dollars 18 19 Year 0 1 2 3 4 5 20 Nominal Dollars 50,000 52,000 54,080 56,243 58,493 60,833 21 Real Dollars 50,000 50,485 50,976 51,470 51,970 52,475 22 23 Formula in cell G20 =FV($B$9,G19,0,-$B$20) 24 Formula in cell G21 =FV($B$15,G19,0,-$B$21) Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 14) Becky made a $30,000 investment in year 1, received a $10,000 return in year 2, $8,000 in year 3, $11,000 in year 4, and $9,000 in year 5. What was her internal rate of return over the five-year period? Excel Solution A B C D 7 Inputs 8 Cash Flow Year 1 ($30,000) 9 Cash Flow Year 2 $10,000 10 Cash Flow Year 3 $8,000 11 Cash Flow Year 4 $11,000 12 Cash Flow Year 5 $9,000 13 14 Solution =IRR(B8:B12) 15 Internal Rate of Return 10% Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.