keown_perfin5_im_03

CHAPTER 3

UNDERSTANDING AND APPRECIATING THE TIME VALUE OF MONEY

CHAPTER CONTEXT: THE BIG PICTURE

This is the third chapter in the four-chapter section of the text entitled “Part 1: Financial Planning.” This section introduces building blocks that are fundamental to the remainder of the text and serve as a foundation for financial planning. The first chapter focused on the importance of the financial planning process, while the second focused on quantitative tools for measuring financial well-being and strategies for developing a budget based on financial goals. This chapter introduces another quantitative concept, the time value of money. Because applications of this concept are made in subsequent chapters on taxes, consumer credit, mortgages, life insurance, investments, and retirement planning, it is critical that the students understand the calculations as well as the logic underlying the time value of money concepts.

CHAPTER SUMMARY

Compound interest and the time value of money are two important factors to consider when developing a financial plan. This chapter explains the concept of the time value of money under three assumptions: 1) a single payment, “lump sum;” 2) a terminating fixed stream of payments, “annuity;” and 3) a never-ending fixed stream of payments, “perpetuity.” Calculations and applications for present value and future value are illustrated using either a financial calculator or the interest factor tables. Using compound interest to generate returns is dependent on three factors: length of the investment period, amount invested, and the rate of return, or interest rate. The use of these factors to determine the return on an investment is discussed and illustrated with calculations. Finally, amortized loans and perpetuities are discussed and calculated.

LEARNING OBJECTIVES AND KEY TERMS

After reading this chapter, students should be able to accomplish the following objectives and define the associated key terms:

1. Explain the mechanics of compounding.a. compound interest b. principalc. present value (PV)d. annual interest rate (i)

Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

40 Chapter 3

e. future value (FV)f. reinvestingg. future-value interest factor (FVIFi,n)h. compound annuallyi. rule of 72

2. Understand the power of time and the importance of the interest rate in compounding.3. Calculate the present value of money to be received in the future.

a. discount rateb. present-value interest factor (PVIFi,n)

4. Define an annuity and calculate its compound or future value.a. annuityb. compound annuityc. future-value interest factor for an annuity (FVIFAi,n)d. present-value interest factor for an annuity (PVIFAi,n)e. amortized loanf. perpetuity

CHAPTER OUTLINE

I. Compound Interest and Future ValuesA. The compound interest equation—FV = PV(1 + i)

1. PV = the present value, in today’s dollars, of a sum of money2. i = the annual interest (or discount) rate3. FV = the future value of the investment at a future point in time

B. Reinvesting—how to earn interest on interestC. Future Value Interest Factor—FVIF = (1 + i)n

1. This is the future-value interest factor for (i) and (n).2. Appendix A factor table

D. The Rule of 72—estimates how many years an investment will take to double in value

E. Compound interest with non-annual periods1. The shorter the compounding period, the quicker the investment grows 2. Effective annual interest rate

II. Using an On-line or Handheld Financial CalculatorA. The TI BAII Plus financial calculator keys

1. N = stores (or calculates) the total number of payments or compounding periods2. I/Y = stores (or calculates) the interest (either compound or discount) rate3. PV = stores (or calculates) the present value4. FV = stores (or calculates) the future value5. PMT = stores (or calculates) the dollar amount of each annuity payment


Understanding and Appreciating the Time Value of Money 41

6. CPT = is the compute key

B. Calculator Clues1. Set calculator to one payment “annual” per year2. Set to display at least four decimals, due to how small percentages are in

decimal format.3. Set to “end” mode, because most annuity payments are made or received at the

end of the period.4. Every problem has at least one positive number and one negative number5. Be sure to enter a zero for any unused variable6. Enter the interest rate as a percent rather than a decimal

III. Compounding and the Power of TimeA. The power of time, money saved now is much more valuable than money saved

later.B. Don’t ignore the bottom line, but also consider the average annual return.

IV. The Importance of the Interest RateA. Historical returnsB. The “Daily Double”

V. Present ValueA. Discount rate, or the interest rate used to bring future dollars back to present dollarsB. Present-value interest factor (PVIFi,n)C. PV = FVn(PVIFi,n) or FVn /(1 + i)n

D. Appendix B factor table

VI. Annuities – a series of fixed (equal) payments recurring at fixed intervalsA. Compound annuities

1. A series of fixed investments for which to find a future value2. Future-value interest factor for an annuity (FVIFAi,n)3. FVn = PMT(FVIFAi,n)4. Appendix C factor table

B. Present value of an annuity1. A series of fixed investments for which to find a present value2. Present-value interest factor for an annuity (PVIFAi,n)3. PV = PMT(PVIFAi,n)4. Appendix D factor table

VII. Amortized loansA. Loan obligation paid off in equal installmentsB. Examples include car loans and home mortgages


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C. Perpetuities—an annuity that last forever



APPLICABLE PRINCIPLES

Principle 3: The Time Value of MoneyMoney invested will generate a return, and if those returns are reinvested, subsequent returns will be generated. This is compounding. Conversely, due to inflation, money will be worth less in the future. To avoid this loss in purchasing power, investment returns must exceed the rate of inflation.

Principle 1: The Best Protection is KnowledgeA little knowledge about compound interest can pay big rewards over time, as well as protect you from inappropriate investment advice.

Principle 10: Just Do It!When looking to compound interest to make money, there is no time like the present. This principle helps remind you that your best ally is time. Don’t forget that the returns generated by reinvested interest will continue to increase over time.

CLASSROOM APPLICATIONS

1. Discuss the difference between simple interest and compound interest. Combine the math of the problem with a graphical illustration to demonstrate how much interest the interest earns when reinvested. Ask the class how the length of time magnifies this effect.

2. Ask students to describe the difference between future value and the future value of an annuity. Discuss why a 10-year annuity started immediately is more valuable in 30 years than a 20-year annuity started 10 years from today, although more money is invested in the 20-year annuity.

3. Have students identify examples of perpetuities. Discuss how the total amount received from the perpetuity increases if the recipient lives longer than anticipated, even though the installment amount remains the same.

4. Ask students to calculate the price of several purchases (e.g., movie ticket, new car, pair of jeans) at several future dates, assuming different rates of inflation. Assuming a 3 percent annual rate of inflation, what would dinner and a movie cost when they return to campus for their twenty-fifth-year college reunion?


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REVIEW QUESTION ANSWERS

1. Compound interest is interest paid on interest. First, interest is earned on the original amount of principal; then, as the interest is reinvested (by leaving it alone and not spending it), additional interest is earned on the amount reinvested plus the interest on the principal. The more frequently interest is added, or compounded; the faster a given sum of money will grow. Compounding can be done annually, semi-annually, quarterly, monthly, or even daily.

Relationship to Time Value of Money: Time value of money calculations are based on the growth of a sum of money over a period of time, also known as compounding. In order to solve a time-value calculation, you must know (or assume) an interest rate at which money will compound.

2. Future value is the value of a certain sum of money at a certain future point in time. This value is of great importance because it allows us to project how much “growth” a certain rate of return will provide. When comparing a portfolio’s estimated future value to the estimated purchasing power of each dollar within the portfolio, the future value serves as a gauge to estimate future standard of living.

3. The Rule of 72 provides a “ballpark estimate” of how long it takes for a sum of money to double in value. To calculate how long doubling will take, divide the expected annual interest rate into 72; e.g., 72 divided by 6 percent = 12 years. The answer tells the approximate time it will take for a given sum (e.g., $5,000) to double (e.g., $10,000). The Rule of 72 can also be used to project how prices will double through inflation. For example, at 4 percent inflation, prices will double every 18 years (72/4 = 18).

4. The four variables are as follows: FV—the future value of a sum of money held for N years. N— the number of periods (e.g., years) of compounding. I/Y— the annual interest rate. Commonly referred to as either the compound (FV) or

discount (PV) rate. PV—the present (current) value of a sum of money.To solve a time value of money problem, you must know (or assume) values for three of the above four variables and solve for the fourth.

When solving for an annuity there is a fifth variable to be used PMT—the periodic payment made or received during the annuity period.

However, when solving for a straight present- or future-value there is no periodic payment being made; therefore, the PMT variable is zero.



Note: To solve a time value of money problem using the interest factor tables rather than the time value of money function keys you must locate the factor using the interest rate and number of periods. This factor replaces both the interest rate and number of period inputs. Once the factor has been determined it is simply multiplied by the investment amount. This alternative may be used whether the investment is a periodic payment, a present value, or a future value by using the various factor tables.

5. The time value of money is the concept that a dollar received today is worth more than a dollar received tomorrow. Therefore, you can’t compare dollar amounts in two separate time periods without adjusting the value of today’s payment forward in time or the later payment backward in time using an interest rate.

Importance of concept answers may vary by student; however, the following are representative.

To determine how money invested today will grow to fund future financial goals: Calculating the growth of a single investment – future value. Calculating the growth of a series of investments – future value of an annuity.

To make an “apples to apples” comparison of money received in different time periods (e.g., past, present, or future): Calculating the present worth of a single sum of money to be received in the future –

present value. Calculating the present worth of a series of payments to be received in the future –

present value of an annuity.

To determine the necessary return on an investment held for a number of years: Determining the present investment requirement to reach a certain goal in the future –

required discount rate.

6. Interest rate: the amount earned on an investment.Time: the number of years during which compounding occurs.

7. Einstein knew that, given sufficient time, compound interest could truly work wonders. Over time, even small dollar amounts can grow into incredible sums. The sooner money is saved, the more time it has to grow. Compound interest (interest earned on interest) will greatly increase the initial sum invested, especially if money is left to grow for several decades or in Peter Minuit’s case centuries.

8. Investors require a greater return because of the increased uncertainty of the future purchasing power of their investment as the time horizon lengthens. It is almost human


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nature to err on the side of having too much; therefore, the farther we try to project our needs, the greater the “average need” becomes.

9. The most common discount rate is the inflation rate. Using the inflation rate as the discount rate allows for “apples to apples” comparison of the future purchasing power or value of today’s dollars. People are typically more concerned with the relative “purchasing power” of money rather than the absolute “value” of the money.

10. The process of using present value of money calculations to convert future sums of money into present dollars by “backing out the interest” is called discounting. Therefore, the interest rate used in present value of money calculations is referred to as the discount rate.

Discounting is also called “inverse compounding” because, instead of adding interest to a present sum to determine its future value, the interest is “backed-out” of a future sum to determine its present value.

11. The future-value interest factor gives a numeric representation of the power of compounding. This shows that as the investment horizon lengthens the total return becomes more a factor of compound interest and less a factor of subsequent investments.

12. An annuity is typically defined as a series of equal dollar payments received, whereas a compound annuity is a series of equal dollar payments paid.

13. Interest factors for annuities are calculated by summing up a number of consecutive present- or future-value interest factors. For example, the future-value interest factor for an annuity of “n” years is simply the sum of individual future-value interest factors for the years 1 through n. Using an annuity factor (rather than individual factors for each year) greatly simplifies what would otherwise be a very tedious math calculation.

14. An amortized loan is one that is paid off in equal periodic payments over time. Two common examples are car loans and home mortgages.

15. The sign, either + or –, used when entering a time value of money equation dictates the direction of the cash flow. Conventionally, all cash outflows or “payments” carry a negative sign, and all cash inflows or receipts carry an implied positive sign. In other words, if the money is flowing away from the investor then a negative sign is used, but when money is flowing towards the investor the sign is positive. When using a calculator to solve for N—number of payments, or I/Y—annual interest rate; the calculator requires at least one cash flow in each direction. If the sign is incorrectly entered in the HP BA-II Plus then the calculator will return an “Error 5” message.



16. A perpetuity is a series of equal dollar payments made at the end of a specified time period for an infinite number of time periods. The best example of a perpetuity is a scholarship.

PROBLEM AND ACTIVITY ANSWERS

1. After 18 years, her prize money will have grown to $999,000. The 8% future-value interest factor is closest to 4 (FVIF = 3.996) at the end of year 18. The Rule of 72 can also be used. By dividing the known variable (8%) into 72, the answer is 9, but since you are quadrupling your money, rather than doubling it, you will need to multiply the answer of 9 by 2.

Using a financial calculator with PV = -$250,000, FV = $1 million, PMT = $0, and I/Y = 8% results in N being 18.01 years.

2. It will take approximately 9 years for Linda’s investment to increase in value to $7,755.

Factor Table A solution Calculator solutionPV $5,000 PV -$5,000PMT n/a PMT $0

(FVIF5 %, 9) 1.551I/Y 5%N ?

FV $7,755 FV $7,755CPT N 8.996 9

3. Paul must earn approximately $30,910 in year ten to maintain equivalent purchasing power.


(FVIF3 %,10) 1.344I/Y 3%N 10

FV $30,912 FV ?CPT FV $30,910


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4. Assuming annual compounding, Anthony and Michelle will have $255,000 in 25 years.


(FVIF12 %,25) 17.000I/Y 12%N 25

FV $255,000 FV ?CPT FV $255,001

Assuming semi-annual compounding (therefore resetting the payments per year [P/Y] = 2 for the calculator solution), they will have approximately $276,300 in 25 years.


(FVIF6%, 50) 18.419I/Y 12%N 25

FV $276,285 FV ?CPT FV $276,302

Using the factor tables the difference attributable to the increased compounding frequency is $21,285 = $276,285 – $255,000. Using the calculator the difference attributable to the increased compounding frequency is $21,302 = $276,302 – $255,000. Therefore, the shorter the compounding period the larger the resultant value if all other variables are held constant.

5. Assuming a one-year period the future value is $5,500


(FVIF10%,1) 1.100I/Y 10%N 1

FV $5,500 FV ?CPT FV $5,500



Assuming a 20-year period the future value is approximately $33,635


(FVIF10%, 20) 6.727I/Y 10%N 20

FV $33,635 FV ?CPT FV $33,637.50

6. Based on a 5% return Ahmed will fall short of his goal with only $8,862 saved.


(FVIF5%,8) 1.477I/Y 5%N 8

FV $8,862 FV ?CPT FV $8,864.73

Ahmed requires an annual rate of approximately 7% to achieve his goal.


(FVIF7%,8) 1.718I/Y ?N 8

FV $10,308 FV $10,283CPT I/Y 6.97% 7%

Note: Using a calculator to determine the required rate is easiest; however, in this instance the time value of money factor can be determined by using the equation ($10,283 / $6,000). This results in a factor of 1.714, and is approximately equal to the (FVIF7%,8) 1.718.

7. Again, solving for I/Y is easiest using a calculator; however, with two single sums (one PV and one FV) using Appendix A and the formula FV = PV(FVIFi %, n years) is also an option.

$1,000,000 (FVIFi %, n years) = $7,039,988.71(FVIFi %, n years) = $7,039,988.71 / $1,000,000 = 7.039

Looking at the 40-year period row in Appendix A, 7.040 is the factor for 5%. Therefore, Dr. Evil is requiring a 5% inflation rate on his 40-year old, $1,000,000 ransom demand.


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Factor Table A solution Calculator solutionPV $1,000,000 PV -$1,000,000PMT n/a PMT $0

(FVIF5%,40) 7.040I/Y ?N 40

FV $7,039,988.71 FV $7,039,988.71CPT I/Y 5.0%

Alternatively, this question can be used to introduce Appendix B and requires the formula PV = FV (PVIFi %, n years) to solve:

$7,039,988.71(PVIFi %, n years) = $1,000,000(PVIFi %, n years) = $1,000,000 / $7,039,988.71 = 0.14205

Looking at the 40-year period row in Appendix B, This time 0.142 is the factor for 5%.

8. Derek’s trust is worth approximately $8,880 today if he is to receive $20,000 at age 35.

Factor Table B solution Calculator solutionFV $20,000 PV ?PMT n/a PMT $0

(PVIF7%, 12) 0.444I/Y 7%N 12

PV $8,880 FV $20,000CPT PV -$8,880

Derek’s trust is worth approximately $6,340 today if he is to receive $20,000 at age 40.

Factor Table B solution Calculator solutionFV $20,000 PV ?PMT n/a PMT $0

(PVIF7%, 17) 0.317I/Y 7%N 17

PV $6,340 FV $20,000CPT PV -$6,331

9. Your immediate lottery payout would be approximately $573,500 (50,000 * 11.470).

Factor Table D solution Calculator solutionFV n/a PV ?PMT $50,000 PMT -$50,000

(PVIFA6%, 20) 11.470I/Y 6%N 20

PV $573,500 FV $0



CPT PV $573,496.06

10. Richard would be able to withdrawal approximately $54,897 ($500,000 / 9.108) at the end of each year.

Factor Table D solution Calculator solutionFV n/a PV -$500,000PMT $54,896.79 PMT ?

(PVIFA7%, 15) 9.108I/Y 7%N 15

PV $500,000 FV $0CPT PMT $54,897.31

11. Step 1: You would have approximately $374,220 ($2,275 * 164.491) at the time you retire. Note: Since the match is 50% on only the first 3% the matching contribution is only 1.5%, rather than 2.5% if there was no limit on matching contributions.Total annual contribution = (0.05 * $35,000) + (0.015 * $35,000) = $2,275

Factor Table C solution Calculator solutionPV n/a PV $0PMT $2,275 PMT -$2,275

(FVIFA10%, 30) 164.491I/Y 10%N 30

FV $374,217.03 FV ?CPT FV $374,223.90

Step 2: The approximate monthly payment is using the online factor table D is $3,308.03. Using the calculator and assuming monthly compounding (therefore resetting the payments per year [P/Y] = 12), results in a PMT of $3,284.02.

Factor Table D solution Calculator solutionFV n/a PV -$374,223.90PMT $3,308.03 PMT ?

(PVIFA10%, 30) 113.951I/Y 10%N 30

PV $374,217.03 FV 0CPT PMT $3,284.08

12. Using Appendix C and the formula FV = PMT(FVIFAi %, n years) solves for the following: Annual investment answers (Table C factors)

20 years—$3,000 x 72.052 (FVIFA12%, 20 years) = $216,156 30 years—$3,000 x 241.330 (FVIFA12%, 30 years) = $723,990 40 years—$3,000 x 767.080 (FVIFA12%, 40 years) = $2,301,240


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Annual investment answers (using calculator) N = 20, PMT = -$3,000, PV = 0, I/Y =12% Therefore, FV = $216,157 N = 30, PMT = -$3,000, PV = 0, I/Y =12% Therefore, FV = $723,990 N = 40, PMT = -$3,000, PV = 0, I/Y =12% Therefore, FV = $2,301,240

Monthly investment answers (multiply Table C factor by 12 for estimate) 20 years—$250 x 989.255 (FVIFA12%, 20 years) = $247,314 30 years—$250 x 3,494.964 (FVIFA12%, 30 years) = $873,741 40 years—$250 x 11,764.773 (FVIFA12%, 40 years) = $2,941,193

Monthly investment answers (using calculator - resetting the compounding periods to 12) N = 20 (x12), PMT = -$250, PV = 0, I/Y =12% Therefore, FV = $247,313 N = 30 (x12), PMT = -$250, PV = 0, I/Y =12% Therefore, FV = $873,741 N = 40 (x12), PMT = -$250, PV = 0, I/Y =12% Therefore, FV = $2,941,193

The difference in dollar amount saved between an annual investment and a monthly investment is $31,158 in 20 years, $149,751 in 30 years, and a whopping $639,953 in 40 years. Reducing the compounding period and increasing the number of investments made during the year can greatly magnify the power compound interest has to drive total returns.



DISCUSSION CASE 1 ANSWERS

1. If Jinhee contributes just 5 percent of her salary ($1,850) and earns another $1,850 (100 percent) from her employer’s match, she will save a total of $3,700 a year.

At 8 percent she’d have $958,510.90 = ($3,700 x 259.057 (FVIFA8%, 40 years)) At 10 percent she’d have $1,637,594.10 = ($3,700 x 442.593 (FVIFA10%, 40 years)).

The employer match is “free money” that shouldn’t be passed up. It is in Jinhee’s best interest to start saving immediately.

2. Approximately $2,198.49 is required to be saved per year.

Factor Table C solution Calculator solutionPV n/a PV $0PMT $2,198.49 PMT ?

(FVIFA6%, 3) 3.184I/Y 6%N 3

FV $7,000 FV $7,000CPT PMT -$2,198.76

Total interest earned = $7,000 – ($2,198.49 x 3) = $404.53

Approximately $177.95 is required to be saved per month

Online Factor Table C solution Calculator solutionPV n/a PV $0PMT $177.95 PMT ?

(FVIFA6%, 3) 39.336I/Y 6%N 3

FV $7,000 FV $7,000CPT PMT $177.95

Total interest earned = $7,000 – ($177.95 x 36) = $593.80

By making monthly payments Jinhee will earn $189.27 ($593.80 – $404.53) more interest.

3. The value of Jinhee’s trust fund value at age 60 is approximately $190,300

Factor Table C solution Calculator solutionPV $25,000 PV -$25,000PMT n/a PMT 0

(FVIF7%, 30) 7.612I/Y 7%N 30

FV $190,300 FV ?CPT FV $190,306.38


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4. Paul’s annual payment would be approximately $12,950.

Factor Table D solution Calculator solutionPV $100,000 PV $100,000PMT $12,950 PMT ?

(FVIF5%, 10) 7.722I/Y 5%N 10

FV n/a FV 0CPT PMT -$12,950.46

Paul’s monthly payment would be approximately $1,060.66.

Online Factor Table D solution Calculator solutionPV $100,000 PV $100,000PMT $1,060.66 PMT ?

(FVIF5%, 10) 94.281I/Y 5%N 120

FV n/a FV 0CPT PMT -$1,060.66

Paying the loan on a monthly basis would result in an interest savings of $2,225.40 [($12,950.46 x 10) – ($1,060.66 x 120)] over the life of the loan.

5. Student answers will vary, but these are representative: “Max out” tax-deferred savings in employer plans. Increase interest rate earned on investments. Repay high interest debt as soon as possible. Shop for a low-rate mortgage.

DISCUSSION CASE 2 ANSWERS

1. The approximate monthly payment is using the online factor table D is $2,147.28. Using the calculator and assuming monthly compounding (therefore resetting the payments per year [P/Y] = 12), results in a PMT of $2,147.29.

Online Factor Table D solution Calculator solutionPV $400,000 PV -$400,000PMT $2,147.28 PMT ?

(FVIF5%, 30) 186.282I/Y 5%N 30

FV n/a FV 0CPT PMT $2,147.29



2. His retirement income will last for 195 months (16.3 years) or until approximately age 73. His portfolio needs to generate $3,000 in income per month ($5,800 – $2,800 retirement annuity). Using the formula $400,000 = [$3,000 * (PVIFA 5%, X months)] from the Appendix to Chapter 3 the PVIFA can be calculated as 133.333.

Online Factor Table D solution Calculator solutionPV $400,000 PV -$400,000PMT $3,000 PMT $3,000

(FVIF5%, 16.3) 133.333I/Y 5%N ?

FV n/a FV 0CPT N 195 months

Now his retirement income will last for 945 months (78.8 years) or until approximately age 135. His portfolio only needs to cover $1,700 per month ($4,500 – $2,800). Again using the formula $400,000 = [$1,700 * (PVIFA 5%, X months)] the PVIFA can be calculated as 235.294.

Online Factor Table D solution Calculator solutionPV $400,000 PV -$400,000PMT $1,700 PMT $1,700

(FVIF5%, 78.8) 235.294I/Y 5%N ?

FV n/a FV 0CPT N 945 months

3. Note: This problem poses a more difficult variation for using the factor tables; therefore the table and calculator solutions will be handled separately.

Factoring in the Social Security benefit, and using his current projected expense level of $5,800 he will deplete his portfolio at age 80.

Step 1: The value of the portfolio remaining at age 67 is $165,992.44

Calculator solutionPV -$400,000PMT $3,000I/Y 5%N 11FV ?CPT FV $165,992.44


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Step 2: Using the future value from above as the new present value calculates that Doug’s portfolio will last an additional 156 months or 13 years.

Calculator solutionPV -$165,992.44PMT $1,450I/Y 5%N ?FV 0CPT N 156 months

Because there is a PV and PMT the calculation to determine the remaining value at age 67 utilizing the factor tables must be handled in two pieces.

Online Factor Table C solution Online Factor Table C solutionPV $400,000 PV n/aPMT n/a PMT $3,000

(FVIF5%, 11) 1.731 (FVIFA5%, 11) 175.506

FV $692,400 FV $526,518

The difference between the two future values is the value of the remaining portfolio $165,882. Using that as the new present value, the factor can be determined as 114.401.

Online Factor Table D solutionFV n/aPMT $1,450

(PVIFA5%, 13) 114.540

PV $165,8824. Yes, prices will double by the time he is 80 years old. Using the Rule of 72 you determine

that, at 3% inflation, it takes 24 years for prices to double. The can of soda will cost $2.43 in 30 years assuming 3% annual inflation.

Factor Table A solution Calculator solutionPV $1 PV -$1PMT n/a PMT 0

(FVIF3%,30) 2.427I/Y 3%N 30

FV $2.43 FV ?CPT FV $2.43


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