# Applying Time Value Concepts

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1 Applying Time Value Concepts C H A P T E R 3 based on the value of two packs of cigarettes per day and a modest rate of return? Let s assume that Lou will save an amount equivalent to the cost of two packs per day at \$3.00 per pack. Each year for 49 years Lou would have available \$2,190 to invest (2 packs \$3.00/pack 365 days). Assuming a rate of return of 8 percent per year, Lou would accumulate \$1,161,450 by age 67! Lou Hollander has stumbled upon a fundamental financial planning tool that will benefit him: The time value of money. Lou Hollander is a million-dollar kid. He has pledged not to smoke from his present age of 18 until retirement at age 67. When asked how much money he could accumulate by age 67 if he saved and invested the equivalent of what he would have spent on two packs per day, Lou guessed, Ten thousand dollars? Lou has no idea of the economic value of his decision, nor does he yet have a grasp of the time value of money. Just how much could he accumulate The time value of money is a powerful tool that can be harnessed for your benefit. A dollar received today is worth more than a dollar received tomorrow because it can be saved and invested. Lou s situation has demonstrated that one way to accumulate a million dollars is to save and invest a mere \$6 per day. Personal financial planning offers many opportunities to apply time value of money calculations. Should you accept a lump sum at retirement or take a monthly payment for life? Should you buy a more expensive car with few maintenance problems or a less expensive car with higher 59

4 62 A P P L Y I N G T I M E V A L U E C O N C E P T S Exhibit 3.1 Future Value of \$1 (FVIF) Year 1% 2% 3% 4% 5% 6% 7% 8% 9% EXAMPLE (continued) Suppose you want to know how much money you will have in five years if you invest \$5,000 now and earn an annual return of 9 percent. The present value of money (PV) is the amount invested, or \$5,000. The FVIF for an interest rate of 9 percent and a time period of five years is (look down the column 9%, and across the row for 5 years). Thus, the future value (FV) of the \$5,000 in five years will be: FV PV FVIF i,n FV PV FVIF 9%,5 \$5, \$7,695.

5 FUTURE VALUE OF A SINGLE DOLLAR AMOUNT 63 Exhibit 3.1 (continued) 10% 12% 14% 15% 16% 18% 20% 25% 30% EXAMPLE USING A FINANCIAL CALCULATOR A variety of financial calculators are available for purchase. These calculators greatly simplify time value calculations as the following example shows. Suppose you have \$5,687 to invest in the stock market today. You like to invest for the long term and plan to choose your stocks carefully. You will invest your money for 12 years in certain stocks on which you expect a return of 10 percent annually. Although financial calculators can vary slightly in their setup, most would require inputs as shown on the following page.

7 PRESENT VALUE OF A DOLLAR AMOUNT Financial Planning Online: Paying Your Bills Online Go to: Click on: Bill pay, under the heading Banking This Web site provides: an illustration of how you can pay your bills online. To determine the present value of an amount of money received in the future, you need to know: the amount of money to be received in the future, the interest rate to be earned on your deposit, and the number of years the money will be invested. present value interest factor (PVIF) A factor multiplied by a future value to determine the present value of that amount. The present value can be calculated by using a present value interest factor (PVIF), which is a factor multiplied by the future value to determine the present value of that amount. It is dependent on the interest rate and the number of years the money is invested. USING THE PRESENT VALUE TABLE Exhibit 3.2 shows the PVIF for various interest rates (i) and time periods (n). Each column in Exhibit 3.2 lists an interest rate, while each row lists a time period. Reviewing any column of Exhibit 3.2, you will notice that the PVIF is lower as the number of years increases. This means that less money is needed to achieve a specific future value when the money is invested for a greater number of years. Reviewing any row of Exhibit 3.2, you will notice that the PVIF is lower as the interest rate increases. This means that less money is needed to achieve a specific future value when you earn a higher rate of return.

8 66 A P P L Y I N G T I M E V A L U E C O N C E P T S Exhibit 3.2 Present Value of \$1 (PVIF) Year 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% (continued) EXAMPLE You would like to accumulate \$50,000 in five years by making a single investment today. You believe you can achieve a return from your investment of 8 percent annually. What is the dollar amount that you need to invest today to achieve your goal? The PVIF in this example is.681 (look down the column 8% and across the row for 5 years). Using the present value table, the present value (PV) is:

9 PRESENT VALUE OF A DOLLAR AMOUNT 67 Exhibit 3.2 (continued) Year 12% 14% 15% 16% 18% 20% 25% 30% PV FV PVIF i,n PV FV PVIF 8%,5 \$50, \$34,050. Thus, you need to invest \$34,050 today to have \$50,000 in five years if you expect an annual return of 8 percent.

10 68 A P P L Y I N G T I M E V A L U E C O N C E P T S USING A FINANCIAL CALCULATOR Using a financial calculator, present values can be obtained quickly by inputting all known variables and solving for the one unknown variable. EXAMPLE Loretta Callahan would like to accumulate \$500,000 by the time she retires in 20 years. If she can earn an 8.61 percent return annually, how much must she invest today to have \$500,000 in 20 years? Since the unknown variable is the present value (PV), the calculator input will be: Input Function N I Where: N 20 years I 8.61%? PV 0 PMT FV Solution \$95, PV present value, or the amount that would have to be deposited today PMT payment, which is not applicable in this problem FV amount of money desired at a future point in time Thus, Loretta would have to invest \$95, today to accumulate \$500,000 in 20 years if she really earns 8.61 percent annually. Use a financial calculator to determine the present value of a single sum by calculating the present value of \$50,000 in five years if the money is invested at an interest rate of 8 percent. This is the example used earlier to illustrate the present value tables. Your answer should be \$34,050. Your answer may vary slightly due to rounding. FUTURE VALUE OF AN ANNUITY annuity due A series of equal cash flow payments that occur at the beginning of each period. timelines Diagrams that show payments received or paid over time. As noted earlier, an annuity (also called an ordinary annuity) is a series of equal cash flow payments that occur at the end of each period. An alternative to an ordinary annuity is an annuity due, which is a series of equal cash flow payments that occur at the beginning of each period. Thus, an annuity due differs from an ordinary annuity in that the payments occur at the beginning instead of the end of the period. The best way to illustrate the future value of an ordinary annuity is through the use of timelines, which show the payments received or paid over time. EXAMPLE You plan to invest \$100 at the end of every year for the next three years. You expect to earn an annual interest rate of 10 percent on the funds that you invest. Using a timeline, the cash flows from this annuity can be represented as follows:

11 FUTURE VALUE OF AN ANNUITY % 10% 10% \$100 \$100 \$100 You would like to know how much money will be in your investment account at the end of the third year. This amount is the future value of the annuity. The first step in calculating the future value of the annuity is to treat each payment as a single sum and determine the future value of each payment individually. Next, add up the individual future values to obtain the future value of the annuity. Since the first payment will be invested from the end of year 1 to the end of year 3, it will be invested for two years. Since the second payment will be invested from the end of year 2 to the end of year 3, it will be invested for one year. The third payment is made at the end of year 3, the point in time at which we want to determine the future value of the annuity. Hence, the third-year payment will not accumulate any interest. Using the future value in Exhibit 3.1 to obtain the future value interest factor for two years and 10 percent (FVIF 10%,2 1.21) and the future value interest factor for one year and 10 percent (FVIF 10%,1 1.10), the future value of your annuity can be determined as follows: % 10% 10% \$100 \$100 \$100 FV \$ FV \$ \$100 \$110 \$121 \$331 Adding up the individual future values shows that the future value of this annuity is \$331 (i.e., you will have \$331 in your account at the end of the third year). Notice that \$300 of the \$331 represents the three \$100 payments. Thus, the remaining \$31 of the \$331 is the combined interest you earned on the three payments. USING THE FUTURE VALUE ANNUITY TABLE Computing the future value of an annuity by looking up each individual singlesum future value interest factor (FVIF) is rather tedious. Consequently, Exhibit

12 70 A P P L Y I N G T I M E V A L U E C O N C E P T S Exhibit 3.3 Future Value of a \$1 Ordinary Annuity (FVIFA) Year 1% 2% 3% 4% 5% 6% 7% 8% (continued) future value interest factor for an annuity (FVIFA) A factor multiplied by the periodic savings level (annuity) to determine how the savings will accumulate over time. 3.3 lists the factors for various interest rates and periods (years). These factors are referred to as future value interest factors for an annuity (FVIFA i,n ), where i is the periodic interest rate and n is the number of payments in the annuity. The annuity payment (PMT) can be multiplied by the FVIFA to determine the future value of the annuity (FVA PMT FVIFA). Each column in Exhibit 3.3 lists an interest rate, while each row lists the period of concern.

13 FUTURE VALUE OF AN ANNUITY 71 Exhibit 3.3 (continued) 9% 10% 12% 14% 16% 18% 20% 25% 30% , , , , , EXAMPLE Suppose that you have won the lottery and will receive \$150,000 at the end of every year for the next 20 years. As soon as you receive the payments, you will invest them at your bank at an interest rate of 7 percent annually. How much will be in your account at the end of 20 years (assuming you do not make any withdrawals)?

14 72 A P P L Y I N G T I M E V A L U E C O N C E P T S 3.2 Financial Planning Online: Estimating the Future Value of Your Savings Go to: app/ main.asp Click on: Savings Calculator This Web site provides: an estimate of the future value of your savings, based on your initial balance, the amount saved per period, the interest rate, and the number of periods. To find the answer, you must determine the future value of an annuity. (The stream of cash flows is in the form of an annuity since the payments are equal and equally spaced in time.) Using Exhibit 3.3 to determine the factor, look in the i 7% column and the n 20 periods row. Exhibit 3.3 shows that this factor is The next step is to determine the future value of your lottery annuity: FVA PMT FVIFA i,n PMT FVIFA 7,20 \$150, \$6,149,250. Thus, after 20 years, you will have \$6,149,250 if you invest all your lottery payments in an account earning an interest rate of 7 percent. As an exercise, use the future value annuity table to determine the future value of five \$172 payments, received at the end of every year, and earning an interest rate of 14 percent. Your answer should be \$1,137. USING A FINANCIAL CALCULATOR TO DETERMINE THE FUTURE VALUE OF AN ANNUITY Using a financial calculator to determine the future value of an annuity is similar to using the calculator to determine the future value of a single dollar amount. As before, the known variables must be input in order to solve for the unknown variable.

15 PRESENT VALUE OF AN ANNUITY 73 The following example illustrates the use of a financial calculator to determine the future value of an annuity. EXAMPLE You have instructed your employer to deduct \$80 from your paycheck every month and automatically invest the money at an annual interest rate of 5 percent. You intend to use this money for your retirement in 30 years. How much will be in the account at that time? Input Function 360 N I 0 PV 80 PMT? FV Solution \$66, This problem differs from the problems we have seen so far, in that the payments are received on a monthly (not annual) basis. You would like to obtain the future value of the annuity and consequently need the number of periods, the periodic interest rate, the present value, and the payment. Because there are 12 months in a year, there are periods. Furthermore, since the annual interest rate is 5 percent, the monthly interest rate is 5/ percent. Also, note that to determine the future value of an annuity, most financial calculators require an input of 0 for the present value. The payment in this problem is 80. The input for the financial calculator would be as shown at the left. Thus, you will have \$66,630 when you retire in 30 years as a result of your monthly investment. PRESENT VALUE OF AN ANNUITY Just as the future value of an annuity can be obtained by compounding the individual cash flows of the annuity and then adding them up, the present value of an annuity can be obtained by discounting the individual cash flows of the annuity and adding them up. Referring to our earlier example of an ordinary annuity with three \$100 payments and an interest rate of 10 percent, we can graphically illustrate the process as follows: % 10% 10% \$100 \$100 \$ PV \$ PV \$ PV \$

16 74 A P P L Y I N G T I M E V A L U E C O N C E P T S Exhibit 3.4 Present Value of \$1 Ordinary Annuity (PVIFA) Year 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 12% (continued) present value interest factor for an annuity (PVIFA) A factor multiplied by a periodic savings level (annuity) to determine the present value of the annuity. Adding up the individual present values leads to the conclusion that the present value of this annuity is \$ Therefore, three \$100 payments received at the end of each of the next three years are worth \$ to you today if you can invest your money at an interest rate of 10 percent. USING THE PRESENT VALUE ANNUITY TABLE Exhibit 3.4 shows the present value interest factors for an annuity (PVIFA i,n ) for various interest rates (i) and time periods (n) in the annuity. Each column in Exhibit 3.4 lists an interest rate, while each row lists a time period.

17 PRESENT VALUE OF AN ANNUITY 75 Exhibit 3.4 (continued) Year 14% 16% 18% 20% 25% 30% EXAMPLE You have just won the lottery. As a result of your luck, you will receive \$82,000 at the end of every year for the next 25 years. Now, a financial firm offers you a lump sum of \$700,000 in return for these payments. If you can invest your money at an annual interest rate of 9 percent, should you accept the offer? This problem requires you to determine the present value of the lottery annuity. If the present value of the annuity is higher than the amount offered by the financial firm, you should reject the offer. Using Exhibit 3.4 to determine the factor, we look

18 76 A P P L Y I N G T I M E V A L U E C O N C E P T S in the i 9% row and the n 25 periods column. Exhibit 3.4 shows that this factor is The next step is to determine the present value of the annuity: PVA PMT PVIFA i,n PMT PVIFA 9,25 \$82, \$805,486. Thus, the 25 payments of \$82,000 each are worth \$805,486 to you today if you can invest your money at an interest rate of 9 percent. Consequently, you should reject the financial firm s offer to purchase your future lottery payments for \$700,000. As an exercise, use the present value annuity table to determine the present value of eight \$54 payments, received at the end of every year and earning an interest rate of 14 percent. Your answer should be \$250.50, which means that the eight payments have a present value of \$ USING A FINANCIAL CALCULATOR TO DETERMINE THE PRESENT VALUE OF AN ANNUITY Determining the present value of an annuity with a financial calculator is similar to using the calculator to determine the present value of a single-sum payment. Again, the values of known variables are inserted in order to solve for the unknown variable. EXAMPLE Input Function 240 N I? PV 600 PMT 0 FV Solution \$62, A recent retiree, Dave Buzz receives his \$600 pension monthly. He will receive this pension for 20 years. If Dave can invest his funds at an interest rate of 10 percent, he should be just as satisfied receiving this pension as receiving a lump-sum payment today of what amount? This problem requires us to determine the present value of the pension annuity. Because there are months in 20 years, n 240. The monthly (periodic) interest rate is 10/ percent. Thus, i Using these inputs with a financial calculator, we obtain the inputs shown to the left. The present value is \$62,192. If Dave is offered a lump sum of \$62,192 today, he should accept it if he can invest his funds at an interest rate of 10 percent. USING TIME VALUE TO ESTIMATE SAVINGS Now that you understand the various time value calculations, you can apply them to financial planning. The key time value tools for building your financial plan are estimating the future value of annual savings and determining the

19 USING TIME VALUE TO ESTIMATE SAVINGS 77 amount of annual savings necessary to achieve a specific amount of savings in the future. Recognizing how much savings you can accumulate over time can motivate you to save money every month. ESTIMATING THE FUTURE VALUE FROM SAVINGS The future value of an annuity is especially useful when determining how much money you will have saved by a future point in time if you periodically save a specific amount of money every year. You can apply this process when you are saving for a large purchase (such as a down payment on a home) in the near future or even for your retirement in the distant future. EXAMPLE Stephanie Spratt believes that she may be able to save about \$5,000 per year. She wants to know how much she will have in 30 years if she earns 6 percent annual interest on her investment. The annuity in this example is \$5,000. The future value annuity factor based on a 30-year period and a 6 percent interest rate is Thus, the future value is: \$5, \$395,290. If she could earn 7 percent instead of 6 percent on her savings, the future value annuity factor would be , and the future value would be: \$5, \$472,305. EXAMPLE ESTIMATING THE ANNUAL SAVINGS THAT WILL ACHIEVE A FUTURE AMOUNT The future value of annuity tables are also useful for determining how much money you need to save each year to achieve a specific amount of savings at a designated future point in time. Thus, you can estimate the size of the annuity that is necessary to achieve a specific future value of savings that you desire. Because FVA PMT FVIFA, the terms can be rearranged to solve for the annuity: FVA/FVIFA PMT. Stephanie Spratt now wants to know how much money she must save every year to achieve \$600,000 in 30 years, based on a 7 percent interest rate. In this example, the future value is \$600,000, and the future value interest factor is The unknown variable is the annuity. PMT FVA/FVIFA \$600,000/ \$6,352. Thus, Stephanie would need to invest \$6,352 each year to accumulate \$600,000 in 30 years.

20 78 A P P L Y I N G T I M E V A L U E C O N C E P T S Exhibit 3.5 shows how Stephanie Spratt uses the time value tools to develop a financial plan. Stephanie developed a tentative plan to save \$5,000 per year. After applying the time value tools, however, she recognizes that she could accumulate \$600,000 in 30 years by saving \$6,352 per year. She decides to strive for this higher level of annual savings. Because she realizes that this goal is ambitious, she sets a minimum goal of saving \$5,000 per year. HOW A SAVINGS PLAN FITS WITHIN YOUR FINANCIAL PLAN The key savings decisions for building your financial plan are: 1. How much should I attempt to accumulate in savings for a future point in time? 2. How much should I attempt to save every month or every year? These decisions require an understanding of the time value of money. Exhibit 3.5 shows how these savings decisions apply to Stephanie Spratt s financial plan. Exhibit 3.5 How Time Value of Money Decisions Fit Within Stephanie Spratt s Financial Plan Goals for a Savings Plan 1. Calculate how much savings I will accumulate by various future points in time. 2. Determine how much I need to save each year to ensure a comfortable living upon retirement. Analysis Present Situation: Expected Savings per Year \$5,000 Expected Annual Rate of Return 6% or 7% Estimated Amount of Savings to Be Accumulated: Assume Annual Assume Annual Savings Accumulated over: Return 6% Return 7% 5 years \$28,185 \$28, years 65,905 69, years 116, , years 183, , years 274, , years 395, ,305

21 SUMMARY 79 Annual Savings Needed to Achieve a Specific Savings Goal: Savings Goal \$80,000 in 10 years, \$200,000 in 20 years, \$500,000 in 30 years Expected Annual Rate of Return 6% or 7% Savings Goal Assume Annual Return 6% Assume Annual Return 7% \$80,000 in 10 years \$6,069 \$5,790 \$200,000 in 20 years 5,437 4,879 \$500,000 in 30 years 6,324 5,293 To achieve a savings goal of \$80,000 in 10 years, I would need to save \$6,069 per year (assuming an annual return of 6 percent on my money). To achieve a goal of \$200,000 in 20 years, I would need to save \$5,437 per year (assuming a 6 percent annual return). Decisions Decision on My Savings Goal in the Future: If I can save \$5,000 a year, I should accumulate \$28,185 in 5 years and \$65,905 in 10 years. These estimates are based on an assumed annual return of 6 percent. If my annual return is higher, I should accumulate even more than that. The estimated savings for longer time periods are much higher. A comparison of the third column with the second column in the table shows how much more savings I could accumulate if I can earn an annual return of 7 percent instead of 6 percent. Decision on My Savings Goal per Year: Although my initial plan was to develop a budget for saving about \$5,000 a year, I will try to save more so that I can achieve my savings goals. I will use a minimum savings goal of \$5,000, but will try to save about \$6,000 per year. DISCUSSION QUESTIONS 1. How would Stephanie s savings decisions be different if she were a single mother of two children? 2. How would Stephanie s savings decisions be affected if she were 35 years old? If she were 50 years old? SUMMARY You can estimate the future value of a single dollar amount to determine the future value of a bank deposit or a fund established for retirement. It is determined by estimating the compounded interest that is generated by the initial amount. The future value can be determined by using a future value table or a financial calculator. You can estimate the present value of a single dollar amount so that you know what a future payment would be worth if you had it today. The present value of a single dollar amount to be received in the future is determined by discounting the future value. The present value of a future amount to be received can be determined by using a present value table or a financial calculator.

22 80 A P P L Y I N G T I M E V A L U E C O N C E P T S You can estimate the future value of an annuity so that you can determine how much a stream of payments will be worth at a specific time in the future. This involves determining the future value of every single dollar amount contained within the annuity, which is easily estimated by using a future value annuity table or a financial calculator. You can estimate the present value of an annuity so that you can determine how much a stream of future payments is worth today. This involves determining the present value of every single dollar amount contained within the annuity, which is easily estimated by using a present value annuity table or a financial calculator. INTEGRATING THE KEY CONCEPTS Time value tools can be applied to all parts of your financial plan. When you borrow money for liquidity reasons or to finance your use of credit cards, the amount of debt grows over time if it is not paid off. When you save money for liquidity reasons or for investment purposes, your money will grow over time if you earn a positive return. The time value tools are especially useful for retirement planning, because they allow you to estimate the future value of retirement funds if you follow a particular savings pattern over time. 6. Retirement and Estate Planning (Retirement Planning, Estate Planning) 5. Investing (Stocks, Bonds, Mutual Funds) 1. Financial Planning Tools (Budgeting, Time Value, Tax Planning) Future Value of Savings Future Value of Debt 4. Insurance Planning (Property, Liability, Health, Life) 2. Liquidity Management (Banking, Money Management, Credit Management) 3. Financing (Personal Loans, Mortgages) REVIEW QUESTIONS 1. What is the time value of money? How is it related to opportunity costs? 2. To what types of cash flows is the time value of money concept most commonly applied? 3. What is an annuity? 4. Define compounding. How is it used in financial planning? 5. What two methods can be used to calculate future values? 6. What is the formula for determining the future value of a single sum when using the future value interest factor table? What information must be known in order to find the correct future value interest factor? 7. What is discounting?

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