# Chapter 5: Time value analysis.

THE IMPORTANCE OF TIME VALUE CALCULATIONSUnderstanding the time value of money is essential in financial planning. It has often been said that time is money. When we postpone the use of our money by lending or investing it, we expect to be compensated by some form of return on our money. Spending and investing decisions involve tradeoffs. If we defer receiving a payment there is an opportunity cost, measured as the highest-valued investment alternative that is given up (i.e., we have lost an opportunity to currently invest or otherwise use the funds). Interest is used to quantify opportunity cost. For example, given the choice of being paid $1.00 today, or $1.50 four years from now, you would need to quantify the opportunity cost of not being paid $1.00 today. The minimum opportunity cost is that amount that you could earn by investing the $1.00 in a virtually risk-free 3-month U.S. Treasury bill. Were you to invest in a vehicle that involved any risk of loss, you would demand a higher rate of return. The difference between this higher rate or return and the risk-free rate of return is known as a risk premium.

Time value of money calculations also help us to make a variety of daily decisions, from how much needs to be saved each month in order to have money for a down payment for a home purchase in 5 years; to deciding which purchase and payment plan is best when buying a new automobile (e.g., zero percent financing for 3 years or taking a $500 discount from purchase price). Interest and annuity calculations are essential in making less frequent, but even more critical, cash flow decisions such as deciding between various payout options offered by a retirement plan (lump sum or annuity payout) or how much can be safely withdrawn from retirement savings.

THE MAGIC OF COMPOUND INTEREST

American history tells us that, in the year 1626, a Mr. Peter Minuit purchased the island of Manhattan from the Indians for approximately $24 worth of trinkets and beads. Based upon the obvious value of Manhattan today, we might conclude that this was a very savvy investment. However, what if Mr. Minuit could have converted his trinkets and beads into $24 and deposited it in a bank account paying 7% compound interest net after taxes? With the magic of compound interest, and more than just a little bit of time, in 2011 the value of this deposit would be $4,931,347,990,957. That is to say, after three hundred and eighty five years, his twenty-four dollars has grown to four trillion, nine hundred and thirty one billion, three hundred and forty seven million, nine hundred and ninety thousand, nine hundred and fifty seven dollars. It has been said that examples like this will make you want to live forever. Needless to say this is a rather fanciful example of the magic of compounding interest.

Now let's apply the magic of compound interest to the real world. Assume an individual, age 25, receives an inheritance of $10,000 and puts it aside for retirement in an account accruing at 7% compound interest net after taxes. In twenty years, the account will grow to $38,597. Although his money has almost quadrupled in just 20 years, it will hardly secure him in his retirement years. Notice the rather flat upwards curve in the diagram; it is the interest being credited to his account.

[GRAPHIC OMITTED]

Since this individual does not intend to retire at age 45, he is now able to leave the $38,697 on deposit, and it continues to grow at 7% compound interest. By age 65, he will have $149,745 in his account. During the first 20 years his original $10,000 grew by only $28,697, but during the next 20 years his money grew by an additional $111,048 (149,745 - 38,697 = 111,048). This escalation in the amount of growth of the original $10,000 is represented by the increasingly steep curve in the following chart.

[GRAPHIC OMITTED]

COMPOUNDING VERSUS DISCOUNTING

In the above example it is compound interest that serves as a link between the present value of money and the future value of money. As we used interest to determine the future value of a present sum of money, in the same way we can use interest to calculate the present value of a future sum of money. When we know the present value we can use interest and compounding to move up the curve in order to determine future value. On the other hand, when we know the future value, we can use interest and discounting to move down the curve to determine present value.

COMPOUND INTEREST & FUTURE VALUE OF MONEY

The future value of money is a very important concept in financial planning. For instance, given the choice of being paid $1.00 today, or $1.00 four years from now, you would obviously choose to be paid $1.00 today (i.e., the earlier the better). Likewise, given the choice of being paid $1.00 today, or $4.00 four years from now, you would probably choose to be paid $4.00 four years from now (i.e., a four-fold increase in only four years). However, given the choice of being paid $1.00 today, or $1.50 four years from now, your decision would likely require a careful consideration of the future value of money.

[check] $1.00 Today or [] $1.00 In 4 Years [] $1.00 Today or [check] $4.00 In 4 Years [] $1.00 Today ? [] $1.50 In 4 Years

The future value of money can be best understood by referring to the Future Value Table on pages 500-501. If you took the $1.00 today and could invest it at 8%, you would have $1.36 in four years (read down the 8% rate column until year 4). By waiting four years, you could receive $1.50, which is 14 cents more than you would get by taking the money and investing it at 8%. However, if you believed that you could invest your funds at 15%, you would expect to have $1.75 in four years ($1.00 x 1.7490 = $1.75). In this case you would probably take the $1.00 today rather than the $1.50 in four years.

But what if your choice was as follows: being paid $1.00 each year for the next 4 years, or being paid $5.50 4 years from now. The Future Value Table on pages 502-503 will help you make the decision, since it shows the sum to which one dollar per year will grow if placed in accounts at varying rates of interest. If you took the $1.00 each year and were able to invest it at 8%, you would have $4.87 in four years ($1.00 x 4.8666 = $4.87). By waiting four years, you could receive $5.50, which is 63 cents more than you would have by taking the money and investing it at 8%. However, if you believed that you could invest your funds at 15%, you would expect to have $5.74 in four years ($1.00 x 5.7424 = $5.74). In this case you would probably take the $1.00 each year for the next four years.

COMPOUND INTEREST & PRESENT VALUE OF MONEY

Present value concepts are also very important in financial planning. For instance, given the choice of being paid $1.00 three years from now, or $1.00 four years from now, you would obviously choose to be paid $1.00 three years from now (i.e., the earlier the better). Likewise, given the choice of being paid $1.00 three years from now, or $2.00 four years from now, you would probably choose to be paid $2.00 four years from now (i.e., double your money in only one year). However, given the choice of being paid $1.00 three years from now, or $1.15 four years from now, again your decision would likely require a careful consideration of the present value of money.

The present value of money can be calculated by referring to the Present Value Table on pages 506-507. If you viewed money as having a time value of 8%, the present value of that $1.00 payment due in three years from now is 79 cents ($1.00 x .7938 = $.7938). The present value of a $1.15 payment four years from now is 84 cents ($1.15 x .7350 = $.8453). You would accept payment of $1.15 four years from now.

[check] $1.00 In 3 Years or [] $1.00 In 4 Years [] $1.00 In 3 Years or [check] $2.00 In 4 Years [] $1.00 In 3 Years ? [] $1.15 In 4 Years

However, if you viewed money as having a time value of 18%, the present value of that $1.00 payment due in three years from now is 61 cents ($1.00 x .6086 = $.6086). The present value of a $1.15 payment four years from now is only 59 cents ($1.15 x .5158 = $.5932). You would not accept the offer to pay $1.15 four years from now, but would rather demand payment of $1.00 three years from now.

But what if your choice was as follows: being paid $1.25 at the end of each year for the next 3 years, or being paid $1.00 at the end of each year for the next 4 years. The Present Value Table on pages 510-511 will help you make the decision. If you viewed money as having a time value of 8%, the present value of that $1.25 payment at the end of each year for the next three years is $3.22 ($1.25 x 2.5771 = $3.2214). The present value of a $1.00 payment each year for four years is $3.31 ($1.00 x 3.3121 = $3.3121). You would accept payment of $1.00 per year for the next four years. However, if you viewed money as having a time value of 18%, the present value of that $1.25 payment at the end of each year for the next three years is $2.72 ($1.25 x 2.1743 = $2.7179). The present value of a $1.00 payment each year for four years is $2.69 ($1.00 x 2.6901 = $2.6901). You would not accept payment of $1.00 per year for four years, but would rather demand payment of $1.25 per year for the next three years.

Compounding Periods Can Make A Difference. Simple interest is the payment of interest on the initial principal sum. In contrast, compound interest is the payment of interest on the initial principal sum plus payment of interest on interest accrued in prior periods. As can be seen from the following table, as between simple and compound interest there is a substantial difference in the interest paid. More frequent compounding will result in a higher effective interest rate. While the difference is not as dramatic between the various frequencies of compounding, the amount of additional interest becomes more meaningful with higher rates of interest and longer periods of compounding.

Total Interest Paid On $1,000 Deposit 7% Rate Of Interest 5 Years 10 Years 20 Years Simple Interest 350.00 700.00 1,400.00 Compounded Annually 402.55 967.15 2,869.68 Compounded Quarterly 414.78 1,001.60 3,006.39 Compounded Monthly 417.63 1,009.66 3,038.74

Rule Of 72. The rule of 72 offers a handy method of determining the approximate number of years it will take an asset to double in value (the known rate of interest is divided into 72). Thus, at 8% interest an investment of $10,000 will double in value to $20,000 in 9 years (72 / 8 = 9). At only 6% interest it will take 12 years for the investment to double in value (72 / 6 = 12). Conversely, given the period of an investment, the rule of 72 can also be used to determine the interest rate that would have to be obtained in order for the investment to double in value (the known number of years is divided into 72). For example, over 9 years an investment of $10,000 will double in value to $20,000 provided it pays 8% interest (72 / 9 = 8). However, over 12 years it will require only 6% interest for the investment to double in value (72 / 12 = 6).

FUTURE VALUE OF A SINGLE SUM

Example 1: Calculate the value after one year of a single deposit of $5,000 made in a savings account yielding 6% interest compounded annually.

Solution using table: 5,000 x 1.0600 = 5,300

Formula behind the table: [FV.sub.SS] = [PV.sub.SS] [(1 + i).sup.n]

[FV.sub.SS] = future value of a single sum

[PV.sub.SS] = present value of a single sum

i = interest rate (annual)

n = number of years

Application of formula: [FV.sub.SS] = 5,000 [(1 + .06).sup.1] = 5,000 (1.06) = 5,300

Example 2: Calculate the value after three years of a single deposit of $7,500 made in a savings account yielding 8% interest compounded annually.

Solution using table: 7,500 x 1.2597 = 9,447.75

Application of formula: [FV.sub.SS] = 7,500 [(1 + .08).sup.3] = 7,500 ((1.08)(1.08)(1.08)) = 9,447.84

Because the values in the table are rounded off, the answers are slightly different, in this example by nine cents (i.e., $9,447.84 - $9,447.75 = $.09).

FUTURE VALUE TABLE The Sum To Which One Dollar Principal Will Increase (complete table is located on pages 500=501) Rate Years 3% 4% 5% 6% 1 1.0300 1.0400 1.0500 1.0600 2 1.0609 1.0816 1.1025 1.1236 3 1.0927 1.1249 1.1576 1.1910 4 1.1255 1.1699 1.2155 1.2625 5 1.1593 1.2167 1.2763 1.3382 Rate Years 7% 8% 9% 10% 1 1.0700 1.0800 1.0900 1.1000 2 1.1449 1.1664 1.1881 1.2100 3 1.2250 1.2597 1.2950 1.3310 4 1.3108 1.3605 1.4116 1.4641 5 1.4026 1.4693 1.5386 1.6105

Example 3: Calculate the value after three years of a single deposit of $7,500 made in a savings account yielding 8% interest compounded semiannually (i.e., two times each year).

Formula: [FV.sub.SS] = [PV.sub.SS] [(1 + i/m).sup.mn]

[FV.sub.SS] = future value of a single sum

[PV.sub.SS] = present value of a single sum

i = interest rate (annual)

m = number of compounding periods

n = number of years

Application of formula: [FV.sub.SS] = 7,500 [(1 + .08/2).sup.2x3] = 7,500 ((1.04) (1.04)(1.04)(1.04)(1.04)(1.04)) = 9,489.89

Compared to example 2, semiannual compounding has produced an additional $42.05 of savings ($9,489.89 - $9,447.84 = $42.05).

PRESENT VALUE OF A SINGLE SUM

Example: Calculate the amount necessary to deposit in a savings account yielding 5% interest, compounded annually, so that four years from now there will be $10,000 in the account.

Solution using table: 10,000 x .82270 = 8,227.00

Formula behind the table: [PV.sub.SS] = [FV.sub.SS]/[(1 + i).sup.n]

[PV.sub.SS] = present value of a single sum

[FV.sub.SS] = future value of a single sum

i = interest rate (annual)

n = number of years

Application of formula: [PV.sub.SS] = 10,000/[(1 + .05).sup.4] =10,000/((1.05) (1.05) (1.05) (1.05)) = 8,227.02

Because the values in the table are rounded off, the answers are slightly different, in this example by two cents (i.e., $8,227.02-$8,227.00 = $.02).

PRESENT VALUE TABLE The Worth Today of One Dollar Due In The Future (complete table is located on pages 506-507) Rate 3% 4% 5% 6% Years 1 .9709 .9615 .9524 .9434 2 .9426 .9246 .9070 .8900 3 .9151 .8890 .8638 .8396 4 .8885 .8548 .8227 .7921 5 .8626 .8219 .7835 .7473 Rate 7% 8% 9% 10% Years 1 .9346 .9259 .9174 .9091 2 .8734 .8573 .8417 .8264 3 .8163 .7938 .7722 .7513 4 .7629 .7350 .7084 .6830 5 .7130 .6806 .6499 .6209

FUTURE VALUE OF AN ORDINARY ANNUITY

Example: Calculate the value at the end of four years of a series of $500 annual deposits made at the end of each year to an account paying 7% interest, compounded annually (an ordinary annuity is a series of equal payments made at the end of each period).

Solution using table: 500 x 4.4399 = 2,219.95

Formula behind the table: [FV.sub.A] = [[(1+i).sup.n] - 1/1] A

[FV.sub.A] = future value of an annuity

i = interest rate (annual)

n = number of years

A = amount of annual payment

Application of formula: [FV.sub.AD] = [[(1 + .07).sup.4] - 1/.07] 500

= [(1.07) (1.07) (1.07) (1.07) - 1/1] 500

= 2,219.97

Because the values in the table are rounded off, the answers are slightly different, in this example by two cents (i.e., $2,219.97-$2,219.95 = $.02).

FUTURE VALUE TABLE The Sum To Which One Dollar Principal Will Increase (complete table is located on pages 504=505) Rate 3% 4% 5% 6% Years 1 1.0000 1.0000 1.0000 1.0000 2 2.0300 2.0400 2.0500 2.0600 3 3.0909 3.1216 3.1525 3.1836 4 4.1836 4.2465 4.3101 4.3746 5 5.3091 5.4163 5.5256 5.6371 Rate 7% 8% 9% 10% Years 1 1.0000 1.0000 1.0000 1.0000 2 2.0700 2.0800 2.0900 2.1000 3 3.2149 3.2464 3.2781 3.3100 4 4.4399 4.5061 4.5731 4.6410 5 5.7507 5.8666 5.9847 6.1051

FUTURE VALUE OF AN ANNUITY DUE

Example: Calculate the value at the end of three years of a series of $1,000 annual deposits made at the beginning of each year to an account paying 8% interest, compounded annually (an annuity due is a series of equal payments made at the beginning of each period).

Solution using table: 1,000 x 3.5061 = 3,506.10

Formula behind the table: [FV.sub.AD] = (1 + i)[[(1 + i).sup.n] - 1/i] A

[FV.sub.AD] = future value of an annuity due

i = interest rate (annual)

n = number of years

A = amount of annual payment

Application of formula: [FV.sub.AD] = (1 + .08)[[(1 + .08).sup.3] - 1/.08] 1,000

= 1.08 [((1.08) (1.08) (1.08)) - 1/.08] 1,000

= 3,506.11

Because the values in the table are rounded off, the answers are slightly different, in this example by one cent (i.e., $3,506.11 - $3,506.10 = $.01).

FUTURE VALUE TABLE The Sum To Which One Dollar Principal Will Increase (complete table is located on pages 502=503) Rate 3% 4% 5% 6% Years 1 1.0300 1.0400 1.0500 1.0600 2 2.0909 2.1216 2.1525 2.1836 3 3.1836 3.2465 3.3101 3.3746 4 4.3091 4.4163 4.5256 4.6371 5 5.4684 5.6330 5.8019 5.9753 Rate 7% 8% 9% 10% Years 1 1.0700 1.0800 1.0900 1.1000 2 2.2149 2.2464 2.2781 2.3100 3 3.4399 3.5061 3.5731 3.6410 4 4.7507 4.8666 4.9847 5.1051 5 6.1533 6.3359 6.5233 6.7156

PRESENT VALUE OF AN ORDINARY ANNUITY

Example: Calculate the present value of a series of $2,000 annual payments to be made at the end of each year for the next four years, assuming a discount rate of 5% (an ordinary annuity is a series of equal payments made at the end of each period).

Solution using table: 2,000 x 3.5460 = 7,092.00

Formula behind the table: [PV.sub.A] = [1 - [[1/[(1 + i).sup.n]]/i] A

[PV.sub.A] = present value of an annuity

i = interest rate (annual)

n = number of years

A = amount of annual payment

Application of formula: [PV.sub.A] = [1 - [[1/[1 + .05).sup.4]]/.05] 2,000

= [1 - [1/((1.05)(1.05)(1.05)(1.05))]/.05] 2,000

= 7,091.90

Because the values in the table are rounded off, the answers are slightly different, in this example by 10 cents (i.e., $7,092.00 - $7,091.90 = $.10).

PRESENT VALUE TABLE The Worth Today Of One Dollar Per Annum, Paid At The End Of Each Year (complete table is located on pages 510-511) Rate 3% 4% 5% 6% Years 1 .9709 .9615 .9524 .9434 2 1.9135 1.8861 1.8594 1.8334 3 2.8286 2.7751 2.7232 2.6730 4 3.7171 3.6299 3.5460 3.4651 5 4.5797 4.4518 4.3295 4.2124 Rate 7% 8% 9% 10% Years 1 .9346 .9259 .9174 .9091 2 1.8080 1.7833 1.7591 1.7355 3 2.6243 2.5771 2.5313 2.4869 4 3.3872 3.3121 3.2397 3.1699 5 4.1002 3.9927 3.8897 3.7908

PRESENT VALUE OF AN ANNUITY DUE

Example: Calculate the present value of a series of $10,000 annual payments to be made at the beginning of each year for the next three years, assuming a discount rate of 9% (an annuity due is a series of equal payments made at the beginning of each period).

Solution using table: 10,000 x 2.7591 = 27,591.00

Formula behind the table: [PV.sub.AD] = (1 + i)[1 - [[1/[(1 + i).sup.n]]]/i] A

[PV.sub.AD] = present value of an annuity due

i = interest rate (annual)

n = number of years

A = amount of annual payment

Application of formula: [PV.sub.AD] = (1 + .09) [1 - [[(1/[(1 + .09).sup.3]]/.09] 10,000

= 1.09 [1 - [[1/((1.09)(1.09)(1.09))]]/.09] 10,000

= 27,591.11

Because the values in the table are rounded off, the answers are slightly different, in this example by 11 cents (i.e., $27,591.11 - $27,591.00 = $.11).

PRESENT VALUE TABLE The Worth Today Of One Dollar Per Annum, Paid At The Beginning Of Each Year (complete table is located on pages 508-509) Rate 3% 4% 5% 6% Years 1 1.0000 1.0000 1.0000 1.0000 2 1.9709 1.9615 1.9524 1.9434 3 2.9135 2.8861 2.8594 2.8334 4 3.8286 3.7751 3.7232 3.6730 5 4.7171 4.6299 4.5460 4.4651 Rate 7% 8% 9% 10% Years 1 1.0000 1.0000 1.0000 1.0000 2 1.9346 1.9259 1.9174 1.9091 3 2.8080 2.7833 2.7591 2.7355 4 3.6243 3.5771 3.5313 3.4869 5 4.3872 4.3121 4.2397 4.1699

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Title Annotation: | 2011 FIELD GUIDE TO FINANCIAL PLANNING |
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Author: | Cady, Donald F. |

Publication: | Field Guide to Financial Planning |

Date: | Jan 1, 2011 |

Words: | 3911 |

Previous Article: | Chapter 4: Measuring economic health & market performance. |

Next Article: | Chapter 6: Cash reserves & equivalents. |

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