The Time Value of Money

Chapter F8 The Time Value of Money Electronic Presentationby Douglas Cloud Pepperdine University

Objectives 1.Define future and present value. 2.Determine the future value of a single amount invested at the present time. 3. Determine the future value of an annuity. 4. Determine the present value of a single amount to be received in the future. 5. Determine the present value of an annuity. 6. Determine investment values and interest expense or revenue for various periods. Once you have completed this chapter, you should be able to:

Objective 1 Determine future and present value.

Future Value The future value of an amount is the value of that amount at a particular time in the future.

Present Value The present value of an amount is the value of that amount on a particular date prior to the time the amount is paid or received.

Interest Rate Future Value Future Value = Present Value(1 + R) If $1,000 is invested on January 1, 2004, at 5% interest, what will be the future value (the amount that will accumulate) by December 31, 2004? Future Value = $1,000(1.05) Future Value = $1,050

Objective 2 Determine the future value of a single amount invested at the present time.

Compound Interest Earning interest in one period on interest earned in an earlier period is known as compound interest.

Compound Interest If the accumulated amount ($1,050 from Slide 6) is left in the savings account for a second year, until December 31, 2005, how much would the investment be worth at that time? $1,050(1.05) = $1,102.50

FV = PV(1 + R) t Compound Interest Assume you invest $500 for three years at 8% interest. How much would your investment be worth at the end of three years? FV = $500(1.08)³ FV = $629.86

Compound Interest Recall that the future value of an amount is the value of that amount at a particular time in the future.

Compound Interest You can use Excel to determine the future value of $500 that earns 8% interest compounded annually for three years.

Compound Interest Insert =500*(1.08^3) in a cell and press Enter.

Compound Interest The amount shown in the cell represents the future value, which is $629.86.

Compound Interest Excel also contains built-in functions for calculating present and future values.

Compound Interest When you see USING EXCELin the margin of the textbook, follow the instructions to learn how to use the built-in function.

Compound Interest To calculating a future value, a future value of a single amount table, such as the one in the next slide, can be used.

Compound Interest Interest Rate Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1 2 3 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.020 1.040 1.061 1.082 1.103 1.224 1.145 1.664 1.188 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295

Compound Interest Interest Rate Period 0.08 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1 2 3 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.020 1.040 1.061 1.082 1.103 1.224 1.145 1.664 1.188 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295

Compound Interest Interest Rate Period 0.08 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1 2 3 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.020 1.040 1.061 1.082 1.103 1.224 1.145 1.664 1.188 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 1.260 FV = $500 x 1.260 = $630 (rounded)

Interest Table for an Investment of $500 for Three Years at 8% Exhibit 1 A B C D Value at Interest Earned FV at End Year Beginning of Year (B x Interest Paid) (B + C) 1 500.00 40.00 540.00 2 540.00 43.20 583.20 3 583.20 46.66 629.86 Total 129.86

Objective 3 Determine the future value of an annuity.

Future Value of an Annuity An annuity is a series of equal amounts received or paid over a specified number of equal time periods.

Future Value of an Annuity If $500 is invested at the end of each year for three years, how much would the investment be worth at the end of three years if the interest earned is 8% per year?

Future Value of an Annuity Interest Rate Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1 2 3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2.010 2.020 2.030 2.040 2.050 2.060 2.070 2.080 2.090 3.030 3.060 3.091 3.122 3.153 3.184 3.215 3.246 3.278 3.246 FVA = Amount invested (A) x Interest factor (IF) FVA = $500 x 3.246 (rounded to three decimal places) FVA = $1,623 (rounded)

Interest Table for an Annuity of $500 at End of Each Year for Three Years at 8% Exhibit 2 A B C D E Value Interest Earned Amount FV at at Beginning (Column B x Invested at End of Year of Year Interest Rate) End of Year Year 1 0.00 0.00 500.00 500.00 2 500.00 40.00 500.00 1,040.00 3 1,040.00 83.20 500.00 1,623.20 Total 123.20 1,500.00

Future Value of an Annuity How much would you need to invest each year to accumulate $1,000 at the end of three years to take a trip to Mexico after you graduate from college? Assume you can earn 6% on your investment.

Future Value of an Annuity Interest Rate Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.08 1 2 3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2.010 2.020 2.030 2.040 2.050 2.060 2.070 2.080 2.090 3.030 3.060 3.091 3.122 3.153 1.191 1.225 1.260 1.295 1.000 2.080 3.030 3.060 3.091 3.122 3.153 3.184 3.215 3.246 3.278 3 3.184 FVA = Amount invested (A) x Interest factor (IF) $1,000 = A x 3.184 (rounded to three decimal places) A = $1,000 ÷ 3.184 A = $314 (rounded)

Future Value of an Annuity We can calculate the amount of the payment in Excel using the payment function. Insert =PMT(0.06,3,,1000) in a cell and press Enter.

Objective 4 Determine the present value of a single amount to be received in the future.

Present Value of a Single Amount Using Excel, the present value of an investment that pays $3,000 at the end of three years at 8% can be calculated by inserting =3000*(1/(1.08^3)) in a cell and pressing Enter.

Present Value of a Single Amount The present value of a single amount table also could be used to determine the present value of the $3,000.

Present Value of a Single Amount Interest Rate Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.08 1 2 3 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.980 0.961 0.943 0.925 0.907 0.890 0.873 0.857 0.842 0.926 0.857 0.794 0.971 0.942 0.915 0.889 0.864 0.840 0.816 0.794 0.772 3 PV = FV x IF PV = $3,000 x 0.794 (rounded to three decimal places) PV = $2,382 (rounded)

Interest Table for a Present Value of $2,381.49 for Three Years at 8% Exhibit 3 A B C D Present Value at Interest Earned Value at End Year Beginning of Year (B x Interest Rate) (B + C) 1 2,381.49 190.52 2,572.01 2 2,572.01 205.76 2,777.77 3 2,777.77 222.23* 3,000.00 Total 618.51 *Adjusted due to rounding

Objective 5 Determine the present value of an annuity.

Present Value of an Annuity Assume that you are considering the purchase of an investment that would pay $1,000 at the end of each year for three years. The investment is expected to earn a return of 8%. How much would you have to invest now?

Present Value of an Annuity Present Value at Beginning of Year 1 $ 925.93 = $1.000 x (1.08)¹ (table value of 0.92593)

Present Value of an Annuity Present Value at Beginning of Year 1 $ 925.93 857.34 = $1,000 ÷ (1.08)² (table value of .= 0.85734)

Present Value of an Annuity Present Value at Beginning of Year 1 $ 925.93 857.34 793.83 = $1,000 ÷ (1.08)³ (table value of 0.79373)

Present Value of an Annuity Present Value at Beginning of Year 1 $ 925.93 857.34 793.83 $2,577.10 Required investment now

$3,000.00 Total amount received over three years 2,577.10 Present value of total investment $ 422.90 Interest earned over three years Present Value of an Annuity Present Value at Beginning of Year 1 $ 925.93 857.34 793.83 $2,577.10 Required investment now

Present Value of an Annuity The PV function in Excel can be used to calculate the present value of an annuity. The function can be entered in the pop-up box or directly into the cell.

Present Value of an Annuity If you purchase an investment that paid $1,000 each year for three years at 8% interest, insert =PV(0.08,3,–1000) in a cell and press Enter.

Present Value of an Annuity Or, you can use the present value of an annuity table.

Present Value of an Annuity Interest Rate Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.08 1 2 3 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 1.970 1.942 1.913 1.886 1.860 1.833 1.808 1.783 1.759 0.926 1.783 2.941 2.884 2.829 2.775 2.723 2.673 2.624 2.577 2.531 3 3 2.577 PVA = FV x IF PVA = $1,000 x 2.577 (table value read to three decimal places) PVA = $2,577 (rounded)

$2,783.27 – $1,000.00 Interest Table for an Annuity of $1,000 Each Year for Three Years at 8% Exhibit 4 A B C D E Present Value Interest Earned Total Amount Value at at Beginning (Column B x Invested End of Year of Year Interest Rate) (B + C) Year 1 2,577.10 206.17 2,783.27 1,783.27

Interest Table for an Annuity of $1,000 at the End of Each Year for Three Years at 8% Exhibit 4 $1,925.93 – $1,000.00 A B C D E Present Value Interest Earned Total Amount Value at at Beginning (Column B x Invested End of Year of Year Interest Rate) (B + C) Year 1 2,577.10 206.17 2,783.27 1,783.27 2 1,783.27 142.66 1,925.93 925.93

Interest Table for an Annuity of $1,000 at the End of Each Year for Three Years at 8% Exhibit 4 A B C D E Present Value Interest Earned Total Amount Value at at Beginning (Column B x Invested End of Year of Year Interest Rate) (B + C) Year 1 2,577.10 206.17 2,783.27 1,783.27 2 1,783.27 142.66 1,925.93 925.93 3 925.93 74.07 1,000.00 0.00 Total 422.90

Objective 6 Determine investment values and interest expense or revenue for various periods.

Loan Payments and Amortization You negotiate with a dealer to purchase a car for $5,000, which you arrange to borrow from a local bank.

The Time Value of Money