Personal & Family Finance

Personal & Family Finance Section 1 Time Value of Money

You have won the lottery and can either take $1,000,000 as a lump sum today, or $100,000 over the next 15 years. Which option would you take, and why?

The Time Value of Money • A dollar today is worth more than a dollar in the future • Time value terminology • Future value (FV) • Present value (PV) • Interest rate per compounding period (i) • Number of compounding periods (n) • Payment or annuity (PMT) Case Reading 4-7

Simple Interest • I = P * R * T • I = Interest earned or interest paid • P = Principal sum of money • R = Annual rate of interest • T = Time period (where 1 year is the maximum)

Simple Interest Example You deposit $100 in a savings account on October 1, 2006 which pays 6% annual interest. How much interest will you earn at the end of the first year? 100 x .06 x 1 = $6

Simple Interest Example You deposit $100 in a savings account on October 1, 2006 which pays 6% annual interest. How much interest will you earn at the end of the first month? 100 x .06 x 1/12 = $.50 100 x .06 x (1/12) * 12 = $6

COMPOUNDING (FINDING FUTURE VALUES) • The process of accumulating value over time. Investing money to earn interest will facilitate the growth of our investment. • A single amount (or payment) can accumulate value over time. • A series of equal payments (annuity) can accumulate value over time. • The amount that the investment grows to is referred to as the future value. It includes both the amount invested as well as the interest earned.

Compound Interest Example You deposit $100 in a savings account on October 1, 2006 which pays 6% interest compounded monthly. How much interest will you earn at the end of the first month? 100 x .06 x (1 / 12) = $.50 100.50 x .06 x (1 / 12) = .50 101.00 x .06 x (1 / 12) = .51 101.51 x .06 x (1 / 12) = .51 102.02 x .06 x (1 / 12) = .51 etc. After one year the total value will be 106.17

Compounding Example: Suppose you invest $500 today, hold the investment for 3 years, and earn 8% each year. How much will you accumulate at the end of 3 years (the future value)? __________________________________________ Year Beginning-of- Interest End-of-Year Year Amount Earned Amount 1 $500.00 $40.00 $540.00 2 540.00 43.20 583.20 3 583.20 46.66 Answer $659.86

Compound versus Simple Interest • Compound interest -- the investment accumulates by earning interest on interest. • Interest in year 2 is greater than the interest in year 1 due to compound interest. • Simple interest does not assume that interest is earned on interest. • If the previous example had assumed simple interest, the interest earned in years 1, 2, and 3 would only be $40 each year. Therefore, the future value would be $620 (not $629.86).

Present Value of a Single Amount PV = FV [1 / (1 + I)n] To determine the current value of an amount (e.g., of an investment or to be deposited) based on the future value at a specified rate over a designated period of time. Example: How much must you deposit today in order to have $3,000 at the end of 3 years when compounded annually at 8%? PV = 3,000 [1 / (1.08)3] PV = 3,000 (1 / 1.26) PV = 3,000 (.794) PV = $2,381.50

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 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Period 0.08 1 2 3 0.990 0.981 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 0.970 0.942 0.915 0.889 0.864 0.837 0.816 0.794 0.772 0.794 3 PV = FV x IF PV = $3,000 x 0.794 (0.79383 rounded to fit screen) PV = $2,381.49

Present Value of an Annuity PVA = PMT x ((1 – (1 / (1 + I)n)) / I) To determine the current value of an amount (e.g., to be invested or deposited) of an annual payment or receipt over a specified number of periods compounded at a specified rate. Example: How much should you invest today if you wish to receive $1,000 at the end of each of the next three years if you expect to earn 8%? PVA = 1,000 {1 - [1 / (1.08)3]} / .08 PVA = 1,000 {1 - [1 / 1.26]} / .08 PVA = 1,000 {1 - .794} / .08 PVA = 1,000 (.0206) / .08 PVA = 1,000 (2.577) PVA = 2,577.10

Present Value of an Annuity Assume that you could purchase an investment that would pay $1,000 at the end of each year for three years, and that you expect to earn a return of 8%.

Present Value at Beginning of Year 1 End of Yr. 1 Present Value of an Annuity $ 925.93 = $1,000 ÷ (1.08) $1,000 1

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

Present Value of an Annuity Present Value at Beginning of Year 1 End of Yr. 3 $ 925.93 857.34 793.83 = $1,000 ÷ (1.08) $1,000 3

$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 End of Yr. 3 $ 925.93 857.34 793.83 $2,577.10 Present value of total investment

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

Present Value of an Annuity Interest Rate 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Period 0.08 1 2 3 0.990 0.981 0.971 0.962 0.952 0.943 0.935 0.926 0.917 1.970 1.942 1.913 1.886 1.859 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 0.794 2.531 2.577 3 PVA = FV x IF PVA = $1,000 x 2.577 (2.57710 rounded to fit screen) PVA = $2,577.10

Future Value Calculations • Future value calculations can be done with any of the following: • A financial calculator • The financial functions in Excel • Financial calculators on the Internet • Use of financial formulas and a calculator • Use of financial tables • To facilitate calculations, tables have been created that assume an initial investment of $1. These tables then specify various time periods and interest rates. Detailed tables appear in the appendix of many financial textbooks.

Future Value FV = PV(1 + I)n To determine the value of an amount at a particular time in the future. Example: If you deposit $500 now into a certificate of deposit at 8%, how much will you have at the end of 3 years? FV = 500 (1 + .08)3 FV = 500 (1.08)3 FV = 500 (1.26) FV = $629.86

Future Value Another approach to calculating a future value is to use the future value of a single amount table.

Future Value Interest Rate 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Period 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 1.124 1.145 1.166 1.188 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295

Future Value Interest Rate 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Period 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 1.124 1.145 1.166 1.188 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 3 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295

Future Value Interest Rate 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Period 0.08 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 1.124 1.145 1.664 1.188 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 1.080 1.166 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 1.260 3 FV = PV x IF FV = $500 x 1.260 (1.25971 rounded to fit screen) FV = $629.86

The Rule of 72 • Popular in estimating the time that it takes for an investment to double • Using the interest rate for an investment the rule of 72 estimates how long it will take for the money to double • Doubling Time (DT) = 72/interest rate • Example: if the interest rate percent is 12 • DT = 72/12 = 6 years • How could you verify this?

What is an Annuity? • A series of equal payments. • rent payments • mortgage payments • car payments • Bonds will often involve interest payments that are annuities. • There are two types of annuities: • An ordinary annuity (OA) assumes the payments occur at the end of the period. • An annuity due (AD) assumes the payments occur at the beginning of the period.

Future Value of an Annuity FVA = PMT x (((1 + I)n – 1) / I) To determine the future value of a series of equal amounts received or paid over a specified number of years. Example: How much will you have at the end of three years if you make annual deposits of $500 each compounded annually at 8%? FVA = 500[(1 + .08)3 - 1] / .08 FVA = 500 [(1.08)3 - 1] / .08 FVA = 500 (1.26 - 1) / .08 FVA = 500 (.26) / .08 FVA = 500 (3.25) FVA = $1,623.20 (rounding factor)

Future Value of an Annuity Similar to the example given of the future value of 1, the amount can be determined by using a future value of an annuity table.

Future Value of an Annuity Interest Rate 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Period 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 1.260 3.278 3.246 3 FVA = A x IF FVA = $500 x 3.246 (3.24640 rounded to fit screen) FVA = $1,623.20

Converting an Ordinary Annuity (OA) Into an Annuity Due (AD) • The conversion formula is: FV(AD) = FV(OA) × (1 + i ) • This formula accounts for the fact that each payment earns interest for one extra period.

Goal Planning • Time value of money techniques are useful for estimating required funds to meet future goals. • Identify specific goals. • What is the particular goal? • When do you plan to achieve the goal? • How much will need to be saved currently or each year? • Inflation must be considered since most goals are long-term. • Determine how much must be saved each year to achieve these goals.

Personal & Family Finance