Accounting

The Time Value of Money: Future Value and Present Value Computations "If I deposit $10,000 today, how much will I have for a down payment on a house in five years?" "Will $2,000 saved a year give me enough money when I retire?" "How much must I save today to have enough for my children's post-secondary education?" As introduced in Chapter 1 and used to measure financial opportunity costs in other chapters, the time value of money, more commonly referred to as interest, is the cost of money that is bor- rowed or lent. Interest can be compared to rent, the cost of using an apartment or other item.

The time value of money is based on the fact that a dollar received today is worth more than a dollar that will be received one year from today because the dollar received today can be saved or invested and will be worth more than a dollar a year from today Similarly, a dollar that will be received one year from today is currently worth less than a dollar today.

The time value of money has two major components: future value and present value. Future value computations, which are also referred to as compounding, yield the amount to which a current sum will increase based on a certain interest rate and period of time, Present value, which is calculated through a process called discounting, is the current value of a future sum based on a certain interest rate and period of time.

In future value problems, you are given an amount to save or invest and you calculate the amount that will be available at some future date. With present value problems, you are given the amount that will be available at some future date and you calculate the current value of that amount Both future value and present value computations are based on basic interest rate calculations.

FINANCIAL CALCULATORS Currently, financial calculators, with time value of money functions built in, are widely used to calculate future value, present values, and annuities, For the following examples, we will use the Texas Instruments BA II Plus financial calculator, which is recommended by the Canadian Institute of Financial Planning.

When using the BA II Plus calculator to solve time value of money problems, you will be working with the TVM keys that include:

CPT — Compute key used to initiate financial calculations once all values are inputted — Number of periods — Interest rate per period — Present value PMT — Amount of payment, used only for annuities — Future value Enter values for PV, PMT, and FV as negative if they represent cash outflows (e.g., investing a sum of money) or as positive if they represent cash inflows (e.g., receiving the proceeds of an investment). To convert a positive number to a negative number, enter the number and then press the +/— key.

37 Part 1 PLANNING YOUR PERSONAL FINANCES The examples that are shown in this chapter assume that interest is compounded annually and that there is only one cash flow per period. To reflect this, we must set the number of payments and compounding per period to 1 (the default setting is 12). To do this, press in turn the I 2!'°) button (yellow), the 1 I/Y button (for the PLY function shown above it), the number 1, the ENTER button, the 1 2ND , button again, and finally the I CPT button (for the quit function above it). Before using any financial calculator, we strongly recommend that you consult the instruction manual that accompanies it and attempt the examples shown there.

Now let's try a problem. What is the future value of $100 after three years at a 10 percent annual interest rate? Remember that an investment of money is considered to be an outflow of cash; therefore, the present value of $100 should be entered as a negative number.

First, you must enter the data. Remember that an investment of money is considered to be an outflow of cash; therefore, the $100 should be entered as a negative number.

3 10 100 + / — I IN PV 0 PMT (optional if registers are cleared) To find the solution, the future value, press , and the future value of 133.1 is displayed.

FUTURE VALUE OF A SINGLE AMOUNT The future value of an amount consists of the original amount plus compound interest. This calculation involves the following elements:

FV = Future value PV = Present value i = Interest rate n = Number of time periods The formula and financial calculator computations are as follows:

(Note: These financial calculator notations may require slightly different keystrokes when using various brands and models.) Future Value of a Single Amount Formula Table Financial Calculator FV = PV(1 + Using Exhibit 1B-1:

FV = PV (Table Factor) PV L IN N PMT , 1 CPT FV1 Example A: The future value of $1 at 10 percent after three years is $1.33. This amount is calculated as follows:

$1.33 = $(1.001 + 0.10) 3 Using Exhibit 1B-1: 1 1 PV , 10E 3 N ,OLPMT ,I CPT 1.33 $1.33 = $1.00(1.33) Future value tables are available to help you determine compounded interest amounts (see Exhibit 1B-1).

Looking at Exhibit 1B-1 for 10 percent and three years, you can see that $1 is worth $1.33 at that time. For other amounts, multiply the table factor by the original amount. This process may be viewed as follows:

Formula Table Financial Calculator Example B: If your savings of $400 earns 12 percent, compounded monthly, over a year and a half, use the table factor for 1 percent for 18 time periods; the future value is:

$478.46 = $400(1 + 0.01) 18 -$478.40-= $400(1.196) 400 PV 12/12 = 1.5 x 12 = 18 I N 01 PMT 1 I CPT', - :FV 478.46 FV = Annuity - FV = Annuity (Table Factor) I PMT I N IIY [ PV CPT L FV (1 + - 1 Using Exhibit 1B-2:

Appendix 1B The Time Value of Money: Future Value and Present Value Computations Sample Problem 1 What is the future value of $800 at 8 percent after six years?

Sample Problem 2 How much would you have in savings if you kept $200 on deposit for eight years at 8 percent, compounded semi-annually?

FUTURE VALUE OF A SERIES OF EQUAL AMOUNTS (AN ANNUITY) Future value may also be calculated for a situation in which regular additions are made to sav- ings. The formula and financial calculator computations are as follows:

Future Value of a Series of Payments Formula Table Financial Calculator This calculation assumes that: (1) each deposit is for the same amount; (2) the interest rate is the same for each time period; and (3) the deposits are made at the end of each time period.

Example C: The future value of three $1 deposits made at the end of the next three years, earning 10 percent interest, is $3.31. This is calculated as follows:

(1 + 0.10) 3 - 1 Using Exhibit 1B-2:

$3.31 - 0.10 $3.31 = $1 x 3.31 - 1 PMT, 31 N ,i0( IN , FV 3.31 This may be viewed as follows:

Example D: If you plan to deposit $40 a year for 10 years, earning 8 percent compounded annually, the future value of this amount is:

$40(1 + 0.08)" - 1 $579.46 = 0.08 Using Exhibit 1B-2:

$579.48 = $40(14.487) - 40 PMT 10 [ CPT_[ J L 579.46 01 IN ,o( PV , Sample Problem 3 What is the future value of an annual deposit of $230 earning 6 percent for 15 years?

Sample Problem 4 What amount would you have in a retirement account if you made annual deposits of $375 for 25 years earning 12 percent, compounded annually?

PRESENT VALUE OF A SINGLE AMOUNT If you want to know how much you need to deposit now to receive a certain amount in the future, the formula and financial calculator computations are as follows:

Present Value of a Single Amount Formula Table Financial Calculator FV PV - (1 + On Using Exhibit 1B-3:

PV = FV (Table Factor) I N IN , I PMT t CPT_ PV V Example E: The present value of $1 to be received three years from now based on a 10 percent interest rate is calculated as follows:

$1 Using Exhibit 1B-3:

$0.75 - (1 + 0.10) 3 $0.75 = $1(0.751) 1 FV , 10[ IN ,0 I PMT , I CPT I PV —75131 (continued) 1 1 (1 + 0.14)10 0.14 _ Using Exhibit 1B-4: 100 PMf, 101 N , 141 IN .

0 1 FV , $521.60 = $100(5.216) 1 CPT I PV ' 521.61156 $521.61 = $100 Part 1 PLANNING YOUR PERSONAL FINANCES This may be viewed as follows:

Present value tables are available to assist you in this process (see Exhibit 1B-3). Notice that $1 at 10 percent for three years has a present value of $0.75. For amounts other than $1, multiply the table factor by the amount involved.

Example F: If you want to have $300 seven years from now and your savings earn 10 percent, compounded semi-annually (5 percent for 14 time periods), finding how much you would have to deposit today is calculated as follows:

$300 Using Exhibit 1B-3:

$151.52 = ( 1 + 0.05)14 $151.50 = $300(0.505) 300 FV , 7 x 2 = 141 N , 10/2 = 1 I/Y , 0 t PMT ,1 CPT; PV -151.52 Sample Problem 5 What is the present value of $2,200 earning 15 percent for eight years?

Sample Problem 6 To have $6,000 for a child's education in 10 years, what amount should a parent deposit in a savings account that earns 12 percent, compounded quarterly? PRESENT VALUE OF A SERIES OF EQUAL AMOUNTS (AN ANNUITY) The final time value of money situation allows you to receive an amount at the end of each time period for a certain number of periods. The formula and financial calculator computations are as follows:

Present Value of a Series of Payments Formula Table Financial Calculator 1 1 Using Exhibit 1B-4:

(1 + O n PV = Annuity (Table Factor) PV = Annuity )< Example G: The present value of a $1 withdrawal at the end of the next three years would be $2.49, for money earning 10 percent. This would be calculated as follows:

1 (1 + 0.10 ) 3 0.10 _ Using Exhibit 1B-4:

$2.49 = $1(2.487) 1IPMT,31 N , 101 ,O1 FV , CPT PV -2.48685 $2.49 = $1 This may be viewed as follows:

This same amount appears in Exhibit 1B-4 for 10 percent and three time periods. To use the table for other situations, multiply the table factor by the amount to be withdrawn each year.

Example H: If you wish to withdraw $100 at the end of each year for 10 years from an account that earns 14 percent, compounded annually, what amount must you deposit now?

Sample Problem 7 What is the present value of a withdrawal of $200 at the end of each year for 14 years with an interest rate of 7 percent?

P T' L IN 1 1 -FV 1 CPT j TV ample Problem 8 $650 at the end of e ow much would you have to deposit now to be able to withdraw year for 20 years from an account that earns 11 percent? Appendix 1B The Time Value of Money: Future Value and Present Value Computations- Present value tables can also be used to determine instalment payments for a loan as follows:

USING PRESENT VALUE TO DETERMINE LOAN PAYMENTS Present Value to Determine Loan Payments Table Financial Calculator Example I: If you borrow $1,000 with a 6 percent interest rate to be repaid in three equal payments at the end of the next three years, the payments will be $374.11. This is calculated as follows:

Sample Problem 9 What would be the annual payment amount for a $20,000, 10-year loan at 7 percent?

$1,000 — $374.11 2,673 1000 6 3 374.10981