### Time Value of Money

**Time Value of Money (Basics)**

A dollar now is worth more than a dollar in the future. Why is this a fact? We have basically two decisions we can do with our money: we can save it now, or we can spend it now. If we spend it now, this comes with an opportunity cost (money foregone); if we save it now, we must earn interest (financing cost for using our money). What controls our motivation to spend results from a need to consume now or to consume later. While what to buy and when to buy are much more complex decisions, the basic premise of the time value of money in a business is that if one decides to consume now, the money generated later from the consumption must be worth more now that what is required to consume. We must use capital budgeting (the process managers use to make capital investment decision) in order to make the decision whether to buy and consume now, save, or look for other investments in order to maximize returns.

In order to use the concept of time value of money, we have to determine whether we are looking at cash flow now or into the future.

- · Computing Future and Present Values of a Single Amount – If one needs to know the dollar amount of cash flow that occurs in the future and will need to determine its value now, we will need to look at the
**present value**of something. If one needs to know dollar amount of cash flow today and what it will be worth in the future, this is an example of a future value problem.

- · Present a future value problems may involve two types of cash flow: a single amount or an annuity (a series of equal cash payments). We will need to know how to deal with four different situations related to the time value of money: (1) Future value of a single payment; (2) Present value of a single payment, Future value of an annuity, and a (4) present value of an annuity.

- o Future Value of a single amount – In computing the future value of a single amount, you will asked to calculate how much money you will have in the future as the result of investing a certain amount right now. You will need to know:
- · The amount to be invested
- · Interest rate (i) the amount will earn
- · The number of periods (n) in which the amount will earn interest.

The future value concept is based on compound interest, which simply means that interest is calculated on top of interest. Thus, the amount of interest for each period is calculated using the principal plus any interest earned in prior periods. For example, if we take $1000 for one period in the future at 10% interest, we would earn $100 in interest ($1,000 *.10 = $100). Our new amount of total money in the future is $1,100 ($1,000 +$100). Now, if we take the $1,100 for another period and multiply it by 10%, we would earn $110 dollars in interest ($1,100*10= $110). Our total new for this period is $1,210 ($1,100 + $110). The power of compounding allows one to have $20 more after two periods in the future. Because of compounding, capital budgeting has to consider this concept in order to make decisions about future cash flows.

- o Present value of a single amount – The present value of a single amount is the value to you today of receiving some amount of money in the future. For instance, you might be offered an opportunity to invest in a financial instrument that would pay you $1,000 in a year. What you will need to determine is how much you would need today to get to the $1,000 in one year. Remember, a dollar now is worth less than a dollar in the future, so one would need less money now to get to the $1,000 one year from now.

Present value is the reciprocal of future value, or 1/1+i. To get to the present value in this situation, take the $1,000 and multiply by 1/1+i, or $1,000*1/1.10 = $1,000*.9091 = $909.10

- · Computing Future and Present Value of an Annuity – Instead of a single payment, many situations involve multiple cash payments over a number of periods. An annuity is a series of consecutive payments characterized by: (1) An equal dollar amount each interest period; (2) Interest periods of equal length (year, half a year, quarter, or month); and (3) An equal interest rate each interest period.

Examples of annuities include monthly payments on a car or house, yearly contributions to a savings account, and monthly pension benefits.

- o Future value of an Annuity – If you are saving money for some purpose, such as remodeling your home or taking vacation (if you are a business, saving for a capital project), you might decide to deposit a fixed amount of money in a savings account each month. The future of an annuity tells you how much money will be in your savings account at some point in the future. The formula for the future value of an annuity is (1+ i)^n -1/i, where n is the number of periods and I is the interest rate. You could figure the future value of an amount by get getting the future value of each cash flow by the interest rate and the number of period compounded, which would result in two, three or more calculations. The annuity formula allows you to do it one time. For example, if you invested $1,000 each year over three years at 10%, what would the future amount be? Deductive reasoning and knowledge of the time value money lets us know that it would be more than $3,000. First, we need to get the annuity factor, and we get this by the formula ((1+.10)^4 – 1/.10 = 1.4641-1)/.10 = .4641/.10 = 4.641)). We multiply $1,000*4.641 = $4,641.

- o Present value of an Annuity – The present value of an auunity is the value now of a series of equal amounts to be received (or paid out) for some specified number of period in the future. Examples of the are retirement programs (for a business, this could cash streams from a project). For example, what amount would you need to put away now to receive $1,000 for three consecutive periods? Remember the present value is the reciprocal of future value, which means that it will take less money is the present to get more money in the future. The formula for present value is (1-(1/(1+i)^n))/I where n is the number of periods and I is the interest rate. For our example, the calculations are (1-(1/(1.10)^3))/.1 = (1-.7513)/.1 = .2487/.10=2.487. We take $1,000*2.487 = $2,487.

- o One will get the same answer whether he/she use the formula or the table. A student can use either one or the other.