## Time Value of Money

### Learning Outcomes

• Understand the time value of money

The capital budgeting process is rooted in the concept of time value of money, (sometimes referred to as future value/present value) and uses a present value or discounted cash flow analysis to evaluate the investment opportunity.

The time value of money recognizes that a dollar received or spent in the future is less valuable than a dollar received or spent in the present. Calculations such as the internal rate of return, net present value, and excess present value include adjustments for the time value of money. In these calculations, present value factors, financial calculators, or computer software are used to discount the cash flows to their present values.

Essentially, money is said to have time value because if invested, over time it can earn interest. For example, $1.00 today is worth$1.05 in one year, if invested at 5.00%. Subsequently, the present value is $1.00, and the future value is$1.05.

Conversely, $1.05 to be received in one year’s time is a Future Value cash flow. Yet, its value today would be its Present Value, which again assuming an interest rate of 5.00%, would be$1.00.

The problem with comparing money today with money in the future is that it’s an apples-to-oranges comparison. We need to compare both at the same point in time. Likewise, the difficulty when investing capital is to determine which is worth more: the capital to be invested now or the value of future cash flows that an investment will produce. If we look at both in terms of their present value, we can compare values.

The time value of money draws from the idea that rational investors prefer to receive money today rather than the same amount of money in the future because of money’s potential to grow in value over a given period of time.

Assume you have the option to choose between receiving $10,000 now versus$10,000 in two years. It’s reasonable to assume most people would choose the first option. Receiving $10,000 today has more value and utility than receiving it in the future because of the opportunity costs associated with the wait. In addition to the fact that if you had the$10,000 right now, you could invest it in something that would grow to twice or three times that, there is an additional cost to waiting—inflation. In 1970, the average price of a dozen eggs was $0.62. In 2010 it was$1.47. In 2015 it was $2.09. (U.S. Bureau of the Census, Historical Statistics of the United States, Colonial Times to 1970, Bicentennial Edition, Part 2., Bureau of Labor Statistics, 2011; 2015) Here is another way to look at it. If you invested$10,000 on June 26, 1992, in the newly publicly traded Starbucks, Inc., you would have 29,411.77 shares of stock in 2020 (due to stock splits and stock dividends). If you had sold those shares on January 17, 2020, when each one was worth $93.62, you would have$2.754 million

If you’d taken the same $10,000 and invested it in a savings bond that returned 7% per year, you would have approximately$67,000.

If a dozen eggs in 1992 cost $0.86, you could have bought almost 12,000 cartons of eggs with your$10,000. If you stuck the $10,000 in a coffee can and hit it under the sink for 23 years, by 2015 that same$10,000 would only buy about 4,800 cartons. That’s the effect of inflation. If you’d invested that $10,000 in Starbucks, by 2015 you could have sold your Starbucks stock at about$60 per share and then have bought over 860,000 cartons of eggs.

The point here is that investors are looking for growth. A savings bond is fairly low-risk, but the trade-off is a low rate of return. Investing in Starbucks is very risky, so the potential for gain is much greater, as is the potential for loss. That is the nature of the marketplace.

There are two basic ways to assess the time value of money: present value and future value.

The present value of our $10,000 investment in a 7% savings bond for 28 years is$10,000. The future value is $67,000 (actually$66,488.38).

## Time Value of Money Formula

Depending on the exact situation in question, the time value of money (TVM) formula may change slightly. For example, in the case of annuity payments, the generalized formula has additional factors—or fewer factors. But in general, the most fundamental TVM formula takes into account the following variables:

FV = Future value of money

PV = Present value of money

i = interest rate

n = number of compounding periods per year

t = number of years

Based on these variables, the formula for TVM is:

FV = PV x [ 1 + (i / n) ] (n x t)

### Time value of money examples

Assume a sum of $10,000 is invested for one year at 10% interest. The future value of that money is$11,000.

Excel note: in Excel and most other spreadsheet programs, the formula would look like this:

FV = $10,000 x (1 + (10% / 1) ^ (1 x 1) =$11,000

The formula can also be rearranged to find the present value of the future sum. In other words, if someone offered to give us $5,000 a year from now, what would that be worth to us in present day dollars? Instead of multiplying the Present Value by the formula, we would divide the Future Value by that same formula. For example, the present value of$5,000 one year from today, compounded at 7% interest, is:

PV = $5,000 / (1 + (7% / 1) ^ (1 x 1) =$4,673.

The present value of $11,000 at 10% for one year is 11,000 / 1.1^1 = 10,000. This reverse use of the formula is often called discounting. Accountants have done the math for a variety of time and interest combinations and created tables, such as the one below, that use factors based on the above formula. The factors are calculated using$1, so in order to apply the factor to our example, we multiply it by $5,000. Find the factor for one period, 7% on the table, and do the math. What do you get? (You should get$4,675—it’s off a bit because the factors are rounded to the nearest thousandth.)

Present Value of $1 Periods 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 12% 14% 15% 16% 18% 20% Period 1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909 0.893 0.877 0.870 0.862 0.847 0.833 Period 2 0.980 0.961 0.943 0.925 0.907 0.890 0.873 0.857 0.842 0.826 0.797 0.769 0.756 0.743 0.718 0.694 Period 3 0.971 0.942 0.915 0.889 0.864 0.840 0.816 0.794 0.772 0.751 0.712 0.675 0.658 0.641 0.609 0.579 Period 4 0.961 0.924 0.888 0.855 0.823 0.792 0.763 0.735 0.708 0.683 0.636 0.592 0.572 0.552 0.516 0.482 Period 5 0.951 0.906 0.863 0.822 0.784 0.747 0.713 0.681 0.650 0.621 0.567 0.519 0.497 0.476 0.437 0.402 Period 6 0.942 0.888 0.837 0.790 0.746 0.705 0.666 0.630 0.596 0.564 0.507 0.456 0.432 0.410 0.370 0.335 Period 7 0.933 0.871 0.813 0.760 0.711 0.665 0.623 0.583 0.547 0.513 0.452 0.400 0.376 0.354 0.314 0.279 Period 8 0.923 0.853 0.789 0.731 0.677 0.627 0.582 0.540 0.502 0.467 0.404 0.351 0.327 0.305 0.266 0.233 Period 9 0.914 0.837 0.766 0.703 0.645 0.592 0.544 0.500 0.460 0.424 0.361 0.308 0.284 0.263 0.225 0.194 Period 10 0.905 0.820 0.744 0.676 0.614 0.558 0.508 0.463 0.422 0.386 0.322 0.270 0.247 0.227 0.191 0.162 Period 11 0.896 0.804 0.722 0.650 0.585 0.527 0.475 0.429 0.388 0.350 0.287 0.237 0.215 0.195 0.162 0.135 Period 12 0.887 0.788 0.701 0.625 0.557 0.497 0.444 0.397 0.356 0.319 0.257 0.208 0.187 0.168 0.137 0.112 Period 13 0.879 0.773 0.681 0.601 0.530 0.469 0.415 0.368 0.326 0.290 0.229 0.182 0.163 0.145 0.116 0.093 Period 14 0.870 0.758 0.661 0.577 0.505 0.442 0.388 0.340 0.299 0.263 0.205 0.160 0.141 0.125 0.099 0.078 Note that if you want to get the factor for 12% compounded quarterly for two years, you would have divided the interest rate by four (four quarters in a year) and multiply the years by the number of quarters, so you would use the factor in row “Period 8” and column “3%”. There is an opportunity cost to waiting for money. If you have$5,000 today, and you could invest it at 7%, then a year from now, that $5,000 will have grown to$5,350. So, if someone offers you $5,000, but you have to wait a year, the waiting will cost you$350. In the meantime, the person offering you the money has to put only $4,673 into a 7% investment in order to give you$5,000 one year from now ($4,673 * 1.07 =$5,000.11).

## Effect of Compounding Periods on Future Value

The number of compounding periods can have a drastic effect on the TVM calculations. Taking the $10,000 example above, if the number of compounding periods is increased to quarterly, monthly, or daily, the ending future value calculations are: Quarterly Compounding: FV =$10,000 x (1 + (10% / 4) ^ (4 x 1) = $11,038 Monthly Compounding: FV =$10,000 x (1 + (10% / 12) ^ (12 x 1) = $11,047 Daily Compounding: FV =$10,000 x (1 + (10% / 365) ^ (365 x 1) = \$11,052

This shows TVM depends not only on the interest rate and time horizon but also on how many times the compounding calculations are computed each year.

In addition to calculating the present and future value of a lump sum, there are ways to calculate the present and future values of a stream of cash flows (annuity), and there are tables and online calculators as well as spreadsheet functions to help determine rates of return (yield), present values, and future values of all kinds of scenarios.

As we explore different ways to compare investment options, we’ll address the time value of money with regard to each example. Here is a brief overview of time value of money:

Now, check your understanding of the time value of money.