## Time Value of Money

### Learning Outcomes

• Understand the implication of the time value of money

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 due to 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 formula may change slightly. For example, in the case of annuity payments, the generalized formula has additional or less 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 × [ 1 + (i / n) ] (n × 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: FV =$10,000 × (1 + (10% / 1) ^ (1 × 1) = $11,000 The formula can also be rearranged to find the value of the future sum in present day dollars. For example, the value of$5,000 one year from today, compounded at 7% interest, is:

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

## 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 × 1) = $11,038 • Monthly Compounding: FV =$10,000 x (1 + (10% / 12) ^ (12 × 1) = $11,047 • Daily Compounding: FV =$10,000 x (1 + (10% / 365) ^ (365 × 1) = \$11,052

This shows TVM depends not only on 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.

This relates to long-term debt and financing because often, as in bonds and leases, the carrying value of the liability is based on the present value of the total future obligations, rather than say the total of the payments.