## Finance 101: The Time Value of Money

By Jeff Greenspan, Managing Director, Financial Modeling Services

Which would you prefer, $100 today are $101 one year from today? (I would take the $100 today!) How about $100 today versus $200 one year from now? Let's **GET STARTED** thinking about the concept of the Time Value of Money (TVM).

You probably already have an intuitive understanding that things tend to become more expensive over time. We can see this clearly when we look at groceries and cars. A one pound loaf of bread cost $0.75 in 1990 but $2.99 in 2010. Similarly a Ford F150 truck cost $10,366 in 1990 but over $22,000 in 2010. The average inflation rate is about 2.5 percent over time.

In addition to inflation we have uncertainty. No one can predict what will happen in the future, therefore $100 today is more valuable to me than $100 a year from now. The combination of inflation plus uncertainty forces us to recognize that the value of money in the future is not the same as the value of money today.

To calculate the future value of money we use the following formula: FV = PV * (1 + r) ^{n} , where

- FV = Future Value
- PV = Present Value
- r = Rate of Return
- n = number of periods

To calculate the present value of money we use this formula: PV = FV / (1 + r) ^{n}

We can also use a simple heuristic called the “Rule of 72” to help us estimate how long it will take before our money doubles at a given rate of return. You divide the rate of return into 72 and that gives you the number of years needed before your money doubles. For example if your rate of return is 10% your money will double every 7.2 years.

Now things get a bit more complicated. If we know that money decreases in value over time, how much should we discount a future cash flow to understand its value to me today? That value is called your discount rate, and to calculate your personal discount rate you have to think about your life. If you have non-mortgage debt like a credit card that you do not pay in full every month, or a car loan at a high rate of interest, then your discount rate equals the highest interest rate that you regularly pay. If on the other hand you pay your credit cards in full each month, your discount rate equals the expected earnings over time on your investments; most people in this category use the average return of the stock market, somewhere between 6 and 10%. To make matters more difficult, your discount rate will change over time. It is typically higher for young people just starting out.

Once we know your discount rate, we can calculate the net present value (NPV) to you of an expected series of cash flows over time. For example, if your discount rate is 7%, the NPV of an investment that costs $10,000 at time 0 but returns $2500/year for 5 years is $250.49.

Discount Rate | 7% | |||||
---|---|---|---|---|---|---|

Time | 0 | 1 | 2 | 3 | 4 | 5 |

Amount | -$10,000.00 | $2,500 | $2,500 | $2,500 | $2,500 | $2,500 |

PV | -$10,000.00 | $2336.45 | $2183.60 | $2,040.74 | $1,907.24 | $1,782.47 |

NPV | $250.49 |

When analyzing an investment, if the NPV is greater than $0, then this is an investment that will enhance your net worth and it should be undertaken. But note that there are often multiple projects that you might consider, and you don't have unlimited funds, so you may have to choose the best project from among a set of projects.

You may also have heard about something called the internal rate of return (IRR). The IRR is a special case of the NPV which tells you what discount rate makes the NPV equal $0. The IRR should never be used by itself to make investment decisions.

### Resources:

For a video review of basic TVM concepts: https://www.youtube.com/watch?v=gkp-7yhfreg

Average inflation rate: https://inflationdata.com/Inflation/Inflation_Rate/Long_Term_Inflation.asp