Overview – How the Value of Money Changes Over Time
Imagine you just won $1 million from the lottery. Assuming you’re a prudent person, you decide not to spend the money right now and instead decide how exactly you plan to use it.
Will you choose to take the money all at once right now, or do you plan to invest all of it today and intermittently withdraw small amounts from this large sum over a span of several years? Perhaps you could even take a certain portion of your winnings today and leave the rest in an interest-bearing bank account.
Assuming you don’t keep all this money under your mattress and decide to put it to work in some way, its value today may vary considerably from its value in the future. It all depends on how exactly you deploy this money.
This idea of the value of the money you have changing over time is known as, unsurprisingly, the time value of money. This relatively simple idea has all sorts of applications, from something as simple as personal financial decisions all the way to deciding which major corporate investments to pursue.
*The time value of money is an extremely broad and detailed topic, filled with countless details, variations, and other nuances. Therefore, this article will only provide a very high-level look at this topic.*
Interest Rate/Rate of Return: The Cornerstone of the Time Value of Money
In order for the value of money to change, it must be affected by some sort of externality. If you have $100 and you keep it tucked away in a safe for 10 years, after 10 years you’ll still have the same $100 that you started with.
The major assumption being made when it comes to the time value of money is that the funds are being put to work in some way, whether it’s being kept in an interest-bearing bank account, used to make investments, or serving as the capital outlay for a major project.
Without any sort of interest rate/rate of return acting upon the money over some period of time, then the time value of money will not apply.
Many people may already have some experience with the time value of money, such as dealing with the interest accrued on outstanding debt. In this common scenario, the time value of money (in this case, the amount of interest you owe) is calculated on a forward-looking basis, that is, by knowing how much time the money has left to grow.
However, the time value of money works both ways, that is, if you know how much money you expect to receive in the future based on a given interest rate/rate of return, then you can work backwards and calculate how much it will be worth today.
Therefore, the time value of money can be used to calculate the future value of money given an initial amount, or how much money you expect to receive in the future at a given interest rate/rate of return would be worth if you received all of it today. Let’s go over these two scenarios.
Understanding Future Value (FV)
By far the simplest form of the time value of money is future value (FV).
As the name implies, the future value is simply the value of an initial amount of money after a set number of years at a given interest rate/rate of return. The equation is as follows:
The variables are as follows:
- FV – Future Value
- PV – Present Value
- r – annual interest rate/rate of return
- n – number of compounding periods
- t – time (usually in years)
If you’ve had the chance to go through the Credit article before, then this equation may look similar. That’s because this is the same equation used to calculate compound interest. After all, compound interest is a classic example of finding the future value of a sum of money.
To see this equation in action, let’s go over a simple example.
Let’s say you have $100 in cash you wish to put in an interest-bearing savings account. The annual interest rate is 1% and is compounded every month (i.e., 12 times a year). After 3 years, what would the value of your money be?
Plugging all this information into the above equation, we get the following solution:
There are other variants to future value, such as regular payments being made to increase the current balance, but these variants, and the explanations behind them, are beyond the scope of this introductory article.
Net Present Value (NPV)
Knowing the future value of your money today is certainly nice to know, but just because you’ll receive more money in the future doesn’t mean every given project or investment is worth pursuing. You probably want to make sure that the money you expect to earn in the future is worth more than the amount you gave up today.
If receiving more money in the future isn’t enough to justify an investment or project, then what is? One way to determine that is to calculate what all future cash flows you expect to receive from a given project or investment will be worth today, or in more technical terms, you “discount” them to their present value.
This idea of finding out what the value of all future cash flows would be worth today is known as net present value (NPV). The reason why it’s called “net” present value is that the original cash outlay is subtracted from the sum of all future cash flows. NPV is essentially just the inverse of FV.
The equation is as follows:
Don’t let this equation intimidate you. It can simply be broken down into two terms.
The first term, the one in the brackets, simply means the sum of all future cash flows. C(t) is the cash flow at a given time, r is the discount rate, and t is the time, usually in years. The second term, C(0), is the initial cash outlay, hence the reason it’s being subtracted.
Most of the variables in the NPV equation are quite straightforward, except for when it comes to the discount rate, r. To understand why let’s backtrack a little.
It’s important to note that when calculating NPV you don’t know what your future cash flows will be. The cash flows you use in your calculations are simply projections, but these could vary considerably from the real-world data.
Because you don’t know for certain what your future cash flows will be, it’s also difficult to determine what rate of return will lead to those specific cash flows. For example, your cash flow for year 1 of a project could be $100,000, translating to a 5% rate of return, whereas in year 2 it could jump to $250,000, which could represent an 8% rate of return.
Because of this, when discounting these projected cash flows to the present you don’t have the luxury of using a specific discount rate for a specific cash flow. This leaves us with having to use a single discount rate for every cash flow we expect to receive.
Therefore, choosing which discount rate to use usually comes down to a matter of making some educated assumptions and referring to past experience. Generally speaking, a low discount rate means the NPV calculation is very conservative, whereas a high discount rate means the NPV is quite optimistic.
A more technical approach to finding the appropriate discount rate is to calculate something known as the Weighted Average Cost of Capital (WACC), but this is beyond the scope of this article.
When calculating NPV, we are left with three possible outcomes:
- NPV > 0: The investment/project adds value, and is, therefore, worth pursuing.
- NPV < 0: The investment/project subtracts value, and is, therefore, not worth pursuing.
- NPV = 0: The investment/project neither adds or subtracts value. It is up to the investor/company to decide if it is still worth pursuing or not.
Let’s go over a simple example.
Imagine you run your own engineering company, and you give the green light to a project that’s expected to produce the following cash flows over the next 4 years:
- Year 1: $50,000
- Year 2: $65,000
- Year 3: $72,000
- Year 4: $80,000
The capital outlay needed at the start of the project is $200,000, and the discount rate is set to 5%. Is this project worth pursuing or not?
Plugging this data into the above equation we can find out:
We end up with an NPV > 0, meaning this project is worth pursuing.
The Time Value of Money and MARR/IRR
Investing is all about giving up money today in anticipation of receiving more in the future, so it’s not surprising that the time value of money is a tool many investors use to assess whether certain investments are worth pursuing or not. Discounted Cash Flow (DCF) Analysis, a common valuation tool used by investors, is based upon the time value of money.
Although FV and NPV are helpful tools, they’re seldom used on their own.
Going back to our FV example from earlier, going from $100 to $103 in three years represents a 3% return on your initial deposit – not bad. However, how does an investor know if this 3% return is cause for celebration or worry?
One way is to implement a Minimum Acceptable Rate of Return (MARR). Prospective investments must meet this minimum criterion in order to be considered by you. Implementing a MARR leaves us with three cases:
- Rate of Return > MARR: The project/investment is worth pursuing.
- Rate of Return < MARR: The project/investment is not worth pursuing.
- Rate of Return = MARR: Indifferent whether the project/investment will be pursued or not.
Earlier in the NPV section, we discussed how a discount rate is chosen before completing the calculation. However, it’s possible to calculate the actual rate of return provided by the cash flows by setting NPV = 0 and then solving for r. The value you get for r when NPV = 0 is known as the Internal Rate of Return (IRR), and is usually compared with the MARR to determine if a project/investment is worth pursuing. There are three outcomes that could happen:
- IRR > MARR: The project/investment is worth pursuing.
- IRR < MARR: The project/investment is not worth pursuing.
- IRR = MARR: Indifferent whether the project/investment will be pursued or not.
Again, the time value of money can be a very helpful tool for investors, but very rarely is it used in isolation – the numbers are only as good as the context they’re placed in.
Wrapping Up
A dollar today may not be worth the same tomorrow. Assuming you put your money to work in some way, then its value will change over time.
The time value of money explores how money is affected when it’s subject to some sort of interest rate/rate of return over time. What’s interesting is that the time value of money can work both ways, that is, it can determine how much a sum of money will be worth in the future, or how much future money you expect to receive will be worth today.
While the time value of money can be of great help to investors, very rarely is it used on its own. Specific criteria such as a MARR or IRR are implemented along with tools such as FV and NPV.
