In this article, we’re going to explore how to calculate future value. Let’s get into it.
The Future Value formula (for a single cash flow) is:
Where reflects the Future Value, represents the cash flow at time , and reflects the discount rate (aka cost of capital, hurdle rate).
If this equation is freaking you out, or if you want to know how it really works (and why it works that way), then keep reading!
What is Future Value
Firstly, what is Future value?
Ultimately, Future value is the value of something in the future.
That’s literally all it is.
And we could be talking about the Future Value for a company, or a stock, or your bank balance, or the value of your house.
It’s just the value of something as at some point in the future. So as at some particular date in the future.
How to calculate Future Value
Calculating the Future Value is actually incredibly straightforward.
Let’s see what this looks like with an example.
Future Value Example #1
Imagine that you were to deposit $10,000 into a savings account today, and suppose that the bank pays you an annual interest rate of 5%.
If that’s the case, how much will your bank balance be worth in a year’s time?
In other words, how much money will you have in your bank account in precisely one year’s time?
As with most things when you’re dealing with timeframes, it’s a good idea to work with or create a timeline.
Start with a timeline
So if we were to think about this question / problem in a sort of timeframe, then it might look like this…
We’re here at time , so today. And we’re looking at a one-year timeframe going up to .
So is precisely one year from today, and we’re depositing $10,000 into our bank, which is going to pay us 5% interest for the whole year.
Stick to the fundamentals
And then we’re asking where, how much is the deposit going to be worth in exactly a year’s time?
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Of course, it’s literally just going to be worth what we had at the beginning, plus the interest that we earn over the course of the year!
(Obviously assuming we don’t withdraw the money and we don’t put in any extra money into the account).
In other words, we’ll have…
Here represents the cash we’ll have in our bank account in exactly 1 year’s time, at .
Plugging in our numbers – the fact that we have $10,000 available to deposit, and the bank is going to pay us 5% interest, we have…
And so we’re going to end up with a balance of $10,500…
Because we earn $500 in interest (being 5% of $10,000) which is added to the $10,000 that we had to begin with.
So our deposit will be worth $10,500 in exactly a year’s time, if we deposit $10,000 today, and the bank pays us 5% interest over the course the year.
Now, the Future Value here is nothing but $10,500, because that’s the value in the future!
So we’ve calculated the future value. And hopefully you can see that it really is very simple and really straightforward.
Formal notations for Future Value
Now that you’ve understood how to calculate future value, let’s think about how we can write this out in slightly more formal notations.
Remember, we start with the fact that the cash at time is going to be the cash that you had to begin with (cash at time ), plus the interest that you earn over that time period.
Now, generally speaking, when we want to denote “cash”, or strictly, when we’re trying to denote the “cash flow”, we tend to use the notation . Strictly speaking, cash is not the same as cash flow, but we’re going to use the notation to refer to cash, just for simplicity.
And so rather than writing , we can write this as .
Thus, that’s going to denote or reflect the cash (strictly, the cash flow, hence the in ).
Strictly then, it’s going to denote the cash flow at time , and that’s going to be equal to…
Note that we’re using the letter to denote the interest rate.
Some people prefer to use instead, to denote the interest rate.
It’s just a personal preference; many people in finance just work with to denote the interest rate, aka:
- required rate of return
- cost of capital
- discount rate
They’re all essentially the same thing.
And so, just to try and minimise complications, and to minimise notations and jargons, we’re working with to reflect the interest rate. Note that we use the same notation for our post on How to Calculate Present Value.
Now, implementing a little bit of factorisation simplifies the equation above to this…
And now it’s just a simple case of plugging in our numbers again:
And again, this is of course our future value, so we can substitute for and write…
Thus, the future value of our deposit is $10,500.
Does that make sense?
If any part of the example is not quite clear, please read it again before moving on any further.
We’re going to assume that you’re more or less alright. So let’s step things up a little bit now, and consider another example.
Future Value Example #2
A similar sort of setup…
You’re still putting in an of $10,000 into your savings account, and the bank is still going to pay you an of 5%.
But this time, we’re explicitly telling you that the interest is compounded annually.
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Now, in case you’re not familiar with “compounding interest”, it just means that you’re going to earn interest on whatever the balance is at a given point in time.
Or put differently, you’re going to earn interest on the “principal amount” (i.e., the amount that you deposited), as well as the interest that you’ve earned, and that has been paid into your bank account.
Again, importantly, with compound interest, you earn interest on the principal (which is the amount that you deposited), as well as the interest.
In simple interest – the opposite of compound interest – you only earn interest on the deposit. In other words, you don’t earn interest on interest.
Thus, if this question said it was a “simple interest”, then you would earn $500 every year; as simple as that.
Because it’s compound interest however, you’re going to earn $500 in the first year, but in the second year, you’re going to earn more than $500.
And specifically in this case, you’re going to earn $525.
That’s because again, you’re going to earn interest on the principal, as well as the interest.
So given the initial deposit of $10,000 and the 5% annual compounded interest, what we’re interested in finding out is how much cash you’re going to have in your bank account in three years time.
In other words, we’re interested in the Future Value as of three years later. Put differently, we’re looking at the given a three year .
As always, it’s a good idea to draw out a timeline because we’re working with a timeframe here.
Start with a timeline
So in our case, we could plot it out like this…
We’re looking at three different timeframes: (one year later), (two years later), and (three years later)
The one we’re particularly interested in is , because that’s three years later, and that’s the Future Value we’re trying to get to.
Put differently, we’re depositing $10,000 today, we’re going to earn 5% interest every year for the next three years, compounded annually. And we’re interested in figuring out what our deposit will be worth precisely three years down the line.
How do we do this?
Well, the approach you can take is actually quite similar to the single year case!
Stick to the fundamentals
We know that the cash at time is the cash at time , plus any interest that you earn, right? Mathematically, we know…
Which of course is identical to…
The key point is that the interest that you earn is based on the:
- interest rate, and
- amount of cash that you have available at the start of the period.
Thus, if the cash at time is the cash at time , plus any interest that you earned for the cash balance as of that time, then the cash at time must be equal to the cash at time , plus the interest that you’ve earned based on the cash at time . Mathematically…
All we’ve done is we’ve replaced with .
Hopefully you can see that the method, the approach we’re taking, is in fact identical to the one year case.
Now of course, we’re interested in , not in . So guess what is going to be equal to!
It’s a good idea to just pause reading right now, and try working this out on your own.
We’re going to assume you did that. So let’s consider it together now.
Naturally, we use the same sort of approach as before. Thus, we have…
Simplify and solve!
Now, given that it is in fact the same process for , , and , we can actually rewrite the equation in a single line, as opposed to having these multiple lines, in order to get to .
What does that look like? Well, we can rewrite it like this…
And this is equivalent to us writing…
We’re raising to the power of 3 here because we’ve got (i.e., we’ve got something being multiplied by itself three times, which is the same as taking that “something” and raising it to the power of 3.
That’s the mathematical rule, which we discuss in great detail in our course on Financial Math Primer for Absolute Beginners. [INSERT LINK TO RELATED COURSE]
The way we think about this in finance, is we want to get to the Future Value, three years down the line.
We want to compound the cashflow over three years. And thus we’re looking at a three year compounding period.
And that’s why we’re raising it to the power of three.
If we now just plug in our numbers, then we have…
Solve for that, and you’ll find that the is equal to $11,576.25, or approximately $11,576.
And this in fact is the Future Value.
Therefore, if we now think about a generalised equation for the Future Value of a single cash flow, it can look like this…
refers to the Future Value, reflects the cash flow at the time
And reflects the interest rate, aka rate of return. And it’s also called the discount rate. The reason for that will actually become far more apparent once you learn how to calculate the present value. But in the context of Future Value, you can just think of it as an interest rate.
Thus, is the rate that you’re compounding the cash flows by, in order to get to the Future Value.
In summary, you learnt that the Future Value, or reflects the value of something in the future.
You learnt that the Future Value can be calculated by…
Hopefully, all of this makes sense. If any part of this article is not quite clear, please do watch it again before moving on any further.
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