In this article, we’re going to explore how to calculate Present Value. It’s a pretty in-depth article, so you might want to grab a cup of tea. If you’re just looking for the Present Value formula, we’ve included it just below.
Okay, let’s get into it.
The Present Value is calculated as…
Where represents the Present Value, reflects the Cash flow at time , and reflects the discount rate (aka cost of capital).
Now, if you want to understand how it really works (and why it works), then keep reading.
If the equation above is freaking you out, calm down. We’ve got your back. Stay with us.
What is Present Value?
Firstly, what is Present Value?
Ultimately, the Present Value (or ) is the value of something today in the present.
So it’s the value of future expectations or future cash flow, expressed in today’s terms.
Thus, the Present Value ultimately just reflects how much something is worth right here, right now, in the present. Hence the term “Present” value.
The Present Value Function
The Present Value is ultimately a function of two things, including:
- future expectations, and
Uses of the Present Value
The Present Value is probably the most important concept in Finance. Approximately 70%-80% of concepts in Finance end up relying on the PV in one way or another.
Broadly speaking though, the PV can be used for:
- Investment Appraisal / Capital Budgeting
- Valuation (using the Discounted Cash Flow (DCF) valuation method)
Let’s now briefly consider these two.
Investment Appraisal / Capital Budgeting
We have a separate post on what capital budgeting is if you’re interested in learning more. But one of the most popular investment appraisal tools is the Net Present Value (NPV).
And the Net Present Value is heavily reliant on the PV. We can even see this in the name!
The Net Present Value is just the Present Value, net of the investment.
The value of a company, or a stock, a business, etc, is all fundamentally based on the Present Value of future expectations.
And while the specific cash flows vary depending on the valuation model, for instance:
- Free cash flow
- Free cash flow to Equity (aka Flow to Equity, FTE)
It’s still fundamentally about “discounting” those future cash flows back to the present. Much more on “discounting” further down, but we do also have a separate article on discounting future cash flows if you’re interested.
For now, let’s think about how to calculate Present Value.
How to Calculate Present Value
Calculating the Present Value is actually incredibly straightforward.
Present Value of a Single Cash flow
Let’s start with the simplest case, of estimating the Present Value of a single cash flow.
The formula for PV of a single cash flow is as follows…
Now, as with most things in math, stuff makes so much more sense when we look at examples.
So before we get into explaining the PV formula above, let’s consider an example.
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Present Value Calculation Example #1
Imagine that you want to have $12,500 in your bank account exactly 1 year from today. Assume that your bank pays 5% interest. Assuming that you don’t have anything in your bank account right now, how much would you need to deposit today in order to have $12,500 in your account next year?
To figure this out, as with most things, when you’re working with different timeframes, it’s a good idea to work with the timeline.
Setup a timeline
So in our case, we’re looking at a timeline starting with , so today. That’s where we’re at — right here, right now.
And we’re looking at a timeline or a timeframe of one year, to .
And we’re saying that we want to have exactly $12,500 in our bank account in precisely one year’s time.
We can think of all of this information like this…
The question is, how much do we need to deposit today?
To solve this, we can actually start with something that we already know.
Stick to the fundamentals
At this stage, you should know how to calculate future value. If you don’t, then don’t worry – just have a quick read of our sister article and then come back here.
We’re going to assume that you (at least roughly) know how to calculate the FV.
So we can actually start with something that looks like the future value.
In other words, we can take this timeline and transform it into an equation, and that would look something like this…
Why can we express it like this?
Well, we know (from having learned how to calculate the future value) that the cash flow at time is in fact the cash flow that you had the year before, multiplied by
And we’re raising to the power of 1 because we’re compounding this cash flow over one year.
And so in our specific case, we don’t actually need to work out the future value () because we already know that!
We know that we want the cash to be worth $12,500 in a year’s time. Thus we can say…
We also know that our interest rate is 5% because we know that the bank is paying us 5%. So is 5% or 0.05, meaning we can write…
Now, all we need to figure out is what the is equal to! So we need to rearrange this equation to solve for .
Solve for the Cash Flow
Rearranging the equation gives us…
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Solve this and you’ll find that the exact value we need to deposit is approximately equal to $11,904.76
And because this particular cash flow represents the cash in the present, we can essentially see this as the present value.
Thus, we can say…
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 actually just think about that equation in a little more detail.
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Present Value Formula (for a single cash flow)
Here’s the generalised equation for the present value for single cash flow we saw earlier on…
Components of the PV Formula
Where is the Present Value, is the cash flow at time , and is the discount rate, aka:
- interest rate,
- hurdle rate
- cost of capital
- required rate
- the required rate of return
- the opportunity cost of capital
- we could go on, really…
The point is, there are a whole host of different terms for . But generally speaking, when we talk about it in present value terms, we tend to call it the discount rate.
And it’s called the discount rate because this is the rate that we’re using to discount the future cash flow.
The Discounting Process
“Discounting” is the process of taking a future cash flow expressing it in present terms by “bringing it back” to the present day.
So when you take a cash flow that you expect to earn in the future and you discount it (i.e., when you divide it by for instance) you discount it, or you bring it back to the present terms, expressing the future cash flow in present value terms.
Now, recall that earlier we said that the Present Value is ultimately a function of two things, including:
- future expectations, and
Looking at the equation for the of a single cash flow, you can see precisely how and why the present value is just a function of two things.
So you can see that it’s a function of the future cash flow — that’s what the reflects or represents.
And you can see that it’s also a function of risk — which is captured by (the discount rate).
The discount rate is actually a proxy for risk, and therefore, it’s how we penalise future cash flows for their level of risk.
That’s how we incorporate the risk of not earning future expectations, into our estimate for the present value.
Hopefully, you kind of understand the intuition behind the present value formula.
If you haven’t quite understood it just yet, then please pause for a moment now. Take your time to think about the equation and think about how it is actually a function of two things — future expectations and risk.
And take your time to see how we’re discounting future cash flows to get to the present value. Because this really is very important.
Okay. We’re going to assume that you’re more or less alright. So let’s go ahead now and step things up just a little bit by considering the case with multiple cash flows.
Present Value of Multiple Cash flows
Calculating the Present Value of multiple cash flows is actually very similar to the single cash flow case.
In a sense, you can think of it as calculating the PV of a single cash flow, multiple times.
This will make more sense when you see it work in an example, so let’s go ahead and do that now.
Present Value Calculation Example #2
Consider the following cash flow stream. You expect to earn $10,000; $15,000; and $18,000 in 1, 2, and 3 years’ times respectively.
Assume the discount rate is 10%.
What is the Present Value of these cash flows?
As always, because we’re working with timeframes over here, it’s a good idea to start with the timeline.
Setup a timeline
In our case, we’re looking at a 3-year timeframe, starting from time (which is where we’re at), and going all the way up to and including time (or 3 years later).
And this is the same as expressing it as times and .
It’s just a different way of writing it. But the point is that we’ve got three different timeframes.
And at , we’ve got a cash flow of $10,000; at , we’ve got $15,000. And finally, at , we’ve got a cash flow of $18,000.
The approach to discount these 3 cash flows is actually identical to the case of the single cash flow we saw earlier.
Stick to the fundamentals
We know that for a single cash flow, the present value is equal to…
Thus, if we think of these 3 cash flows as 3 separate and individual cash flows, then we can say that the first cash flow can be discounted as…
The cash flow at time can be discounted as…
Note that we’re raising to the power of 2 here instead of 1. That’s because this particular cash flow needs to be discounted over 2 years, to bring it back to the present.
Put differently, we need to discount this cash flow over 2 years in order to express it in present terms.
Finally, the cash flow at time can be discounted as…
Solve for the PV
Combining all 3 individual discounted cash flows we have…
Solving for each item gives us…
Finally, we just need to add the 3 numbers to get our answer!
And now that we know how to estimate the Present Value of multiple cash flows, we can think about what the Present Value formula actually looks like.
So we can say that the formula for the Present Value of multiple cash flows is…
And because the process of discounting the cash flows is identical across all cash flows, we can actually summarise this further as…
In a nutshell, then, we can say that the Present Value is nothing but the sum of the discounted future cash flows.
And with that, you now know how to calculate Present Value!
Hopefully, all of this makes sense.
In summary, you learned that the Present Value or is the value of something, today, in the present.
So it’s the value of something expressed in today’s terms or in present terms.
Furthermore, you learned that the Present Value can be calculated by using this particular formula…
In a nutshell, it’s just sum of the discounted cash flows.
If any part of the article isn’t quite clear, please read it again.
The Present Value is an incredibly important concept – it’s what approximately 70-80% of Finance is based on in one way or another.
If the math in this article was a bit too advanced for you, then we strongly recommend exploring and enrolling on our Financial Math Primer course (see below).
The course is designed for the absolute beginner, and will really help you finally get the mathematical foundation you’ve always needed.
Finally get the mathematical foundation you’ve always needed.
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