Finally! A Simple Explanation of the Present Value

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The Present Value is one of the many (beautiful) concepts in Finance.

Unfortunately a lot of students dread it thinking it’s too complicated. And perhaps that’s because their first sight of it looks something like this:

\[
\sum_{t=1}^{n}\frac{CF_{t}}{(1+r)^{t}}=\frac{CF_{1}}{(1+r)^{1}}+\frac{CF_{2}%
}{(1+r)^{2}}+\frac{CF_{3}}{(1+r)^{3}}+…+\frac{CF_{n}}{(1+r)^{n}}
\]

Which in my opinion, does indeed look beautiful, but can seem quite overwhelming to those who aren’t the biggest fans of maths. So here’s a simple (and fun) explanation of the Present Value concept!

As the name suggests, the Present Value is the value (of something) in the present (i.e right here, right now).

It’s based on future expectations, just like every single thing you do. Literally.

Every single thing we all do in the present is based on our expectations of the future. For example, here are some things that most of you do at some point or another…

Present Action Future Expectation
Go to the cinema To watch a good movie and relax
Go clubbing To have a good time and socialise
Go for a walk in the park To feel rejuvenated
Go to the gym To keep fit and healthy
Drink a glass of water To quench thirst
Eat food To stop feeling hungry

You don’t go to the cinema with the expectation of watching a bad movie – do you? I mean, you could end up at a bad movie, but surely you don’t go to the cinema knowing that it’s a bad movie! You go with the expectation of watching a good movie, having a good time, and relaxing, even though that may not actually happen. You take a chance – a ‘risk’.

You go clubbing with the expectation of having a good time and socialising – you may have a really bad time and not socialise at all, but you don’t go clubbing with that expectation. You go with the expectation of having a good time. You take a chance. You take a risk.

Sometimes the risks pay off, other times they don’t. But the important thing is that we consider the risks and rewards of the future when making decisions in the present.

Finance works on exactly this very principle…

The value of anything and everything is always, always, always based on the Present Value of Future Expectations.

But is the future guaranteed? Well of course not!

With the future comes uncertainty and risk.

In the beauty that is Finance, we want to quantify those risks and incorporate them into our estimation of the value.

So what sort of risks are we talking about?

In the context of investments, there are broadly two kinds:

  1. Those that are within the control of the firm, and
  2. Those that aren’t (i.e. they’re beyond the control of the firm)

And of course, there’s a whole host of different terms all of which describe these two kinds of risks. Let’s consider both kinds.

Risks that are outside the firm’s control

These are risks that no firm / company can control by itself. Typical examples include:

  • Inflation (which is a general increase in prices)
  • Deflation (which is a general decrease in prices)
  • Recession (which is when the economy’s facing negative growth and generally not doing well)
  • Depression (which is when the economy’s got major problems)
  • Boom (this is a risk for companies that thrive in a recession)
  • Changes in interest rates (a change either way (i.e. increase or decrease) can have significant positive and negative impacts).

Oh and since we really like jargon, here’s a list of different names/terms for exactly the same thing (risks that are outside the firm’s control):

  • Systematic risk
  • Non-diversifiable risk
  • Market risk
  • Market specific risk

Risks that are within the firm’s control

These are risks that most firms will, to a considerable extent, be able to control / impact / influence. Examples would typically include:

  • Idiotic managers (we’ve used this term for a reason, you’ll see why in a bit).
  • Poor decision making
  • Default (aka bankruptcy, insolvency)
  • Death of the founder / CEO (sure, they can’t make the founder immortal, but they can put in steps to ensure this doesn’t adversely affect the firm).
  • Scandals (accounting, ethical)

You’re going to love the different terms we use to define this risk:

  • Unsystematic risk
  • Unique risk
  • Firm specific risk
  • Idiosyncratic risk (now you see why we referred to ‘idio’tic managers)
  • Diversifiable risk

Every investment opportunity you’re evaluating must incorporate both sets of risks.

We want to incorporate all of these risks when estimating the value. Dear student, we do this beautifully, by using the ‘discount rate’.

The discount rate (aka hurdle rate, cost of capital opportunity cost of capital, required rate, “r”) beautifully incorporates all of these risks. We’ll be doing a blog post later on that will show you exactly how all of these risks are incorporated into the discount rate.

For now though, you need to know this one fundamental law:

Risk and Value are inversely related to each other, such that…

  • The Higher The Risk, The LOWER The Value
  • The Lower The Risk, The HIGHER The Value

You don’t need to memorise that. Please don’t. It really is very logical and simple. If you had 2 investment opportunities, which we’ll conveniently call ‘A’ and ‘B’, and they looked like this:

A

B

Expected Return

10%

10%

Risk

24%

12%

Price

£100

?

A is twice as risky as B. If A is priced at (valued at) £100, would you be willing to pay…

  • More for B
  • Less for B
  • Exactly the same as A?

You’d of course be willing to pay more! If it’s less risky, it automatically becomes more valuable. And on the flip side, if it’s more risky, then it’s worth less (less valuable).

Money Loses Value Over Time. Here’s Why.

Money today is worth more than money in the future because having it today:

  • Allows you to do whatever you want with it right now
  • Allows you to invest your money and grow it
  • Prevents you from suffering a loss in the value of money because of the effect of inflation.

Inflation decreases the value of money over time.

Think back to when your parents and /or grandparents said something along the lines of ‘Back in our days, things were so much cheaper!’.

Here’s the fact of the matter – they weren’t exactly cheaper. It’s just that money lost its value over time. Consider a simple example.

Let’s say your favourite fruit is strawberries and that you can get 1 box for £1. Further let’s say that you have £1,000 today. Given your love for strawberries, you can (and you actually do) buy 1,000 boxes of strawberries with your £1,000.

Now let’s say the general rate of inflation is 5%. That would imply that prices of goods, on average, are increasing by 5%. Take yourself a year into the future and the price of a box of strawberries would’ve increased by 5%.

That would mean that the price per box of strawberries would be

\[
\pounds 1+\pounds 1\text{x}5\%=\pounds 1(1+5\%)=\pounds 1.05
\]

If you still only had £1,000 you’d be able to buy

\[
\text{{}Number of boxes of strawberries}=\frac{\pounds 1,000}{\pounds 1.05}%
=952
\]

That’s 48 fewer boxes! Your money essentially lost 4.8% of its value over 1 year.

Imagine how much value it would lose over 10 years, 20 years, 50 years.

We refer to the fact that money loses value over time and estimate the impact of that loss as the ‘Time Value of Money

Let’s look at an example!

Consider an investment that offers the following cashflows

Year 0
(Right Now)
Year 1 Year 2 Year 3 Year 4 Year 5
Cashflows  £0 £10,000 £10,000 £10,000 £10,000 £10,000

Assuming inflation is greater than 0%, we know that…

The £10,000 in Year 1 is worth more than the £10,000 in Year 2, which is worth more than the £10,000 in Year 3.

The £10,000 in Year 3 is worth more than the £10,000 in Year 4, which is worth more than the £10,000 in Year 5 🙂

In addition to that, all of the cashflows from Year 1 through until Year 5 face the market specific and firm specific risks.

To incorporate all of these risks (including inflation), and to see how much those cashflows are worth to us today (given all of those risks), we’re going to DISCOUNT them back to the present (i.e. to right here, right now).

We’re going to discount the cashflow in year 1 (£10,000) by the discount rate. We do this by dividing the cashflow (£10,000) by (1+r) raised to the power of t or n, both of which refer to the time at which the cashflow occurs.

This will look like this:

\[
\text{Present Value of the Cashflow in Year 1}=\frac{\pounds 10,000}{%
(1+r)^{1}}
\]

r is simply the discount rate. (1+r) is raised to the power of 1 because we’re discounting the cashflow back by 1 YEAR.

The cashflow in Year 2 will therefore be discounted over 2 YEARS to the ‘bring’ it back to the present value. This will take the form:

\[
\text{Present Value of the Cashflow in Year 2}=\frac{\pounds 10,000}{%
(1+r)^{2}}
\]

The cashflows in Years 3, 4, and 5 will be discounted over 3, 4, and 5 YEARS respectively and take the form:

\[
\text{Present Value of the Cashflow in Year 3}=\frac{\pounds 10,000}{%
(1+r)^{3}}
\]
\[
\text{Present Value of the Cashflow in Year 4}=\frac{\pounds 10,000}{%
(1+r)^{4}}
\]
\[
\text{Present Value of the Cashflow in Year 5}=\frac{\pounds 10,000}{%
(1+r)^{5}}
\]

More generally, we can identify each individual £10,000 as a “cashflow”, and abbreviate it as CF.

Our equation would then take the form:

\[
\text{PV of a Cashflow occurring in Year t}=\frac{CF_{t}}{(1+r)^{t}}
\]

The Year ‘t’ could refer to any year (in our case, 1, 2, 3, 4, or 5). If we were discounting the Cashflow in Year 4, we’d write it like this:

\[
\text{PV of the Cashflow in Year 4}=\frac{CF_{4}}{(1+r)^{4}}
\]

The Present Value of all 5 future cashflows would look like this:

\[
\text{PV of Future Cashflows}=\frac{\pounds 10,000}{(1+r)^{1}}+\frac{\pounds %
10,000}{(1+r)^{2}}+\frac{\pounds 10,000}{(1+r)^{3}}+\frac{\pounds 10,000}{%
(1+r)^{4}}+\frac{\pounds 10,000}{(1+r)^{5}}
\]

Or in letter-form (i.e. a more generalised form) would look like:

\[
\text{PV of Future Cashflows}=\frac{CF_{1}}{(1+r)^{1}}+\frac{CF_{2}}{%
(1+r)^{2}}+\frac{CF_{3}}{(1+r)^{3}}+\frac{CF_{4}}{(1+r)^{4}}+\frac{CF_{5}}{%
(1+r)^{5}}
\]

We could summarise that long equation like this:

\[
\text{PV of Future Cashflows}=\sum_{t=1}^{5}\frac{CF_{t}}{(1+r)^{t}}
\]

We’d read the above equation like this:

We’re going to add up all the values of (CF/(1+r)^t) starting from t = 1 (the first year) up until the t = 5 (the last year). In our case, n would be equal to 5 (because we have 5 sets of ‘observations’ or data).

We refer to ∑ as ‘sigma’ or ‘sigma summation’.

More generally, estimating the Present Value of n number of Future Cashflows would look like this:

\[
\text{PV}=\frac{CF_{1}}{(1+r)^{1}}+\frac{CF_{2}}{(1+r)^{2}}+\frac{CF_{3}}{%
(1+r)^{3}}+…+\frac{CF_{n}}{(1+r)^{n}}
\]

We could generalise and summarise that like this:

\[
\text{PV}=\sum_{t=1}^{n}\frac{CF_{t}}{(1+r)^{t}}
\]

We’d read that equation like this:

We’re going to add up all the values of (CF/(1+r)^t) starting from t = 1 (the first year or period) up until the t = n (the last year or the nth period). n could be any year or period, including infinity! That would make it what we call a Perpetuity. Don’t worry, we’ll be looking at the PV of a Perpetuity in a bit too!

Applying the Discount Rate

Let’s say that we did the maths and found that the appropriate discount rate is 8%. Discounting the cashflow in Year 1 would then take the form:

\[
\text{PV of the cashflow in Year 1}=\frac{\pounds 10,000}{(1+0.08)^{1}}
\]

\[
\text{PV of the cashflow in Year 1}=\frac{\pounds 10,000}{1.08^{1}}
\]

Solving for that would give us…
\[
\text{PV of the cashflow in Year 1}=\pounds 9,259.26
\]

You could interpret that as follows:

Assuming the appropriate discount rate is indeed 8%, we should be indifferent between receiving £9,259.26 today, or £10,000 in exactly 1 year’s time.

Let’s consider the 2nd cashflow (the one that occurs in Year 2). This would be discounted to the present by…

\[
\text{PV of the cashflow in Year 2}=\frac{\pounds 10,000}{(1+0.08)^{2}}
\]
\[
\text{PV of the cashflow in Year 2}=\frac{\pounds 10,000}{1.08^{2}}
\]
\[
\text{PV of the cashflow in Year 2}=\pounds 8,573.39
\]

This could be interpreted by saying that assuming the appropriate discount rate is indeed 8%, we should be indifferent between accepting £8,573.39 today or £10,000 in two year’s time.

We could now combine the cashflows of Years 1 and 2 and estimate the Present Value of both cashflows like this:

\[
\text{PV of the cashflows in Years 1 and 2}=\frac{\pounds 10,000}{%
(1+0.08)^{1}}+\frac{\pounds 10,000}{(1+0.08)^{2}}
\]
\[
\text{PV of the cashflows in Years 1 and 2}=\frac{\pounds 10,000}{1.08^{1}}+%
\frac{\pounds 10,000}{(1.08)^{2}}
\]
\[
\text{PV of the cashflows in Years 1 and 2}=\pounds 9,259.26+\pounds 8,573.39
\]
\[
\text{PV of the cashflows in Years 1 and 2}=\pounds 17,832.65
\]
We could interpret that by saying that if the appropriate discount rate is indeed 8%, we should be indifferent between receiving £17,832.65 today or £10,000 exactly 1 year from now AND another £10,000 exactly 2 years from today!

Now let’s look at the PV of all our cashflows.

\[
\text{PV of the cashflows occurring in Years 1 through to Year 5}=~\text{“PV”%
}
\]
\[
\text{PV}=\sum_{t=1}^{5}\frac{CF_{t}}{(1+r)^{t}}
\]

The summarised equation would ‘open up’ in the form…

\[
\text{PV}=\frac{CF_{1}}{(1+r)^{1}}+\frac{CF_{2}}{(1+r)^{2}}+\frac{CF_{3}}{%
(1+r)^{3}}+…+\frac{CF_{5}}{(1+r)^{5}}
\]
\[
\text{PV}=\frac{\pounds 10,000}{(1+0.08)^{1}}+\frac{\pounds 10,000}{%
(1+0.08)^{2}}+\frac{\pounds 10,000}{(1+0.08)^{3}}+\frac{\pounds 10,000}{%
(1+0.08)^{4}}+\frac{\pounds 10,000}{(1+0.08)^{5}}
\]

This could also be written as…

\[
\text{PV}=\frac{\pounds 10,000}{(1.08)^{1}}+\frac{\pounds 10,000}{(1.08)^{2}}%
+\frac{\pounds 10,000}{(1.08)^{3}}+\frac{\pounds 10,000}{(1.08)^{4}}+\frac{%
\pounds 10,000}{(1.08)^{5}}
\]

Solving for each of the individual cashflows would yield the following Present Values…

\[
\text{PV}=\pounds 9,259.26+\pounds 8,573.39+7,938.32+7,350.30+6,805.83
\]

Adding up all the individual discounted cashflows gives us…

\[
\text{PV}=\pounds 39,927.10
\]

In other words, assuming the appropriate discount rate is 8%, we should be indifferent between receiving / accepting £39,927.10 today or receiving / accepting £10,000 every year for 5 years (starting 1 year later).

While that’s all well and good, some of you are perhaps (and rightly so) thinking that this method is quite tedious, long, and painful because of the amount of repetitive calculations one has to do. Ah dear student, it’s time to learn another beautiful aspect of Finance!

Generally in Finance, there’s almost always an easier way to do something (HINT: Annuities!)

In this case, because the cashflows remained constant throughout the project/period (at £10,000 per year, every year, for 5 years) and the discount remained constant throughout the period too (at 8%), and that the period of the project was finite (i.e. 5 years), there is indeed a much easier way to compute the Present Value.

At this stage you should be well excited to find out how exactly you can do that!

We’re first going to formally set out the 3 conditions that MUST BE MET in order for us to use this simple method. They are…

  1. Condition 1: The cashflows remain the same (i.e. remain constant) throughout the entire project’s period,
  2. Condition 2: The discount rate remains constant throughout the project’s period, AND
  3. Condition 3: The period of the project is defined (i.e. that it’s finite)

If and ONLY if all 3 conditions are met, we have what we call an ‘Annuity’. We can then apply the annuity formula, which makes life so much easier!

The PV of an Annuity takes the form:

\[
\text{PV of an Annuity}=\frac{CF}{r}\left( 1-\frac{1}{(1+r)^{n}}\right)
\]

If you’re feeling overwhelmed by the look of that (beautiful) equation, please don’t. We promise it’s really easy to understand and apply.

The formula is the result of the annuity taking the form of a ‘geometric progression’. If that freaks you out, breathe! Breathe! If it doesn’t, and you’re interested in seeing why it works, we’ll be posting the ‘Proof of the PV of an Annuity’ soon! For now…

Let’s see what applying the numbers from our equation would look like.

Firstly, we want to test to see whether all 3 conditions of the annuity are met. In this case, we know all 3 conditions are met, but it’s a good idea to conduct that test for all your exam questions.

  1. Condition 1: The cashflows remain the same (i.e. remain constant) throughout the entire project’s period.

This holds, because our cashflows equate to £10,000 every year throughout the 5 year period.

  1. Condition 2: The discount rate remains constant throughout the project’s period

This holds too, since the discount remains constant at 8% throughout the period.

  1. Condition 3: The period of the project is defined (i.e. that it’s finite)

This condition holds as well, since our project / investment’s period is 5 years.

Surprise, Surprise! All 3 conditions are met, so we do indeed have an Annuity.

Now we can apply our numbers to the equation for the PV of an Annuity.

\[
\text{PV of an Annuity}=\frac{CF}{r}\left( 1-\frac{1}{(1+r)^{n}}\right)
\]

\[
\text{PV of an Annuity}=\frac{\pounds 10,000}{0.08}\left( 1-\frac{1}{%
(1+0.08)^{5}}\right)
\]

**If you know how to solve this from a mathematical standpoint, scroll down to the next section by**

We solve for this equation using the rules of ‘BODMAS’ which takes the form:

B: Brackets
O: Of (orders of Powers and Square roots, cube roots, nth roots)
D: Division
M: Multiplication
A: Addition
S: Subtraction

Applying BODMAS would mean we first solve for the part inside the bracket, i.e. solve for…

\[
\left( 1-\frac{1}{(1+0.08)^{5}}\right)
\]

We’d need to solve for the order of the power (Of), i.e. solve for

\[
\left( 1+0.08\right) ^{5}
\]

That equates to 1.4693280768

So our equation would not look like:

\[
\left( 1-\frac{1}{1.4693280768}\right)
\]

That equates to 0.680583197.

Now there’s no multiplication nor addition, so we subtract 1 from 0.680583197

1 – 0.680583197 equates to 0.319416803

So our equation went from:

\[
\text{PV of an Annuity}=\frac{\pounds 10,000}{0.08}\left( 1-\frac{1}{%
(1+0.08)^{5}}\right)
\]

To this…

\[
\text{PV of an Annuity}=\frac{\pounds 10,000}{0.08}\text{x }0.319416803
\]

Dividing £10,000 by 0.08 yields £125,000 so our equation takes the form:

\[
\text{PV of an Annuity}=\pounds 125,000\text{ x }0.319416803
\]

\[
\text{PV of our Annuity}=\pounds 39,927.10
\]

*****************Students who know how to solve join back here*******************

\[
\text{PV of an Annuity}=\frac{\pounds 10,000}{0.08}\left( 1-\frac{1}{%
(1+0.08)^{5}}\right)
\]

\[
\text{PV of our Annuity}=\pounds 39,927.10
\]

Which of course is exactly the same result we got when we took the long, tedious, and painful approach of discounting the individual cashflows one by one.

The equation and approach remain unchanged for any sort of Annuity, as long as all 3 conditions are met.

Let’s work together on another example.

Welcoming Cat Inc is evaluating an investment that offers £45,000 every year for 18 years (starting one year from now). The company’s appropriate discount rate is 12.5%. Estimate the Present Value of the investment.

We want to start by collecting our ‘raw data’.

  • Cashflows from year 1 until Year 18 = £45,000
  • Discount rate = 12.5%
  • n = 18 years.

We now want to test whether all 3 conditions of the Annuity are met.

  • Condition 1: The cashflows remain the same (i.e. remain constant) throughout the entire project’s period.

This holds, because our cashflows equate to £45,000 every year throughout the 18 year period.

  • Condition 2: The discount rate remains constant throughout the project’s period

This holds too, since the discount remains constant at 12.5% throughout the period.

  • Condition 3: The period of the project is defined (i.e. that it’s finite)

This condition hold as well, since our project / investment’s period is 18 years.

 Wahayyy! All 3 conditions are met, so we do indeed have an Annuity.

Recall that the Present Value of an Annuity takes the form:

\[
\text{PV of an Annuity}=\frac{CF}{r}\left( 1-\frac{1}{(1+r)^{n}}\right)
\]

Inputting our numbers into the equation takes the form:

\[
\text{PV of an Annuity}=\frac{\£45,000}{0.125}\left( 1-\frac{1}{%
(1+0.125)^{18}}\right)
\]

Solving for this using the law of BODMAS yields a PV equal to £316,792.70

What if cashflows occur every year, FOREVER?

Oh you inquisitive student – I love thy questions!

While in reality it is quite unlikely to encounter an investment that pays a fixed amount of money every year, forever, let’s consider if that were to be true.

In that case, our cashflows would look something like this…

\[
CF_{1}+CF_{2}+CF_{3}+…\rightarrow \infty
\]

If we were to discount these cashflows individually to the present, it would look something like this…

\[
\text{PV}=\frac{CF_{1}}{(1+r)^{1}}+\frac{CF_{2}}{(1+r)^{2}}+\frac{CF_{3}}{%
(1+r)^{3}}+~…~\rightarrow \infty
\]

This would essentially (rather, literally) become…

\[
\text{PV}=\frac{CF}{r}
\]

But I know you’re almost certainly going, ‘Huh?!’ How on earth did we go from this…

\[
\text{PV}=\frac{CF_{1}}{(1+r)^{1}}+\frac{CF_{2}}{(1+r)^{2}}+\frac{CF_{3}}{%
(1+r)^{3}}+~…~\rightarrow \infty
\]

To this…

\[
\text{PV}=\frac{CF}{r}
\]

?!?!?

Loving your curiosity dear student! Let’s break the rules a little bit 😛

If we dared to use the Annuity equation we just learned for our perpetual stream of cashflows, it would look like this…

\[
\text{PV}=\frac{CF}{r}\left( 1-\frac{1}{(1+r)^{\infty }}\right)
\]

Notice that we’ve raised (1+r) to the power of infinity (since n is equal to infinity, because the cashflows occur every year, forever!). Now of course, your calculator doesn’t have a function for infinity. Go on, check it – I dare you 😛

But we know that infinity is a pretty big number. Way bigger than Googol (which is 1 x 10100)

Googol = 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

That’s 1 followed by 100 zeroes, before any decimal places.

I’m not making this up. I would not make this up even if I could, and since I could not make it up, I clearly would not make it up either!

But here’s the point… Infinity is even greater than Googol. But let’s consider if we did raise (1+r) by Googol – a proxy for infinity, if you will.

Our equation would look like this…

\[
\text{PV}=\frac{CF}{r}\left( 1-\frac{1}{(1+r)^{Googol}}\right)
\]

\[
\text{PV}=\frac{CF}{r}\left( 1-\frac{1}{(1+r)^{10^{100}}}\right)
\]

That would make the denominator even bigger than Googol. Yikes.

But then we’re dividing 1 by that number. What happens there?

When we divide 1 by 10, we get 0.1
1 divided by 100 = 0.01
1 divided by 1,000 = 0.001
1 divided by 10,000 = 0.0001
1 divided by 100,000 = 0.00001

Notice that we’re getting closer and closer to 0?

We can therefore say that 1 divided by Googol = 1 divided by 10100 = 0. And hence, we can also say that…

\[
\frac{1}{(1+r)^{10^{100}}}=0
\]

So coming back to our equation…

\[
\text{PV}=\frac{CF}{r}\left( 1-\frac{1}{(1+r)^{Googol}}\right)
\]

\[
\text{PV}=\frac{CF}{r}\left( 1-\frac{1}{(1+r)^{10^{100}}}\right)
\]

1 minus 0 is of course 1, so our equation becomes…

\[
\text{PV}=\frac{CF}{r}(1-0)
\]

\[
\text{PV}=\frac{CF}{r}\text{x}~1
\]

Since anything multiplied by 1 is equal to itself, our equation becomes…

\[
\text{PV}=\frac{CF}{r}=\text{ Present Value of a Perpetuity}
\]

Fascinating, isn’t it?! 🙂