Understanding Time Value of Money

Time Value of Money (TVM) describes that money in hand today is worth more than the same amount at a future date due to its present earning potential. This concept becomes the building block for further advanced finance topics such as NPV and IRR.

Let’s say you have money with you which are of the following denomination – $100, €100 and ₹100. If I ask you how much net money you have, what would your answer be? You can’t answer that question just like that. You would first have to convert the different currencies into one common currency (as per the prevalent currency exchange rate at that time) and then only you will be able to give a conclusive answer. In this case, if you want the answer in Rupees, you would first have to convert the Euro and Dollar currency into the Rupee equivalent. Once everything is in rupees, you will add them up to get the net money in hand.

Converting Euro and Dollar into Rupee helps us get an apple to apple comparison (in this case addition). With that out of the way, we can apple-y this knowledge to understand Time Value of Money by looking at the three rules of time travel.

Rule 1 – Combining and Comparing Money

Let’s consider another situation now – say you write a book and the publisher agrees to give you royalties of $100 per year for the next three years (let’s call them cashflows). How much money will you have at the end of the third year? The answer is $300, right? Well, not really. This situation is similar to the previous example where different currencies that you possessed had different intrinsic values. Similarly, the cashflows in different periods have different intrinsic values despite being in the same currency unit.

Why is it different?

It differs due to various reasons such as inflation (increase in prices of commodities year on year), interest rate (earning interest on money invested or interest amount to be paid on loans), etc. Thus, a dollar today is not the same as a dollar in the past or the future.

And this makes our Rule number 1 – we can’t add/subtract cash flows received in different periods, we would first have to bring them to a common ground of comparison. In the case of the currency example, we used the exchange rate to reach a common base currency. Similarly, all we need now is an exchange rate for time which helps bring cash flows to a common period of time, as we did for comparing different currencies (more on this later). Is there a way to actually do this? Yes, there are two actually. One looks at moving forward in time and the other going back in time. I’d take up time travel now, but there’s no future in it!

The Timeline

Before we delve further into how cash flows can be transformed, we need to understand the concept of a timeline. When we pay or receive cash flows in different time periods, we call it a stream of cash flows. It’s a good practice to draw these cash flows on a timeline. To illustrate this, let’s assume you lent $300 to your friend and he agrees to pay you back the money in three equal installments at the end of 1^st, 2^nd, and 3^rd year.

0 represents the present day, 1 represents one year later, 2 represents two years later, and so on. The cash mentioned below the year, states the cash inflow or outflow. When you are paying cash, it is going out of your pocket and hence, known as cash outflow and when you receive cash, it is coming into your pocket and thus, is known as cash inflow. Cash outflows are represented with a negative sign or by using parenthesis around the cash outflow. In this example, there is a cash outflow of $300 at time period 0, and $100 is received by the end of 1^st, 2^nd, and 3^rd year.

Note – End of 1^st year is the same as starting of 2^nd year.

Now that we know how to create a timeline, let’s look at how cash flows can be transformed to a future date.

Rule 2 – Taking it forward

Let’s suppose you have an excess of $100 with you (lucky you!) and you wish to invest it somewhere. When you invest your money somewhere, you would expect to gain a premium on your invested amount. You decide to deposit the amount in a bank that promises to pay you an interest of 10% per annum. What the bank is essentially doing is that it is taking the excess money that you decided to invest and lending it to someone who needs it (organizations for investment in property, plant, equipment, or individuals taking home loans, car loans, etc.).

In return, the bank promises to increase the money that you deposited by 10% at the end of the 1^st year (known as per annum interest rate). So, you will have $100 + 10% of $100 at the end of the 1^st year which equals $110. Well, not too bad. The things that you can buy with that extra $10.

Let’s take this spaceship forward. Say at the end of the 1^st year, you have an option to withdraw the $110 or invest it for another two years. Let’s say you decide to invest the money accrued ($110) for two more years at the same interest rate of 10% offered by the bank.

At the end of the 2^nd year, you will have:

$110 Balance at the end of 1^st year + (10% of $110$) = $121
which can also be written as:
$110 * (1 + 10%) = $121

It’s important to note that you don’t have:
$100 initial investment + (2 years * $10) = $120

But you do have:
100$ initial investment * (1 +10%) * (1+10%) or $100 * (1.10) * (1.10) = $121

$100 initial investment is multiplied by (1+10%) for every year the amount stays invested in the bank (which in the above example is 2 years) to get the final value of the initial deposit.

Or $100 initial deposit * (1.10)² (I know this seems like I am playing games here but I have a point, I swear)

Let’s see what happens at the end of the 3^rd year:

$121 Balance at the end of 2^nd year * (1+10%) = $133.1

Or $100 initial deposit * (1.10)³ = $133.1

This $133.1 is known as the Future value (in 3 years), the initial deposit/investment of $100 is termed as Present Value and the 10% interest rate is termed as R. If the money was invested for n number of years, the future value can be computed by:

Future Value in n years (FV) = $100 * (1 + 10%)³

To further generalize the equation for any value of PV and R, it will look something like:

Future Value in n years (FV) = (PV) * (1 + R)ⁿ

Let’s summarize what we learned from this whole exercise:

a) You deposited $100 in the bank at the beginning of the 1st year and that money became $133.1 by the end of the 3rd year due to a 10% per annum interest rate promised by the bank. This difference in value between money in the future (at the end of 3^rd year) and money today is known as the time value of money.

b) This means $100 today is worth a lot more in the future because you can invest that money today and gain interest on it over time.

c) The interest gain for the 1^st, 2^nd and 3^rd year were $10 ($110 – $100), $11 ($121 – $110) and $12.1 ($133.1 – $121) respectively. What does that mean? It means you were gaining interest on interest in the 2^nd and 3^rd year. This explains the increase in gains in the 2^nd and 3^rd years. This interest on interest is also known as Compound Interest.

d) This exercise of moving cash flows forward in time is called Compounding. E.g. – Interest getting compounded, problems getting compounded (except for Jay-Z), etc.

We just saw what happens to a single cash flow when it is compounded to the future. But what happens when we have multiple cashflows at different periods? Let’s suppose you have lots of excess money (I really really envy you!) and you decide to invest $100 today followed by $100 each subsequent year for the next two years. Let’s create a timeline and try to understand how we can find out the future value of all these cashflows at the end of the 3^rd year.

As shown in the picture above, $100 at period 0 (initial deposit) will be transformed to period 3 with a jump of 3 places (0 to 1, 1 to 2 and 2 to 3), hence, we will compound it by a power of 3. Similarly, $100 at periods 1 and 2 will be compounded by the power of 2 and 1 respectively. Now that all cashflows are in the same period, we can add or subtract them to get the final value since they are in the same time period.

Rule 3 – Travelling back in time

As the name suggests, this rule describes how to move cash flows backward in time. Remember that friend who borrowed $300 from you and promised to pay you back in three annual installments of $100 each. Let’s find out the value of those future cash flows, today.

In the previous rule, we had generalized the Time Value of Money equation as:

FV = PV * (1 + R)ⁿ

Now, in this case, you know the future value – which is $100 at the end of 1^st year – and you would like to know the present value (at period 0) of that $100.

Thus, rearranging the equation, we can write:

PV = FV / (1 + R)ⁿ

Let’s assume the interest rate to be 10% per annum and n, in this case, is 1 year.

PV₁ = $100 / (1 + 10%)¹ = $90.90

If I have completely and absolutely confused you, think of it this way – you would have to invest $90.90 today at 10% p.a. interest rate to get $100 a year from today. In the above example, we did some reverse engineering and found out the present value of future cash flow – $100 at the end of 1st year – which in this case is $90.90. This act of transforming a future cash flow to its corresponding present value is called Discounting.

Similarly, the present value of $100 payment done at the end of 2^nd year can be written as

PV₂ = 100 / (1 + 10%)² = $82.64

and the present value of the final payment of $100 done at the end of 3^rd year can be written as

PV₃ = 100 / (1 + 10%)³ = $75.13

Notice that the longer the discounting period, the lower the present value. To understand why this happens, go back to the investment example – an investment made for a longer duration will naturally result in larger gains. Similarly, a cash flow discounted back over a longer duration, will have a lower present value.

Now that you have transformed all the future cashflows to present-day (day 0), we can now add them up to find out the Net Present Value.

Net Present Value (NPV) shows that the cash inflows at the end of 1^st, 2^nd, and 3^rd year is worth $248.67 today. Or to put it differently, you could have invested $248.67 today in a bank at 10% p.a. to get three equal cash flows of 100$ each year for the next three years.

This means that you lent $300 to your friend and essentially, he/she paid you back only $248.67. There is an apparent loss here that you are facing. This apparent loss can be attributed as a lost opportunity, and hence is also known as Opportunity Cost. But having said, there is only one key takeaway from this example – your friends can really bank on you!

Why is Time Value of Money such a big deal?

Time Value of Money is frequently used in the world of Finance in making investment decisions, valuing assets, or evaluating long-term projects. For instance, let’s suppose your organization is considering taking up a project which requires an initial investment of $100 million today and the expected cash flow at the end of Year 1, 2, and 3 is $60 million. Let’s say your organization requires a minimum of 10% per annum return on investment (also knows as Cost of Equity). Your company is counting on your advice regarding whether it is worth taking up the project? As a general rule, an organization would take up a project only when there is a profit expected out of it. To put it in Time Value of Money terms – take up projects which have NPV greater than 0.

In this case, the NPV is $49.21 million > 0 and hence, you should advise your organization to take up the project.  

Now that you know key concepts such as Compounding, Discounting, FV, and NPV, you can apply these concepts for making sound investment decisions, and conduct project feasibility analyses. I was going to end this article with a joke about time travel but nobody laughed!

Finance Essentials Reading:
1. Make it Ac-count: Everything you need to know about Accounting