Present Value (PV)

If you want to learn about the present value, please make sure that you already know the time value of money concept. For those who do, welcome. For those who don’t, I recommend you to spend some time going over the Time Value of Money section. Why? As you can see from the subject, this section will involve the idea of time.

Now. From the time value of money concept, you should know that when it comes to finance, time and timing matters a lot. Remember? $100 today is more valuable than $100 in Year 5.

Frankly speaking, none of the values you calculate and come up with typically has no meaning at all, unless they are “compared”. It’s like your health checkup. Your weight is 70kg. Is it heavy? or is it light? You can’t tell. But once you compare with average weight of your classroom, which is, say, 63kg. Then you know you are on a heavy side. If you compare with the average weight of your age, which is, say, 75kg, then you know you are quite fit. Finance is the same. Your numbers tell almost nothing unless you have something to compare it with.

And, to compare something, it’s always helpful and useful if they are on an apple-to-apple basis, meaning they should be measured by the same scale. (what’s the point of comparing your dog’s weight with that of a……tree, right?)

For the same token, $100 “today” and $100 in “Year 5” are not apple-to-apple because they are measured on different time horizon, ignoring the time value attached to them.

The Present Value (PV) is the key to converting the amount in different time horizon into “today’s (present)” value by accounting for the timing effect.

For an easy understanding, let me revisit the $100 case I used in the time value of money section, where (for those who skipped that section) you are depositing your cash in a fictional bank with 3% interest.

Case 1: Receive $100 today with 3% compound interest for 5 years.

(NOTE: if you don’t know why we are using compound interest and not a simple interest, I advise you to take a quick glance at Assumption section)

In Case 1, you are receiving $100 today (present), so the present value is the same as what you received today, $100. However, in Case 2, you are receiving $100 in the future (Year 5), so that $100 is on a different horizon than Case 1. In order for you to compare the value of those two figures, you need to convert the Case 2 amount to a present value to align them on the same time horizon, hence apple-to-apple.

Well, how do you do that? There is a formula to convert future amount to a present value. But, because I know there are so many people who runs away once seeing a formula, let me describe to you in words what the formula will play out. (In fact, that’s way easier to interpret the formula)

Recall that your goal is “to convert $100 you receive in Year 5 to today’s value”. To do so, you need to find out an amount that “will be $100 in Year 5, if received today”. It may sound confusing, but I’m only rephrasing them with the same meaning. In other word, because money grows over time, how much do you need to start today so that the amount will be $100 in 5 years. (…It’s like “you are at a school and it will take 10 minutes to a library. If you are leaving your house and go to library via the school, at what time you need to leave the house?” kinda thing……more confusing…?)

By converting $100 in Year 5 into today’s value, you will be able to compare the value of Case 1 and Case 2, because they both are now on the same time horizon; today (Year 0).

Next. Let’s plug in the assumption (“whenever you have cash, you deposit it in a bank to enjoy compound interests”) to this idea, which will be like this:

“find out an amount that is deposited today to earn 3% compound interest so that the ending balance in Year 5 will be $100”.

So, if you can figure out this amount, that is the present value of $100 you are receiving in Year 5. All good?

Now. Let me bring up the compound interest formula I introduced to explain the time value of money concept:

In the time value of money section, we wanted to calculate the final balance at Year 5, or the “Amount at Year X” part in this formula. But, this time you know that amount is $100 in Year 5. Instead, you now want to know the original amount, or the present value, so need to calculate for the “Present Value” part. What do you do?

Yes, you flip the formula like this:

And you can plug in the numbers you have, which are the interest rate 3% and year 5:

The result will be $86.3. What’s this? Yes, this is the answer to your question of “an amount that is deposited today to earn 3% compound interest so that the ending balance in Year 5 will be $100”. In other words, “if you deposit $86.3 today and enjoy 3% compound interest for 5 years, you will end up with $100 at the end of Year 5”

Good job! This $86.3 is the present value for Case 2, or today’s value of $100 receiving in Year 5. Because now both cases are on the same scale (=today), you can finally compare them: today’s value of Case 1 is $100, and today’s value of Case 2 is $86.3. See? Now you can tell which case is better off.

For those people who really want to make sure what I’m saying is true, you can use the same cash flow table or the compound interest formula to prove it.

You see? If you deposit $86.3 and enjoy 3% compound interest for 5 years, the ending balance will be $100 in Year 5. If you wish to say in more professional-ish way, you say “the present value of $100 in Year 5 with 3% interest is $86.3”.

By the way, this process of “converting the future amount into today’s value” is known in finance as discounting, because you are “discounting” the effect of compound interest. Hence, in Case 2, you have just “discounted” $100 to the present value”.