Previously, I wrote about the concept of risk and return and linked this concept to interest rates. Now, I will take this concept a step further and use rates of return to evaluate different cash flows across time.

Let’s return to our savings account. Recall that the money you’ve placed in a savings account is being loaned to a bank, and in return you will be paid an interest rate. Let’s say in this case you initially deposited $100 and you’re paid 5% annually. So at the end of Year 1, assuming you never withdrew the $100, you earn ($100 x 5% (or 0.05)) = $5. At the end of the year, you have $100+$5 = $105 in your savings account.

Apologies for the math. Although I am keeping things simple and focusing more on concepts, sometimes we just can’t avoid numbers! Anyway let’s continue:

Over Year 2, you earn another $5 on your initial deposit of $100. However, you *also* earn another 5% on $5 interest you earned over the previous year. So at the end of Year 2, you have:

$100 + $5 interest +$5 interest from last year + $0.25 (5% interest on $5 from last year) = $110.25.

So aside from earning interest from your initial investment, you are now also earning interest on interest. This is called compounding.

Interest rates make compounding possible. While we have already talked about how interest rates represent a rate of return for risk taken, in the example above, interest rates also serve a second purpose.

Let’s say you’re curious to know how much you’ll have in your savings account by the end of Year 5. We can calculate that by using the following equation:

$100 x (1+0.05)^{5 } = $127.63.

(We use exponents to arrive at this number because exponents take into account compounding.)

Now for an interesting question: What if you never checked your account balance until the end of Year 5, but now you want to know how much it was at the end of Year 3?

Yes, you *could* use the equation above and just change the exponent to 3 instead of 5. But, you could also work backwards by doing the opposite of multiplying – dividing:

$127.63/(1+0.05)^{2}= $110.25.

(Note that we use an exponent of 2 instead of 3 since we are starting at Year 5 and we want to get to Year 3 (5-2=3).)

This process of determining a past or present value from a future value is called discounting. Discounting is a fundamental concept in Finance and is used in many evaluations and transactions.

Here’s a simple example: Would you take $100 today or $115 in five years? While $115 is the larger amount, what happens if you could invest $100 at a 5% annual interest rate? You need to compare apples to apples, and you can do this in one of two ways:

- Compound $100 at 5%. $100 x (1+0.05)
^{5 }= $127.63.

or

- Discount $115 at 5%. $115/(1+0.05)
^{5}= $90.11

Either way, it’s better to take the $100 now than wait five years for $115.

This all works because of a fundamental concept in Finance: the time value of money. Essentially, money is worth more today than it is in the future, so to compare a future amount against a present amount requires discounting. Why is money worth more today than in the future? Think about all the uncertainties in waiting for money to be paid out tomorrow versus it actually being in your hands today. Think about the risks we went over in the previous post.

The saying “time is money” takes on a whole new meaning when it comes to Finance.

In this post, we have gone over the concepts of compounding and discounting. Notice that the interest rate makes this all possible. In such cases, the interest rate is often referred to as the discount rate.

We have gone over interest rates as required rates of returns and discount rates. There is one further definition of interest rates which we will look at in the next post.

Ramon Rodrigo Cuenca, CFA

Source: CFA Program Curriculum Level I, 2009, vol. 1: Ethical and Professional Standards and Quantitative Methods, pp. 172-173

Categories: Art, Artwork, Finance, Intro to Finance, Paintings, Time Value of Money, Watercolor

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