Chapter 4

THE MECHANICS OF PRESENT VALUE — WHAT IS A DISCOUNT RATE?

Before we move on to a detailed discussion of what each of the five steps in using Present Value entails, I want to be a little bit more rigorous in describing the term “Present Value” and then go through the basics of a Present Value calculation. Critical to understanding that discussion will be the concept of a discount rate, so let’s spend some time talking about discount rates and what Present Value really means.

Present Value Defined—Savings Bonds and What “Discount Rate” Means

The next few paragraphs have a tiny bit of math and are more quantitative than we will generally get. While Present Value math is not at all necessary to understanding the rest of the book, there is one “mathematical process” —using a discount rate—that is required before you can actually use Present Value in your life. Fortunately, however, the math required to use a discount rate is very easy and is just the reverse of a concept almost everyone is familiar with—that of compound interest and the way money accumulates into the future.

For the reader who is interested in getting a more complete understanding of the math behind Present Value, I have included a short primer in the Notes at the end of this book.¹⁰ The derivation of the generalized Present Value formula there will give you all you ever need or want to know about how to actually calculate Present Values. However, for those of you who just want to understand how to use Present Value in your own life, the following should be plenty.

To get started, let’s try and state exactly what Present Value is. As an idea, it is actually pretty straightforward. As we have said before, Present Value is the value today of something that (may) happen in the future. So for example, if I offered you the choice between $1000 today or a promise to give you $1100 a year from now, which would you choose? Well, you might say that $1000 today is not enough because if inflation is running at 2% per year you will only need $1020 a year from now to buy what you can today with $1000, suggesting that waiting for the $1100 is the right thing to do. By this reasoning the promise to pay you $1100 next year is worth about $1080 today, which is more than the $1000 I’m offering. Note also that this is almost, but not quite the same as saying that you can earn 2% on the $1000 risk free. We will come back to this distinction later, but for now it is important just to note that when we focus only on how the value of money changes over time, we (in this case) get a Present Value of that future promise equal to $1080 and therefore opt to defer receipt.

Now let’s explore our example just a little further. What if you said to yourself that you know an investment that you’re pretty sure will give you a 5% return instead of 2%? Should you then ask for $1050? ($1050 invested at a rate of 5% will yield approximately $1100 in one year.) And what about the possibility that if you choose to wait a year, I will renege on my promise, or be unable to pay you the $1100 I promised? If there is even a 10% chance that I will be unwilling or unable to pay you next year, you should take the $1000. Here we have introduced the element of risk into the calculation. And note it is not just the downside risk of not getting the money because I won’t fulfill my promise, but also the upside risk that with the $1000 you might find an investment that will yield you more than the 2% risk-free return noted above. Now you might say that we should ignore this last factor because any riskier investment will also come with an offsetting probability of loss (e.g., if you have a “hot” business investment that will return you $1200 next year, there might be a 15% chance that you will lose the whole investment meaning your “expected” value after a year is still 85% × $1200 = $1020). It turns out, however, that most investment professionals will tell you that if you invest in riskier vehicles (like stocks) you should anticipate getting a greater return simply because you are willing to take on this risk. This is a deep concept and, in my view, not as obvious as the experts suggest. We will talk much more about this in chapters 6 and 7 where we will discuss how to think about the future, what risk really means, and how to evaluate all the possibilities.

But to complete the picture, we need to talk about one further component of Present Value. Turning again to our choice between $1000 today and $1100 a year from now, let’s say you have complete confidence in me and you have no appetite for risk. Even though you know you will earn much less than 10% on the money, you still may want to take the $1000, and that’s because you might need (or maybe just want) to have the money now rather than next year. This is the “liquidity premium,” or “time preference” aspect of present value. This is a mysterious and in my view a fundamentally psychological factor. It is also one that is inseparable from the concept of time and what time means to different entities, to an individual (or different individuals), to a corporation (profitable vs. on the verge of bankruptcy), to a government, or to a foundation (or charity, university, etc.). It is the essential component of the personal rate of discount I referred to in the Introduction, and it is the portion of the Present Value calculation that is often the most difficult for individuals (including actuaries) to fully understand. We saw how this aspect of the calculation plays out practically in chapters 2 and 3. In chapter 8 we will see how to go about setting your own personal rate of discount.

When I’ve tried to explain Present Value to people, it’s the term discount rate that most people find unnatural and confusing, and so it’s worth looking at another example of Present Value that almost everyone should be familiar with to illustrate how “discounting values with a discount rate” is just the reverse of accumulating money with an interest rate. Specifically, let’s consider the purchase of a US savings bond. A savings bond is a very official-looking certificate issued by the US government, which after ten years is worth exactly what it says on the front. When my son was born, he received as gifts several of these. Essentially each $100 savings bond he got was a gift whose future value was $100. Notice that in this case future value is determined at one specific point in the future, and there is essentially no doubt (unless the US Government goes out of business) that this bond will be worth exactly $100 at that time. In this case, determining future value was very easy. As we have already seen, there are situations where the determination of future value is not so easy because we don’t know exactly when the payoff will occur, what the amount will be when it does occur, or even a combination of both. For now though let’s stay with this savings bond and turn our attention to how to use a discount rate to calculate its Present Value.

Again, Present Value answers the question, “what is the value today of something that I will be getting sometime in the future?” For the savings bond, the answer is simply the price that you have to pay to get one. Interestingly, the price of a $100 savings bond changes almost daily and will vary depending on prevailing interest rates (why this is so we will talk about shortly). Today as I write these words, a $100 US savings bond costs about $80, but a few years ago you could get one for a little more than $60, and thirty years ago (when interest rates were at record highs) the price was less than $50. Is this because the price of everything seems to go up over time? Absolutely not. In fact, back in the early 1960s the price of our $100 US Savings Bond was almost the same $80 that it costs today. So what is it that causes the price today of $100 payable ten years from now to be something less than $100? What determines that difference and what causes that difference to go up and down as the environment changes? We all feel that the price today of getting $100 in the future should be less than $100, as getting something now is always better than getting it later, but figuring out why and how much is not something many people think about.

Again, the basic question is “how much would we agree to pay today to make sure we have $100 in ten years when the bond matures?” To answer that we first have to ask at what rate will the money we set aside today grow at (the concept of compound interest that we are all familiar with). That answer of course depends on what we would invest the money in. When we talk about the rate at which money could grow, we call this the interest rate. This is the rate we use to project the future value of the money we have on hand today. When we go back the other way and ask ourselves how much that future value is worth today, we will call this the discount rate. For all practical purposes these two rates are the same thing, just viewed from different vantage points (we will ignore for now the aforementioned “time preference” and “risk” factors in a discount rate that generally makes a discount rate somewhat higher than an interest rate). It turns out that the interest/discount rate inherent in this example is about 2.3%. The way to see this mathematically is to see that:

$80 × 1.023¹⁰ = $100
(the future value of the bond)

(Or, if you are curious about how to solve for the rate, r, and your math is a little rusty, $80 × (1 + r)¹⁰ = $100, which yields (1 + r)¹⁰ = $100/$80 = 1.25, and finally, r = 1.25^1/10 – 1 = 0.0226, which rounds up to 2.3%.)

In the basic Present Value calculation, this equation is reversed, and instead of multiplying by an interest factor of 1 + 2.3% = 1.023 as we move year by year from the present into the future, we divide by a discount factor of 1.023 as we move year by year from the future back to the present, that is,

$100 × (1/1.023)¹⁰ = $80
(the Present Value of the bond)

That is it. The above equation, and in particular the idea of a discount rate is all the math you really need to know about Present Value and all you will ever need to understand everything I have to say in the rest of this book.

And now it’s time to dig a little deeper into actually using Present Value in a systematic way to make better decisions, and in particular to examine more closely each of the five steps of that process that we identified in the Introduction to this book.