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CHAPTER 1
Time Value of Money
A security is a package of cash flows. The cash flows are delivered across time with varying degrees of uncertainty. To value a security, we must determine how much this package of cash flows is worth today. This process employs a fundamental finance principle—the time value of money. Simply stated, one dollar today is worth more than one dollar to be received in the future. The reason is that the money has a time value. One dollar today can be invested, start earning interest immediately, and grow to a larger amount in the future. Conversely, one dollar to be received one year from today is worth less than one dollar delivered today. This is true because an individual can invest an amount of money less than one dollar today and at some interest rate it will grow to one dollar in a year’s time.
The purpose of this chapter is to introduce the fundamental principles of future value (i.e., compounding cash flows) and present value (i.e., discounting cash flows). These principles will be employed in every chapter in the remainder of the book. To be sure, no matter how complicated the security’s cash flows become (e.g., bonds with embedded options, interest rate swaps, etc.), determining how much they are worth today involves taking present values. In addition, we introduce the concept of yield, which is a measure of potential return and explain how to compute the yield on any investment.
FUTURE VALUE OF A SINGLE CASH FLOW
Suppose an individual invests $100 at 5% compounded annually for three years. We call the $100 invested the original principal and denote it as P. In this example, the annual interest rate is 5% and is the compensation the investor receives for giving up the use of his or her money for one year’s time. Intuitively, the interest rate is a bribe offered to induce an individual to postpone their consumption of one dollar until some time in the future. If interest is compounded annually, this means that interest is paid for use of the money only once per year.
We denote the interest rate as
i and put it in decimal form. In addition,
N is the number of years the individual gives up use of his or her funds and
FVN is the future value or what the original principal will grow to after
N years. In our example,
P = $100
i = 0.05
N = 3 years
So the question at hand is how much $100 will be worth at the end of three years if it earns interest at 5% compounded annually?
To answer this question, let’s first determine what the $100 will grow to after one year if it earns 5% interest annually. This amount is determined with the following expression
Using the numbers in our example
In words, if an individual invests $100 that earns 5% compounded annually, at the end of one year the amount invested will grow to $105 (i.e., the original principal of $100 plus $5 interest).
To find out how much the $100 will be worth at the end of two years, we repeat the process one more time
From the expression above, we know that
Substituting this in the expression and then simplifying, we obtain
FV2 = P(1 + i)(1 + i) = P(1 + i)2
Using the numbers in our example, we find that
FV2 = $100(1.05)2 = $110.25
Note that during the second year, we earn $5.25 in interest rather than $5 because we are earning interest on our interest from the first year. This example illustrates an important point about how securities’ returns work; returns reproduce multiplicatively rather than additively.
To find out how much the original principal will be worth at the end of three years, we repeat the process one last time
Like before, we have already determined
FV2, so making this substitution and simplifying gives us
FV3 = P(1 + i)2(1 + i)
FV3 = P(1 + i)3
Using the numbers in our example, we find that
FV3 = $100(1.05)3 = $115.7625
The future value of $100 invested for three years earning 5% interest compounded annually is $115.7625.
The general formula for the future value of a single cash flow
N years in the future given an interest rate
i is
(1.1)
From this expression, it is easy to see that for a given original principal
P the future value will depend on the interest rate (
i) and the number of years (
N) that the cash flow is allowed to grow at that rate. For example, suppose we take the same $100 and invest it at 5% interest for 10 years rather than five years, what is the future value? Using the expression presented above, we find that the future value is
FVN = $100(1.05)10 = $162.8894
Now let us leave everything unchanged except the interest rate. What is the future value of $100 invested for 10 years at 6%? The future value is now
FVN = $100(1.06)10 = $179.0848
As we will see in due course, the longer the investment, the more dramatic the impact of even relatively small changes in interest rates on future values.
PRESENT VALUE OF A SINGLE CASH FLOW
The present value of a single cash flow asks the opposite question. Namely, how much is a single cash flow to be received in the future worth today given a particular interest rate? Suppose the interest rate is 10%, how much is $161.05 to be received five years hence worth today? This question can be easily visualized on the time line presented below:
Alternatively, given the interest rate is 10%, how much would one have to invest today to have $161.05 in five years? The process is called “discounting” because as long as interest rates are positive, the amount invested (the ...
