Both the amount and timing of cash flows are important when comparing financial instruments. To make an accurate comparison among instruments with differences in the amount and timing of cash flows we need to understand future value, present value and the internal rate of return.
I realize many of you are already familiar with future and present value from other accounting/finance courses, so feel free to skip over this.
Future value is based on the fundamental reality that you are not indifferent between getting $100 today versus waiting one year to receive $100. Why? Well in financial markets, you could receive interest on that $100 over the course of one year, and end up with more than $100 at the end of the year. The cost of waiting is the simple interest rate; i.e. the interest rate on a simple loan. You lend me $100 at an interest rate of 5% per year, then at the end of one year you will receive
$100 + (.05 x $100) = $100 x (1 + .05) = $105.
at the end of 2 years you will receive
In general, if the simple interest rate is i and the loans are made for n years you will receive:
The amount above is known as the future value of $100 in n years.
If the holding period is a fraction of a year, then n is a fraction as well: If the holding period is one month, then n = 1/12.
Compounding interest is a bit more complicated. In general i is an annual interest rate. If you have the monthly interest rate im , the annual interest rate is
So future values tells use what an amount TODAY is worth in the future. Working backwards, for any amount received in the future we need to discount it to the present. In other words, if you are getting $100 in one year, how much less would you accept in order to get it today? The answer is the present value and will depend on the interest rate.
Suppose again the interest rate is 5%. If you will receive $100 in one year, what is the present value? We want to solve the equation
PV x (1 + .05) = $100 or
If you will receive $100 in 3 years, what is the present value?
In general, for the PV of $100, n years from now, with a simple interest rate of i, we use the formula
Note that larger values for n and i imply smaller PV. In other words, the longer you wait for the payment, the lower its present value. From the examples above, we see how extending the wait for $100 from 1 year to 3 years lowers the present value by about $9.
Also, the larger the interest rate, the larger the opportunity cost of waiting for the payment, the lower its present value. For example, if the interest rate is i=10%, then the present value of $100 is 1 year is
And in 3 years is
Present value is the single most important concept in pricing financial instruments, so it is important to understand it.
Applying Present Value
First a note about some definitions. The interest rate, i, is also known by other names. Since i is used to reduce future cash values to their present day equivalent it is also known as the discount rate -- i.e. the rate used to discount future payments to the present. In the context of bond markets, the interest rate is often referred to as the bond's yield. So as we progress through the chapters in this module, keep in mind that interest rate, discount rate, and yield mean the same thing.
Internal Rate of Return
Our examples above calculate present value for a single cash flow, but we can apply this concept to multiply payments over time by simply calculating the present value of each cash flow and adding this up for all of the cash flows. One application involving multiple cash flows is calculating and using the internal rate of return or IRR. The IRR is the interest rate the equates the present value of an investment with its cost. Huh? Let's look at this concept through an examples:
example 1: Suppose you are considering paying for a computer course that when completed will lead to a salary bonus over the next 5 years from your employer. (To simplify this example, we are assuming the increase in salary is known. In reality it would be an estimate.) The course costs $1800 and for taking the course (again, let's assume successful completion is known) you will receive a bonus of $500 over each of the next 5 years.
So here the present value is the $1800 cost, and the future payments are a series of 5 $500 bonuses. We would like to know at what interest rate does the cost of the course equal the present value of the of the bonus payments. This interest rate is the IRR. So we want to solve the following equation:
So how do we solve this for i? Well it is not easy, since there is no way to isolate i in this equation. It could be done by trial and error (trying values of i until the right-hand side of the equation is $1800), but that is too time consuming. This problem is solved with the aid of a financial calculator, or spreadsheet programs that do this automatically. A financial calculator is not required for this course, so I always provide the necessary resources when asking you questions like these.
Using an Excel spreadsheet (click here for directions) we get i = 12.05%.
If you don't have Excel, you can use the online calculator here.
So what does this all mean? In this case the IRR tells us that the annual return on the $1800 computer course is about 12%. That is a very good return. If there are other investment opportunities that could return say 15%, then the computer course would not be a good idea. In reality, 12% is pretty tough to beat.
Bond and Bond Valuation
In general, bonds are an IOU. The borrower, also known as the bond seller or bond issuer, is promising a series of payments over a period of time. The bond buyer (or lender) is paying the bond price up front in return for the right to receive those payments. Bonds are a contract with obligations and consequences if those obligations are not met.
Let's consider the specific case of a coupon bond. A coupon purchased at some price and entitles the buyer to fixed interest payments annually (coupon payments) until maturity and a face value payment (or par value) at maturity. The size of the coupon payments is determined by the coupon rate and the face value of the bond.
Your textbook example uses a coupon bond with annual coupon payments on page 79. However, all bonds issued in the United States have coupon payments semi-annually, or every 6 months, including Treasury notes, Treasury bonds, and corporate bonds. So the example below also uses 6-month payments.
example 2: Consider a 2-year Treasury note with a face value of $10,000, a coupon rate of 6%, and a price of $9750.
So the yield to maturity will solve the equation:
bond price = PV(future bond payments)
What are the future payments?
There are coupon payments every 6 months, and a face value payment at maturity.
What are the coupon payments?
The coupon payments are [face value x coupon rate]/2 = $10,000 x .06 x .5 = $300. Note that we divide by 2 because there are 2 coupon payments in a year. So the payment schedule is
So the yield to maturity will solve the following equation:
Note that since payments are every 6 months for 2 years, there are a total of 2 x 2 = 4 payment periods. Also, i, is expressed on an annual basis, so i/2 represents the 6 month discount rate (note 2)
Like the IRR examples, this problem is solved with the aid of a financial calculator, or spreadsheet programs that do the trial-and-error calculations automatically. A financial calculator is not required for this course, so I always provide the necessary resources when asking you questions like these.
Using a spreadsheet, we get a yield of 7.37%
Click below for the "high tech" ways to solve this example:
|Financial Calculator: TI BA II+
Real vs. Nominal Interest Rate
Up until now, we have not accounted for the effects of inflation on the return or interest rate on a bond. While the owner of a bond is entitled to future payments, in an economy with inflation, the purchasing power of those payments is declining over time. It is pretty much a given that $10,000 in 2011 will buy less than $10,000 today.
The interest rate (yield to maturity) we calculate in Part I is specifically the nominal interest rate, which does not consider the impact of inflation. Instead, expected price changes are reflected in the real interest rate. The relationship between the real and nominal interest rate, known as the Fisher equation, is given by:
Where is the expected inflation rate.
So the nominal interest rate is the sum of the real interest and the expected inflation rate. The real interest is a truer measure of the cost of borrowing. Lower real interest rates increase the incentive to borrow (while reducing the incentive to lend). Higher real interest rates decrease the incentive to borrow (while increasing the incentive to lend).
This distinction is important because real and nominal interest rates do not alway move in the same direction. Consider the time series graph in figure 4.3, page 82. Note how in the early 1980s, the real interest rate on the 3-month Tbill is falling, while the nominal interest rate is rising. Looking only at the nominal interest rate gives you a misleading picture of the true cost of borrowing.
FYI: Related Links
Mulitple Choice Quiz, Chapter 4 Test your understanding with a quiz from the textbook website.