WARNING: There is a lot going on in chapter 4. Understanding measurement is the first step towards analyzing the behavior of anything. This chapter is substantially quantitative so take the necessary time to review the formulas and the examples. These notes contain different examples than the textbook so you will have two sets of examples to guide you.
I. Measuring Interest Rates
A. Credit Market Instruments
A good first step is to carefully define what we are going to measure. Interest rates apply to four types of credit market instruments:
B. Present and Future Value
I realize many of you are already familiar with present value from other accounting/finance courses, but let's review..
Present 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.
So, 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.
C. Yield to Maturity
Now that we understand present value, we have the tools to calculate the most important measure of interest rates, the yield to maturity. The yield to maturity is the interest rate that makes the discounted value of the future payments from a debt instrument equal to its current value (market price) today. Let's look at the yield to maturity for the 4 credit market instruments discussed above.
This is the easiest case, because there is only one cash flow at the end of the loan to discount.
The value of the loan today is $1500. The future payments on the loan
are $1500(1+.06) = $1590.
So the yield to maturity is the i that solves the equation
Solving for i,
(1+ i) = 1590/1500
i = .06 6%
This case is more complicated due to the multiple payments through the life of the loan. Your textbook example uses a loan with annual payments on page 71. However, the most common forms of this type of loan are for monthly payments, like a mortgage, student loans or an auto loan. Loans with multiple payments during the year are a bit more complicated, as shown in the example below:
example 2: Suppose you take out a $15,000 car loan for 5 years, with monthly payments of $300.
The value of the loan today is $15,000. The future payments are $300
payments over the next 60 months.
The yield to maturity is the i that solves the following equation:
Note that since payments are monthly for 5 years, there are a total
of 5 x 12 = 60 payment periods. Also, the yield to maturity, i, is expressed
on an annual basis, so i/12 represents the monthly discount rate. This
assumes that interest charges compound annually instead of monthly.
If interest charges compounded monthly, then the appropriate monthly discount
rate is where
Your textbook is cavalier with this point, but the distinction is important.
In this application, though, it makes very little numerical difference
in the answer.
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 $15,000), but that is too time consuming. This problem is solved with the aid of a table, financial calculator, or spreadsheet programs that do this automatically. A financial calculator is not required for this course, so I provide loan or bond table when needed.
Consider the following loan table:
We are looking for a 5 year loan (shaded yellow), and a monthly payment of $300. Looking at the table above we see that at 7.5% yield to maturity, the payment is $300.57. So the yield to maturity is slightly under 7.5% (7.42% to be more precise).
Click below for the "high tech" ways to solve this example:
With the multiple interest payments involved, this case is similar to the fixed payment loan in its complexity. Again, your textbook example uses a coupon bond with annual coupon payments on page 72. 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.
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, the yield to maturity, i, is
expressed on an annual basis, so i/2 represents the 6 month discount rate.
This assumes that interest charges compound annually instead of semiannually.
If interest charges compounded semi-annually, then the appropriate
discount rate is where
Your textbook is cavalier with this point, but the distinction is important. In this application, though, it makes very little numerical difference in the answer.
Like the fixed payment loan, this problem is solved with the aid of a table, financial calculator, or spreadsheet programs that do the trial-and-error calculations automatically. A financial calculator is not required for this course, so I provide loan or bond table when needed.
Consider the following bond table:
We are looking for a 2 year bond (shaded yellow), and a price of $9750. Looking at the table above we see that at 7.5% yield to maturity, the price is $9726.15. So the yield to maturity is slightly under 7.5% (7.37% to be more precise).
Click below for the "high tech" ways to solve this example:
Looking at the bond table above, there are 3 important points to be made about the relationship between bond prices, maturity, and the yield to maturity:
The current value is $9850, and the only future payment is $10,000 at
maturity. However, we do not wait a year for this payment but only 90 days
so we need to adjust the discounting for this.
The yield to maturity solves the following equation:
Solving for i,
This method is the convention in financial markets, known as the bond equivalent basis.
If you use a financial calculator you may come up with a different answer:
If you do the following on your financial calculator,
9850 [+/-] [PV]
you come up with i = 6.32%. Which is greater than our lecture notes
calculation of 6.18%. Why? Because you instructed you calculator
to annualize i by compounding every 91 days. The calculator solved
While the method above makes sense and is a legitimate measure of an
interest rate, the method in the lecture notes, known as a bond equivalent
basis, is what we use by tradition in financial markets.
where F is the face value, P is the bond price, and d is the days to maturity
D. Current Yield
The yield to maturity is the truest measure of the interest rate. However there are other measure out there developed for their computational convenience. It this day of cheap computing, it is easy to forget that calculators were not available until 1975 (and then cost $200 for one that could just do arithmetic!). Bonds traded long before that, so traders used yield measures that approximated the yield to maturity but were easier to calculate.
The current yield is an approximation used for coupon bonds.
It is simply the annual coupon payment divided by the price of the bond:
where C is the coupon payment and P is the bond price. This is obviously a lot simpler that the yield to maturity
The current yield is a better approximation
the current yield is
Recall that the true yield to maturity, from example 3, is 7.37%. So
in this example, the approximation is lousy because it is only a 2-year
bond and it is selling at 25% below its face value.
Also known as the yield on a discount basis, the discount yield is used by dealers to quote the interest rates on U.S. Treasury bills. Again, this a computationally convenient approximation of the yield to maturity.
Compare this to the formula for the yield to maturity:
Note there are two major differences:
(1) the yield to maturity takes the discount (F-P) as a proportion of the bond price, while the discount yield takes the discount as a proportion of the face value.
(2) the yield to maturity uses a 365-day year while the discount yield uses a 360-day year.
Both of these differences make the arithmetic easier in the case of the discount yield, but the also cause the discount yield to understate the true yield to maturity (F is always greater than P and 365 is always greater than 360). The discount yield will always be less than the yield to maturity for any zero coupon bond.
This is slightly less than the yield to maturity which is 6.18% (example
Wow, this is a lot of stuff to think about. What next? I suggest if you want some more practice with calculating various interest rates, try the problems at the end of chapter 4, page 90. There are solutions in the back of the book for the odd numbered problems.
II. Other Measurement Issues
Understanding what the interest rate does and does tell you is as important as measuring the interest rate in the first place. Here are a couple of issues in interest rate measurement.
A. Interest Rates vs. Returns
The yield to maturity assumes that the bond is held until maturity.
If that is not true, then fluctuations in the bond price (which occur with
interest rate fluctuations) will affect the return, or the gain
to the investor from holding this security.
The return for holding a bond between periods t and t+1 is
where Pt is the initial price and Pt+1 is the price at the end of the holding period.
We can rewrite this formula as
The last term is the rate of capital gain, g, or the change in
the bond price relative to the initial bond price. So a bond's return can
be rewritten as
A bond's return is identical to the yield to maturity if the holding period is identical to the time left to maturity.
B. Maturity and Bond Price Volatility
Any bond price moves in the opposite direction of interest rates, but what determines how much a bond's price fluctuates, or in other words, it volatility? Let's reconsider the bond table from part I, example 3:
Look at each bond's price (the 2-year, 5-year, and 10-year bonds) as the yield to maturity rises from 6% to 8%. The prices fall for all of the bonds, but by different amounts. The price on the 2-year bond falls less than $400 or less than 4%. The price on the 10-year bond falls by more than $1300 or more than 13%. This brings to the principle bond characteristic that affects price volatility: Prices (and thus returns) are more volatile for long-term bonds than short-term bonds. In other words, long-term bonds have greater interest-rate risk.
Why is this the case? Intuitively, with a long-term bond, you are "locked in" to a coupon rate for a longer period of time. So if newer bonds are issued with lower coupon rates, your long-term bond becomes much more valuable. If new bonds have higher coupon rates, your long-term bond becomes much less valuable. For a bond with less than 1 year left until maturity, the change in interest rates will not matter that much. The consequences of changing interest rates are much more serious for bonds with longer times left until maturity.
C. Real vs. Nominal Interest Rates
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 below:
Note how in the 1970s, 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
ABC's of Figuring Interest A Federal Reserve publication that
discusses of basics of calculating alternative interest rate measures.