Interest Rates and the Pricing of Bonds
Types of Loans
We will examine four types of loan agreements; simple loan, fixed payment loan, coupon bond, and a discount bond.
Simple Loan
A simple loan is when the borrow repays the amount borrowed plus interest when the loan matures. For example, suppose one borrows 100 at the interest rate of 10%, and the loan matures in one year. Then in one year the borrower pays the lender the principle of 100, plus an interest payment of 10%(100) = 10, for a total payment of 110.
This is called a simple loan because there is only one payment when the loan matures, and it includes both the principle and interest.
Fixed Payment Loan
A fixed payment loan is when the borrower makes periodic payments (e.g. monthly payments) on a loan until the loan is repaid. The fixed payment includes both an interest and principle component. House and car loans are fixed payment loans. For example, if you borrow 6000 to buy a car, you may negotiate with the bank to repay the loan over 36 months, paying the bank 200 per month. At the end of the 36 months you would have repaid the bank 7200, implying interest and principle were repaid.
How can one find the fixed payment? To do so we use the present value formula. Recall,
In this case we know the interest rate i and we know the present value of the loan, which is simply the amount borrowed. Finally, since this is a fixed payment loan (meaning every payment in the future is the same amount), it must be . We will refer to these constant fixed payments as FP.
Now suppose the borrowed amount (i.e. the PV of the loan) is 10000. Also suppose the interest rate is 1%, so that 1+i = 1.05. Furthermore assume that the loan is repaid in 4 years. Then from the present value formula we have
Now we must simply solve the equation for FP.
Solving for FP we have
Coupon and Discount Bonds
Recall that a coupon bond has a face value repaid to the bondholder when the bond matures. The coupon bond also makes regular interest payments to the bondholder. This is called the coupon payment. The coupon payment is equal to the face value times the coupon rate, as stated on the bond. That is,
Recall a discount bond promises to pay the bondholder the face value when the bond matures, but there are no coupon payments of interest. Hence a discount bond is sometimes called a “zero-coupon” bond.
Determining the Price of a Discount Bond
The market price of a bond is simply the present value of all its future payments. To demonstrate this we begin with a discount bond, since this has only one future payment.
Note the present value of a discount bond maturing in n years is given by
where FV is the face value and P is the price (or present value).
As an example, suppose the face value is 5000, the interest rate is 7%, and the bond matures in 10 years. Then the price of the bond is given by
That is, a discount bond paying 5000 in ten years when the current market interest rate is 7% is worth 2542 today.
Effects of changes in interest rate and years to maturity
Now let us consider how a change in the current market interest rate affects bond prices. Consider again the above example in which the face value is 5000 and the bond matures in 10 years. But now suppose the interest rate is 15%. Then the price of the bond is given by
Thus we see that an increase (decrease) in the interest rate will cause a decrease (increase) in the price of a bond.
Now consider again when the bond has a face value of 5000 and the interest rate is 7%, but let the years to maturity be 20. Then the price of the bond is given by
Thus we see that an increase in the years to maturity will lower the bond price.
Determining the Price of a Coupon Bond
As in the case of the discount bond the price of a bond is the present value at current market interest rates. That is,
Where n is the years to maturity. Note that the interest rate i is the current market rate, not the coupon rate.
As an example, suppose the face value is 1000, the coupon rate is 7%, and the years to maturity is 3. In this case the coupon payment is 70 and the pricing formula is given by
Now we will find the price of the coupon bond for alternative values of i, the market interest rate. We will consider three cases; when the market interest rate equals the coupon rate, when the market interest rate is less than the coupon rate, and when the market interest rate is more than the coupon rate.
Case 1:Market Interest Rate = Coupon Rate
In this case i = 7%. Hence the pricing formula becomes
Hence, when the when the market interest rate equals the coupon rate the price of the bond is equal to its face value.
Case 2:Market Interest Rate Coupon Rate
In this case let i = 3%. Hence the pricing formula becomes
Hence, when the when the market interest rate is less than the coupon rate the price of the bond is greater than its face value.
Case 3:Market Interest Rate Coupon Rate
In this case let i = 10%. Hence the pricing formula becomes
Hence, when the when the market interest rate is more than the coupon rate the price of the bond is less than its face value.
Yield to Maturity
In the above analysis we assumed we knew the market interest rate, the coupon payments, the face value, the years to maturity, and the market interest rate. However, in the real world we observe the price of the bond and infer from it what the implied market interest rate is. This implied interest rate is called the yield to maturity.
For example, suppose a discount bond has a face value of 10000, matures in 2 years, and has a current price of 9000. Then by the present value formula we have
We must simply solve the equation for i to find the yield to maturity. That is,
So the yield to maturity on a two year discount bond is 5.4%.
In general, the yield to maturity on a discount bond is given by
The yield to maturity on a coupon bond is found in a similar way. Suppose the coupon payment is 100, the face value is 1000, and there are 5 years to maturity. Also suppose the price is 800. Then by the present value formula we have
From this equation one must solve for i to find the yield to maturity. However, because the interest rate is raised to different exponents there is no simple formula, though there are methods to find the yield to maturity from the above equation.
Nominal vs. Real Interest Rates
When discussing interest rates it is important to understand the difference between nominal and real interest rates.
- The nominal interest rate refers to the percentage increase in money one receives/pays on a loan.
- The real interest rate refers to the percentage increase in purchasing power one has after making a loan.
Of the two, the interest rate that is agreed to on a loan or bond is the nominal interest rate. However, it is clear that it is the real interest rate that is important to borrowers and lenders.
One may wonder why the two interest rates are different. The two interest rates will differ if there is a change in the price of goods/services.
For example, suppose you make a loan today of RO100 at the nominal interest rate of 10%. This implies in one year you will receive RO110. Now also suppose that today the price of a good is RO5. Hence the real value of today’s loan of RO100 is 20 goods. That is, at the price of RO5, RO100 will buy 20 goods (this is found by finding RO100/RO5).
Now suppose the price in one year is still RO5. Then the RO110 received in one year will buy RO110/RO5 = 22 goods. Hence one could say you loaned 20 goods today and received 22 goods in the future. So the purchasing power of your money increased by 10%, the same as the nominal interest rate. Thus we can conclude the following:
If the price of goods is constant over time (i.e. inflation is zero) then the real interest rate will equal the nominal interest rate.
Now suppose the price in one year rises by 10% to RO5.5. In this case the RO110 received in one year will buy RO110/RO5.5 = 20 goods. Hence one could say you loaned 20 goods today and received 20 goods in the future. So the purchasing power of your money increased by 0%. In fact, 10% increase in money combined with a 10% rise in prices, cause the purchasing power to remain constant. Thus we can conclude the following:
If the price of goods rises over time (i.e. inflation is positive) then the real interest rate will be less than the nominal interest rate. In fact, the real interest rate will be equal to the nominal interest rate minus the rate of inflation.
Letting r be the real interest rate, i be the nominal interest rate, and π be the rate of inflation, we have the following formula
Inflation Risk
The fact that the real interest rate (which is the one borrowers and lenders care about) depends on inflation implies that if inflation changes unpredictably the real interest rate will change unpredictably; that is, inflation risk creates risk in the real interest rate.
To understand the risk better, consider the following definitions:
- Let πe be the expected inflation rate
- Let πa be the actual inflation rate
- Let re be the expected real interest rate
- Let ra be the actual real interest rate
In this case we have
And
Hence the difference between the actual real interest rate and the expected real interest rate is given by
Which implies
This says that if the actual inflation rate is greater than expected inflation, then the actual real interest rate will be less than the expected real interest rate. Hence the more difficult it is to predict inflation, the more risk one is exposed to. For this reason, uncertainty in inflation can have a negative effect on bond markets.
Interest Rate Risk
The above explains that bondholders face a risk due to inflation. But there is another risk bondholders face. When considering one year rates of return on bonds, bondholders face a risk in their annual rate of return that derives from the fact that the current market interest rate can change. Recall that if the market interest rate rises (falls), the price of an existing bond will fall (rise). Since future interest rates cannot be perfectly predicted, this means the future price of one’s bond holdings cannot be perfectly predicted, and hence the risk.
To understand this better, let us consider the rate of return on a bond. The rate of return is given by the following:
Where I is the interest payment, P0 is the price one pays for the bond, and P1 is the price of the bond after one year. If we split this into two parts we have
The first terms is the interest payment divided by the purchase price of the bond. This is simply the interest rate, and is known at the time of the bond purchase.
The second term is the capital gain or loss on the bond. It is not known at the time the bond is purchased and thus represents a source of risk.
Now recall that the price of a bond falls with an increase in the interest rate. This is implies if one buys a bond and then the interest rate rises, the price will fall and P1 – P0will be negative; that is, the bondholder will experience a capital loss.
To understand the effect better let us consider an example. Suppose P0 = 1000, i = 10%, and there are two years to maturity. Now suppose after one year the market interest rate rises to 20%. Using the present value formula, the new price of the bond (now with only one year to maturity) is given by
Thus the rate of return on the bond is given by
Hence the rate of return is 10% minus the capital loss experienced. Now note that the longer the time to maturity the greater is the price change in a bond following an interest rate increase.
To illustrate this consider again the example above in which P0 = 1000, i = 10%, and after one year the market interest rate rises to 20%. However, assume that at the time of purchase the bond has three years to maturity. This implies that after one year the bond has two years to maturity. So using the present value formula, the new price of the bond (now with only two year to maturity) is given by
.
So notice that while the two year bond fell in price from 1000 to 917, the three year bond fell in price from 1000 to 846. The reason is simply due to the fact that denominator in the present value formula is being raised to a higher exponent.
Given the fall in price to 846, the rate of return on the bond is given by
Given that the longer the time to maturity the greater is the price change in a bond following an interest rate increase. This implies that bonds of longer maturities will have a greater capital loss when the interest rate increases. The following table shows the results for bonds of different maturities. In all cases, the initial interest rate is 10%, the market interest rate rises to 20%, and the initial price is 1000.
Years to Maturity when Purchased / P1 / Initial Interest Rate / Capital Loss / Rate of Return2 / 917 / 10% / -8.3% / 1.7%
5 / 741 / 10% / -25.9 / -15.9
10 / 597 / 10% / -40.3 / -30.3
20 / 516 / 10% / -48.4 / -38.4
30 / 503 / 10% / -49.7 / -39.7
As one can see, the longer the time to maturity, the greater the risk associated with an interest rate change.
Term Structure of Interest Rates
As explained above, the yield to maturity is the implied interest rate from a bond with a particular maturity date. The term structure compares the interest rate of bonds of different maturities. Often this is expressed graphically, as below, in what is known as the Yield Curve.
In this graph we see that a 1 year bond has a current yield of 4%, a 5 year bond has a current yield of 6%, and a 10 year bond has a current yield of 8%.
Three Facts about Yield Curves
- Yields on bonds of different maturities move together. That is, when yields on short-term bonds rise, we also observe that yields on long-term bonds rise.
- When the yields on short-term bonds are low, the yield curve is more upward sloping.
- Yield curves are usually upward sloping, as in the graph above.
Expectations Hypothesis
We now attempt to explain the above three facts regarding yield curves by a theory called the Expectations Hypothesis.
Before we begin let us introduce some notation. Let
- be the annual interest rate on a 1-year bond at time t
- be the annual interest rate on a 2-year bond at time t
- be the annual interest rate on a n-year bond at time t
- be the expected annual interest rate on a 1-year bond at time t+j
The basic idea of the expectations hypothesis is that to see bonds of different maturities as substitutes. For example, suppose one wants to invest in bonds for two years. There are two methods of doing this.
Method 1
Invest in one 2-year bond. Such a bond will pay an annual interest rate of . Hence one receives this interest rate two times. So the total interest received with this method is 2.
Method 2
Invest in two 1-year bonds. That is, at time t invest in a one year bond that pays . Then at time t+1 reinvest in another 1-year bond, which pays an expected interest rate of . Hence the total interest paid with this method is .
Since these two methods are substitutes we expect their interest rates to be the same. That is, we expect
Or, dividing by 2 we have
This is our equation of the expectations hypothesis. It links the annual interest rate on a long-term bond (i.e. 2-year bond) to the annual interest rates on 1-year bonds, both current and expected future. In fact, a 2-year bond’s annual interest rate is just the average of current and expected future annual interest rates on 1-year bonds over the next two years.
As an example, suppose and , then
Now what is true for the two-year bond is true for three-year bonds, four-year bonds, etc. That is, an n-year bond’s annual interest rate is just the average of current and expected future annual interest rates on 1-year bonds over the next n years. Hence we have