If the yield curve is upward sloping, or normal, short-term rates are higher than longer-term rates. If the market rate of interest required return or YTM on a bond is less than the bond's coupon rate the bond's current market price will be less than par face. All other things being equal, the higher a bonds yield to maturity YTM , the lower will be its price. A basic characteristic of all bonds is that both the Coupon Interest Rate and the Yield to Maturity YTM can be expected to change during the life of the bond as economic conditions and investors' expectations change. Assume the coupon interest payments are paid annually.
To help, we have below a good overview of the term structure, interest rates and yield curves. The term structure of interest rates refers to the relationship between the yields and maturities of a set of bonds with the same credit rating. Typically, the term structure refers to Treasury securities but it can also refer to riskier securities, such as AA bonds. A graph of the term structure of interest rates is known as a yield curve.
For example, the following table shows the term structure of interest rates for Treasury securities as of June 28, Historically, the U. The yields of longer-maturity bonds tend to be higher than the yields of shorter-maturity bonds since the longer maturity bonds are riskier. This is because:. The shape and level of the yield curve can change over time.
If economic activity is expected to accelerate in the future, the yield curve tends to become steeper , since future rates are expected to be higher than they are now. If the economy is expected to slow in the future, the yield curve tends to become flatter , since future rates are expected to be lower than they are now. During periods of rising inflation, the entire yield curve shifts up as lenders require higher rates of return to compensate for the loss of purchasing power.
During periods of falling inflation, the entire yield curve shifts down. The prices of Treasury securities may be used to compute discount factors, spot rates, forward rates and yields. Discount factors can be computed directly from the prices of Treasuries. Spot rates can be computed from discount factors; forward rates can be computed from spot rates. A discount factor represents the present value of a sum. For example, if the one-year discount factor is 0.
Discount factors cannot exceed 1 and will fall continuously as their maturity increases. Discount factors may be computed from the prices and coupons of Treasury securities. As an example, the following table shows the prices and coupon rates of off-the-run Treasury securities with maturities of 6 months, 12 months, 18 months and 24 months. Off-the-run securities are those that have been issued in the past; the most recently issued securities are said to be on-the-run.
For example, if a one-year Treasury bill was issued six months ago, it is now considered to be an off-the-run six month Treasury bill. A newly-issued six-month Treasury bill is considered to be on-the-run. The annual coupon rate is 0. Since the price of a bond equals the present value of its promised future cash flows, the following relationship holds for the six-month bond:. Since this cash flow occurs in six months, its present value is obtained by multiplying it by the six month discount factor:.
With a 1. This bond will make a coupon payment in six months, and then a final coupon payment and repayment of principal when the bond matures in twelve months. These results can be confirmed by pricing the 24 month bond using these discount factors, as follows. The 24 month bond has four remaining cash flows. Therefore, the price of the bond equals:. A spot rate of interest is the yield to maturity of a zero-coupon bond.
Spot rates may be derived directly from discount factors using the following formula:. A forward rate of interest is a rate that applies to a future time interval. For example, the rate of interest that applies to a five-year loan beginning in one year is known as a five-year rate one year forward. Forward rates are uniquely determined by the pattern of spot rates observed in the market. At the end of two years, the investor will have:. The amount that the investor will have in two years cannot be determined since the one-year rate that will apply in one year is unknown.
The rate of interest that equalizes the return under both scenarios is known as the one-year rate one year forward. In other words, this is the one year rate that applies one year in the future. The one year rate one year forward can be determined by equalizing the returns to investing at the two year spot rate for two years and investing at the one year spot rate for one year and then reinvesting the proceeds for another year at the one year rate one year forward:.
The forward rate of interest can be written as f t,T , where t is the time at which the rate begins and T is the time at which the rate matures. In this example, the one-year rate one-year forward is written as f 1,2. The one-year rate two years forward is written as f 2,3 ; the two-year rate one-year forward is written as f 1,3. Forward rates can be computed directly from spot rates. Based on the previous example, the forward rates can be computed directly from the spot rates:.
For example, the six month forward rate six months forward, designated f 0. These results can be confirmed by pricing the 24 month bond using these spot rates, as follows. When pricing bonds with forward rates, each cash flow is discounted by a sequence consisting of the short-term spot rate followed by the appropriate forward rates.
The yield to maturity YTM is the rate of interest at which the market price of a bond equals the present value of its expected future cash flows. Yields cannot be determined algebraically; they must be computed with a specialized financial calculator or Microsoft Excel. Note that the settlement and maturity dates must be entered with quotes.
These numerical values may be used for settlement and maturity. These results can be confirmed by pricing the 24 month bond using these yields, as follows. Each cash flow is discounted by the two-year yield since this is the maturity of the bond. Unlike pricing a bond with discount factors, spot rates and forward rates, when pricing a bond with yields, each cash flow is discounted with the same yield.
Also note that the price of the 24 month bond is the same whether it is priced using discount factors, spot rates, forward rates or yields. The following chart summarizes the discount factors, yields, spot rates and forward rates for the previous set of examples. The spot rates shown are S 0. The forward rates are f 0, 0. Note that f 0, 0. The yields are y 0.
In this case, the yields are increasing with maturity; i. When this happens:. Several theories have been proposed to explain the relationship between the maturities of bonds and the rates paid by these bonds. These include:. According to this theory, if the yield curve is upward sloping, this indicates that investors expect short-term rates to be higher in the future. If the yield curve is downward sloping, this indicates that investors expect short-term rates to be lower in the future.
If the yield curve is flat, this indicates that investors expect short-term rates to be unchanged in the future. One of the key assumptions of the Expectations Theory is that investors do not have any preferences for bond maturities; they are indifferent between bonds of all maturities. For example, an investor who needs to save for five years would be indifferent between:. The following numerical example illustrates this idea. According to the Expectations Theory, the fact that the two-year rate is higher than the one-year rate implies that investors expect the one-year rate to be higher next year.
In order for the investor to be indifferent between the two strategies, he must expect to earn the same return from both. When long-term rates exceed short-term rates, this indicates that investors expect future short-term rates to be greater than they are today. Equivalently, if long-term rates are below short-term rates, according to the Expectations Theory, investors expect short-term rates to be lower in the future than they are today.
According to the Liquidity Premium Theory, a long-term rate of interest is an average of short-term rates plus a liquidity premium. In other words, investors expect to be compensated for holding long-term bonds instead of short-term bonds as long-term bonds are perceived to be riskier. This causes the yield curve to be steeper than it would be under the Expectations Theory. What are the implied two-year and five-year rates under the:.
Under the Market Segmentation Theory, rates are determined by supply and demand conditions in each maturity range of the yield curve. For example, year rates are influenced by the supply and demand for mortgages. According to this theory, investors have a strong preference for a specific maturity and would require a premium to invest in other bonds with different maturities. As a result, there can be a risk premium for different maturities along the yield curve, but they do not necessarily increase with maturity as they do under the Liquidity Premium Theory.
As a result, the yield curve may not rise or fall continuously. This article is one part of a series on fixed income portfolios. Other articles in this series include:. We have provided you with a good overview of the term structure, interest rates and yield curves. Call us: Term Structures, Interest Rates and Yield Curves The term structure of interest rates refers to the relationship between the yields and maturities of a set of bonds with the same credit rating.
This is because: Changes in market conditions have a greater impact on the prices of longer maturity bonds than shorter maturity bonds; and There is more uncertainty over market conditions that take place further in the future. Discount factors Spot rates Forward rates Yields The prices of Treasury securities may be used to compute discount factors, spot rates, forward rates and yields. Maturity Coupon Rate Price 6 months 0.
Since the price of a bond equals the present value of its promised future cash flows, the following relationship holds for the six-month bond: Since this cash flow occurs in six months, its present value is obtained by multiplying it by the six month discount factor: Since d 0. Therefore, the price of the bond equals:
Yield Curve Slope, Theory, Charts, Analysis (Complete Guide)
A par yield curve is a graphical representation of the yields of hypothetical Treasury securities with prices at par. On the par yield curve, the coupon rate will equal the yield-to-maturity of the security, which is why the Treasury bond will trade at par. The yield curve is a graph that shows the relationship between interest rates and bond yields of various maturities, ranging from 3-month Treasury bills to year Treasury bonds. The graph is plotted with the y-axis depicting interest rates, and the x-axis showing the increasing time durations.
Yields-to-maturity of zero-coupon government bonds could be analyzed for a full range of maturities, which is called the government bond spot curve or zero or strip curve.
Fixed Income Relations
To help, we have below a good overview of the term structure, interest rates and yield curves. The term structure of interest rates refers to the relationship between the yields and maturities of a set of bonds with the same credit rating. Typically, the term structure refers to Treasury securities but it can also refer to riskier securities, such as AA bonds. A graph of the term structure of interest rates is known as a yield curve. For example, the following table shows the term structure of interest rates for Treasury securities as of June 28, Historically, the U.WATCH THE VIDEO ON THEME: Explaining Bond Prices and Bond Yields
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par yield, YTM, and spot rate
By Amruth Sundarkumar 4 Comments. Fixed Income Tutorials. Before diving into it, I presume you must be knowing what a bond is. You can google more about the basics of bonds like par bonds, discount bonds etc. The second point to note is that bond prices and their yields in most cases move in the opposite direction.
Par Yield Curve
Given an upward sloping par yield curve, the spot rate curve will be: Upward sloping and higher. Upward sloping and lower. Downward sloping and higher. Downward sloping and lower. An upward sloping forward rate curve always lies above the upward sloping spot curve from which it was calculated. Likewise, the upward sloping spot curve always lies above the upward sloping par curve from which the spot curve was calculated. The spot rate curve will be higher than the upward sloping par yield curve.
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In finance , the yield curve is a curve showing several yields or interest rates across different contract lengths 2 month, 2 year, 20 year, etc. The curve shows the relation between the level of the interest rate or cost of borrowing and the time to maturity , known as the "term", of the debt for a given borrower in a given currency. Treasury securities for various maturities are closely watched by many traders, and are commonly plotted on a graph such as the one on the right which is informally called "the yield curve". The shape of the yield curve indicates the cumulative priorities of all lenders relative to a particular borrower such as the US Treasury or the Treasury of Japan , or the priorities of a single lender relative to all possible borrowers. With other factors held equal, lenders will prefer to have funds at their disposal, rather than at the disposal of a third party. The interest rate is the "price" paid to convince them to lend. As the term of the loan increases, lenders demand an increase in the interest received. In addition, lenders may be concerned about future circumstances, e. Occasionally, when lenders are seeking long-term debt contracts more aggressively than short-term debt contracts, the yield curve "inverts", with interest rates yields being lower for the longer periods of repayment, because borrowers find it easier to attract long-term lending. The yield of a debt instrument is the overall rate of return available on the investment.
.VIDEO ON THEME: Why Bond Prices and Yields are Inversely Related