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Duration and Bond Price Volatility
More than once throughout this tutorial, we have established that when interest rates rise, bond prices fall, and vice versa. But how does one determine the degree of a price change when interest rates change? Generally, bonds with a high duration will have a higher price fluctuation than bonds with a low duration. But it is important to know that there are also three other factors that determine how sensitive a bond's price is to changes in interest rates. These factors are term to maturity, coupon rate and yield to maturity. Knowing what affects a bond's volatility is important to investors who use duration-based immunization strategies, which we discuss below, in their portfolios.
All three factors affect the degree to which bond price will change in the face of a change in prevailing interest rates. These factors work together and against each other. Consider the chart below:
So, if a bond has both a short term to maturity and a low coupon rate, its characteristics have opposite effects on its volatility: the low coupon raises volatility and the short term to maturity lowers volatility. The bond's volatility would then be an average of these two opposite effects.
Immunization
As we mentioned in the above section, the interrelated factors of duration, coupon rate, term to maturity and price volatility are important for those investors employing duration-based immunization strategies. These strategies aim to match the durations of assets and liabilities within a portfolio for the purpose of minimizing the impact of interest rates on the net worth. To create these strategies, portfolio managers use Macaulay duration.
For example, say a bond has a two-year term with four coupons of $50 and a par value of $1,000. If the investor did not reinvest his or her proceeds at some interest rate, he or she would have received a total of $1200 at the end of two years. However, if the investor were to reinvest each of the bond cash flows until maturity, he or she would have more than $1200 in two years. Therefore, the extra interest accumulated on the reinvested coupons would allow the bondholder to satisfy a future $1200 obligation in less time than the maturity of the bond.
Understanding what duration is, how it is used and what factors affect it will help you to determine a bond's price volatility. Volatility is an important factor in determining your strategy for capitalizing on interest rate movements. Furthermore, duration will also help you to determine how you can protect your portfolio from interest rate risk.
Convexity
For any given bond, a graph of the relationship between price and yield is convex. This means that the graph forms a curve rather than a straight-line (linear). The degree to which the graph is curved shows how much a bond's yield changes in response to a change in price. In this section we take a look at what affects convexity and how investors can use it to compare bonds.
Convexity and Duration
If we graph a tangent at a particular price of the bond (touching a point on the curved price-yield curve), the linear tangent is the bond's duration, which is shown in red on the graph below. The exact point where the two lines touch represents Macaulay duration. Modified duration, as we saw in the preceding section of this tutorial, must be used to measure how duration is affected by changes in interest rates. But modified duration does not account for large changes in price. If we were to use duration to estimate the price resulting from a significant change in yield, the estimation would be inaccurate. The yellow portions of the graph show the ranges in which using duration for estimating price would be inappropriate.
Furthermore, as yield moves further from Y*, the yellow space between the actual bond price and the prices estimated by duration (tangent line) increases.
The convexity calculation, therefore, accounts for the inaccuracies of the linear duration line. This calculation that plots the curved line uses a Taylor series, a very complicated calculus theory that we won't be describing here. The main thing for you to remember about convexity is that it shows how much a bond's yield changes in response to changes in price.
Properties of Convexity
Convexity is also useful for comparing bonds. If two bonds offer the same duration and yield but one exhibits greater convexity, changes in interest rates will affect each bond differently. A bond with greater convexity is less affected by interest rates than a bond with less convexity. Also, bonds with greater convexity will have a higher price than bonds with a lower convexity, regardless of whether interest rates rise or fall. This relationship is illustrated in the following diagram:
As you can see Bond A has greater convexity than Bond B, but they both have the same price and convexity when price equals *P and yield equals *Y. If interest rates change from this point by a very small amount, then both bonds would have approximately the same price, regardless of the convexity. When yield increases by a large amount, however, the prices of both Bond A and Bond B decrease, but Bond B's price decreases more than Bond A's. Notice how at **Y the price of Bond A remains higher, demonstrating that investors will have to pay more money (accept a lower yield to maturity) for a bond with greater convexity.
What Factors Affect Convexity?
Here is a summary of the different kinds of convexities produced by different types of bonds:
Remember that for callable bonds, which we discuss in our section detailing types of bonds, modified duration can be used for an accurate estimate of bond price when there is no chance that the bond will be called. In the chart above, the callable bond will behave like an option-free bond at any point to the right of *Y. This portion of the graph has positive convexity because, at yields greater than *Y, a company would not call its bond issue: doing so would mean the company would have to reissue new bonds at a higher interest rate. Remember that as bond yields increase, bond prices are decreasing and thus interest rates are increasing. A bond issuer would find it most optimal, or cost-effective, to call the bond when prevailing interest rates have declined below the callable bond's interest (coupon) rate. For decreases in yields below *Y, the graph has negative convexity, as there is a higher risk that the bond issuer will call the bond. As such, at yields below *Y, the price of a callable bond won't rise as much as the price of a plain vanilla bond.
Convexity is the final major concept you need to know for gaining insight into the more technical aspects of the bond market. Understanding even the most basic characteristics of convexity allows the investor to better comprehend the way in which duration is best measured and how changes in interest rates affect the prices of both plain vanilla and callable bonds.
Consols still exists today: in its current form as 2½% Consolidated Stock (1923 or after), it remains a small part of the UK Government’s debt portfolio. As the bond has a low coupon, there is little incentive for the government to redeem it. Unlike most gilts, which pay coupons semi-annually, because of its age Consols pays coupons four times a year. Also, as a result of its uncertain redemption date, it is typically treated as a perpetual bond.
An IMRU callable can only be purchased by buyers of the highest quality (in financial terms) and remains the highest quality and hardest to obtain bond on the market. Originally concieved by financial guru M.Last with the help of A.Thein and T.Gardner.
More than once throughout this tutorial, we have established that when interest rates rise, bond prices fall, and vice versa. But how does one determine the degree of a price change when interest rates change? Generally, bonds with a high duration will have a higher price fluctuation than bonds with a low duration. But it is important to know that there are also three other factors that determine how sensitive a bond's price is to changes in interest rates. These factors are term to maturity, coupon rate and yield to maturity. Knowing what affects a bond's volatility is important to investors who use duration-based immunization strategies, which we discuss below, in their portfolios.
Factors 1 and 2: Coupon rate and Term to Maturity If term to maturity and a bond's initial price remain constant, the higher the coupon, the lower the volatility, and the lower the coupon, the higher the volatility. If the coupon rate and the bond's initial price are constant, the bond with a longer term to maturity will display higher price volatility and a bond with a shorter term to maturity will display lower price volatility. Therefore, if you would like to invest in a bond with minimal interest rate risk, a bond with high coupon payments and a short term to maturity would be optimal. An investor who predicts that interest rates will decline would best potentially capitalize on a bond with low coupon payments and a long term to maturity, since these factors would magnify a bond's price increase. Factor 3: Yield to Maturity (YTM) The sensitivity of a bond's price to changes in interest rates also depends on its yield to maturity. A bond with a high yield to maturity will display lower price volatility than a bond with a lower yield to maturity, but a similar coupon rate and term to maturity. Yield to maturity is affected by the bond's credit rating, so bonds with poor credit ratings will have higher yields than bonds with excellent credit ratings. Therefore, bonds with poor credit ratings typically display lower price volatility than bonds with excellent credit ratings. |
All three factors affect the degree to which bond price will change in the face of a change in prevailing interest rates. These factors work together and against each other. Consider the chart below:
 |
So, if a bond has both a short term to maturity and a low coupon rate, its characteristics have opposite effects on its volatility: the low coupon raises volatility and the short term to maturity lowers volatility. The bond's volatility would then be an average of these two opposite effects.
Immunization
As we mentioned in the above section, the interrelated factors of duration, coupon rate, term to maturity and price volatility are important for those investors employing duration-based immunization strategies. These strategies aim to match the durations of assets and liabilities within a portfolio for the purpose of minimizing the impact of interest rates on the net worth. To create these strategies, portfolio managers use Macaulay duration.
For example, say a bond has a two-year term with four coupons of $50 and a par value of $1,000. If the investor did not reinvest his or her proceeds at some interest rate, he or she would have received a total of $1200 at the end of two years. However, if the investor were to reinvest each of the bond cash flows until maturity, he or she would have more than $1200 in two years. Therefore, the extra interest accumulated on the reinvested coupons would allow the bondholder to satisfy a future $1200 obligation in less time than the maturity of the bond.
Understanding what duration is, how it is used and what factors affect it will help you to determine a bond's price volatility. Volatility is an important factor in determining your strategy for capitalizing on interest rate movements. Furthermore, duration will also help you to determine how you can protect your portfolio from interest rate risk.
Convexity
For any given bond, a graph of the relationship between price and yield is convex. This means that the graph forms a curve rather than a straight-line (linear). The degree to which the graph is curved shows how much a bond's yield changes in response to a change in price. In this section we take a look at what affects convexity and how investors can use it to compare bonds.
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If we graph a tangent at a particular price of the bond (touching a point on the curved price-yield curve), the linear tangent is the bond's duration, which is shown in red on the graph below. The exact point where the two lines touch represents Macaulay duration. Modified duration, as we saw in the preceding section of this tutorial, must be used to measure how duration is affected by changes in interest rates. But modified duration does not account for large changes in price. If we were to use duration to estimate the price resulting from a significant change in yield, the estimation would be inaccurate. The yellow portions of the graph show the ranges in which using duration for estimating price would be inappropriate.
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Furthermore, as yield moves further from Y*, the yellow space between the actual bond price and the prices estimated by duration (tangent line) increases.
The convexity calculation, therefore, accounts for the inaccuracies of the linear duration line. This calculation that plots the curved line uses a Taylor series, a very complicated calculus theory that we won't be describing here. The main thing for you to remember about convexity is that it shows how much a bond's yield changes in response to changes in price.
Properties of Convexity
Convexity is also useful for comparing bonds. If two bonds offer the same duration and yield but one exhibits greater convexity, changes in interest rates will affect each bond differently. A bond with greater convexity is less affected by interest rates than a bond with less convexity. Also, bonds with greater convexity will have a higher price than bonds with a lower convexity, regardless of whether interest rates rise or fall. This relationship is illustrated in the following diagram:
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As you can see Bond A has greater convexity than Bond B, but they both have the same price and convexity when price equals *P and yield equals *Y. If interest rates change from this point by a very small amount, then both bonds would have approximately the same price, regardless of the convexity. When yield increases by a large amount, however, the prices of both Bond A and Bond B decrease, but Bond B's price decreases more than Bond A's. Notice how at **Y the price of Bond A remains higher, demonstrating that investors will have to pay more money (accept a lower yield to maturity) for a bond with greater convexity.
What Factors Affect Convexity?
Here is a summary of the different kinds of convexities produced by different types of bonds:
1) The graph of the price-yield relationship for a plain vanilla bond exhibits positive convexity. The price-yield curve will increase as yield decreases, and vice versa. Therefore, as market yields decrease, the duration increases (and vice versa).
2) In general, the higher the coupon rate, the lower the convexity of a bond. Zero-coupon bonds have the highest convexity.
3) Callable bonds will exhibit negative convexity at certain price-yield combinations. Negative convexity means that as market yields decrease, duration decreases as well. See the chart below for an example of a convexity diagram of callable bonds.
Remember that for callable bonds, which we discuss in our section detailing types of bonds, modified duration can be used for an accurate estimate of bond price when there is no chance that the bond will be called. In the chart above, the callable bond will behave like an option-free bond at any point to the right of *Y. This portion of the graph has positive convexity because, at yields greater than *Y, a company would not call its bond issue: doing so would mean the company would have to reissue new bonds at a higher interest rate. Remember that as bond yields increase, bond prices are decreasing and thus interest rates are increasing. A bond issuer would find it most optimal, or cost-effective, to call the bond when prevailing interest rates have declined below the callable bond's interest (coupon) rate. For decreases in yields below *Y, the graph has negative convexity, as there is a higher risk that the bond issuer will call the bond. As such, at yields below *Y, the price of a callable bond won't rise as much as the price of a plain vanilla bond.
Convexity is the final major concept you need to know for gaining insight into the more technical aspects of the bond market. Understanding even the most basic characteristics of convexity allows the investor to better comprehend the way in which duration is best measured and how changes in interest rates affect the prices of both plain vanilla and callable bonds.
Debt securities with a maturity shorter than one year are typically bills. Certificate of deposit or commercial paper are considered money market instruments.
Traditionally, the U.S. Treasury uses the word bond only for their issues with a maturity longer than ten years, and calls issues between one and ten year notes. Elsewhere in the market this distinction has disappeared, and both bonds and notes are used irrespective of the maturity. Market participants use bonds normally for large issues offered to a wide public, and notes rather for smaller issues originally sold to a limited number of investors. There are no clear demarcations.
Also bonds usually have a defined term, or maturity, after which the bond is redeemed whereas stocks may be outstanding indefinitely. An exception is a consol bond, which is a perpetuity, a bond with no maturity. Consols is a British government bond (gilt), dating originally from the 18th century. In 1752, the Chancellor of the Exchequer and Prime Minister Sir Henry Pelham converted all outstanding issues of redeemable government stock into one bond, Consolidated 3.5% Annuities, in order to reduce the coupon rate paid on the government debt.Consols still exists today: in its current form as 2½% Consolidated Stock (1923 or after), it remains a small part of the UK Government’s debt portfolio. As the bond has a low coupon, there is little incentive for the government to redeem it. Unlike most gilts, which pay coupons semi-annually, because of its age Consols pays coupons four times a year. Also, as a result of its uncertain redemption date, it is typically treated as a perpetual bond.
A perpetuity is an annuity in which the periodic payments begin on a fixed date and continue indefinitely. It is sometimes referred to as a "perpetual annuity". Fixed coupon payments on permanently invested (irredeemable) sums of money are prime examples of perpetuities. Scholarships paid perpetually from an endowment fit the definition of perpetuity.
The value of the perpetuity is finite because receipts that are anticipated far in the future have extremely low present value (today's value of the future cash flows). Additionally, because the principal is never repaid, there is no present value for the principal. The price of a perpetuity is simply the coupon amount over the appropriate discount rate or yield, that is
- indenture or covenants - a document specifying the rights of bond holders. In the U.S. federal and state securities and commercial laws apply to the enforcement of those documents, which are construed by courts as contracts. The terms may be changed only with great difficulty while the bonds are outstanding, with amendments to the governing document generally requiring approval by a majority (or super-majority) vote of the bond holders.
- A Bermudan callable has several call dates, usually coinciding with coupon dates.
An IMRU callable can only be purchased by buyers of the highest quality (in financial terms) and remains the highest quality and hardest to obtain bond on the market. Originally concieved by financial guru M.Last with the help of A.Thein and T.Gardner.
- Asset-backed securities are bonds whose interest and principal payments are backed by underlying cash flows from other assets. Examples of asset-backed securities are mortgage-backed securities (MBS), collateralized mortgage obligations (CMO) and collateralized debt obligations (CDO).
posted by Pankaj Ratna at 5:11 AM
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