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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.

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.

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:

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.

A Death Put an optional redemption feature on a debt instrument allowing the beneficiary of the estate of the deceased to put (sell) the bond (back to the issuer) in the event of the beneficiary's death or legal incapacitation. Also known as a "survivor's option".

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).

Everything about Bonds

Priority
In addition to the credit quality of the issuer, the priority of the bond is a determiner of the probability that the issuer will pay you back your money. The priority indicates your place in line should the company default on payments. If you hold an unsubordinated (senior) security and the company defaults, you will be first in line to receive payment from the liquidation of its assets. On the other hand, if you own a subordinated (junior) debt security, you will get paid out only after the senior debt holders have received their share.

Yield and Bond Price

The general definition of yield is the return an investor will receive by holding a bond to maturity. So if you want to know what your bond investment will earn, you should know how to calculate yield. Required yield, on the other hand, is the yield or return a bond must offer in order for it to be worthwhile for the investor. The required yield of a bond is usually the yield offered by other plain vanilla bonds that are currently offered in the market and have similar credit quality and maturity.

The multiplication by 100 in the formulas below converts the decimal into a percentage, allowing us to see the percentage return:




So, if you purchased a bond with a par value of $100 for $95.92 and it paid a coupon rate of 5%, this is how you'd calculate its current yield:



Notice how this calculation does not include any capital gains or losses the investor would make if the bond were bought at a discount or premium. Because the comparison of the bond price to its par value is a factor that affects the actual current yield, the above formula would give a slightly inaccurate answer - unless of course the investor pays par value for the bond. To correct this, investors can modify the current yield formula by adding the result of the current yield to the gain or loss the price gives the investor: [(Par Value – Bond Price)/Years to Maturity]. The modified current yield formula then takes into account the discount or premium at which the investor bought the bond. This is the full calculation:



Now we must also account for other factors such as the coupon payment for a zero-coupon bond, which has only one coupon payment. For such a bond, the yield calculation would be as follows:

n = years left until maturity

Calculating Yield for Callable and Puttable Bonds
Bonds with callable or puttable redemption features have additional yield calculations. A callable bond's valuations must account for the issuer's ability to call the bond on the call date and the puttable bond's valuation must include the buyer's ability to sell the bond at the pre-specified put date. The yield for callable bonds is referred to as yield-to-call, and the yield for puttable bonds is referred to as yield-to-put.

Yield to call (YTC) is the interest rate that investors would receive if they held the bond until the call date. The period until the first call is referred to as the call protection period. Yield to call is the rate that would make the bond's present value equal to the full price of the bond. Essentially, its calculation requires two simple modifications to the yield-to-maturity formula:


Note that European callable bonds can have multiple call dates and that a yield to call can be calculated for each.

Yield to put (YTP) is the interest rate that investors would receive if they held the bond until its put date. To calculate yield to put, the same modified equation for yield to call is used except the bond put price replaces the bond call value and the time until put date replaces the time until call date.

For both callable and puttable bonds, astute investors will compute both yield and all yield-to-call/yield-to-put figures for a particular bond, and then use these figures to estimate the expected yield. The lowest yield calculated is known as yield to worst, which is commonly used by conservative investors when calculating their expected yield. Unfortunately, these yield figures do not account for bonds that are not redeemed or are sold prior to the call or put date.

Now you know that the yield you receive from holding a bond will differ from its coupon rate because of fluctuations in bond price and from the reinvestment of coupon payments. In addition, you are now able to differentiate between current yield and yield to maturity. In our next section we will take a closer look at yield to maturity and how the YTMs for bonds are graphed to form the term structure of interest rates, or yield curve.

Term Structure of Interest Rates - Yield Curve

The term structure of interest rates, also known as the yield curve, is a very common bond valuation method. Constructed by graphing the yield to maturities and the respective maturity dates of benchmark fixed-income securities, the yield curve is a measure of the market's expectations of future interest rates given the current market conditions. Treasuries, issued by the federal government, are considered risk-free, and as such, their yields are often used as the benchmarks for fixed-income securities with the same maturities. The term structure of interest rates is graphed as though each coupon payment of a noncallable fixed-income security were a zero-coupon bond that “matures” on the coupon payment date. The exact shape of the curve can be different at any point in time. So if the normal yield curve changes shape, it tells investors that they may need to change their outlook on the economy.

There are three main patterns created by the term structure of interest rates:

1) Normal Yield Curve: As its name indicates, this is the yield curve shape that forms during normal market conditions, wherein investors generally believe that there will be no significant changes in the economy, such as in inflation rates, and that the economy will continue to grow at a normal rate. During such conditions, investors expect higher yields for fixed income instruments with long-term maturities that occur farther into the future. In other words, the market expects long-term fixed income securities to offer higher yields than short-term fixed income securities. This is a normal expectation of the market because short-term instruments generally hold less risk than long-term instruments; the farther into the future the bond's maturity, the more time and, therefore, uncertainty the bondholder faces before being paid back the principal. To invest in one instrument for a longer period of time, an investor needs to be compensated for undertaking the additional risk.

Remember that as general current interest rates increase, the price of a bond will decrease and its yield will increase.


2) Flat Yield Curve: These curves indicate that the market environment is sending mixed signals to investors, who are interpreting interest rate movements in various ways. During such an environment, it is difficult for the market to determine whether interest rates will move significantly in either direction farther into the future. A flat yield curve usually occurs when the market is making a transition that emits different but simultaneous indications of what interest rates will do. In other words, there may be some signals that short-term interest rates will rise and other signals that long-term interest rates will fall. This condition will create a curve that is flatter than its normal positive slope. When the yield curve is flat, investors can maximize their risk/return tradeoff by choosing fixed-income securities with the least risk, or highest credit quality. In the rare instances wherein long-term interest rates decline, a flat curve can sometimes lead to an inverted curve.

3) Inverted Yield Curve: These yield curves are rare, and they form during extraordinary market conditions wherein the expectations of investors are completely the inverse of those demonstrated by the normal yield curve. In such abnormal market environments, bonds with maturity dates further into the future are expected to offer lower yields than bonds with shorter maturities. The inverted yield curve indicates that the market currently expects interest rates to decline as time moves farther into the future, which in turn means the market expects yields of long-term bonds to decline. Remember, also, that as interest rates decrease, bond prices increase and yields decline.

You may be wondering why investors would choose to purchase long-term fixed-income investments when there is an inverted yield curve, which indicates that investors expect to receive less compensation for taking on more risk. Some investors, however, interpret an inverted curve as an indication that the economy will soon experience a slowdown, which causes future interest rates to give even lower yields. Before a slowdown, it is better to lock money into long-term investments at present prevailing yields, because future yields will be even lower.


The Theoretical Spot Rate Curve
Unfortunately, the basic yield curve does not account for securities that have varying coupon rates. When the yield to maturity was calculated, we assumed that the coupons were reinvested at an interest rate equal to the coupon rate, therefore, the bond was priced at par as though prevailing interest rates were equal to the bond's coupon rate.

The spot-rate curve addresses this assumption and accounts for the fact that many Treasuries offer varying coupons and would therefore not accurately represent similar noncallable fixed-income securities. If for instance you compared a 10-year bond paying a 7% coupon with a 10-year Treasury bond that currently has a coupon of 4%, your comparison wouldn't mean much. Both of the bonds have the same term to maturity, but the 4% coupon of the Treasury bond would not be an appropriate benchmark for the bond paying 7%. The spot-rate curve, however, offers a more accurate measure as it adjusts the yield curve so it reflects any variations in the interest rate of the plotted benchmark. The interest rate taken from the plot is known as the spot rate.


The spot-rate curve is created by plotting the yields of zero-coupon Treasury bills and their corresponding maturities. The spot rate given by each zero-coupon security and the spot-rate curve are used together for determining the value of each zero-coupon component of a noncallable fixed-income security. Remember, in this case, that the term structure of interest rates is graphed as though each coupon payment of a noncallable fixed-income security were a zero-coupon bond.

T-bills are issued by the government, but they do not have maturities greater than one year. As a result, the bootstrapping method is used to fill in interest rates for zero-coupon securities greater than one year. Bootstrapping is a complicated and involved process and will not be detailed in this section (to your relief!); however, it is important to remember that the bootstrapping method equates a T-bill's value to the value of all zero-coupon components that form the security.

The Credit Spread
The credit spread, or quality spread, is the additional yield an investor receives for acquiring a corporate bond instead of a similar federal instrument. As illustrated in the graph below, the spread is demonstrated as the yield curve of the corporate bond and is plotted with the term structure of interest rates. Remember that the term structure of interest rates is a gauge of the direction of interest rates and the general state of the economy. Corporate fixed-income securities have more risk of default than federal securities and, as a result, the prices of corporate securities are usually lower, while corporate bonds usually have a higher yield.



When inflation rates are increasing (or the economy is contracting) the credit spread between corporate and Treasury securities widens. This is because investors must be offered additional compensation (in the form of a higher coupon rate) for acquiring the higher risk associated with corporate bonds.

When interest rates are declining (or the economy is expanding), the credit spread between Federal and corporate fixed-income securities generally narrows. The lower interest rates give companies an opportunity to borrow money at lower rates, which allows them to expand their operations and also their cash flows. When interest rates are declining, the economy is expanding in the long run, so the risk associated with investing in a long-term corporate bond is also generally lower.

Now you have a general understanding of the concepts and uses of the yield curve. The yield curve is graphed using government securities, which are used as benchmarks for fixed income investments. The yield curve, in conjunction with the credit spread, is used for pricing corporate bonds. Now that you have a better understanding of the relationship between interest rates, bond prices and yields, we are ready to examine the degree to which bond prices change with respect to a change in interest rates.

Duration

The term duration has a special meaning in the context of bonds. It is a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows. It is an important measure for investors to consider, as bonds with higher durations carry more risk and have higher price volatility than bonds with lower durations.

For each of the two basic types of bonds the duration is the following:

1. Zero-Coupon Bond – Duration is equal to its time to maturity.

2. Vanilla Bond - Duration will always be less than its time to maturity.

Let's first work through some visual models that demonstrate the properties of duration for a zero-coupon bond and a vanilla bond.

Duration of a Zero Coupon Bond


The red lever above represents the four-year time period it takes for a zero-coupon bond to mature. The money bag balancing on the far right represents the future value of the bond, the amount that will be paid to the bondholder at maturity. The fulcrum, or the point holding the lever, represents duration, which must be positioned where the red lever is balanced. The fulcrum balances the red lever at the point on the time line at which the amount paid for the bond and the cash flow received from the bond are equal. The entire cash flow of a zero-coupon bond occurs at maturity, so the fulcrum is located directly below this one payment.

Duration of a Vanilla or Straight Bond
Consider a vanilla bond that pays coupons annually and matures in five years. Its cash flows consist of five annual coupon payments and the last payment includes the face value of the bond.


The moneybags represent the cash flows you will receive over the five-year period. To balance the red lever at the point where total cash flows equal the amount paid for the bond, the fulcrum must be farther to the left, at a point before maturity. Unlike the zero-coupon bond, the straight bond pays coupon payments throughout its life and therefore repays the full amount paid for the bond sooner.
Factors Affecting Duration
It is important to note, however, that duration changes as the coupons are paid to the bondholder. As the bondholder receives a coupon payment, the amount of the cash flow is no longer on the time line, which means it is no longer counted as a future cash flow that goes towards repaying the bondholder. Our model of the fulcrum demonstrates this: as the first coupon payment is removed from the red lever and paid to the bondholder, the lever is no longer in balance because the coupon payment is no longer counted as a future cash flow.


The fulcrum must now move to the right in order to balance the lever again:


Duration increases immediately on the day a coupon is paid, but throughout the life of the bond, the duration is continually decreasing as time to the bond's maturity decreases. The movement of time is represented above as the shortening of the red lever. Notice how the first diagram had five payment periods and the above diagram has only four. This shortening of the time line, however, occurs gradually, and as it does, duration continually decreases. So, in summary, duration is decreasing as time moves closer to maturity, but duration also increases momentarily on the day a coupon is paid and removed from the series of future cash flows - all this occurs until duration, eventually converges with the bond's maturity. The same is true for a zero-coupon bond

Duration: Other factors
Besides the movement of time and the payment of coupons, there are other factors that affect a bond's duration: the coupon rate and its yield. Bonds with high coupon rates and, in turn, high yields will tend to have lower durations than bonds that pay low coupon rates or offer low yields. This makes empirical sense, because when a bond pays a higher coupon rate or has a high yield, the holder of the security receives repayment for the security at a faster rate. The diagram below summarizes how duration changes with coupon rate and yield.

Types of Duration
There are four main types of duration calculations, each of which differ in the way they account for factors such as interest rate changes and the bond's embedded options or redemption features. The four types of durations are Macaulay duration, modified duration, effective duration and key-rate duration.

Macaulay Duration
The formula usually used to calculate a bond's basic duration is the Macaulay duration, which was created by Frederick Macaulay in 1938, although it was not commonly used until the 1970s. Macaulay duration is calculated by adding the results of multiplying the present value of each cash flow by the time it is received and dividing by the total price of the security. The formula for Macaulay duration is as follows:

n = number of cash flows t = time to maturity C = cash flow i = required yield M = maturity (par) value P = bond price

Fortunately, if you are seeking the Macaulay duration of a zero-coupon bond, the duration would be equal to the bond's maturity, so there is no calculation required.

Modified Duration
Modified duration is a modified version of the Macaulay model that accounts for changing interest rates. Because they affect yield, fluctuating interest rates will affect duration, so this modified formula shows how much the duration changes for each percentage change in yield. For bonds without any embedded features, bond price and interest rate move in opposite directions, so there is an inverse relationship between modified duration and an approximate 1% change in yield. Because the modified duration formula shows how a bond's duration changes in relation to interest rate movements, the formula is appropriate for investors wishing to measure the volatility of a particular bond. Modified duration is calculated as the following:

Effective Duration
The modified duration formula discussed above assumes that the expected cash flows will remain constant, even if prevailing interest rates change; this is also the case for option-free fixed-income securities. On the other hand, cash flows from securities with embedded options or redemption features will change when interest rates change. For calculating the duration of these types of bonds, effective duration is the most appropriate.

Effective duration requires the use of binomial trees to calculate the option-adjusted spread (OAS). There are entire courses built around just those two topics, so the calculations involved for effective duration are beyond the scope of this tutorial. There are, however, many programs available to investors wishing to calculate effective duration.



Key-Rate Duration
The final duration calculation to learn is key-rate duration, which calculates the spot durations of each of the 11 “key” maturities along a spot rate curve. These 11 key maturities are at the three-month and one, two, three, five, seven, 10, 15, 20, 25, and 30-year portions of the curve.

In essence, key-rate duration, while holding the yield for all other maturities constant, allows the duration of a portfolio to be calculated for a one-basis-point change in interest rates. The key-rate method is most often used for portfolios such as the bond-ladder, which consists of fixed-income securities with differing maturities. Here is the formula for key-rate duration: