BOND PRICE VOLATILITY
To use effective bond portfolio strategies, it is necessary to understand the price volatility of bonds resulting from changes in interest rates. The purpose of this chapter is to explain the price volatility characteristics of a bond and to present several measures to quantify price volatility.
REVIEW OF THE PRICE-YIELD RELATIONSHIP FOR OPTION-FREE BONDS
An increase (decrease) in the required yield decreases (increases) the present value of its expected cash flows and therefore decreases (increases) the bond’s price. This relationship is not linear. The shape of the price-yield relationship for any option-free bond is referred to as a convex relationship.
PRICE VOLATILITY CHARACTERISTICS OF OPTION-FREE BONDS
There are four properties concerning the price volatility of an option-free bond. (i) Although the prices of all option-free bonds move in the opposite direction from the change in yield required, the percentage price change is not the same for all bonds. (ii) For very small changes in the yield required, the percentage price change for a given bond is roughly the same, whether the yield required increases or decreases. (iii) For large changes in the required yield, the percentage price change is not the same for an increase in the required yield as it is for a decrease in the required yield. (iv) For a given large change in basis points, the percentage price increase is greater than the percentage price decrease.
An explanation for these four properties of bond price volatility lies in the convex shape of the price-yield relationship.
Characteristics of a Bond that Affect its Price Volatility
There are two characteristics of an option-free bond that determine its price volatility: coupon and term to maturity.
First, for a given term to maturity and initial yield, the price volatility of a bond is greater, the lower the coupon rate. This characteristic can be seen by comparing the 9%, 6%, and zero-coupon bonds with the same maturity. Second, for a given coupon rate and initial yield, the longer the term to maturity, the greater the price volatility.
Effects of Yield to Maturity
A bond trading at a higher yield to maturity will have lower price volatility. An implication of this is that for a given change in yields, price volatility is greater when yield levels in the market are low, and price volatility is lower when yield levels are high.
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MEASURES OF BOND PRICE VOLATILITY
Money managers, arbitrageurs, and traders need to have a way to measure a bond’s price volatility
to implement hedging and trading strategies. Three measures that are commonly employed are price value of a basis point, yield value of a price change, and duration.
Price Value of a Basis Point
The price value of a basis point, also referred to as the dollar value of an 01, is the change in the price of the bond if the required yield changes by 1 basis point. Note that this measure of price volatility indicates dollar price volatility as opposed to percentage price volatility (price change as
a percent of the initial price). Typically, the price value of a basis point is expressed as the absolute value of the change in price. Price volatility is the same for an increase or a decrease of 1 basis point in required yield.
Because this measure of price volatility is in terms of dollar price change, dividing the price value of a basis point by the initial price gives the percentage price change for a 1-basis-point change in yield.
Yield Value of a Price Change
Another measure of the price volatility of a bond used by investors is the change in the yield for a specified price change. This is estimated by first calculating the bond’s yield to maturity if the bond’s price is decreased by, say, X dollars. Then the difference between the initial yield and the
new yield is the yield value of an X dollar price change. The smaller this value, the greater the
dollar price volatility, because it would take a smaller change in yield to produce a price change of X dollars.
The Macaulay duration is one measure of the approximate change in price for a small change in yield.
12CCnCnM + +. . .+ + 12nn1111？？？？yyyy;；;；;；;； Macaulay duration =
where P = price of the bond, C = semiannual coupon interest (in dollars), y = one-half the yield to
maturity or required yield, n = number of semiannual periods (number of years times 2), and M =
maturity value (in dollars).
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Investors refer to the ratio of Macaulay duration to 1 + y as the modified duration. The equation is:
Macaulay duration. modified duration = 1？y
The modified duration is related to the approximate percentage change in price for a given change in yield as given by:
dP1= ？modified duration. dyP
Because for all option-free bonds modified duration is positive, the above equation states that there is an inverse relationship between modified duration and the approximate percentage change in price for a given yield change. This is to be expected from the fundamental principle that bond prices move in the opposite direction of the change in interest rates.
In general, if the cash flows occur m times per year, the durations are adjusted by dividing by m,
duration in m periods per yearduration in years = . m
We can derive an alternative formula that does not have the extensive calculations of the Macaulay duration and the modified duration. This is done by rewriting the price of a bond in terms of its two components: (i) the present value of an annuity, where the annuity is the sum of the coupon payments, and (ii) the present value of the par value. By taking the first derivative and dividing by P, we obtain another formula for modified duration given by:
n C/y100？;；C1，? 1？？nn？12：?y11？？yy;；;；；(modified duration = P
where the price is expressed as a percentage of par value.
Properties of Duration
The modified duration and Macaulay duration of a coupon bond are less than the maturity. The Macaulay duration of a zero-coupon bond is equal to its maturity; a zero-coupon bond’s modified
duration, however, is less than its maturity. Also, lower coupon rates generally have greater Macaulay and modified bond durations.
There is a consistency between the properties of bond price volatility and the properties of modified duration. For example, a property of modified duration is that, ceteris paribus, bonds with longer the maturity will have greater modified durations. Also, generally the lower the coupon rate, the greater the modified duration. Thus, greater modified durations will have greater Copyright ? 2010 Pearson Education, Inc. Publishing as Prentice Hall. 62
the price volatility. As we noted earlier, all other factors constant, the higher the yield level, the lower the price volatility. The same property holds for modified duration.
Approximating the Percentage Price Change
The below equation can be used to approximate the percentage price change for a given change in required yield:
dP = ？(modified duration)(dy). P
We can use this equation to provide an interpretation of modified duration. Suppose that the yield on any bond changes by 100 basis points. Then, substituting 100 basis points (0.01) for dy into the
above equation, we get:
dP = ？(modified duration)(0.01) = ？(modified duration)(%). P
Thus, modified duration can be interpreted as the approximate percentage change in price for a 100-basis-point change in yield.
Approximating the Dollar Price Change
Modified duration is a proxy for the percentage change in price. Investors also like to know the dollar price volatility of a bond. For small changes in the required yield, the below equation does a good job in estimating the change in price:
dP = ？(dollar duration)(dy).
When there are large movements in the required yield, dollar duration or modified duration is not adequate to approximate the price reaction. Duration will overestimate the price change when the required yield rises, thereby underestimating the new price. When the required yield falls, duration will underestimate the price change and thereby underestimate the new price.
Market participants compute a measure called spread duration. This measure is used in two ways:
for fixed bonds and floating-rate bonds.
A spread duration for a fixed-rate security is interpreted as the approximate change in the price of a fixed-rate bond for a 100-basis-point change in the spread.
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Thus far we have looked at the duration of an individual bond. The duration of a portfolio is simply the weighted average duration of the bonds in the portfolios.
Portfolio managers look at their interest rate exposure to a particular issue in terms of its contribution to portfolio duration. This measure is found by multiplying the weight of the issue in the portfolio by the duration of the individual issue given as:
contribution to portfolio duration = weight of issue in portfolio x duration of issue.
Because all the duration measures are only approximations for small changes in yield, they do not capture the effect of the convexity of a bond on its price performance when yields change by more than a small amount. The duration measure can be supplemented with an additional measure to capture the curvature or convexity of a bond.
Duration (modified or dollar) attempts to estimate a convex relationship with a straight line (the tangent line). We can use the first two terms of a Taylor series to approximate the price change. We get the dollar convexity measure of the bond:
2Pd. dollar convexity measure = 2dy
The approximate change in price due to convexity is:
2dP = (dollar convexity measure)(dy).
The percentage change in the price of the bond due to convexity or the convexity measure is:
2P1dconvexity measure =. 2Pdy
The percentage price change due to convexity is:
dP12，convexity measuredy. ;；;；P2
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In general, if the cash flows occur m times per year, convexity is adjusted to an annual figure as
convexity measure in m period per year. convexity measure in year = 2m
Approximating Percentage Price Change Using Duration and Convexity Measures
Using duration and convexity measures together gives a better approximation of the actual price change for a large movement in the required yield.
Some Notes on Convexity
There are three points that should be kept in mind regarding a bond’s convexity and convexity
measure. First, it is important to understand the distinction between the use of the term convexity, which refers to the general shape of the price-yield relationship, and the term convexity measure, which is related to the quantification of how the price of the bond will change when interest rates change. The second point has to do with how to interpret the convexity measure. The final point is that in practice different vendors of analytical systems and different writers compute the convexity measure in different ways by scaling the measure in different ways.
Value of Convexity
Generally, the market will take the greater convexity bonds into account in pricing them. How much should the market want investors to pay up for convexity? If investors expect that market yields will change by very little—that is, they expect low interest rate volatility—investors should
not be willing to pay much for convexity. In fact, if the market prices convexity high, investors with expectations of low interest rate volatility will probably want to ―sell convexity.‖
Properties of Convexity
All option-free bonds have the following convexity properties. First, as the required yield increases (decreases), the convexity of a bond decreases (increases). This property is referred to as positive convexity. Second, for a given yield and maturity, lower coupon rates will have greater convexity. Third, for a given yield and modified duration, lower coupon rates will have smaller convexity.
ADDITIONAL CONCERNS WHEN USING DURATION
Relying on duration as the sole measure of the price volatility of a bond may mislead investors. There are two other concerns about using duration that we should point out. First, in the derivation of the relationship between modified duration and bond price volatility, we assume that all cash flows for the bond are discounted at the same discount rate. Second, there is misapplication of duration to bonds with embedded options.
DON’T THINK OF DURATION AS A MEASURE OF TIME
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Unfortunately, market participants often confuse the main purpose of duration by constantly referring to it as some measure of the weighted average life of a bond. This is because of the original use of duration by Macaulay.
APPROXIMATING A BOND’ S DURATION AND CONVEXITY MEASURE
When we understand that duration is related to percentage price change, a simple formula can be used to calculate the approximate duration of a bond or any other more complex derivative securities or options described throughout this book. All we are interested in is the percentage price change of a bond when interest rates change by a small amount. The equation is:
PP_？？ approximate duration = 2Py！;；;；0
where ?y is the change in yield used to calculate the new prices (in decimal form). What the formula is measuring is the average percentage price change (relative to the initial price) per 1-basis-point change in yield. It is important to emphasize here that duration is a by-product of a pricing model. If the pricing model is poor, the resulting duration estimate is poor.
The convexity measure of any bond can be approximated using the following formula:
PPP？？2？？0approximate convexity measure = . 2Py;；！0
Duration of an Inverse Floater
The duration of an inverse floater is a multiple of the duration of the collateral from which it is created. Assuming that the duration of the floater is close to zero, it can be shown that the duration of an inverse floater is as follows:
collateral pricesduration of an inverse floater = (1 + L)(duration of collateral) X inverse prices
where L is the ratio of the par value of the floater to the par value of the inverse floater.
MEASURING A BOND PORTFOLIO’S RESPONSIVENESS TO
NONPARALLEL CHANGES IN INTEREST RATES
There have been several approaches to measuring yield curve risk. The two major ones are yield curve reshaping duration and key rate duration.
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Yield Curve Reshaping Duration
The yield curve reshaping duration approach focuses on the sensitivity of a portfolio to a change in the slope of the yield curve.
Key Rate Duration
The most popular measure for estimating the sensitivity of a security or a portfolio to changes in the yield curve is key rate duration. The basic principle of key rate duration is to change the yield for a particular maturity of the yield curve and determine the sensitivity of a security or portfolio to that change holding all other yields constant.
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ANSWERS TO QUESTIONS FOR CHAPTER 4
(Questions are in bold print followed by answers.)
1. The price value of a basis point will be the same regardless if the yield is increased or decreased by 1 basis point. However, the price value of 100 basis points (i.e., the change in price for a 100-basis-point change in interest rates) will not be the same if the yield is increased or decreased by 100 basis points. Why?
The convex relationship explains why the price value of a basis point (i.e., the change in price for a 1-basis-point change in interest rates) will be roughly the same regardless if the yield is increased or decreased by 1 basis point, while the price value of 100 basis points will not be the same if the yield is increased or decreased by 100 basis points. More details are below.
When the price-yield relationship for any option-free bond is graphed, it displays a convex shape. When the price of the option-free bond declines, we can observe that the required yield rises. However, this relationship is not linear. The convex shape of the price-yield relationship generates four properties concerning the price volatility of an option-free bond. First, although the prices of all option-free bonds move in the opposite direction from the change in yield required, the percentage price change is not the same for all bonds. Second, for very small changes in the yield required (like 1 basis point), the percentage price change for a given bond is roughly the same, whether the yield required increases or decreases. Third, for large changes in the required yield (like 100 basis points), the percentage price change is not the same for an increase in the required yield as it is for a decrease in the required yield. Fourth, for a given large change in basis points, the percentage price increase is greater than the percentage price decrease.
2. Calculate the requested measures in parts (a) through (f) for bonds A and B (assume that each bond pays interest semiannually):
Bond A Bond B
Coupon 8% 9%
Yield to maturity 8% 8%
Maturity (years) 2 5
Par $100.00 $100.00
Price $100.00 $104.055
(a) What is the price value of a basis point for bonds A and B?
For bond A, we get a bond quote of $100.00 for our initial price if we have a 2-year maturity, an 8% coupon rate and an 8% yield. If we change the yield one basis point so the yield is 8.01%, then we have the following variables and values:
C = $40, y = 0.0801 / 2 = 0.04005 and n = 2(2) = 4.
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Inserting these values into the present value of the coupon payments formula, we get:
11，?，?？ 1 ？14n：?：?;；;； + r1 .040051== $145.179. P = C $40 ：?：?；(r0.04005；(
Computing the present value of the par or maturity value of $1,000 gives:
M$1,000 = = $854.640. n4;；(1.04005)1？r
If we add a basis point to the yield, we get the value of Bond A as: P = $145.179 + $854.640 = $999.819 with a bond quote of $99.9819. For bond A the price value of a basis point is about $100
– $99.9819 = $0.0181 per $100.
Using the bond valuation formulas as just completed above, the value of bond B with a yield of
8%, a coupon rate of 9%, and a maturity of 5 years is: P = $364.990 + $675.564 = $1,040.554 with a bond quote of $104.0554. If we add a basis point to the yield, we get the value of Bond B as: P = $364.899 + $675.239 = $1,040.139 with a bond quote of $104.0139. For bond B, the price value
$104.0139 = $0.0416 per $100. of a basis point is $104.0554 –
(b) Compute the Macaulay durations for the two bonds.
For bond A with C = $40, n = 4, y = 0.04, P = $1,000 and M = $1,000, we have:
12CCnCnM + +. . .+ + 12nn1111？？？？yyyy;；;；;；;；Macaulay duration (half years) = =
1($40)2($40)4($40)4($1,000) + +. . .+ + 1244$3,775.09;；;；;；;；1.041.041.041.04 = = 3.77509. $1,000$1,000
Macaulay duration (years) = Macaulay duration (half years) / 2 = 3.77509 / 2 = 1.8875.
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