Bond Convexity Floyd Vest, August 2011“Bond Duration,” and “Bond Pricing Theorems.”) Bond Convexity August, 2011 5 It can be proved that, in general, the higher the coupon

Bond Convexity August, 2011 1

Bond Convexity

Floyd Vest, August 2011 In financial literature, bond convexity is an important concept for understanding bond price and yield. (See www.investopedia.com/university/advancedbond.) The following Example 1 will demonstrate this application. Example 1 For a 20-year, annual pay bond, with value at maturity of $100 and yearly coupon payments of $10, a 10% bond, the price P is

(1) 20

201 (1 )10 100(1 )iP ii

−−⎡ ⎤− += + +⎢ ⎥

⎣ ⎦, where i is interest rate or yield.

To do a Price/Yield graph on the TI83/84, write in the Y= functions, Y= 10((1–(1+x)∧(-)20)÷x)+100(1+x)∧(-)20 where x = i. In Window, write Xmin = 0, Xmax = .3, Xscl = .01, Ymin = 0, Ymax = 400, Yscl = 100 and graph. You see the graph in Figure 1(a). You see that the curve is convex, and that as i approaches zero, price approaches $300. For example, let i = 0.001 and price = $297.72. (See the Side Bar Notes for how to use Y= on the TI83/84 to evaluate P at a point.) P is undefined at i = 0. As i increases, the curve approaches the i-axis asymptotically from above. (See the Side Bar Notes for the general formula for pricing a bond on coupon date.) One way to describe a convex curve is to say that all open line segments connecting points on the curve are on the same side of the curve. But in this context, positive convexity seems to mean that in the first quadrant, as yield i decreases, the absolute value of slope increases. In Example 1, the rate of increase of the absolute value of slope is increasing. Figure 1(b) also displays the duration line, which is tangent to the Price/Yield curve at the point

ic , P(ic )( ) and has the same slope as the curve at that point.

(a) (b) Figure 1: TI83/84 Price/yield graph for the bond in Example 1.


To investigate the convexity of the price/yield curve, we will use several concepts and symbols including bond duration (See the article entitled “Bond Duration,” 2011, Formula 3, in this course.), Δi (change in i), ΔP (change in P), actual price and actual changes in price and i, approximations of changes based on bond duration, and |e| which is the absolute value of the difference between actual values and approximations. (See the Side Bar Notes for a formula for MacD (duration) for an annual pay bond.) The approximation for ΔP is

(2) !P " #DP

1+ i$%&

'()!i and

!PP

" #D1+ i

!i . If duration is high there is more down side risk.

Duration Actual Approximation |e|

D(0.01) = 13.239608 From 0.01 to 0.02

ΔP = 230.81 – 262.41 = –$31.60

ΔP ≈ –$34.40 |e(0.01)| = $2.80

D(0.06) = 11.040694 From 0.06 to 0.07

ΔP = 131.78 – 145.88 = –$14.10

ΔP ≈ –$15.19 |e(0.06)| = $1.09

D(0.11) = 8.8203693 From 0.11 to 0.12

ΔP = 85.06 – 92.04 = –$6.98

ΔP ≈ –$7.31 |e(0.11)| = $0.34

Table 1: for Example 1- Accuracy of approximations. In Table 1 starting at D(0.06), bond duration D calculated at 0.06 = 6% is denoted by D(0.06). Under Actual, prices are given for i = 0.06 and 0.07 and the change in P (ΔP) for Δi = 0.01. Under Approximation is the approximation for ΔP calculated by the above formula. In the column headed |e| is the absolute value of the difference between actual ΔP and the approximation to ΔP calculated from bond duration. This is done at three points, i = 0.01, i = 0.06, and i = 0.11. Each Δi is an increase in i of 0.01. For example, in the 0.06 row, the increase in i from 0.06 to 0.07 is 0.01 = Δi. Examination of Table 1 reveals that |e(0.01)| = $2.80, |e(0.06)| = $1.09, and |e(0.11)| = $0.34. Of these three values, |e| is greatest where the curve bends the most away from the duration line at the point, where the curve has the greatest positive convexity, and where the rate of change of the slope of the curve is greatest. As you can see, the convexity of the curve increases as you move along the i-axis from right to left. So |e| is proportional to the rate of change of slope of the curve in the neighborhood where D is evaluated. Thus the more curved the price function of a bond, the more inaccurate the estimate of interest rate sensitivity. If you have had calculus, you know that the first derivative evaluated at a point gives the slope of the curve at the point, and the second derivative give the rate of change of slope.


You notice that the slope of the curve is negative, and that D(0.01) is greater than D(0.11). Negative duration (–D) is proportional to slope.

(3) In fact, slope at ic = ! 1

1+ icD(ic )( ) P(ic )( ) in dollars per unit of one. To get

dollars per percentage point, divide by 100. The ratio ( )P i ii+ΔΔ

is also an approximation

for the slope of the curve at a point. (See Ho Kuen Ng, “The Duration of a Bond”, 1988, in this course for a derivation of the bond Duration concept and formula.) By examining the duration line and the convexity of the yield curve, as you move farther from the point ic, the space between the actual bond price and the duration line increases, and the error of the estimate of price P, the change in price ΔP, and percent change PPΔ increases. The main thing is that we see the convexity of the Price/Yield curve, and

how it affects the changes in price P in response to changes in yield i, and how it affects the accuracy of approximations based on bond duration. In general, the higher the convexity the more sensitive the bond price is to decreasing interest rates (increase in price) and the less sensitive it is to increasing rates (decrease in price). The price sensitivity to interest rate changes is highest for zero-coupon bonds. Comparing different bonds It is possible for two bonds to have the same yield, price, and duration, but different convexity. Consider Figure 2.

Figure 2: Convexity of Bond A and Bond B. As you can see in Figure 2, Bond A has greater convexity than Bond B. They both have the same price and yield at ic. If yield changes from this point by a very small amount, both bonds would have approximately the same price. But when increased by a large amount, both bonds decrease in price but Bond B’s price decreases more than Bond A’s price. So a price of a bond with greater convexity has less sensitivity to interest rate increases and Bond A’s market price runs higher than Bond B’s price. Investors will


often pay more for a bond with greater convexity because it has less down side risk and more up side advantage. They are more likely to sell a bond with less convexity because it has more down side risk. Bond convexity is a risk measurement figure. For a callable bond with yield ic, if market interest rates fall below ic, the issuer is likely to call the bond (pay it off.) because they can borrow money at a lower rate. As market yields get greater there is little chance that the bond will be called, thus positive convexity. We know that for a "plain vanilla” bond, as market yields decline, the market price rises. But for a callable bond, as market yields decline below ic, the bond's price can have a slow down in increase in price and thus negative convexity. See Figure 3.

Figure 3: Convexity for a callable bond. More about the duration line in Figure 1(b) The duration line in Figure 1(b) is tangent to the Price/Yield curve at i = 0.10. Referring to Formula 3 above.

The slope of the duration line at i = ! 1

1+ iD(i)P(i) in dollars per unit of one.

To apply slope to change in dollars per one percentage point in i, divide by 100. (See Ho Kuen Ng in this course for a proof of Formula 3.) If the duration line in Figure 1(b) were tangent to the Price/Yield curve at i = 0.06 as in our Example 1, D(0.06) = 11.040694 years and P(0.06) = $145.88. This makes slope of the duration line at 0.06 =

! 1

1.06(11.040694)(145.88) = –$1519.4461 per one unit or –$15.194461 per one

percentage point. We can check this with an approximation, for this version of slope,

such as P(0.06)! P(0.07)

0.01 = –$14.10 per one percentage point. When writing the

equation of the tangent line, use slope in dollars per one unit. Summary The Price/Yield graph of a “plain vanilla” bond exhibits positive convexity, that is as yield decreases, the price of the bond increases, the absolute value of the slope increases, and the rate of change of the slope increases. The price of the bond decreases as market rate increase. In general, the longer the time to maturity, the greater the volatility (See “Bond Duration,” and “Bond Pricing Theorems.”)


It can be proved that, in general, the higher the coupon rate, the lower the convexity of the bond. (See Malkiel.) Zero coupon bonds have the greatest convexity. Callable bonds will have negative convexity at certain price-yield combinations. With negative convexity as market yields decrease, duration (slope) decreases. (See Figure 3.) For the more common semiannual pay bonds, see the Side Bar Notes.


Exercises 1. Investigate the Formula 1 Price/Yield curve for negative and positive values of i. On a TI83/84, use Graph, Table, Trace, and Solver. Report and draw what you see in terms of asymptotes, intercepts, minimum points, and points where P is undefined. (a) For P in the first quadrant, Put P = into a Y= and graph for x = i from 0 to .5 with x scale of .1, for P from 0 to 350 with y scale of 50. Draw and report. (b) To see another horizontal asymptote and a minimum point, graph for x from –10 to –1.5 with scale of 1, and y from –10 to 10, scale 1. Use Trace to estimate the coordinates of the minimum point and the coordinates of the i-intercept point. Use the Solver to calculate the coordinates of the i-intercept point. (c) Use Table and Graph to investigate the vertical asymptote at x= i = –1. Draw and report. (d) Do the mathematics to predict the graphical behavior such as asymptotes. 2. Discuss the slope and changing slope of Example 1, Price/Yield curve in quadrant I. (a) Is the slope at points along the curve positive or negative? (b) As i increases, how does the slope change? (c) As i decreases, how does the absolute value of the slope change? (d) Do you think the rate of change of the slope is constant? Why? Describe it as i gets larger, and as i gets smaller. 3. Theorem 4 in Malkiel (referred to as the Asymmetry Theorem) says that “A decrease in yield raises the bond price more than the same increase in yield lowers the price.” Do the calculations at i = 0.06 and Δi = 0.01, examining ΔP for i from 0.06 to 0.07 and for i from 0.06 to 0.05. Use the bond in Formula 1 and some of the numbers you need are in Table 1. 4. Not all bonds are highly convex.

(a) Graph the annual pay bond P = 105

51 (1 ) 100(1 )i ii

−−⎡ ⎤− + + +⎢ ⎥

⎣ ⎦ in the first quadrant.

Discuss where it approaches the P-axis, the asymptote, and the convexity. (b) Describe the terms of the bond in the general formula in terms of value at Maturity M, coupon C, and time to maturity, K. (c) Find the Price at i = 6%.

(d) For this bond D(0.06) = 4.235 years. Use Formula 2 to estimate ΔP and PPΔ for a

change in i of one percentage point. By how much and by what percent would the bond price change from a decline in interest rates of two percentage points?


5. For the bond in Figure 1, the slope of the duration line (tangent line) at (0.06, 145.88)

is ! 1

1.06D(0.06)( ) P(0.06)( ) = –1519.516

(a) Write the equation of this duration line. (b) Find the intercepts with the P-axis and the i-axis. (c) On a TI83/84, graph both the Price/Yield curve and this tangent line (duration line) on the same axes. Use Trace to check these calculations. Use Trace on the line to check the slope. Discuss. 6. For the graph in Figure 1, as i increases, the negative slope increases. Prove that if slope s is negative, and slope is increasing from i to i + Δi, where i > 0 and Δi > 0, then |s| is increasing from i + Δi to i. Assume that "s is increasing" means for i and Δi, s(i + Δi) > s(i). 7. (a) For the bond in Example 1, use a larger Δi = 0.04 from i to i + Δi to show that if we use duration to estimate price and percentage change in price for a larger change in yield, the estimate would be more inaccurate than those in Table 1. Do this for D(0.01), D(0.06), and D(0.11). Where is the estimate more accurate and why? (b) Show that if we use D(0.06) all along the curve, the approximations are more inaccurate than for D evaluated at the different points where the D(0.06) estimates are applied. You can use the numbers in Table 1. Summarize. 8. Consider Bond C, which is a five-year annual pay bond with coupon of $60 per year, and value at maturity of $700. Bond D is a five-year zero coupon bond with $1000 at maturity. (a) On a TI83/84, graph both bonds on the same axes for i from 0 to 0.15, and P from 0 to 1100. The graph of Bond C will be above that of Bond D. As interest rates rise, which bond has price decline faster (It is the most sensitive to interest rate increases.)? (b) Which bond has the greatest convexity? (c) What can you say about bond convexity and price sensitivity? (d) What can you say about a zero coupon bond? (d) Give a general definition of “Bond C has greater convexity than Bond D.” Answer in complete sentences.


9. (a) Use the formula R i( ) = ! 1

1+ iD i( )P i( ) / 100 (which gives the approximate dollar

change in P for a one percentage point change in i) and Table 1 to show that as i increases the absolute value of rate of change of P with respect to i decreases and as i decreases, it increases. Build a table for i, R(i), and |R(i)|. Write a summary. (b) You can illustrate the rate of change of slope with respect to i (Convexity, D2) by calculating where |D2| is proportional to |R2(0.06 to 0.01)| =

| R(0.06)! R(0.01) |

0.06! 0.01= |!15.20! (!30.26) |

0.06! 0.01= 15.07

0.05 = 301.4. For rate of change in |R| for

change in i of one percentage point, divided by 100 to get $3.14 per (percentage point)2. Calculate |R2(0.11 to 0.06)| as change to change in |R| per percentage point change in i and compare. Summarize. 10. Derive the closed form formula for D1 in the Side Bar Notes from the expanded form D1(i) in the Side Bar Notes.

11. Given a semiannual pay bond, which pays $100 at maturity in 122

years with

semiannual coupons of $2.50, and which sold on a coupon date at $95.91, what is the adjusted semiannual coupon interest rate, what is the annual quoted yield?


Side Bar Notes Since you have Formula 1 in Y=, you can use the TI83/84 to evaluate P at a point. Example: Evaluate P at 0.06. Code and commentary: Vars > to Y-vars Enter ∨ ∨ to your Y= Enter You see your Y on the home screen (.06) Enter You read 145.8796849 Conclusion P(.06) = $145.88 (For the sake of accuracy, it is better to let the decimal places on your calculator float.) To use DRAW for a tangent line at a point: An Example for Problem 5: Code and commentary: Have the graph of the curve drawn on the calculator. Code: 2nd Draw 5 To select tangent( .06 Enter This draws the tangent line to the point on the curve at i = 0.06 and writes the slope-intercept equation Y = -1519.516X + 237.0 for P = –1519.516i + 237.0 . The large |–1519.516| can be explained. For i = 0, you get the point (0, 237.0), For P = 0

you get the point (0.15136, 0). The slope =

0! 237.00.1559707 ! 0

= –1519.516. The slope

represents a family of parallel lines, all having the same slope, one line going from (1, 0)

to (0, 1519.516) has a slope of 0 1519.5161 0

−−

= –1519.516. Slope of –$1519.516 can be

thought of as a drop of $1519.516 per increase of one unit on the i-axis. The general expanded sum for the price of the bond in Formula 1 is

1(1 ) (1 )

Km K

mP C i M i− −

=

= + + +∑ , where K is the number of coupon periods to maturity, C is

the coupon, M is the value at maturity, and i is the interest rate. The closed form formula for P is illustrated by Formula 1 in this article. For an annual pay bond, the formula for D1(i) with respect to i, which gives the slope of

the Price/Yield curve at i is 1 1

11( ) ( )(1 ) ( )(1 )

Km K

mD i m i M K i− − − −

=

= − + + − +∑ , which shows

that for all i > 0, slope is negative.


The expanded formula that gives rate of change of slope with respect to i is

2 2

12( ) ( )( 1)(1 ) ( 1) (1 )

Km K

mD i C m m i K K M i− − − −

=

= + + + + +∑ , which shows that for

all i>0, D2(i) is positive so that as i increases, the negative slope is increasing, and as i decreases absolute value of slope is increasing. D2(i) (the second derivative with respect to i) evaluated at a point is the convexity of the curve at that point. Examination of D3(i) would reveal that it is negative for all i > 0, that as i decreases the absolute value of slope is increasing faster. Any student that has had calculus can see that to get D1 and D2, all we did was to differentiate term by term. Below is a closed form formula for MacD for the bond duration of an annual pay bond where i = yield, C = coupon, M = value at maturity, K is the number of coupon periods and years to maturity, for MacD calculated on a coupon date and at the value i,

MacD = 1+ ii

!(1+ i)+ K C

M! i

"#$

%&'

CM

(1+ i)K !1() *+ + i years. (See “Bond Duration.” in this course.)

The following closed form formula for D1 with respect to i gives the slope of the Price/Yield curve:

11 1 (1 ) 11 ( )( )1

KKi iD C K C Mii i i

+−⎡ ⎤+ − +⎛ ⎞ ⎛ ⎞= + − +⎜ ⎟ ⎢ ⎥ ⎜ ⎟+⎝ ⎠ ⎝ ⎠⎣ ⎦. In our discussion this gives

slope in dollars per unit of one. To apply slope to dollars per percentage point, divide by 100. The next closed form formula is for D2 with respect to i, and gives bond convexity:

D2=

2Ci3 + CK

(1+ i)K+1

!(K +1)i(1+ i)

! 1i2

"

#$

%

&' +

C(1+ i)K

!Ki(1+ i)!1 ! 2i3

"

#$

%

&' +

MK(K +1)(1+ i)K+2 , in dollars

per unit of one, per unit. To get an application of convexity in dollars per percentage point, divide by 100 × 100.


For the more common semiannual pay bond, see the article “Bond Duration” for MacD for a semiannual pay bond, and D1 in expanded form. The below closed form formula

gives D1 with respect to y, where y is the annual yield and 2y is the semiannual rate and K

is the number of coupon periods, and M is the value at maturity. 12 1

2

(1 ) 2(1 )1 2 21 ( ) 12 2 2 2

K K Ky yKy KM yD Cy y

− − −− − −⎡ ⎤+ +⎢ ⎥− ⎛ ⎞ ⎛ ⎞= + + − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

. In our examples,

slope is in dollars per unit of one. To get “slope” in dollars per percentage point, divided

by 100. Slope at i = 2y is equal to – 1

1 i+MacD(i)P(i). Use MacD for a semiannual bond

given in “Bond Duration.” The following closed form gives D2 with respect to y, for the semiannual pay bond.

D2 =

1 2

2

3 2

1 (1 ) ( 1) 14 4(1 ) ( 1)2 22 12 4 2

K K

K Ky yy K K K

K K M yCy y y

− − − −

− − −⎡ ⎤⎛ ⎞ ⎛ ⎞+ − + +⎢ ⎥− + ⎜ ⎟ ⎜ ⎟ + ⎛ ⎞⎝ ⎠ ⎝ ⎠⎢ ⎥+ − + +⎜ ⎟⎢ ⎥ ⎝ ⎠⎢ ⎥⎣ ⎦

.

In our examples, this give rate of change is in dollars per unit of one, per unit. To apply D2 in dollars per (percentage point)2, divide by 100 × 100. D2(y) is the convexity of the curve at y.


References Malkiel, Burton G., “Expectations, Bond Prices, and the Term Structure of Interest Rates,” Quarterly Journal of Economics, 76, No.2 (May, 1962), pp. 197-218. For background for this article, see in this course: Vest, Floyd and Reynolds Griffith, “The Mathematics of Bond Pricing and Interest Rate Risk,” HiMAP Pull-Out, Consortium 59, Fall 1996. Vest, Floyd, “Bond Duration,” Spring 2011. Vest, Floyd, “Bond Pricing Theorems,” Spring 2011. Ho Kuen Ng, “The Duration of a Bond” UMAP Module 685, 1989. Does the derivation of bond duration by using the first derivative of price. For a copy of the TI 84 manual, see http://www.ti.com/calc. See http://en.wikipedia.org/wiki/Bond_convexity. For the basics of mathematics of finance, see in this course: Kasting, Martha, “Concepts of Math for Business: The Mathematics of Finance,” UMAP Modules 99370-99372, 1982. 34 pages. Provides the basics of mathematics of finance. Starts with a review of algebra. Does mathematics of finance on pages 26-43. Has Compound Interest, Annuities, and Amortization with derivations, examples, exercises, sample test, and answers. Luttman, Frederick W, “Selected Applications of Mathematics to Finance and Investment,” UMAP Module 99381, 1983. (51 pages of the basics of mathematics of finance, 24 formulas, derivations, examples, exercises, model test, and answers). Has some financial mathematics of real estate investments, and some calculus. Does Compound Interest, Continuous Compound Interest, Annuities, Mortgages, Present Value of an Annuity, and APR. The calculus can be left out.


Answers to Exercises 1. (b) You see as i → –∞, P approaches the i-axis from below. By Trace, you see a minimum point at approximately (–2.4, –4.04). By using Solver, you get an i-intercept of i = –2.16894. Code and commentary: Go to Y= and highlight your Y. Math 0 for Solver ∧ ∧ for Equation Solver You see 0 = Vars > to Y-vars ∨ ∨ to your Y Enter You see 0 = Y Enter For first estimate, put in –2.2 Enter For bound, put in –2.2, –2.1 ∧ ∧ to 0 = Y in Equation Solve Alpha Solve Alpha Solve and read –2.16894. 5. (a) An equation of the tangent line (duration line) at (0.06, 145.88) with slope –1519.516 is P = –1519.516i + 237.05. See the Side Bar Notes for an example of using DRAW on the TI83/84 to draw a tangent line at a point and get the y-intercept equation of the line. 8. (d) Bond C has greater positive convexity than Bond D means for i > 0, as i decrease, the absolute value of slope of Bond C increases faster than the absolute value of slope of Bond D. 9. (a) Table for i, R(i), |R(i)|

i 0.01 0.06 0.11 R(i) –30.255781 –15.194495 –9.154824 |R(i)| 30.255781 15.194495 9.154824

At i = 0.06, Price P declines by about $30.26 for a one percentage point change in i, at i = 0.06, P declines by about $15.19, and at i = 0.11, P declines by $9.15. So, as i increases, the absolute value of rate of decline in P is decreasing. Similarly, as i decreases, the absolute value of rate of appreciation of P increases. These are all approximations (b) |R2(0.11 to 0.06)| = 121. Divide by 100 to get $1.12 per (percentage point)2. Summary: |R2(0.06 to 0.01)|/100 = 3.014 and |R2(0.11 to 0.06)|/100 = 1.21. As i increases, the rate of change of slope and |R| decreases, and as i increases, the rate of change of absolute value of slope and |R| increases. 11. The adjusted semiannual coupon rate is 3.401%, The annual yield quote is 6.802%.

Documents

Bond Convexity Floyd Vest, August 2011“Bond Duration,” and “Bond Pricing Theorems.”) Bond Convexity August, 2011 5 It can be proved that, in general, the higher the coupon