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 −20 ⎡⎤1(1)−+i −20 (1) Pi=++10⎢⎥ 100(1 ) , where i is interest rate or yield. ⎣⎦i 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 i , P(i ) and has the same slope as the curve at that point. ( c c ) (a) (b) Figure 1: TI83/84 Price/yield graph for the bond in Example 1. Bond Convexity August, 2011 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 $ #DP' !P #D (2) !P " !i and " !i . If duration is high there is more down side risk. & 1+ i ) P 1+ i % ( Duration Actual Approximation |e| D(0.01) = 13.239608 ΔP = 230.81 – 262.41 ΔP ≈ –$34.40 |e(0.01)| = $2.80 From 0.01 to 0.02 = –$31.60 D(0.06) = 11.040694 ΔP = 131.78 – 145.88 ΔP ≈ –$15.19 |e(0.06)| = $1.09 From 0.06 to 0.07 = –$14.10 D(0.11) = 8.8203693 ΔP = 85.06 – 92.04 ΔP ≈ –$7.31 |e(0.11)| = $0.34 From 0.11 to 0.12 = –$6.98 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. Bond Convexity August, 2011 2 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. 1 (3) In fact, slope at i = ! D(i ) P(i ) in dollars per unit of one. To get c 1 i ( c )( c ) + c Pi()+Δ i dollars per percentage point, divide by 100. The ratio is also an approximation Δi 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 ΔP increases. The main thing is that we see the convexity of the Price/Yield curve, and P 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 Bond Convexity August, 2011 3 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. 1 The slope of the duration line at i = ! D(i)P(i) in dollars per unit of one. 1+ i 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 ! (11.040694)(145.88) = –$1519.4461 per one unit or –$15.194461 per one 1.06 percentage point. We can check this with an approximation, for this version of slope, P(0.06) ! P(0.07) such as = –$14.10 per one percentage point. When writing the 0.01 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.