Figure 6.2 shows the Bloomberg Yield and Spread Analysis page for the 3.85% Apple (AAPL) bond due May 4, 2043. It is priced at 87.24 for settlement on March 5,2014. This is the market discount bond we saw in Chapter 4, where we worked through the projected after-tax rate of return calculation. Now let's confirm the (interest rate) risk numbers. This bond has a modified duration statistic of 16.285 and a convexity shown to be 3.803.1 get to Bloomberg Risk, reported at 14.418, in a bit. These risk statistics are annualized and relate the total market value, 88.534028 (percent of par value, including accrued interest) to a change in the street convention yield, 4.653675%, which is quoted on a semiannual bond basis for a 30/360 day count.

Let's first get the Macaulay duration statistic for this bond. In equation 6.13, enter y = 0.04654/2 = 0.02327 (I'm going to round off the yield a bit), c = 0.0385/2 = 0.01925, N = 59, and t/T = 121/180. There are 59 semiannual coupon periods between the last payment date on November 4, 2013, and the maturity date.

The annual Macaulay duration is 16.66365 (=33.3273/2). The annual modified duration rounded to three decimals is 16.285 (= 16.66365/1.02327). The Excel function gets the same results, confirming the Bloomberg number.

FIGURE 6.2 Bloomberg Yield and Spread Analysis Page (YA), AAPL 3.85% Bond Due May 4, 2043

Used with permission of © 2014. All rights reserved.

We need to do yield convexity in two steps. First, use the same inputs for y, c, and N in equation 6.16.

Second, substitute this result and t/T = 121/180 and MacDur(t/T = 0) = 33.9995 into equation 6.17.

So, the annual convexity is 380.280 (= 1,521.1210/4). Bloomberg reports the convexity to be 3.803, but that is just our result divided by 100.

Why does Bloomberg scale the yield convexity down by a factor of 100? The answer goes back to how the convexity adjustment improves the estimate of the change in market value given a change in the yield. Suppose the question at hand is how much gain in MV should we anticipate if the yield falls by 25 basis points, from 4.654% to 4.404% ? We can use equation 6.12, knowing that the annual modified duration and convexity statistics are 16.285 and 380.3.

We conclude that modified duration alone estimates a 4.07% increase in market value, but factoring in convexity, we get 4.19%. Duration by itself underestimates gains and overestimates losses. The convexity adjustment adds twelve basis points, bringing the estimate closer to the actual result.

Bloomberg convexity allows you to work with another version of equation 6.12. Multiply both sides by 100 to get the change in percentage terms directly.


Because the second term of the Taylor series expansion entails the change in yield squared, the annual convexity needs to be divided by 100. Now the estimate is figured as:

The answers are the same, of course, but in my opinion the reformulation (the convexity of 380.3 becomes 3.803) is not worth the “convenience” of

using the convexity adjustment in equation 6.20 rather than equation 6.12. Frankly, Bloomberg convexity reflects the olden days when such estimates were made on the back of an envelope or with a handheld calculator; nowadays we use spreadsheets. Moreover, convexity can be used as a summary statistic for bond strategy – more on this in Chapter 10 – and for that purpose, there is no advantage to scaling convexity down to such a small number.

Macaulay and modified durations are measures of percentage price sensitivity. Money duration and its variants indicate the price change. Often this is on a per-basis-point basis; that is, the change in the value of the bond for a one-basis-point change in the yield. These variants go by various names – the BPV (basis point value), the PV01 (price, or present, value of an “01” change in yield), and the DV01 (the dollar value of an “01”). Sometimes these are calculated simply by multiplying the money duration by 0.0001 – that is, the modified duration times the market value, times one basis point. Bloomberg determines the PV01 and DV01 a little differently.

The Bloomberg Risk statistic is the PV01 * 100. On the Apple bond in Figure 6.2, the PV01 is 0.14418 and Risk is 14.418. If the yield goes up by 100 basis points, the flat price of the bond will go down by approximately 14.418 (per 100 of par value), from 87.24 to 72.822, (87.24 – 14.418 = 72.822). For bonds trading at a discount such as this one, Bloomberg Risk is less than modified duration. For bonds trading at a premium, Risk is greater than duration.

I demonstrate the PV01 calculation using Excel because it requires the precise street convention yield to maturity, 0.04653675. We solve for the new flat prices after changing the yield by adding and subtracting one basis point (0.0001).

The new flat prices are 87.384344 for the lower yield and 87.095987 for the higher yield. The PV01 is the difference in these prices divided by two.

Notice that we could include the accrued interest in the numerator, but it just cancels out.

So, what do we know about the interest rate risk of this Apple bond? Its modified duration is 16.285 and its convexity is 3.803 (as scaled by Bloomberg). These relate to changes in the yield to maturity. We can say that if the yield jumps up suddenly by 100 basis points, the market value will fall by approximately 16.285% using duration alone. If we add in the convexity adjustment, we can say that the expected drop will be more like 14.384%. If we actually ran the experiment on a spreadsheet, we would see that the bond price actually would fall by 14.540%. Our estimate works quite well, even for a large jump in the yield.

Yield duration and convexity are impact statistics and not causation factors. The yield might have gone up because of an unexpected downgrade in the issuer's credit rating. Or it might have gone up because all bond yields increased following dramatically revised forecasts for expected inflation. We explore this further in the next chapter. There, understanding the sensitivity of floating-rate notes (floaters) requires that we ask why the yield changes. Let's now step beyond classic yield duration and consider how market value changes when the entire benchmark Treasury yield curve shifts up or down.

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