A number of versions of duration have been introduced since Macaulay first wrote down a formula for the statistic in the 1930s. I always wonder: Does this mean he invented it or discovered it? Anyway, another version that I'll call spot duration sometimes is used in academic fixed-income research (where it is called “Fisher-Weil duration”).

(6.21)

This looks much like the weighted-average formula for Macaulay duration in equation 6.14. The difference is that instead of discounting the cash flows with the yield to maturity, the sequence of spot, or zero-coupon, rates (z1, z2, ..., zN) is used. The price of the bond in the denominator is the same as in equation 6.14 – recall from Chapter 3 that the yield is a “weighted average” of the spot rates. The numerator can be different, however, the more so the greater the slope to the yield curve.

An advantage to spot duration is that it implicitly includes the shape of the yield curve. One can then analyze the impact on market value following nonparallel shifts to the term structure, for instance, a steepening such that the increase in zN is greater than z1. The problem in taking this academic construct into practice is that the requisite sequence of zero-coupon rates generally is not available for corporate, agency, and municipal bonds, only for Treasury securities. Moreover, as we saw in Chapter 4, Treasury STRIPS have significantly different taxation than coupon bonds and usually are less liquid. For that reason, spot durations are not reported on data systems like Bloomberg; nor are they used in practice.

Macaulay and modified durations, however, depend only on an observable and unambiguous input: the bond price and from it the yield to maturity. You might hear the complaint made by some that “Macaulay duration assumes a flat yield curve.” That is just not true, in the same way that the yield to maturity statistic does not assume a flat term structure. It's the same argument as I made in Chapter 3. The yield is a summary statistic about the cash flows on the bond; it's the internal rate of return. Yield duration and convexity estimate the change in market value associated with a change in that summary statistic. Many different shapes to the underlying spot curve can produce the same yield to maturity; many different shifts to that underlying spot curve can produce the same change in the yield. This explains why yield duration and convexity merely estimate the actual change in market value (along with dropping the remaining terms of the Taylor series).

A more significant problem with yield duration is that on some securities it has little meaning. That's because the yield itself is not well defined. Consider a callable bond. What is its yield? Is it the yield to maturity – that is, assuming the embedded call option is and will remain out of the money (so the bond will not be called)? Is it the yield to first call date – that is, assuming the call option is and will remain in the money (so it definitely will be called)? Often on callable bonds you will see the “yield to worst,” which is the lowest of the yields to first call, second call, and so on, out to the yield to maturity. Presumably it's the most conservative measure of rate of return, but I think it's just more bond data, not information. In any case, is the duration of a callable bond its sensitivity to a change in the yield to worst?

Figure 6.3 illustrates this problem. It shows the Bloomberg Yield and Spread Analysis page for the 6% Fannie Mae bond that matures on April 18, 2036. This bond is callable at par value one time on April 18, 2016 (this is called the workout date on Bloomberg). The bond is priced flat at 108.625 for settlement on March 12, 2014. Its street convention yield to maturity is shown to be 1.795488%.

Suppose that you work in the back office of an investment firm, and one of your many jobs is to enter the key statistics on newly acquired bonds into the risk management tracking system. Would you enter 1.795488% for the yield? Would you enter 1.949 for the modified duration for this bond and 0.049 for the convexity (or 4.9)? I hope not! These risk and return statistics assume that the bond definitely will be called. They are calculated using April 18, 2016, as the maturity date. Although it is quite likely that Fannie Mae in fact will call the bond, which is trading at a premium above the strike price on

the option and so is in the money, there is always a chance that interest rates go up unexpectedly and bond is not called. We need to be careful in stating risk statistics based on an optimistic view of future market conditions.

For better numbers, you should go to the Bloomberg Option-Adjusted Spread Analysis page (OAS1) for the same Fannie Mae bond, shown in Figure 6.4, to get more reasonable yield, duration, and convexity statistics for this callable bond. Bloomberg uses an interest rate term structure model to value the callable bond, here a lognormal model. Starting with the underlying CMT (Constant Maturity Treasury) yield curve, shown at the right in Figure 6.4, the implied spot and forward rates are calculated in the same manner as in Chapter 5. Then around that forward curve a range of possible rates is postulated, consistent with the assumed level of interest rate volatility, in this instance, 18.96%. Those rates are then calibrated to value exactly the benchmark Treasury bonds. The principle of no arbitrage allows the model to value other securities.

Given the model and, importantly, the assumed level of rate volatility, the option-adjusted spread (OAS) for the Fannie Mae callable bond is determined to be 102.3 basis points. That is the spread over the chosen Treasury

FIGURE 8.4 Bloomberg Option-Adjusted Spread Analysis Page (OAS1), Fannie Mae Callable Bond

security (the 0.625% T-note due February 15, 2017) after subtracting out the value of the embedded call option. The OAS measures the compensation to the investor for the remaining risk factors – that is, default, liquidity, and taxation – and allows for direct comparison to other callable, as well as noncallable, bonds to identify relative value.

There is a lot of bond data on these two Bloomberg pages. It's up to the fixed-income analyst to identify the information. Consider first the Spread of 142.18 basis points shown in the top left of the Yield and Spread Analysis page. That's the difference between the yield to call, 1.795488%, and the yield on the indicated benchmark Treasury, the 0.25% T-Note that matures on February 29, 2016. Its yield is 0.373665%. The spread is 1.421823% (= 1.795488% – 0.373665%).

Next, look at the Spread of 456.2 basis points shown in the left side of the OAS1 page. Where does it come from? Well, the yield to maturity on the bond (i.e., assuming away the presence of the call option) is 5.330604%. It's shown in the middle of the page rounded to three digits, but you can get the full precision on Excel.

The benchmark Treasury now is the 0.625% note due on February 15, 2017. It's priced at 99.5859375 (= 99 + 18.75/32) to yield 0.768116%.

The spread over the Treasury is 4.562488% (=5.330604% -0.768116%). But what is the meaning to a yield spread on bonds with such divergent times to maturity?

Fortunately, there is useful information on these pages, especially the OAS of 102.3 basis points (given that you are fine with the assumed volatility). Another is the option-adjusted yield (OAY), the yield after including the estimated value of the option. The value of the embedded call is determined to be 16.16 (percent of par value), shown in the Bloomberg OAS1 page as “Option Px Value.” (I don't know why it is shown as a negative amount.) This is added to the flat price of the bond to get the option-adjusted flat price, 124.785 (= 108.625 + 16.16). The OAY is the yield to maturity based on that price. It's shown to be 4.258%, the “Option Free Yld” in the middle of the page. That result is easily confirmed using the Excel YIELD function.

The idea of the OAY is that if the embedded call option, which has a value of 16.16, were somehow to be removed, the bond would trade at a higher price and a lower yield to maturity.

The same option-pricing model that produces the OAS and OAY is used for the interest rate risk statistics. First, all the benchmark Treasury CMT yields used to get the spot and forward rates (i.e., those shown at the right in Figure 6.4) are increased by a certain number of basis points. Then the model is run, generating a new value for the bond, MV(up). Next the benchmark yields are decreased by the same amount, and the model generates MV(down). MV (initial) on this bond is 111.025 (percent of par value, including accrued interest).

Those market values are the inputs to equations 6.22 and 6.23 for the effective duration and convexity, which are curve duration and convexity statistics very similar to those for approximate annual modified yield duration and convexity. The difference is that now in the denominator the change is to the entire benchmark yield curve rather than to the bond's own yield to maturity.

(6.22)

(6.23)

The effective duration for this callable bond is 3.66, reported in Figure 6.4 under OAS method. On Bloomberg effective duration is called OAS duration. The OAS convexity or effective convexity or, as I prefer, curve convexity, is reported to be -2.69. Once again, this is scaled by dividing by 100; I would rather see -269. Negative convexity is a common feature with callable bonds because of the limit to price appreciation as the yield falls.

Curve duration and convexity also can be calculated for bonds that do not have embedded derivatives. In fact, the contrast between yield duration and curve duration has interesting implications for bond strategy and risk management. It becomes most meaningful in assessing the rate sensitivities of long-term, low-coupon bonds. To see this, Figure 6.5 shows the Bloomberg Yield and Spread Analysis page for the zero-coupon Treasury P-STRIPS maturing on May 15, 2042. It is priced at 33.51171875 (= 33 + 16.375/32) to yield 3.918000% (s.a.) on a street convention basis for settlement on March 12, 2014. (Note how strange-looking that Japanese simple yield of 7.041% is. Do not bother looking at the after-tax rate – assuming a zero-coupon bond like this P-STRIPS is issued at par value and that it is a non-OID bond with a market discount is just wrong. Also, I don't understand why the principal is shown to be 335,111.24 in the Invoice section

rather than 335,117.19 given that the bond price is shown to be 33-16 3/8. Yes, I have a love-hate relationship with some Bloomberg pages.)

Let's first confirm the reported yield modified duration and convexity numbers for the P-STRIPS reported under the “Workout” heading. Because the coupon rate is zero, its Macaulay duration (in semiannual periods) is just N – t/T = 57 – 117/181 = 56.3536. There are 57 semiannual periods between the start of the current period on November 15,2013, and the maturity date. Note that the day count is actual/actual because this is a Treasury security, even though there are no coupon payments. Dividing by two gives an annual Macaulay duration of 28.1768 (= 56.353 6/2). Dividing that by one plus the yield per period gives the reported annual modified duration, 27.635 [= 28.1768/(1 + 0.03918/2)]. The yield convexity entails substitution into equations 6.16 and 6.17, here combined because so many terms drop out when c = 0.

Dividing that by the periodicity squared obtains the annual convexity, 777.2685 (= 3,109.074/4). Bloomberg scales that down and reports 7.773.

Now look at right side of the Risk section, which shows results from the Bloomberg OAS1 page for the same zero-coupon Treasury STRIPS. The modified curve duration is 31.415 and the curve convexity is 9.406 (I would prefer 940.6). These are the effective duration and convexity statistics obtained by shifting the benchmark Treasury yield curve. Those risk statistics are considerably higher than the modified yield duration of 27.635 and yield convexity of 7.773 (or 777.3).

Which numbers do you use? It depends on the circumstance. If you own the Treasury P-STRIPS and want an estimate of the change in market value given a change in its yield to maturity, yield duration and convexity provide your answer. However, if you own a portfolio of Treasury notes and bonds and want to aggregate the risk statistics – that is, calculate the average portfolio modified duration and convexity – you should use the curve duration and curve convexity for each. This is a very important point, to which I return in Chapter 9 on bond portfolios.

However, before wrapping up this chapter and moving on to interesting extensions such as floaters, linkers, and swaps, let's look at why the curve duration and convexity on this long-term, zero-coupon bond are so different from the yield duration and convexity. The answer goes to the heart of bond math and the assumption of no arbitrage in financial modeling. In brief, it's because the yield curve is so steep in this example – flat curves simply are not so interesting.

Look at the benchmark Treasury CMT curve at the right of Figure 6.4. These are the yields on coupon Treasury notes and bonds for the same settlement date of March 12, 2014, as the P-STRIPS. When the underlying yield curve for coupon bonds is upward sloping, the implied spot curve as we derived in Chapter 5 will lie above it. It's no surprise at all that the 3.918% STRIPS yield is higher than coupon bonds having similar times to maturity. In Figure 6.5, the spread on the STRIPS is reported to be 19.92 basis points above the 3.718823% yield on the 3.625% Treasury bond due February 15,2044.

The key point is that when we shift the benchmark Treasury curve up and down in the interest rate term structure model to get the effective duration and convexity, the implied spot yields shift as well but not by the same amount. That happens only when the curve is flat. In fact, the implied spot curve shifts by a larger amount because of the upward slope. Suppose the curve duration and convexity statistics are based on a 25-basis point shift to the benchmark par curve. To be consistent with the no-arbitrage assumption, the yields on long-term, zero-coupon bonds will shift by more than 25 basis points. A larger change in the yield leads to a larger change in market value – so the curve duration and convexity statistics are larger as well.

For a numerical example of this important and interesting property of bond math, suppose that 1-year and 2-year annual payment bonds are priced at par value and have coupon rates of 2% and 10%, respectively. This is an incredibly steep par curve but will make the point with minimal bootstrapping. The 0 x 2 implied spot rate is 10.433927%, as expected above the 10% yield on the 2-year coupon bond. The approximation formulas for duration and convexity are very sensitive to rounding so I need to display a high degree of precision.

A 2-year zero-coupon bond is priced at 81.996435 (percent of par value), assuming no arbitrage and no transactions costs.

Its Macaulay duration is 2 and its modified duration is 1.8110 (= 2/1.10433927). Its convexity is 4.9198 [= (2*3)/(1.10433927)2]. Those are the yield duration and convexity statistics.

To get the curve duration and convexity, first shift the underlying yield curve, which in this case is the par curve, up by 25 basis points. The new 0×2 implied spot rate is 10.694755%, an increase of 26.1 basis points (0.10694755 – 0.10433927 = 0.00261).

The no-arbitrage price on the 0×2 zero-coupon bond falls to 81.610476.

Then shift the par curve down by 25 basis points. The new 2-year implied spot rate is 10.173098%, a decrease of 26.1 basis points (0.10433927 – 0.10173098 = 0.00261).

The price goes up to 82.385139.

We now have the inputs for effective duration and convexity in equations 6.22 and 6.23: MV(initial) = 81.996435, MV(up) = 81.610476, MV(down) = 82.385139, and the change in the yield curve is 0.0025. Remember that “up” and “down” here refer to the change in the yield curve, not the price.

If we put 0.00261 in the denominator of each, we would get the approximations for the modified yield duration and yield convexity, but that is not the point. We want to understand the sensitivity with respect to a change in the benchmark yield curve, not with respect to the yield to maturity. Corresponding to a 25-basis point parallel shift in the yield curve, the curve duration of 1.8895 and curve convexity of 5.3563 are more relevant interest rate sensitivities than the yield duration of 1.8110 and yield convexity of 4.9198.

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