A REAL BOND PORTFOLIO
This portfolio is composed of four U.S. Treasury securities, including shortterm and intermediateterm Tnotes, a longterm Tbond, and, just to make things interesting, a position in longterm PSTRIPS. Figures 9.1 to 9.4 show
FIGURE 9.1 Bloomberg Yield and Spread Analysis Page, 0.375% Treasury Note Due 2/15/2016
Used with permission of Bloomberg.com © 2014. All rights reserved.
FIGURE 9.2 Bloomberg Yield and Spread Analysis Page, 2% Treasury Note Due 2/15/2023
Used with permission of Bloomberg.com © 2014. All rights reserved.
FIGURE 9.3 Bloomberg Yield and Spread Analysis Page, 3.125% Treasury Bond Due 2/15/2043
Used with permission of Bloomberg.com © 2014. All rights reserved.
FIGURE 9.4 Bloomberg Yield and Spread Analysis Page, 0% Treasury PSTRIPS Due 2/15/2043
Used with permission of Bloomberg.com © 2014. All rights reserved.
the Bloomberg Yield and Spread Analysis pages for each security backdated for settlement on February 15, 2014, using the historical prices – this is a nice feature to Bloomberg. Conveniently (and intentionally) each Treasury matures on February 15, so the accrued interest is zero and all future cash flows are scheduled for February 15 and August 15 of each year. That makes it easy to build a spreadsheet to illustrate the portfolio statistics.
Table 9.1 summarizes the risk and return characteristics on the individual Treasuries using the Bloomberg data. These numbers are easily verified on Excel, other than curve duration and convexity, which emerge from the “black box” behind the OAS1 pages. Notice that I report the convexity statistics in their more natural and mathematical form, multiplying Bloomberg convexity by 100. Notice also the significant differences between the modified durations and the curve durations, especially on the longterm bonds. The difference between the yield and curve convexities is even larger. This is because of the steepness in the Treasury yield curve at that time. It's no surprise that the yield on the PSTRIPS is considerably higher than the Tbond maturing on the same date, 3.919371% compared to 3.697998%.
Before turning to the portfolio, let's focus a bit on the individual yield and curve durations and convexities. Consider the price sensitivity of a
TABLE 8.1 Individual Treasury Statistics for Settlement on 2/15/14
TNote 
TNote 
TBond 
PSTRIPS 

Coupon Rate 
0.375% 
2.00% 
3.125% 
0% 
Maturity 
2/15/16 
2/15/23 
2/15/43 
2/15/43 
Price 
100.08203125 
95.1875 
89.859375 
32.444999 
Yield to Maturity 
0.333813% 
2.603264% 
3.697998% 
3.919371% 
Macaulay Duration 
1.994 
8.258 
18.679 
29.000 
Modified Duration 
1.991 
8.152 
18.340 
28.443 
Curve Duration 
1.992 
8.441 
19.559 
32.855 
Yield Convexity 
5.0 
74.2 
448.8 
822.9 
Curve Convexity 
5.0 
78.6 
502.4 
1,018.6 
PV01 
0.01993 
0.07760 
0.16480 
0.09228 
Par Value 
$120,000,000 
$100,000,000 
$100,000,000 
$125,000,000 
$100 million (par value) position in the 3.125% longterm Treasury bond in isolation, for instance, resulting from a 25basispoint jump in its yield to maturity. The modified yield duration and convexity are the correct inputs to the estimation. The estimated loss is $3,994 million.
Now consider a $100 million (par value) P$TRIPS holding. If its yield goes up by 25 basis points, the estimated loss is $2,224 million.
On a percentage basis, the loss on the PSTRIP$ is much higher than on the coupon bond because of its much lower price as a percentage of par value.
Let's turn this into a hedging problem. Suppose you own the $100 million par value position in the PSTRIPS and fear an imminent rise in Treasury yields. You decide to hedge fully your position by shortselling the more actively traded comparable maturity Tbond. Your problem is to calculate the hedge ratio, which is the amount of Tbonds you need to short sell. A common
way to get this amount is to use the ratio of the PV01s (or BPVs). A formula for the hedge ratio between two bonds is written in equation 9.14.
(9.14)
From Table 9.1, the PV01 for the PSTRIPS is 0.09228 and 0.16480 for the TBond. Entering those in 9.14 for a par value of $100 million gives the result that you would need to short sell about $56 million in par value of the Tbond to hedge the exposure on the PSTRIPS.
The same hedge ratio would be obtained with a ratio of DV01s, which are just the PV01s scaled up to a given par value, for instance, $1 million.
Do you see the error – or at least the violation of the principle of no arbitrage? It is in the implicit assumption that both bond yields change by the same amount. That would be fine if the Treasury yield curve is flat – but it's not; it's steeply and upwardly sloped. If the yield on the 3.125%, 29 year Tbond goes up by 25 basis points, the yield on the 29year PSTRIPS will go up by more than that and its market value will fall by more than is estimated. You need a bigger hedge; a $56 million short position in the couponbearing Tbond is simply not enough.
A better hedge ratio uses “revised PV01s.” First notice that the given PV01s are essentially equivalent to the modified duration times the full price of the bond, times one basis point.
The revised PVOls use the curve durations instead of the yield durations.
The newandimproved hedge is to short sell about $60.65 million in the Tbond.
The key point is that the curve durations (and the revised PV01) are based on a parallel shift in the entire Treasury yield curve and are calibrated so that there are no arbitrage opportunities.
The noarbitrage assumption is particularly important in the U.S. Treasury market because of the presence of zerocoupon CSTRIPS and P STRIPS. When their yields to maturity veer away from the implied spot curve by enough to compensate for transactions costs, arbitrageurs can and will exploit opportunities for couponstripping and bond reconstitution. If these were corporate bonds, it would be a different story. Our bond math techniques allow us to say, “If a zerocoupon corporate bond exists, its noarbitrage yield would equal the implied spot rate.” But without a series of actual zerocoupon corporate securities to carry out the trades to eliminate any mispricing, it's a weak argument. The zerocoupon corporate bond yield could differ from the theoretical implied spot rate by a lot and for quite a while.
Let's now build a portfolio of the four Treasury securities using the par values indicated in Table 9.1. The market value for settlement on February 15, 2014, is $345,701,561, which easily is calculated when there is no accrued interest to deal with. As usual, the prices are quoted as percentages of par value in Table 9.1.
Table 9.2 shows an abridged version of the spreadsheet I use to calculate the portfolio statistics. The date0 cash flow for the purchase price of the portfolio is $345,701,561 – this amount needs to be made negative for internal rate of return calculations. There are 58 future cash flows, occurring on February 15 and August 15 of each year until February 15, 2043. The first three semiannual payments are coupon interest totaling $2,787,500.
TABLE 9.2 Portfolio Cash Flows and Calculations
Date 
Date 
Cash Flow 
PV of Cash Flow 
Weight 
Date * Weight 
Dispersion 
Convexity 
0 
02/15/14 
345,701,561 

1 
08/15/14 
2,787,500 
2,741,389 
0.007930 
0.007930 
4.217 
0.016 
2 
02/15/15 
2,787,500 
2,696,041 
0.007799 
0.015598 
3.795 
0.047 
3 
08/15/15 
2,787,500 
2,651,443 
0.007670 
0.023009 
3.401 
0.092 
4 
02/15/16 
122,787,500 
114,862,257 
0.332258 
1.329034 
133.690 
6.645 
5 
08/15/16 
2,562,500 
2,357,452 
0.006819 
0.034097 
2.477 
0.205 
6 
02/15/17 
2,562,500 
2,318,455 
0.006707 
0.040239 
2.187 
0.282 
17 
08/15/22 
2,562,500 
1,929,801 
0.005582 
0.094899 
0.278 
1.708 
18 
02/15/23 
102,562,500 
75,961,396 
0.219731 
3.955160 
8.067 
75.148 
19 
08/15/23 
1,562,500 
1,138,099 
0.003292 
0.062551 
0.084 
1.251 
20 
02/15/24 
1,562,500 
1,119,273 
0.003238 
0.064754 
0.053 
1.360 
56 
02/15/42 
1,562,500 
613,966 
0.001776 
0.099456 
1.812 
5.669 
57 
08/15/42 
1,562,500 
603,810 
0.001747 
0.099557 
1.895 
5.774 
58 
02/15/43 
226,562,500 
86,104,181 
0.249071 
14.446109 
286.926 
852.320 
345,701,561 
1.000000 
24.059117 
480.571 
1,083.471 
On February 15, 2016, the first Tnote having a par value of $120 million matures and the total payment is $122,787,500. Between August 15, 2016, and August 15, 2022, the coupon interest is $2,562,500.
After the next Tnote is retired on February 15, 2023, entailing a total payment of $102,562,500, the cash flows are just the coupon interest on the remaining Tbond.
On February 15, 2043, the Tbond and the PSTRIPS mature and the final payment is $226,562,500.
This sequence of evenly spaced, semiannual cash flow is the MV, CF1, CF2, ..., CFn in equation 9.1 needed to get the internal rate of return for the portfolio. I use the IRR function in Excel to get YieldPORT = 1.682%, which is the yield per semiannual period. (Actually, given the full precision of the spreadsheet, YieldPORT = 1.682028892%.) Annualized, the cash flow yield for this portfolio is 3.364% (s.a.).
The fourth column of Table 9.2 is the present value of the cash flow for each date, discounted using the (full precision) YieldPORT. The sum of those present values is $345,701,561, the market value of the portfolio, thereby verifying the internal rate of return calculation. Following equation 9.3, the fifth column divides each present value by the overall market value, giving the weights that sum to one. The sixth column multiplies the time to the receipt of the cash flow (measured in semiannual periods, column 1) times the weight. The sum of column 6 produces the Macaulay duration for the portfolio, MacDurPORT = 24.059. Annualized (by dividing by two), the portfolio Macaulay duration is 12.030. The annualized modified duration is 11.831, the Macaulay duration divided by one plus the cash flow yield per period.
The seventh column follows equation 9.4 to get the cash flow dispersion statistic for the portfolio. Whereas Macaulay duration is the average of the times to receipt of cash flow, dispersion is the variance. For each date, the difference between the time period (column 1) and the Macaulay duration (24.059) is squared and then multiplied by the present value of the cash flow (column 4). Summed over the 58 semiannual periods, DispPORT turns out to be 480.571.
The eighth column gets the convexity of the portfolio using equation 9.6. For each date, the time period (column 1) is multiplied by one plus that time period, and then by the present value of the cash flow (column 4). The sum is 1,083.471. CotwPORTis found to be 1,047.9 by dividing the sum by one plus the portfolio yield (per period) squared.
The portfolio yield, Macaulay duration, dispersion, and convexity statistics are linked by equation 9.7.
Annualized, the portfolio dispersion is 120.1 and the convexity is 262.0, after dividing by four (the periodicity squared).
So, we have portfolio valued at $345,701,561. It has a cash flow yield of 3.364%, a Macaulay duration of 12.030, a modified duration of 11.831, a cash flow dispersion of 120.1, and a convexity of 262.0. Those annualized summary statistics use the bigbundleofcashflow approach and represent the “true” risk and return profile for the portfolio. Now let's see how the statistics commonly used in practice compare.
The market value (MV) and basis point value (BPV) weights for the portfolio are shown in Table 9.3. The MV weights are straightforward (MVj/MV); the BPV weights (BPVj/BPV) entail multiplying the modified duration for each security by its market value, and then by one basis point. The differences between these weights are instructive – the PSTRIPS holding, while less than 12% of market value, represents a bit over 30% of interest
TABLE 9.3 Market Value and Basis Point Value Weights
Position 
MV Weights 
BPV Weights 
0.375% Tnote due 2/15/16 
34.740% 
6.265% 
2.00% Tnote due 2/15/23 
27.535% 
20.332% 
3.125% Tbond due 2/15/43 
25.993% 
43.179% 
0% PSTRIPS due 2/15/43 
11.732% 
30.224% 
100.000% 
100.000% 
rate risk. The shortterm Tnotes are almost onethird of market value but only about 6% of risk.
Due to the steepness of the yield curve, the marketvalueweighted average yield is considerably less than the internal rate of return on the portfolio. AvgYieldMV is only 2.254% whereas YieldPORT is 3.362%.
Suppose that the U.S. Treasury actually did use capital budgeting techniques as taught in introductory finance courses, and it has a longterm (29year) project in mind. If it issues these four bonds at these prices to fund the project, the government has $345.7 million to invest. The point here is that if the project earns only 2.254% annually, the revenues are not sufficient to pay off that debt, especially the large payment due on February 15, 2043. The project would have to earn at least 3.364% to have positive net present value. The BPVweighted average, 3.332%, is a better indicator of the “hurdle rate” for the project than the MVweighted average.
Average yield duration and convexity are also lower than the “true” portfolio statistics. These correspond to changes in the yields to maturity on the individual securities.
The strong upward slope to the yield curve is the reason why these weighted averages understate the bigbundleofcash flow portfolio statistics: 12.030 for Macaulay duration, 11.831 for modified duration, and 262.0 for convexity. Moreover, these are not trivial differences.
The other set of marketvalueweighted averages uses the curve duration and convexity as inputs.
Notice that these are considerably higher than the averages based on the yield duration and convexities. Once again that's due to the upward slope of the Treasury yield curve. Also, these results are even higher than the “true” portfolio modified duration and convexity.
What is right? What numbers should an analyst use to understand the risk and return profile of the bond portfolio? Which statistics best measure risk exposure? In sum, which statistics are just data and which ones provide information? These are excellent questions.