The Australian Commonwealth Government Securities Bond Valuation Project
Part 1:
A: The Australian Commonwealth Government Securities, (AUG01500631) issued on 31st May, 2019 had a yield of 1.6073. From the issue date till the current period the yield curve shows a decreasing trend, with a yield of 0.862 as on 22nd October 2020.
Considering a very short term period of 1 day the yield curve shows a slight increase in its trend from 0.843 leading to a slight decrease in price. The minimum yield till date has been 0.642 while the maximum is 1.6073.
Based on the trends in the yield curve the interest rates have been highly fluctuating between 0.8 to 1% on a daily basis. For the next 6 months the yield of the bond is expected to be fluctuating within this range. This also indicates that post the global health crisis the recovery of the Australian economy is bumpy and uneven. The interest rate i.e. cash rate is expected to increase slightly in the next 6 months from the current all-time low rate of 0.25 %.
The long term (12 month) interest rate is highly dependent on the evolution of the health crisis and the Reserve Bank of Australia’s (RBA) monetary policy. However, the interest rate is expected to increase.
B: Based zero-coupon curve in Australian Commonwealth Government Securities the bond shows an increasing trend. The minimum and maximum z-spread is -29.910 and 14.050.
A Zero coupon curve maps the interest rates on zero coupon bonds to different maturities across time. They have a single payment at maturity. While an yield curve reveals the relationship between interest rates and time to maturity. The slope of yield curve of a bond is a leading indicator of where the economy of a country is heading. This bond’s yield curve clearly indicates the economic slowdown due to the global pandemic. The zero coupon curve on the other hand shows the price that an investor is willing to pay to purchase the bond at a particular time period, which is increasing over here.
Part 2
- Based on the 5 corporate bonds and the benchmark Index fund, iShares Wholesale Australian Bond Index, the annualized tracking error over a 12 month period ending 31st October 2020 is approx. 39%. This high tracking error, which shows an investment’s consistency versus benchmark over a period of time indicates that the portfolio managers are less benchmark aware. However there are situations where employing a high tracking error is a reasonably strategy for the managers. It may also be appropriate to combine low and high tracking error managers within the same overall portfolio, perhaps in a core/satellite approach.
The table below shows the calculation of Tracking error and the weights of bond:
|
Returns |
||||||||
|
Months |
Bond 1 |
Bond 2 |
Bond 3 |
Bond 4 |
Bond 5 |
Portfolio Monthly Return (Rp) |
Index Return (Ri) |
Rp-Ri |
|
31-Oct-2020 |
0.4060% |
3.5380% |
0.2520% |
0.4020% |
0.8720% |
1.1849% |
1.1585% |
0.0264% |
|
30-Sep-2020 |
0.5380% |
3.4830% |
0.3330% |
0.5510% |
0.8830% |
1.2102% |
1.1550% |
0.0551% |
|
31-Aug-2020 |
0.6340% |
3.5790% |
0.3790% |
0.6210% |
1.0530% |
1.2623% |
1.1471% |
0.1152% |
|
31-Jul-2020 |
0.7220% |
3.3450% |
0.4170% |
0.6230% |
0.9010% |
1.2062% |
1.1519% |
0.0543% |
|
30-Jun-2020 |
0.8910% |
3.3740% |
0.5280% |
0.8170% |
0.9510% |
1.2705% |
1.1467% |
0.1237% |
|
31-May-2020 |
1.0540% |
3.1710% |
0.6760% |
0.9540% |
0.9720% |
1.2684% |
1.1713% |
0.0971% |
|
30-Apr-2020 |
1.2960% |
3.3120% |
0.8430% |
1.1180% |
0.9870% |
1.3810% |
1.1680% |
0.2129% |
|
31-Mar-2020 |
1.3740% |
1.7990% |
1.2760% |
1.5570% |
0.8780% |
1.0831% |
1.1677% |
-0.0846% |
|
29-Feb-2020 |
1.2100% |
2.0930% |
1.0830% |
1.2570% |
0.8770% |
1.0849% |
1.1796% |
-0.0947% |
|
31-Jan-2020 |
1.2910% |
2.3420% |
1.1450% |
1.3340% |
1.0070% |
1.1868% |
1.1697% |
0.0171% |
|
31-Dec-2019 |
1.5730% |
2.7770% |
1.4550% |
1.7040% |
1.4190% |
1.4459% |
1.1430% |
0.3029% |
|
30-Nov-2019 |
1.3370% |
2.4450% |
1.2050% |
1.4190% |
1.0670% |
1.2435% |
1.1681% |
0.0754% |
|
Weightage (%) |
||||||||
|
Bond 1 |
5% |
|||||||
|
Bond 2 |
30% |
|||||||
|
Bond 3 |
25% |
|||||||
|
Bond 4 |
10% |
|||||||
|
Bond 5 |
30% |
|
Tracking Error |
0.39% |
- In order to match the benchmark index without changing the portfolio components, the manager will try to change the values of factors that affect the tracking error. Some of the causes of tracking error in individual managers can be diversified away so that the tracking error of the overall portfolio is reduced. For eg. in an ETF where a minimal tracking error is critical, several dimensions that have an impact may include:
- Expense ratio – expenses cause tracking error, since indices typically used as tracking benchmarks do not include fees. ETF sponsors try to keep expenses as low as possible in an effort to minimize tracking error
- Execution – efficient transaction execution within the ETF portfolio reduces tracking error
- Optimization – ETFs will often hold only a subset of the benchmark’s constituents, and the optimization process for constructing the concentrated portfolio will normally result in tracking error.
The managers may also change the portfolio components in order to reduce the tracking error. This can be done by accommodating at least a small portion of cash in the portfolio.
- The most important uses of tracking error is to assess the performance of a portfolio, and the ability of a portfolio manager to generate excessive returns and beat benchmark returns. Thus being used as an input to calculate information ratio.
In part 2 (A) shows a high tracking error which is typically associated with lower r-squared, meaning the benchmark fit is not as tight. A low r-squared, usually accompanied by a high tracking error, reduces the validity of statistics such as alpha and beta.
One major issue to be aware of when evaluating managers performance is that as tracking error increases, so does the difficulty of determining a manager’s skill. The fit of the benchmark drives the confidence one can have in the resulting performance statistics: the better the fit, the higher degree of confidence in the statistics.
|
Bonds |
Mac Duration (as on 8th May, 2020) |
Current Mac Duration |
Weightage |
|
Bond 1 |
0.94 |
0.481 |
5% |
|
Bond 2 |
18.55 |
16.961 |
30% |
|
Bond 3 |
1.84 |
1.388 |
25% |
|
Bond 4 |
3.52 |
3.065 |
10% |
|
Bond 5 |
10.14 |
9.868 |
30% |
|
Mac Duration of the portfolio |
9.465 |
8.72625 |
- The Macaulay Duration as on 8th May,2020 and 5 months later is given in the following table:
Macaulay Duration is the weighted average term to maturity of the cash flows from a bond. This implies that after the given amount of years an investor will receive back its nominal amount. The portfolio duration as on 8th May, 2020 is 9.465 years and that 5 months later is 8.726 years.
As on 8th May, 2020:
Asset Duration 9.465
Liability Duration 9.465
After 5 months :
Asset Duration 8.72625
Liability Duration 9.0481
To immunize, we need to choose to change the weightage of the bonds in our portfolio. We would prefer to keep the same bond mix and not change the existing bonds.
|
Bond 1 |
Bond 2 |
Bond 3 |
Bond 4 |
Bond 5 |
|
|
Weight |
5% |
35% |
25% |
10% |
25% |
|
Duration |
0.481 |
16.961 |
1.388 |
3.065 |
9.868 |
|
Asset Duration |
9.0481 |
We use goal seek to find out the appropriate weight. As the asset duration today is lesser than liability duration, we need to immune our portfolio by shifting weights. We have just changed the weights of 2 bonds, bond 2 and bond 5, since they have a larger duration in comparison to the remaining bonds. As we need to increase the asset duration, we increased the weight in bond 2, which has a higher bond duration, and simultaneously decreased weightage of bond 5.
Maintaining an immunized portfolio means rebalancing the portfolio’s average duration every time interest rates change, so that the average duration continues to equal the investor’s time horizon. When implementing immunization strategies an investor has to deal with the risk of term structure movements that can make possible to achieve the objective these strategies were intended for. Apart from changing the weights the immunization result can be affected by diversification. The main result is that diversification helps, in a greater degree than concentration, in reducing the volatility of the deviations between target and realized returns. Also, this result is independent of the size and sign of interest rate changes. Finally, we find that immunized portfolios outperform those strategies based on minimizing M-Absolute.
List of bonds and index assigned:
|
Bond Code |
Name |
|
|
Bond 1 |
AUTLS0421 |
Telstra Corp. Ltd |
|
Bond 2 |
AU169422338 |
Commonwealth Bank of Australia |
|
Bond 3 |
AUNAB0322 |
National Australia Bank Ltd |
|
Bond 4 |
AUCBA0124 |
Commonwealth Bank of Australia |
|
Bond 5 |
AUG01500631 |
Australia, Commonwealth of Government |
|
Index Fund |
LP65062168 |
iShares Wholesale Australian Bond Index |
Appendices:
Bond 1
|
08-May-20 |
19-Apr-21 |
||||
|
Maturity (in years) |
0.948 |
||||
|
Current market price |
€ 1,011.80 |
||||
|
Market interest rate |
0.20% |
||||
|
Coupon rate |
1.45% |
||||
|
Annual coupon |
€ 14.50 |
||||
|
Nominal value |
€ 1,000.00 |
||||
|
Using goal seek. We solved the value of current market price by changing nominal value. |
|||||
|
Period |
Cash flow |
PV of flow |
PV as % of price |
||
|
0.5 |
€ 14.50 |
€ 14.49 |
0.014104 |
0.0071 |
|
|
0.948 |
€ 1,014.50 |
€ 1,012.55 |
0.985896 |
0.9346 |
|
|
€ 1,027.04 |
0.9416 |
||||
|
Bond duration = 0.9416 years |
|||||
Bond 2
|
11-Oct-47 |
08-May-20 |
|||
|
Maturity (in years) |
27 |
|||
|
Current market price |
€ 1,103.61 |
|||
|
Market interest rate |
1.76% |
|||
|
Coupon rate |
2% |
|||
|
Annual coupon |
€ 22.43 |
|||
|
Nominal value |
€ 1,000.00 |
|||
|
Using goal seek. We solved the value of current market price by changing nominal value. |
||||
|
Period |
Cash flow |
PV of flow |
PV as % of price |
|
|
0.5 |
€ 22.43 |
€ 22.23 |
0.013945757 |
0.0070 |
|
1 |
€ 22.43 |
€ 22.04 |
0.013824424 |
0.0138 |
|
1.5 |
€ 22.43 |
€ 21.84 |
0.013704147 |
0.0206 |
|
2 |
€ 22.43 |
€ 21.65 |
0.013584916 |
0.0272 |
|
2.5 |
€ 22.43 |
€ 21.47 |
0.013466723 |
0.0337 |
|
3 |
€ 22.43 |
€ 21.28 |
0.013349558 |
0.0400 |
|
3.5 |
€ 22.43 |
€ 21.09 |
0.013233413 |
0.0463 |
|
4 |
€ 22.43 |
€ 20.91 |
0.013118278 |
0.0525 |
|
4.5 |
€ 22.43 |
€ 20.73 |
0.013004144 |
0.0585 |
|
5 |
€ 22.43 |
€ 20.55 |
0.012891004 |
0.0645 |
|
5.5 |
€ 22.43 |
€ 20.37 |
0.012778848 |
0.0703 |
|
6 |
€ 22.43 |
€ 20.19 |
0.012667668 |
0.0760 |
|
6.5 |
€ 22.43 |
€ 20.02 |
0.012557455 |
0.0816 |
|
7 |
€ 22.43 |
€ 19.84 |
0.012448201 |
0.0871 |
|
7.5 |
€ 22.43 |
€ 19.67 |
0.012339898 |
0.0925 |
|
8 |
€ 22.43 |
€ 19.50 |
0.012232537 |
0.0979 |
|
8.5 |
€ 22.43 |
€ 19.33 |
0.01212611 |
0.1031 |
|
9 |
€ 22.43 |
€ 19.16 |
0.012020608 |
0.1082 |
|
9.5 |
€ 22.43 |
€ 18.99 |
0.011916025 |
0.1132 |
|
10 |
€ 22.43 |
€ 18.83 |
0.011812352 |
0.1181 |
|
10.5 |
€ 22.43 |
€ 18.67 |
0.011709581 |
0.1230 |
|
11 |
€ 22.43 |
€ 18.50 |
0.011607703 |
0.1277 |
|
11.5 |
€ 22.43 |
€ 18.34 |
0.011506713 |
0.1323 |
|
12 |
€ 22.43 |
€ 18.18 |
0.0114066 |
0.1369 |
|
12.5 |
€ 22.43 |
€ 18.02 |
0.011307359 |
0.1413 |
|
13 |
€ 22.43 |
€ 17.87 |
0.011208981 |
0.1457 |
|
13.5 |
€ 22.43 |
€ 17.71 |
0.01111146 |
0.1500 |
|
14 |
€ 22.43 |
€ 17.56 |
0.011014786 |
0.1542 |
|
14.5 |
€ 22.43 |
€ 17.41 |
0.010918954 |
0.1583 |
|
15 |
€ 22.43 |
€ 17.25 |
0.010823956 |
0.1624 |
|
15.5 |
€ 22.43 |
€ 17.10 |
0.010729784 |
0.1663 |
|
16 |
€ 22.43 |
€ 16.95 |
0.010636431 |
0.1702 |
|
16.5 |
€ 22.43 |
€ 16.81 |
0.010543891 |
0.1740 |
|
17 |
€ 22.43 |
€ 16.66 |
0.010452155 |
0.1777 |
|
17.5 |
€ 22.43 |
€ 16.52 |
0.010361218 |
0.1813 |
|
18 |
€ 22.43 |
€ 16.37 |
0.010271072 |
0.1849 |
|
18.5 |
€ 22.43 |
€ 16.23 |
0.01018171 |
0.1884 |
|
19 |
€ 22.43 |
€ 16.09 |
0.010093126 |
0.1918 |
|
19.5 |
€ 22.43 |
€ 15.95 |
0.010005313 |
0.1951 |
|
20 |
€ 22.43 |
€ 15.81 |
0.009918263 |
0.1984 |
|
20.5 |
€ 22.43 |
€ 15.67 |
0.009831971 |
0.2016 |
|
21 |
€ 22.43 |
€ 15.54 |
0.00974643 |
0.2047 |
|
21.5 |
€ 22.43 |
€ 15.40 |
0.009661633 |
0.2077 |
|
22 |
€ 22.43 |
€ 15.27 |
0.009577573 |
0.2107 |
|
22.5 |
€ 22.43 |
€ 15.13 |
0.009494245 |
0.2136 |
|
23 |
€ 22.43 |
€ 15.00 |
0.009411642 |
0.2165 |
|
23.5 |
€ 22.43 |
€ 14.87 |
0.009329758 |
0.2192 |
|
24 |
€ 22.43 |
€ 14.74 |
0.009248586 |
0.2220 |
|
24.5 |
€ 22.43 |
€ 14.61 |
0.00916812 |
0.2246 |
|
25 |
€ 22.43 |
€ 14.49 |
0.009088354 |
0.2272 |
|
25.5 |
€ 22.43 |
€ 14.36 |
0.009009283 |
0.2297 |
|
26 |
€ 22.43 |
€ 14.24 |
0.008930899 |
0.2322 |
|
26.5 |
€ 22.43 |
€ 14.11 |
0.008853197 |
0.2346 |
|
27 |
€ 22.43 |
€ 13.99 |
0.008776172 |
0.2370 |
|
27.444 |
€ 1,022.43 |
€ 632.89 |
0.397041014 |
10.8963 |
|
€ 1,594.03 |
18.5494 |
|||
|
Bond duration = 18.5494 years |
||||
Bond 3
|
24-Mar-22 |
08-May-20 |
|||
|
Maturity (in years) |
1.877 |
|||
|
Current market price |
€ 1,028.30 |
|||
|
Market interest rate |
0.11% |
|||
|
Coupon rate |
2% |
|||
|
Annual coupon |
€ 16.25 |
|||
|
Nominal value |
€ 1,000.00 |
|||
|
Using goal seek. We solved the value of current market price by changing nominal value. |
||||
|
Period |
Cash flow |
PV of flow |
PV as % of price |
|
|
0.5 |
€ 16.25 |
€ 16.24 |
0.015281592 |
0.0076 |
|
1 |
€ 16.25 |
€ 16.23 |
0.015272843 |
0.0153 |
|
1.5 |
€ 16.25 |
€ 16.22 |
0.015264099 |
0.0229 |
|
1.877 |
€ 1,016.25 |
€ 1,014.07 |
0.954181467 |
1.7907 |
|
€ 1,062.76 |
1.8365 |
|||
|
Bond duration = 1.8365 years |
||||
Bond 4
|
11-Jan-24 |
08-May-20 |
|||
|
Maturity (in years) |
3.679 |
|||
|
Current market price |
€ 1,047.45 |
|||
|
Market interest rate |
0.20% |
|||
|
Coupon rate |
2% |
|||
|
Annual coupon |
€ 15.00 |
|||
|
Nominal value |
€ 1,000.00 |
|||
|
Using goal seek. We solved the value of current market price by changing nominal value. |
||||
|
Period |
Cash flow |
PV of flow |
PV as % of price |
|
|
0.5 |
€ 15.00 |
€ 14.98 |
0.013476 |
0.0067 |
|
1 |
€ 15.00 |
€ 14.97 |
0.013462 |
0.0135 |
|
1.5 |
€ 15.00 |
€ 14.95 |
0.013448 |
0.0202 |
|
2 |
€ 15.00 |
€ 14.94 |
0.013435 |
0.0269 |
|
2.5 |
€ 15.00 |
€ 14.92 |
0.013421 |
0.0336 |
|
3 |
€ 15.00 |
€ 14.91 |
0.013407 |
0.0402 |
|
3.5 |
€ 15.00 |
€ 14.89 |
0.013393 |
0.0469 |
|
3.679 |
€ 1,015.00 |
€ 1,007.41 |
0.905958 |
3.3334 |
|
€ 1,111.98 |
3.5213 |
|||
|
Bond duration =3.5213 years |
||||
Bond 5
|
21-Jun-31 |
08-May-20 |
|||
|
Maturity (in years) |
11.126 |
|||
|
Current market price |
€ 1,062.42 |
|||
|
Market interest rate |
0.42% |
|||
|
Coupon rate |
1% |
|||
|
Annual coupon |
€ 10.00 |
|||
|
Nominal value |
€ 1,000.00 |
|||
|
Using goal seek. We solved the value of current market price by changing nominal value. |
||||
|
Period |
Cash flow |
PV of flow |
PV as % of price |
|
|
0.5 |
€ 10.00 |
€ 9.98 |
0.00846934 |
0.0042 |
|
1 |
€ 10.00 |
€ 9.96 |
0.008451422 |
0.0085 |
|
1.5 |
€ 10.00 |
€ 9.94 |
0.008433542 |
0.0127 |
|
2 |
€ 10.00 |
€ 9.92 |
0.0084157 |
0.0168 |
|
2.5 |
€ 10.00 |
€ 9.89 |
0.008397895 |
0.0210 |
|
3 |
€ 10.00 |
€ 9.87 |
0.008380128 |
0.0251 |
|
3.5 |
€ 10.00 |
€ 9.85 |
0.008362399 |
0.0293 |
|
4 |
€ 10.00 |
€ 9.83 |
0.008344707 |
0.0334 |
|
4.5 |
€ 10.00 |
€ 9.81 |
0.008327052 |
0.0375 |
|
5 |
€ 10.00 |
€ 9.79 |
0.008309435 |
0.0415 |
|
5.5 |
€ 10.00 |
€ 9.77 |
0.008291856 |
0.0456 |
|
6 |
€ 10.00 |
€ 9.75 |
0.008274313 |
0.0496 |
|
6.5 |
€ 10.00 |
€ 9.73 |
0.008256807 |
0.0537 |
|
7 |
€ 10.00 |
€ 9.71 |
0.008239339 |
0.0577 |
|
7.5 |
€ 10.00 |
€ 9.69 |
0.008221908 |
0.0617 |
|
8 |
€ 10.00 |
€ 9.67 |
0.008204513 |
0.0656 |
|
8.5 |
€ 10.00 |
€ 9.65 |
0.008187155 |
0.0696 |
|
9 |
€ 10.00 |
€ 9.63 |
0.008169834 |
0.0735 |
|
9.5 |
€ 10.00 |
€ 9.61 |
0.00815255 |
0.0774 |
|
10 |
€ 10.00 |
€ 9.59 |
0.008135302 |
0.0814 |
|
10.5 |
€ 10.00 |
€ 9.56 |
0.00811809 |
0.0852 |
|
11 |
€ 10.00 |
€ 9.54 |
0.008100915 |
0.0891 |
|
11.126 |
€ 1,010.00 |
€ 963.51 |
0.817755798 |
9.0984 |
|
€ 1,178.23 |
10.1385 |
|||
|
Bond duration = 10.3185 years |
||||
This objective is to guarantee to end portfolio value independently of interest rate changes. Many authors suggested to develop several techniques in order to minimize the effects of adverse term structure changes. estos usual technique simply to concentrate investment in one or two bonds, leading to portfolios with a high degree of idiosyncratic risk. So, we have checked which of these two sources of risk, immunization risk and idiosyncratic risk, has a greater impact on bond returns or more precisely on the deviations of realized bond portfolio returns with respect to the target return. So we have tested four different strategies using linear programming techniques which are shown to be an extremely useful instrument in bond portfolio management. Strategy 1 which leads to immunized portfolios with maximum concentration around the investor horizon, strategy 2 that also
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Articles
Bond Portfolio Immunization, Immunization …selects immunized portfolios but with maximum diversification, strategy 3 which can be considered as a mixture of the two former strategies and finally, strategy 4 as benchmark we test the Nawalkha and Chambers’ immunization strategy. These authors suggest to reduce simultaneously interest risk and immunization risk by minimizing a dispersion measure, M-Absolute. The main contribution of this paper is that we have introduced more realistic features to the portfolio design and its rebalancing process. First, we have used prices of actual transactions, allowing to take into consideration idiosyncratic risk which may disappear if we had used fitted term structure bond prices. Second, we proceed to rebalance portfolio weights each time a cash flow is due instead of assuming periodical portfolio adjustments. The assumptions made about cash flow reinvestments within each rebalancing period may have important effects on the results. Third, short sales are not allowed. To overcome the problems this fact may cause when adjusting portfolio duration during the investor horizon, especially when shorter bonds mature, we let bills and repo operations to enter portfolio. This is a more realistic feature than keeping short positions in bonds (when allowed) for long periods of time. The main result is that diversification helps, in a greater degree than concentration, in reducing the volatility of the deviations between target and realized returns. Also this result is independent of the size and sign of interest rate changes. Finally, we find that immunized outperform portfolios those strategies based on minimizing M-Abso
