You are a professional money manager, and your clients evaluate your performance by comparing your percentage return with the percentage return using some benchmark index. Thus, you measure your risk as deviation between the return on your portfolio versus the return on your benchmark; you attempt to keep the Omega of your portfolio close to the Omega of the benchmark index.
In structuring your portfolio, you may use any of the instruments in the following list. (We will refer to this list as your trading universe.)
ID Price Yield Delta Benchmark
A 10.29 7.87% 0.28 1.00
B 655.53 8.52% 31.34 1.00
C 1,071.20 8.21% 44.93 1.00
D 9,058.66 9.57% 852.23 0.00
E 86.24 7.68% 1.72 1.00
F 93.05 7.47% 0.93 1.00
The column headings in the above table are: “ID” is the identification code for each security in the trading universe. “Price” is the current price of the security. Since we are ignoring trading costs, this price is the price at which the security can be bought. If you decide to short the security, then the price represents the amount of money you receive by shorting one unit of the security. “Yield” is quoted as annualized number using annual compounding. “Delta” is our usual measure of an approximate capital gain (loss) associated with a 100 basis point decrease (increase) in the term structure. “Benchmark” is the number of units of the security held in the benchmark portfolio. The benchmark only consists of securities in the trading universe.
Part a.
Calculate the duration or Omega (Dollar Delta divided by the value) of your benchmark.
Part b.
You believe interest rates are going to increase, and you want to structure your portfolio to exploit this forecast. You have decided to set the Omega of your entire portfolio to 2.00% (i.e., a duration of 2), which is less than that of the benchmark.
Currently, you are managing 100 million dollars. You have decided to keep 50 million dollars in the benchmark. The remaining money you will allocate to securities A and B in such a way to achieve your Omega target.
How should you structure your portfolio? Formulate the proper algebraic solution to achieve your desired strategy. You should not attempt to solve the problem to find the exact combination of A and B. However, make sure your algebraic setup is clear enough so we are confident that you could generate the exact numerical solution if you had enough time to do so.