Fixed Income and Credit Risk

Two Term Structure Assignment

Educational purpose of the assignment: you learn a lot of theories and techniques. Through the examples in the lecture, the problem sets and problems in the text book, you deepen your understanding of the techniques, and learn how to apply them in highly simplified settings. The exam then tests your ability to solve problems which are again posed in a very simplified setting. There is often a right way of solving these problems, and your goal is to get as close as possible to the “model answer”.

But when you leave here and try and make use of what you have learnt, the problems you will face are very different. There will be no model answer. The data you need have to be collected; they are not given to you. The data are messy (in this case, you would have bonds maybe with different coupon rates, more dates than prices, significant dealing costs that mean that you would not expect model and actual prices to match exactly). The actual calculations might be done for you by Bloomberg or by some internal software. You would need to think hard about how reliable their methodology is, and what assumptions and judgements are embedded in the software. The numbers themselves would be much less important than what interesting conclusions (firm or tentative) you can draw from the calculations.

The assignment gives you the opportunity to use your skills and creativity, taking the techniques you have learnt, and applying them to a messy problem, and showing that the concepts allow you to find out things about the real world that are new and interesting.

Details of the assignment: let me reiterate. You do need to find two issuers who have each issued a number of conventional coupon bonds that are traded. The payment dates and the maturities do not need to be the same for the two issuers. But you will find it helpful to have bonds with a reasonable spread of maturities – if you want to estimate a term structure ought to ten years, you will need at least one bond with a maturity of at least ten years. It would be helpful to have bonds with a reasonable spread of maturities (eg a bond which matures in the next year or two, a medium dated bonded maturing in the next five years, and something that matures between the medium and long-dated bond).

Having got the bonds and their (dirty) prices on one day, you need to find the term structure of interest rates that underlies those prices. It is easiest to think of the term structure as a set of discount factors, one for each payment date. It is easy to go from the discount factors to the prices – you just value each bond by taking its cashflow, multiplying each payment by the appropriate discount factor, and adding the numbers together. The difficult thing is to go the other way – from the prices to the discount factors.

One way of thinking about it is shown in the attached spreadsheet (of course, if you have some alternative way of doing it that you prefer, please do use that) . It takes a number of bonds and values them using an arbitrary set of discount factors – the bonds are taken from the gilts market and use the data from week 2. There are six bonds and ten payment dates.

Using an arbitrary set of discount factors, I value the bonds and find I am out by 0.32 on average – which is pretty bad. My original choice was not good.

However it is easy to make a better choice. Using the Solver add-in (under the Data tab), I can simply ask Excel to choose the discount factors to make the errors zero. There will be a solution – because I have more dates than bonds. But the solution may not be a very good one, in that the term structure may be unreasonable, with negative spot or forward rates, spikes etc.

So the ingenuity comes with trying to find a sensible term structure. You can try by hand, putting in a more plausible curve. Or you can get Solver to help by using the facility to impose constraints on the solution. You should use your ingenuity to do this. There are no right answers. You may find you have to make a trade off between how closely you fit the prices and how plausible a term structure you get. Just try it and see.

Further help: I have been getting a number of questions about the assignment. I am slightly worried about fairness – giving some people more help than others. So what I would like to do is to spend a few minutes in class this Friday where you can raise questions and I can answer you in front of everyone else.

If you have some questions that are highly specific to the bond markets you have chosen, I would be happy to deal with them individually.

Regards

Anthony

2 November 2015

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