KTH Matematik

Matematisk statistik

SF2942 Portfolio Theory and Risk Management 7.5 cr, autumn 2016

Current information.
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Thursday 2/2
The exam from Dec 2016 has been graded and the results are reported. General information and an overview of the results is available here. Students with the grade Fx who wish to attempt to obtain a passing grade should contact Pierre.

Tuesday 11/22
All results should now be available in Rapp; Ladok may take an additional day or two. General information and an overview of the results is available here. Students with the grade Fx that have not contacted me yet should do so.

Thank you for particiapting in the course! I hope it was an overall positive experience for you; if you have any further questions please don't hesitate to contact me or Calle.

Thursday 11/17
The final exam has now been graded and the results are being reported. General information and an overview of the results is available here. Students with the grade Fx who wish to attempt to obtain a passing grade should contact Pierre.

Thank you to those of you who have taken part in the course evaluation. For those of you who have yet to do so, the evaluation will stay open a few more days - we greatly appreciate all your feedback.

Thursday 10/27
I hope everyone did well on the final exam today. Here is a (preliminary) set of solutions. Note that these have not been carefully proofread and may contain some minor errors - they will be updated in the upcoming days.

As only 30 of you have taken part in the course evaluation the form will stay open for a few more days. Please consider filling out the form to assist us in making appropriate changes to this and other courses.

Lastly, thank you for taking part in the course. Any updates regarding exam scores and similar will be posted on this page; don't hesitate to contact either of us if you have any questions.

Monday 10/24
Scores on assignments and the number of bonus points are now available. Please send me an email if you want to know your score(s).

Regarding the final exam: The preliminary score needed to pass the exam is 25. Please consult the list on the main page for what you can bring to the exam.

Remember to fill out the course evaluation prior to the exam. We certainly appreciate your feedback and it is an important part of planning for upcoming courses (such as risk management and time series).

Sunday 10/23
The second assignment is being graded and you can find out your grade Monday afternoon. The number of bonus points awarded for the final exam will be available Monday evening; send me an email to find out your result.

Please remember fill out the course evaluation prior to the exam; we certainly appreciate your feedback.

Tuesday 10/18
Regarding Assignment 2: i) You do not need to worry about (or motivate) the choice of the instrument used to construct the zero-rate curve, the default setting is fine (and need not be commented on other than whether you kept it or not), ii) you can submit your report electronically; send me an email with a pdf with the name of one group member as filename (e.g., "Nyquist_Pierre_Sol2.pdf").

As always, a reminder to please fill out the course evaluation prior to the exam; we certainly appreciate your feedback.

Monday 10/17
Update on this week's office hours: Due to a meeting with the program representatives from Engineering Physics, Tuesday's regular office hours have to be re-scheduled. On Tuesday I will be available after lunch instead and then throughout the week; if you know that you want to stop by a certain day/time you can send men an email to make sure I will be available. You are of course also welcome to stop by either Calle's or my office at any time during the week.

I will continue to remind you to please fill out the course evaluation prior to the exam; we certainly appreciate all your feedback.

Friday 10/14
Today we covered parts of Ch. 6 on risk measurement. Specifically, the notion of a risk measure, proposed properties, and their interpretations, that such measures may have (not necessarily all!), leading to the notions of convex risk measures and coherent risk measures. We then discussed Valute-at-Risk and Expected Shortfall at some length, ending with Problem 5 on the exam from Oct 2014 (part b). For more examples on the computation of these two risk measures please see the textbook.

Regarding final exam: Some additional information is available on the main page.

If you want to know your score on Assignment 1 please send me an email.

Wednesday 10/12
A summary of the course is now available: courseSummary.pdf.

Regarding previous exams: Exams given by Henrik Hult were open book. However, all previous exams contain (at least some) good problems that will prepare you for the final. The fact that it will be open book just means that the problems must be formulated slightly different than otherwise, the core principles that are tested of course remain the same.

Tuesday 10/11
Today we covered Ex. 5.3 in some detail, discussing the interpretation of different parameter values in the utility function (polynomial HARA), Ex. 5.5 (illustrating one more link between the contents in Ch. 4 and Ch. 5), and parts of Section 5.3. For the latter, we stated the optimization problem of interest (eq. (5.5), maximizing w.r.t. the payoff function) and corresponding sufficient optimality conditions.

Regarding Assignment 2: In order to open the workspace in Quantlab: Right click on the link to the workspace and choose "Save link as...", then upload to your catalogue "appdata" through home.ug.kth.se

Monday 10/10
In the workspace published on Friday 10/7 there was to quantities missing in the window showing the liability cash flow. This omission is corrected for in the updated version available here and on the main page; the code was included in the original version and you can compile it for yourself if you want to.

Sunday 10/9
For a brief description of the workspace used for the immunization assignment you can have a look at the following pdf.

Important: At some point prior to the final exam please take the time to fill out the course evaluation, available here. The evaluation form is online starting today and will be closed the day before the final exam. Course evaluations are an important part of ensuring the quality of your education so please take the short time it takes to fill out the form. All feedback is greatly appreciated; all answers are of course anonymous.

Friday 10/7
The second assignment is now available on the main page.

In today's class we embarked on Ch. 5 - utility-based investment principles; I encourage anyone who is interested to look at Section 4.5 regarding the drawbacks of the mean-variance approach (only briefly mentioned today), as it motivates the move away from the investment problems we have considered. We covered most of Section 5.1 up to Example 5.4 - the link between the trade-off problem of Ch. 4 and the quadratic utility function - omitting for now Examples 5.2 (Allais' paradox) and 5.3 (fire insurance).

Important: It has been brought to my attention that some students are (severely) allergic to peanuts. Therefore, I ask you to please refrain from bringing foods/snacks that contain penauts to the final exam. If you feel that this is too much of an impostition then let me know and we will figure something out that works for everyone involved.

Tuesday 10/4
Today we covered Section 4.3 - "Investment in the presence of a liability". We discussed the modified investment problem (4.18) and how its solution is a combination of the optimal position from a quadratic investment perspective and the optimal quadratic hedge of a libability (as in Prop 3.3). After a brief discussion of Ex. 4.9 and we spent most of the time on Ex. 4.10. I encourage you to look at pages 110-111 in the book for some aspects of the example that we did not cover in class.

Friday 9/30
We covered Section 4.2.3 on evaluating performance of the optimal portfolios, obtained from the quadratic investment problems, using simulated data.

Thursday 9/29
In today's problem session Calle covered Problems 4.1 and 4.2.

Wednesday 9/28
Important: The Student Affairs Office has sent out a reminder that all students must register for the final exam prior to 10/5. To be able to register for the exam you must first be registered for the course. If the latter is true, or if you are registered for the course but still cannot register for the exam, please contact the student affiars office - elevexp@math.kth.se - immediately.

Tuesday 9/27
Today we continued the discussion of the so-called trade-off problem (4.3), for which we obtained the optimal solution. We then proceeded to the analogous problem but with a risk-free asset included - math display (4.7) in the book - and obtained the associated optimal portfolio. We discussed the Sharpe ratio for such portfolios and considered properties of the optimal portfolio in some explicit settings: Uncorrelated returns (Ex. 4.6) and when there is uncertainty in the model parameters (Ex. 4.7). Lastly we defined and discussed briefly the notion of the efficient frontier in (4.3) and (4.7) (Ex. 4.8).

Monday 9/26
On Friday 9/30 represenatives of Nektar Asset Management AB, a subdivision of Brummer and Partners, will visit us and talk about some current opportunities within their organization. This will take place at approx. 9:00, in case you will not attend the lecture but want to listen to what their presentation. Any other information concerning the visit will be posted here prior to Friday.

Also, note that there has been a change from the preliminary plan in that the second home assignment is not yet out; more information will be available later this week.

Friday 9/23
We first discussed how one can perform immunization using PCA and finished Example 3.16. Then we moved on to Ch. 4: Quadratic investment princples. We discussed the general setting, with initial capital, spot prices of risky assets that we can invest in etc., and formulated three optimization problems of interest (with is no risk-free asset available): The "trade-off problem", the "maximizing-of-expectation problem" and the "minimization-of-variance problem". The session ended with a quick remark on the form of the optimal solution to the trade-off problem (see p.88) and this is where we will continue on Tuesday.

Thursday 9/22
In today's problem session Calle discussed Problem 3.4, as well as some aspects of the content in Chapter 1 (pricing measure and discount factors) in the context of Problem 1.5.

Tuesday 9/20
Today we discussed principla component analysis. First from a general perspective, describing the method, and then in the setting of a specific example (Example 3.16). We only made it through roughly the first half of the example, which included the PCA-part of the problem, and will look at the second half, which includes setting up and solving the linear equations arising in our approach to immunization, on Friday.

Both parts of today's lecture contained illustrations/computations in R. For the first part concerning PCA, there is an R Markdown page available here; the corresponding rmd-file can be downloaded here.

Friday 9/16
In today's session we finished Example 3.8 on non-life insurance and started on Section 3.6 - Immunization of cash flows. We discussed the general setup, matching present value and Taylor expansion of present value when there is a change in the zero-rate curve, summarized in equations (3.11) and (3.12), and the concept of "duration". In particular, we looked at an example of how entering an interest rate swap can change the sensitivity of one's hedging portfolio to changes in the zero-rate curve.

Next time we will look at an approach to finding suitable scenarios for the instantaneous changes to the zero-rates.

Thursday 9/15
In the problem session (8:00-10:00), in addition to some general discussion and recap, Problems 1.5, 3.1 and parts of 3.2 were discussed. Calle will present the remaining parts next session.

In the second session we continued with examples of hedging insurance liabilities. Example 3.7 was covered and the remaining parts of Example 3.8 will be presented on Friday. Some calculations regarding binomial and Poisson distributions were discussed at some length; reviewing the (alternative) calculations in the book for joint conditional distributions in Example 3.8 might be beneficial.

Lastly, note that Calle has moved and his office is now on the 7th floor, room 3750, of the math building.

Wednesday 9/14
Two clarifications regarding Problem 3 on first assignment: You can use Black's formula even though the option is of American type (why?). Most of you have already done this and can thus ignore this announcement. Also, the second part of the problem should read "Plot this together with the implied volatilities ...", not probabilities.

A second reminder for Thursday's problem session: If you feel unsure about futures contracts, it might benefit you to review the beginning of Section 3.2.

Tuesday 9/13
We covered parts of Section 3.3 - hedging of insurance liabilities - up to and including Example 3.6. The remainder of Section 3.3 will be covered during the afternoon session on Thursday (9/15).

For Thursday's problem session, if you feel unsure about the structure of a futures contract you are encouraged to review the beginning of Section 3.2; one of the problems will concern hedging with futures.

Monday 9/12
For those interested in familiarizing themselves with KTH Finance Lab (Quantlab) you can visit Alexander Aurell's webpage where there is some material related to the course SF2701. In particular there is an exercise which serves as an introduction to Quantlab, as well as a user's manual (more advanced than we need). If you are having trouble accessing Quantlab send an email to either me or Alexander. Note that this is strictly for students interested in using KTH's Finance Lab and by no means a requirement for the course. However, it will come in handy later on as well as in future courses.

Friday 9/9
The first (voluntary) homework/assignment is now available. There is also a link to a note on American options on the main page. You can solve the problems without reading the note but for those interested it provides some more details on the this particular topic.

Today's session covered Section 3.1 in the textbook: Results on minimizing the expected squared error for different forms of the portfolio (completely arbitrary function of the underlying assests vs. a linear combination) and two examples - 3.1 and 3.2 - of hedging insurance contratcs.
Two minor errors in the presentation: 1) In Example 2.3, The covariance of between the asset value, i.e. the discount factor from time 2 to time 1, and the liability should indeed be (1-p) times the variance of the asset value. This was claimed during the lecture but I did not go back and correct the calculations - you can check this on your own. 2) In Example 3.1, the position in the 1-year zero-coupon bond should not have the price of the zcb in the denominator - B_0 was accidentally (and erroneously!) used to refer to the growth factor rather than the price in that expression.

Thursday 9/8
During the first session we covered Example 1.4 and the first part of Ch. 2, up to and including stating Proposition 2.1 (The Lagrange multiplier method). The second session covered the remaining parts of Ch. 2 - a proof of Prop. 2.1 and the section on more general convex optimization (including proofs) - and the very beginning of Ch. 3: the minimization problem of interest in quadratic hedging and some basic facts about conditional expectations.

Tuesday 9/6
In today's session Calle covered Problems 1.1, 1.2 and 1.3.
Solutions to (some of) the exercises in the book are available here (there is also a link on the main page).

The week will continue with some additional examples from Chapter 1 - particularly one discussing the notions of subjective and forward probability in a specific example - before we embark on Chapter 2 (Thursday) and Chapter 3 (Friday).

Friday 9/2
Covered implied forward probabilities (Section 1.2.2) .Specific examples on (i) a digital option and (ii) fitting a volatility smile and corresponding distribution and density function given option prices (Example 1.9).

This finishes up Chapter 1 (excluding the problem session on 9/6); those who want more information about financial markets and contracts are encouraged to look at either the course Financial Mathematics, Basic course (SF2701, lecture notes available on the webpage) or any of the books by Hull (see syllabus).

Thursday 9/1
We covered parts of Section 1.2, ``Derivatives and no-arbitrage pricing'', up to Black-Scholes formula for call options. Topics included European derivatives, arbitrage opportunities in markets with derivatives, Theorem 1.2 - the analogue of Thm 1.1 for derivatives instead of deterministic cash flows - and the log-normal model; Examples 1.6 and 1.8.

Some updates regarding the course in general: The exam will be open book, i.e., you are free to bring the main text (Hult et. al.). On the administrative side, a somewhat more detailed plan of the course is now available (see the main page). There may still be some changes in terms of which sessions are theory sessions and which are problem sessions, but such changes will be announced well in advance. Note that the first problem session will be on Tuesday 9/6 .

Tuesday 8/30
The general course information is now available on the main page. Also, my office has been settlead to be room 3443, telephone number 6633. As of Thursday 9/1 this is where all office hours will be held.
The first lecture contained a basic intro to the course - these slides can be found here - and then covered parts of Section 1.1 (interest rates, deterministic cash flows), up to and including stating Theorem 1.1.
The second lecture - 15--17 - we discussed Thm 1.1, complete markets, forward prices, boostrapping zero rates (Example 1.1) and extracting zero rates from forward prices (Example 1.2).

Tuesday 8/23
Welcome to SF2942! The first session is on Tuesday August 30 at 8 am in room B3 (Brinellvägen 23) [Map]. Note that there are two sessions that day, the second at 3 pm.
More information will be available shortly. For now, if you have any questions about the course you can reach out to either me (Pierre) or Carl; see the main page for contact information. For administrative questions please contact the Student Affairs Office.

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Published by: Filip Lindskog
Updated: 22/06-2011