How can I apply for MuSigma
Department of Economics
The subject of decision theory is, as the name suggests, the financial theory of decisions. For us, decisions are so commonplace (for example: »Cafeteria: yes or no?«) That we often no longer think about them extensively. But when financially significant consequences threaten (for example: "Should I make provisions for my pension and if so, how?"), Then we need a substantive and convincing theory that gives us guidelines. We will get to know two of these theories here.
These are the expected utility calculus and the µ-σ theory (portfolio theory). Both approaches should be dealt with in the lecture. Because the approaches are very general theories, this lecture will not be application-oriented. It requires that you are willing to work formally and think abstractly. It is worth the effort if you want to understand the basic principles of financial decisions. The lecture usually takes place in the winter semester.
We have the following exams on the net: WS2006 / 07 with solution, SS2007 with solution, WS2007 / 08 with solution, SS2008 with solution, WS2008 / 09 with solution, the intermediate exam WS2009 / 10 with solution, the final exam WS2009 / 10 with solution and the final exam WS2010 / 11 with solution.
Ideally, you should have read the following literature before the respective lectures:
|1||Basic model under security, preference revealed||Section 1.1|
|2||Preferences under security||Section 1.2|
|3||Basic model expected utility||Section 1.3|
|4||Markowitz Prize||Section 2.1|
Simple portfolio problem
|6||Absolute risk aversion||Section 2.3|
|7||Relative risk aversion||Section 2.4|
|8||Stochastic Dominance (FSD)||Section 2.5|
|9||Stochastic Dominance (SSD)||Section 2.5|
|10||Basic model uncertainty||Section 3|
|11||Expected utility in the event of uncertainty||Section 3|
|12||Applications of uncertainty||Section 3|
|13||Basic model μ-σ theory||Section 4|
|14||Saturation in the μ-σ calculus||Section 4|
Varian, H. "Grundzüge der Mikroökonomik", Munich (Oldenbourg) 1991.
Formal exam questions (last change October 20, 2019)
We have observed in the past that some students rely on memorizing the most important elements of the lecture as well as the solutions to some of the exercises just before the exam. That is not enough to understand decision theory. You may pass the exam using this approach, but you need to be a lot of luck to get a good grade. The subject of this lecture is not simple and it takes some time to understand the relationships. It is certainly not possible to cram this theory in a fast run. We therefore strongly advise you to go through the material regularly during the semester with other fellow students and to try to solve the exercises together. On the one hand, it is more fun with others, and on the other hand, you are more likely to notice where the subject is not understood. And last but not least, you will have less work to do with this lecture shortly before the exam.
Please note that we reserve the right to provide our answers up to two weeks before the exam retrospectively to change.
- What is the exam based on?
The exam will largely consist of arithmetic tasks that are based on the exercises. The script is of course also relevant (definitions, statements of the sentences).
- Is evidence relevant to the exam?
Smaller proofs (i.e. <1 page) are relevant for the exam, but not the harder proofs (> 1 page) in the script.
- Which math skills are necessary?
You should be able to derive and integrate the usual functions such as logarithm and the power functions.
- How can you optimally prepare for the exam?
If you have always solved the exercises and were able to understand the material of the lecture (in this order!), Then you are optimally prepared.
- Will there be choices in the tasks?
No. There are several small tasks for this, I typically assume around 4-5 tasks in the exam.
- Can you give a sample question?
No, but you will have to spend about half to 2/3 of the time on calculations and the rest on theory questions. For example, you should know the definitions and basic meanings of the sentences.
- Do you have to be able to reproduce the sentences verbatim?
No, the content is sufficient.
- I take from your notes on the exam included in the script that smaller pieces of evidence are relevant for the exam, but the "heavy" pieces of evidence in the script are not. Can I conclude from this that only evidence that was provided in the exercise (positive transformation of the utility function, etc.) is potentially tested?
No, this does not apply to the exam.
- For example, the proof of Theorem 2.4 is still a bit difficult in my opinion, even if you made it very understandable in the lecture. So I wouldn't have to provide such proof in the exam if I understood that correctly.
You misunderstood that (see above): "a little difficult" is less than "difficult".
- Furthermore, I know from previous exams that evidence that is longer than one page will not be requested. But I was also told something about "split" evidence. In this context the question arises for me whether the proof for theorem 2.7 for the exam is relevant because it is longer than a page.
Yes. Theorem 2.7 consists of three parts, each under one side.
- My math skills are rather rudimentary. Is this taken into account?
No. We assume that you know the details from our math script (see above).
- Now it was said that "heavy" evidence and evidence that is over 1 page long is not relevant. In the FAQ I found a question about the proof of Theorem 2.7. The FAQ replied that this sentence was relevant. Therefore, I now have difficulty classifying which evidence is relevant for which sentences. For example, is the proof of Theorem 3.4 (Tobin separation) relevant to the exam?
Theorem 2.7 consists of three parts, each under one side. Therefore it is relevant for the exam. But I would rule out the evidence of the Tobin separation. Please understand that a criterion of the form "serious evidence will not be checked" is nonsensical because we would then always have to focus on the student who has the least desire to learn. With such exams you cannot distinguish good from very good female students.
- Can we assume in the exam that the safe title is called Y_1 and that this has the price or the expected value 1?
No, it could also be called Y_2, for example, have an expected (= certain) return flow of 2.5 and a price p (Y_2) = 3. However, it is not our aim to confuse you by using variable names that are as nonsensical as possible.
- Do we have to define variables in the exam or do we have to e.g. B. if we write ω also mean the share of the total portfolio?
If you adapt your notation to the lecture / exercise, you do not have to explicitly document it. Likewise, if it is clear and quickly recognizable from the context what you mean by a certain symbol (for example if you choose θ instead of ω as the share of a security in the total portfolio and this is clearly evident from the equation for the expected value or the variance of the portfolio) . If, on the other hand, you use your own "wild" notation - which we do not recommend - (e.g. θ for the price of a security), then you must document this. In short: we refuse to undertake extensive decryption work when making corrections, but we also do not play stupidly.
- Is the calculation of the Markowitz premium both precise and relevant to the exam using the approximation formula?
- Are follow-up errors taken into account in the exams?
Yes. We therefore recommend that you do not waste too much processing time looking for discovered calculation errors, but instead point out this discovery in a short sentence ("The result must be wrong because ...").
- If I don't have time for a correct solution, should I at least write down the approach to a problem?
If this personal contribution is classified as grade-relevant, there are also points on the mere description of the approach. The same applies to a brief sketch of how an incomplete invoice should be continued. There are no points on trivialities and vague or confused explanations that were obviously only written down to land a chance hit.
- Are exercises relevant to the exam that were not dealt with in the exercise?
No, unless the relevant content has been conveyed extensively elsewhere (e.g. through other exercises).
- Can we use a "cheat sheet"?
Yes, in addition to a pocket calculator, you are allowed to take a DIN A4 sheet of paper with you on both sides of the exam. However, it must not be pasted (otherwise it could become an accordion with 20 pages ...).
- In old exams, tasks were set that are no longer practice tasks today. Can these still be used in the exam?
Even then, these tasks were not exercises. We expect you to be able to provide transfer services.
- Some things (complicated derivations, rule of l'Hospital, calculation of the return on portfolio shares, etc.) we have only dealt with marginally. Are they also relevant for the exam?
- First of all, I am unsure how long the processing time was for the old exams that were made available, are they designed for one or two hours, or will our exam this semester only be one hour?
You can see this from the number of points: One point usually corresponds to a minute. Our exam is one hour.
- I would also be interested in whether, if the comparability, continuity and transitivity of a preference order is to be checked, naming the associated utility function (if available) would suffice as evidence or whether we should nevertheless address the individual criteria.
Yes, that would be sufficient - unless you are explicitly asked to check these properties and you are Not allowed to use Debreu's theorem.
- Can you also check / prove stochastic dominance graphically?
If this is not explicitly excluded, yes.
Content-related exam questions
- What exactly does it mean when dRRA / dx> 0? Does this simply mean that relative risk aversion increases as wealth increases, and therefore the utility function is not appropriate, since the RRA should normally decrease as wealth increases?
- In exercise set 8 / no.2 the following relationship is used: x transposed * Omega * x = Var (x). Where does this connection come from?
In the exercise (repetition of the calculation of probability) we gave you a definition of the expected value for continuous and discrete random variables and, based on this, a definition of the (co) variance. If you insert in this formula for the variance [Var (x) = Cov (x, x) = E (xE (x))] for x the return flow from a portfolio X_Pf = aX + bY + cZ + ..., so After using the definition of the expected value and some transformations, you get the result x'Omega x with the vector of the quantities of securities x '= (abc ...). We dispensed with this calculation in the exercise.
- Is it not enough in exercise b) of Set 6) to simply form the integral from minus infinity to four to prove SSD?
No, so that there is an SSD, it must be for any Upper limit t apply that the integral from minus infinity to t over F_x is less than or equal to the integral within the same limits over F_y.
- In the exercises [on stochastic dominance] we always formed the interval from "lower bound" to s [...]. Would it also be possible to use the concrete numerical values for the integral limits? For example, in part 6b) then the integral from 0 to 0.25, from 0.25 to 0.75 and from 0.75 to 1?
No. The way you suggested does not represent a generally valid solution, but only for certain simple problems as in the exercises (straight line equations).
- Using the formula for SSD, I can only measure the stochastic dominance of F_x over F_y. How do I measure whether F_y is stochastically dominant over F_x?
The designations in Theorem 2.14 are chosen completely arbitrarily. You can also reverse X and Y, but then not only in the distribution but also in the utility functions above and then obtain the condition of the stochastic dominance of the second order of F_y over F_x.
- In the second case (0.25 ≤ s ≤ 0.75) of exercise d) from set 6), why did we choose the value 0.25 not only as the lower limit of the integral over F_x, but also as the lower limit of the integral over F_y? Doesn't the lower limit have to be zero for F_x?
Yes, you are right. In the exercise we have arbitrarily assumed a stricter value of 0.25 as the lower limit of the integral over F_x, so that the calculation is more elegant. However, a lower integral limit of zero would be methodologically correct for F_x.
- In exercise 3/2) the probability of a payout of 8 monetary units should actually be 1/16, I think.
No, because you will not only receive a payout of 8 CU if you throw a coat of arms the fourth time, but also if you throw a coat of arms on the fifth, sixth, seventh, ... time. In short: Whenever you have not thrown a coat of arms when you tossed 3, you will receive a payout of 8 GE, no matter how often you may continue to throw. This is always the case after you have thrown a number the first three times. The probability for this is 1/2 * 1/2 * 1/2 = 1/8.
- In the online exercise exam from WS 07/08, the first task is to transform utility functions. Since this is not possible, a counterexample can be found. In the solution to this exam, numbers have been used for these counterexamples that we would not come to so quickly in the exam. Is there a way to calculate these numbers or do you have to try around forever?
You have to try. But there are a few tricks to find counterexamples. First of all, you should clarify whether and why the two utility functions do not always represent the same preferences. For example, with - (1 / x) this is the jump point or with x ^ 2 the minimum at x = 0. In accordance with these considerations, you should select a few simple numerical values (preferably zeros, ones, etc.) so that you get into the corresponding areas where the two functions behave differently. If you have not yet come across different signs for the two utility functions, you should vary the values of the variables accordingly.
- A question about exam task 2 from the summer semester 08. How should I set the probabilities if the second state and not the first state would occur with a probability of 1/2.
You are completely free to which of the remaining two states (one and three) you then assign the variable q (or z or v ...) as the probability. The remaining last state would then have the probability 1-1 / 2-q = 1/2-q.
- I'm interested in checking the SSD in exercise set 6d. Can't we make things easier for ourselves there by calculating the area from 0 to 0.5 and comparing this to the area from 0.5 to 1?
If you mean the difference in areas, then you are right in this particular case. In general, this procedure only works if the slope of one distribution function in an interval is always greater than the slope of the second distribution function.
- A question about maximizing utility in mu-sigma utility functions: Why is the risk-free portfolio or its EV used when determining E (x) and NOT when determining the variance, i.e. why is there no x1?
The variance of the risk-free title is zero, as are all covariances of this title with risky titles. By multiplying with these zeros, all x1 in the variance component are omitted.
- Exercise set 1, exercise 2 examines whether U * (X) represents the same preference as U (X). In the event that this is not the case, we have always looked for a counter-example in the exercise. Would it also be possible in an exam question not to find a counterexample and to argue that U * (X) can be generated by transformation from U (X), but it is not a strictly monotonous transformation? E.g. subtask f) -> - (..) ^ - 1 monotonic transformation, but not strictly.
This reasoning is not yet proof, but it is an essential step towards it. You would therefore not receive the full number of points, but what we consider appropriate proportion.
- In Set 9 No. 1b): Do I have to buy the risk-free asset or do I get one for free?
According to the task for an additional risk-free asset (amount y) you have to forego shares in the risky portfolio (amount (1-y)). This corresponds to a purchase decision with a limited budget.
- When calculating the exercises, there was a problem with the drawing of the indifference curve in set 2, exercise 1. It is not clear to us how "c" affects the shifting of the curve, or whether we can mark "c" arbitrarily.
"c" is any constant, so you can choose c = 1, 2, 3 or 4, for example. If you choose c = 3, you get the indifference curve for a utility of U = 3 and this runs in Set 2/1) from X_0 = 3 and X_1 = infinite to X_0 = 4 and X_1 = 0 (see Figure 1 with c = 4). You can graphically depict the influence of the utility level c on the indifference curve if you insert c = 1,2,3,4 as sketched above and draw in the resulting indifference curves. As can be seen from Figure 1, this indifference curve always has the same shape and is only shifted to the right.
- In the old exam from WS 07/08 In task 4 B we should calculate both the variance and the expected value. When calculating the variance, it is incomprehensible to me why, firstly, I square the expected value minus 100 at the very back and, secondly, why the 1/20 disappear from the first integral when the antiderivative is formed.
The risk-free component of the total payment is one hundred. 587.5-100 therefore gives the expected value of the risky payment. If these were not subtracted from E (X) when calculating E (X) ^ 2, they would also have to be taken into account in the first term E (X ^ 2) of the formula for the variance. On the second question: the antiderivatives have already been combined . (20S) ^ 2 = 400 S ^ 2. Divided by 20 results in 20 S ^ 2. Broken down you get 20/3 S ^ 3.
- Is it possible in exercise d) of exercise set 6 to use theorem 2.15 for the solution instead of determining the result via the relatively extensive calculation of the integrals?
- How do I know in the exam from WS09 / 10 in task 4 that I only have to check for a possible stochastic dominance of security Y over security X?
For small t, F_Y has a lower value than F_X. Therefore X cannot be stochastically dominant, neither first nor second order.
- Is one of the securities always stochastically dominant?
- In task 9/3) the investor does not exhaust his budget because there is no risk-free asset. Does the investor just keep the remaining money?
No. Without a risk-free asset, the investor has no way of transporting his (monetary) assets into the subsequent period without risk. The unused (monetary) assets are lost.
- In exam task 4 from WS09 / 10, the SSD is integrated over the entire definition range, although the functions intersect at t = 0.5. We did it differently in the exercises.
In the exercises, the functions showed points of discontinuity ("jumps"). Therefore we had to make case distinctions with the integrals. Interfaces, on the other hand, have no influence on the integral limits.
- In Set 4/1 we use the well-known approximation formula for the Markowitz premium. Why do we use 0.5c for x in the last step?
The approximation formula for the Markowitz premium contains ARA (E (x)). We had already determined E (x) = c / 2 and we are now inserting this value into ARA (x).
- Set 9.1a): The problem asks about the standard deviation, but we only calculate the variance. In our opinion, the roots still have to be pulled.
- Set10.4: The investor's utility function contains the portfolio variance at the root. There is a formula for this: X1 ^ 2 * Var (Y1) + X2 ^ 2 * Var (Y2) + 2 * X1 * X2 * Cov (Y1, Y2). Why do we not take the covariance into account in our solution?
The covariance of the payments of a risky security and a risk-free security is always zero. Therefore this term is omitted.
- A question about the final exam WS09 / 10, task 1 (simple portfolio problem). You said that as wealth falls, the statements about wealth increase are simply reversed. For example, if x falls and ARA / RRA falls, the amount or proportion of risky securities decreases. Does this also apply to increasing ARA / RRA, i. H. does the amount / proportion of risky securities increase if x decreases?
Yes. With falling wealth, the statements for increasing wealth are simply reversed. This means that your second statement follows directly from Theorem 2.8. If ARA represents an increasing function in w, n * is a decreasing function in w (increasing wealth) and thus an increasing function in -w (decreasing wealth).
- In exercise set 9 (3) I do not understand why the budget restriction is not binding.
In an optimization under a secondary condition, the secondary condition is always met exactly. In our case, the budget restriction applies in all directions, i. H. You always use your entire budget exactly. If there is a risk-free asset, this restriction is necessary so that you do not buy more risk-free asset than you can pay for from your budget. Without a risk-free asset, however, the secondary condition forces you to exhaust your budget, even if your benefit decreases as a result (saturation point exceeded). We therefore initially optimized the risky portfolio under one secondary condition and then without it. We have come to the conclusion that your maximum benefit is available if you do not use your budget to the full.
- Why do we sell an infinite amount of the risk-free assets empty in exercise set 10 (4)?
In simplified terms, you can imagine a short sale in such a way that you borrow a risk-free asset from another person in t = 0 and sell it on the capital market. So you get money in t = 0. With this money you buy the risky asset. In t = 1 you receive the payment of the purchased risky asset and buy back the risk-free asset from this money. You give this back to the person from whom you borrowed it in t = 0. So you need the short sales of the risk-free asset in t = 0 to finance the purchase of the risky asset.
- A question about the third task of exam SS08 In the solution, the utility function is determined "uniquely" with 1 - e ^ (- 0.2x). Is this really "unambiguous" or could the +1 be omitted because it is a cardinal transformation?
Using U (0) = 0 and U '(0) = 2, the CARA utility function U (x) = (1 - e ^ (- 0.2x)) actually clearly specified. The utility function you proposed - e ^ (- 0.2x) is equivalent in the sense that it leads to identical decisions, but it does not satisfy the condition U (0) = 0.
- If the ARA is constant at zero and the RRA is greater than zero, e.g. at 2 then the RRA increases with growing wealth and the ARA remains constant. Accordingly, as assets grow, the amount of risky stocks consumed remains the same and the proportion of risky stocks in the overall portfolio decreases. But what happens when wealth falls? If I think about it properly, common sense would have to keep the proportion at most the same and the amount would have to decrease. The amount consumed will definitely not stay the same, will it?
But! In the case mentioned, the amount of risky assets remains the same with falling assets and the proportion thus increases until it is 100% at some point. That this seems implausible to you is due to the fact that a falling RRA with growing wealth can hardly be reconciled with "common sense". One can perhaps intuitively imagine the case that with a sinking wealth, risk-free returns are no longer enough to survive and that one therefore increasingly has to put everything on one card.
- Why is f_y (t) = 2 and f_x (t) = 1 in Set 6c)?
In the exercise for Set 4) we derived that a random variable x uniformly distributed over the interval [a, b] has the density function f (x) = 1 / (b-a).
- In my opinion, sentence 2.8 (p. 27) is longer than a page. So is this negligible?
You mean the proof, not the sentence itself: Correct.
- In the mock exams or old exams there was usually a task in which an alternative solution with the Lagrange approach was possible, but the task could also be solved without this. So is this not necessarily relevant for the exam?
No, it could well be that we ask in a task that it can be solved with Lagrange.
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