Category Archives: Assignments

Erratum – new version of problem set 8 now available at http://fin4335.garven.com/spring2018/ps8.pdf

One of your fellow classmates (who will remain unnamed :-)) pointed out to me that as originally written, problem set 8 asked for solving the value of a call option using the delta hedging approach.  Since I did not cover delta hedging, I have modified the problem (just moments ago) so that it now asks for calculating call and put values using replicating portfolio and risk neutral valuation approaches.  So if you have previously downloaded the problem set before now, delete that copy and re-download the corrected version, which looks like this:

Problem Set 7 helpful hints

Problem Set 7 is due at the beginning of class on Tuesday, March 20.  Here are some helpful hints:

  1. The least risky combination of Security A and Security B in Problem 1 is found by calculating {w_A} = \displaystyle\frac{{\sigma _B^2 - {\sigma _{AB}}}}{{\sigma _A^2 + \sigma _B^2 - 2{\sigma _{AB}}}} and {w_B} = 1 - {w_A}.
  2. It will always be the case for 2 security portfolios that by following the minimum variance portfolio weighting scheme in the previous bullet point, such a portfolio must have zero variance if {\rho _{AB}} = 1 or -1.
  3. In part B of Problem 2, the Sharpe Ratio for security j is \displaystyle\frac{{E({r_j}) - {r_f}}}{{{\sigma _j}}}.

Problem Set 6 Hints and Spreadsheet (mea culpa)

My bad… I wrote the following not right before spring break and forgot to post it.  If y’all are having problems solving question 1, part C, you’ll find the following information quite helpful.  Also, I am pushing the due date for problem set 6 back to this Thursday; however, you’re welcome to turn it in if you like…

So here goes – In question 1, part C of Problem Set 6, I ask you to “Find the maximum price which the insurer can charge for the coinsurance contract such that profit can still be earned while at the same time providing the typical Florida homeowner with higher expected utility from insuring and retrofitting. How much profit will the insurer earn on a per policy basis?” Here are some hints which you’ll hopefully find helpful.

For starters, keep in mind that in part A, insurance is compulsory (i.e., required by law), whereas, in parts B and C, insurance is not compulsory. In part B of question 1, you are asked to show that the homeowner retrofits while choosing not to purchase insurance. However, in part C, the homeowner might retrofit and purchase coverage if the coinsurance contract is priced more affordably. Therefore, the insurer’s problem is to figure out how much it must reduce the price of the coinsurance contract so that the typical Florida homeowner will have higher expected utility from insuring and retrofitting compared with the part B’s alternative of retrofitting only. It turns out that this is a fairly easy problem to solve using Solver. If you download the spreadsheet located at http://fin4335.garven.com/spring2018/ps6.xls, you can find the price at which the consumer would be indifferent between buying coinsurance and retrofitting compared with self-insurance and retrofitting. You can determine this “breakeven” price by having Solver set the D28 target cell (which is expected utility for coinsurance with retrofitting) equal to a value of 703.8730 (which is the expected utility of self-insurance and retrofitting) by changing cell D12, which is the premium charged for the coinsurance. Once you have the “breakeven” price, all you have to do to get the homeowner to buy the coinsurance contract is cut its price slightly below that level so that the alternative of coinsurance plus retrofitting is greater than the expected utility of retrofitting only. The insurer’s profit then is simply the difference between the price that motivates the purchase of insurance less the expected claims cost to the insurer (which is $1,250).

Relative risk aversion and the demand for insurance (addendum to Problem Set #5)

In Problem Set #5, we studied the effect that premium loadings have upon the demand for insurance (see http://fin4335.garven.com/spring2018/ps5solutions.pdf for the solutions to this problem set).  One of the important takeaways from Problem Set #5 was that other things equal, a more risk averse decision-maker will select a higher coinsurance rate for a given premium loading than a less risk averse decision-maker.  Thus, if you have logarithmic utility; i.e., U = ln W, then your optimal coinsurance rate for a 20% premium loading is \alpha  = 5/6, but if your utility is U = - {W^{ - 1}}, then your optimal coinsurance rate for a 20% premium loading is \alpha  = .9139.

The fact that U = - {W^{ - 1}} is a more risk averse utility than U = ln W is apparent by comparing Arrow-Pratt risk aversion coefficients for these utilities (see pp. 16-26 of http://fin4335.garven.com/spring2018/lecture6.pdf for more on this topic).  Specifically, the Arrow-Pratt risk aversion coefficient for U = ln W is equal to 1/W, and the Arrow-Pratt risk aversion coefficient for  U = - {W^{ - 1}} is equal to 2/W.  Both of these utilities feature decreasing absolute risk aversion, which implies that if we were to increase initial wealth without making any further changes, then both of these utilities would imply lower coinsurance rates (although the decision-maker whose  U = - {W^{ - 1}}  will still select a higher coinsurance rate then the decision-maker whose U = ln W).  For example, suppose that we double initial wealth for both utilities without making any other changes; i.e., increase initial wealth from $1,000 to $2,000.  Then, the optimal coinsurance rate for a 20% premium loading for = ln W falls to \alpha  = .614, and if U = - {W^{ - 1}}, then the optimal coinsurance rate for a 20% premium loading falls to \alpha  = .8007.

However, suppose that instead of only doubling initial wealth, we also double the state-contingent loss from $1,000 to $2,000, holding everything else constant.  This will cause the coinsurance rates for both utilities to revert back to their original values of 5/6 and .9139 respectively.  The reason why this occurs is because not only are these utilities characterized by decreasing absolute risk aversion, they also feature constant relative risk aversion (see the discussion on p. 25 of the http://fin4335.garven.com/spring2018/lecture6.pdf lecture note as to how relative risk aversion is calculated).  Constant relative risk aversion implies that the proportion of  wealth which the decision-maker is willing to put at risk does not change as wealth changes.

Clarification of Problem Set 5 requirements

Problem Set 5 is due at the beginning of class on Thursday.  You may solve it analytically (via the calculus) or by making appropriate modifications to the Bernoulli and Mossin Spreadsheet (or better yet, writing your own spreadsheet code from scratch).  Even better yet, solve the problem set analytically and confirm your results with a spreadsheet model.

If you decide to use a spreadsheet rather than calculus to solve this problem set, then you are required to email a copy of your spreadsheet to risk@garven.com; your problem set answers should reference the spreadsheet so that the source for your answers can be verified.

Problem Set 2 helpful hints

Problem Set 2 is now available from the course website at http://fin4335.garven.com/spring2018/ps2.pdf; its due date is Thursday, January 25.

Problem Set 2 consists of two problems.  The first problem requires calculating expected value, standard deviation, and correlation, and using this information as inputs into determining expected return and standard deviation for 2-asset portfolios.  We covered these concepts during last Thursday’s statistics tutorial; also see pp. 10-20 of the http://fin4335.garven.com/spring2018/lecture3.pdf lecture note.  The second problem involves using the standard normal probability distribution to calculate probabilities of earning various levels of return by investing in risky securities and portfolios.  We will devote next Tuesday’s class meeting to this and related topics.

Problem Set 1 hint…

Problem Set 1 is due at the beginning of class on Tuesday, January 16. Here is a hint for solving the 4th question on problem set 1.

The objective is to determine how big a hospital must be so that the cost per patient-day is minimized. We are not interested in minimizing total cost; if this were the case, there would be no hospital because marginal costs are positive, which implies that total cost is positively related to the number of patient-days.

The cost equation C = 4,700,000 + 0.00013X2 tells you the total cost as a function of the number of patient-days. This is why you are asked in part “a” of the 4th question to derive a formula for the relationship between cost per patient-day and the number of patient days. Once you have that equation, then that is what you minimize, and you’ll be able to answer the question concerning optimal hospital size.

How to know whether you are on track with Finance 4335 assignments

At any given point in time during the upcoming semester, you can ensure that you are on track with Finance 4335 assignments by monitoring due dates which are published on the course website. See http://fin4335.garven.com/readings/ for due dates pertaining to reading assignments, and http://fin4335.garven.com/problem-sets/ for due dates pertaining to problem sets. Also keep in mind that short quizzes will be administered in class on each of the dates indicated for required readings. As a case in point, since the required readings entitled “Optimization” and ” How long does it take to double (triple/quadruple/n-tuple) your money?” are listed for Thursday, January 11, this means that a quiz based upon these readings will be given in class on that day.

Important assignments due on the second class meeting of Finance 4335 (scheduled for Thursday, January 11) include: 1) filling out and emailing the student information form as a file attachment to risk@garven.com, 2) subscribing to the Wall Street Journal, and 3) subscribing to the course blog. A completed Student information form is graded as a problem set and receives 100 points; if you don’t turn in a Student information form, then you will receive a 0 for this “problem set”. Furthermore, tasks 2 and 3 listed above count toward your class participation grade in Finance 4335.