Category Archives: Helpful Hints

Helpful hints for Problem Set 9

A student asked me whether it’s okay to solve Problem Set 9 by using Excel. While I generally encourage students to use Excel for the purpose of validating their work (especially for computationally challenging problem sets such as the present one), I also expect students to demonstrate understanding and knowledge of the logical framework upon which any given problem is based. In other words, I expect you to show and explain your work on this problem set just as you would have to show and explain your work if this was an exam question.

By all means, create your own spreadsheet model of Problem Set 9 to validate your answers for this problem set. But start out by devising you own computation strategy using a piece of paper, pen or pencil, and calculator. Since you know that the value of risky debt is equal to the value of safe debt minus the value of the limited liability put option, one approach to solving this problem set would be to start out by calculating the value of a riskless bond, and the value of the limited liability put option. The value of a riskless bond is V(B) = B{e^{ - rT}}, where B corresponds to the promised payment to creditors. The value of the option to default (V(put)) can be calculated by applying the BSM put equation (see the second bullet point on page 8 of http://fin4335.garven.com/spring2018/lecture16.pdf); this requires 1) calculating {d_1} and {d_2}, 2) using the Standard Normal Distribution Function (“z”) Table to find 1-N({d_1}) and 1- N({d_2}) , and inputting these probabilities into the BSM put equation, where the exercise price corresponds to the promised payment to creditors and the value of the underlying asset corresponds to the value of the firm’s assets (keep in mind that the (risk neutral) probability of default corresponds to  1- N({d_2}) for reasons explained during yesterday’s class meeting). Once you obtain the value of the safe bond (V(B)) and the value of the option to default (V(put)) for each firm, then the fair value for each firm’s debt is simply the difference between these two values; i.e., V(D) = V(B) – V(put). Upon finding V(D) for firm 1 and firm 2, then you can obtain these bonds’ yields to maturity (YTM ) by solving for YTM in the following equation: V(D) = B{e^{ - YTM(T)}}; the credit risk premium is equal to the difference between the yield to maturity and the riskless rate of interest.

Since the value of equity corresponds to a call option written on the firm’s assets with exercise price equal to the promised payment to creditors, you could also solve this problem by first calculating the value of each firm’s equity (V(E)) using BSM call equation (see the second bullet point on page 7 of http://fin4335.garven.com/spring2018/lecture16.pdf and substitute the value of assets (V(F)) in place of S and the promised payment of $B in place of K in that equation). Once you know V(E) for each firm, then the value of risky debt (V(D)) is equal to the difference between the value of assets V(F) and the value of equity V(E) Then YTM and credit risk premium follow in the manner described in the previous paragraph.

Midterm exam 2 information…

Midterm 2 will be given during class on Tuesday, April 3. This test consists of 4 problems. You are only required to complete 3 problems. At your option, you may complete all 4 problems, in which case I will throw out the problem on which you receive the lowest score.

The questions pertain to topics which we have covered since the first midterm exam. Topics covered include 1) demand for insurance, 2) adverse selection, 3) portfolio/capital market theory, and 4) financial derivatives (calls and puts specifically).  I have posted the formula sheet that will appear as the back page of the exam booklet at the following location: http://fin4335.garven.com/spring2018/formulas_part2.pdf.

Tomorrow’s class meeting will be devoted to a review session for the midterm exam. If you haven’t already done so, I highly recommend that you review Problem Sets 5-8 and also try working the Sample Midterm 2 Exam (solutions are also provided) prior to coming to class tomorrow.  I will come to class prepared to work through the solutions for Problem Set 8 and the sample exam, as well as address any questions or concerns that y’all may have.

On the role of replicating portfolios in the pricing of financial derivatives

Replicating portfolios play a central role in terms of pricing financial derivatives. Here is a succinct summary from yesterday’s class meeting:

  1. Buying forward is equivalent to buying the underlying on margin, and selling forward is equivalent to shorting the underlying and lending money. Like options, forwards and futures are priced by pricing the replicating portfolio and invoking the “no-arbitrage” condition. If the forward/futures price it too low, then one can earn positive returns with zero risk and zero net investment by buying forward, shorting the underlying and lending money. Similarly, if the forward futures price is too high, one can earn positive returns with zero risk and zero net investment by selling forward and buying the underlying with borrowed money. This is commonly referred to as “riskless arbitrage”; it’s riskless because you’re perfectly hedged, and it’s arbitrage because you are buying low and selling high.
  2. The replicating portfolio for a call option is a margined investment in the underlying. During yesterday’s class meeting, we priced a one timestep call option where the price of the underlying asset is $100, the exercise price is also $100, u = 1.05, d = .95, the interest rate r = 5%, and the timestep \delta t = 1/12. Given these parameters, the payoff on the call is $5 at the up (u) node and $0 at the down (d) node.  The replicating value consists of half a share that is financed by a margin balance of $47.30; thus the “arbitrage-free” price of the call option is (.5(100) – 47.30) = $2.70.
  3. Since the replicating portfolio for a call option is a margined investment in the underlying, it should come as no surprise that the replicating portfolio for a put option consists of a short position in the underlying combined with lending. Thus, in order to price the put, we need to determine and price the components of the replicating portfolio. We will begin class tomorrow by completing our analysis of the replicating portfolio approach to pricing calls and puts, and move on to other pricing methods such as delta hedging and risk neutral valuation.

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.

Analytic and Numerical Proofs of the Bernoulli Principle and Mossin’s Theorem

Last week, we discussed two particularly important insurance economics concepts: 1) the Bernoulli Principle, and 2) Mossin’s Theorem. The Bernoulli Principle states that if an actuarially fair full coverage insurance policy is offered, then an arbitrarily risk averse individual will purchase it, and Mossin’s Theorem states that if insurance is actuarially unfair, then an arbitrarily risk averse individual will prefer partial insurance coverage over full insurance coverage.

The Bernoulli Principle is graphically illustrated in the following figure (taken from p. 4 in the Insurance Economics lecture note):

Capture

Here, the consumer has initial wealth of $120 and there is a 25% chance that a fire will occur which (in the absence of insurance) will reduce her wealth from $120 to $20. The expected loss is E(L) = .25(100) = $25, which is the actuarially fair price for a full coverage insurance policy. Thus if she remained uninsured, she would have E(W) = .25(20) + .75(120) = $95, and her utility would be EU. Note that the (95, EU) wealth/utility pair corresponds to point C in the figure above. Now suppose the consumer purchases an actuarially fair full coverage insurance policy. Since she is fully insured, she is no longer exposed to any risk; her net worth is $95 irrespective of whether the loss occurs, her utility is U(95), and her wealth/utility pair is located at point D in the figure above. Since U(95) > EU, she will fully insure. More generally, since this decision is optimal for an arbitrarily risk averse consumer, it is also optimal for all risk averse consumers, irrespective of the degree to which they are risk averse.

Next, we introduced coinsurance. A coinsurance contract calls for proportional risk sharing between the consumer and the insurer. The consumer selects a coinsurance rate \alpha , where \alpha represents the proportion of loss covered by the insurer. By definition, \alpha is bounded from below at 0 and from above at 1. Thus, if the consumer selects \alpha = 0, this means that she does not purchase any insurance (i.e., she self-insures). If she selects \alpha = 1, this implies that she obtains full coverage. Furthermore, the price of a coinsurance contract is equal to the price of a full coverage insurance contract (P) multiplied by \alpha . On pp. 6-7 of the Insurance Economics lecture note, I analytically (via the calculus) confirm the Bernoulli principle by showing that if the consumer’s utility is U = W.5 and insurance is actuarially fair, then the value for \alpha which maximizes expected utility is \alpha = 1. An Excel spreadsheet called the “Bernoulli and Mossin Spreadsheet” is available from the course website which enables students to work this same problem using Solver. I recommend that you download this spreadsheet and use Solver in order to validate the results for a premium loading percentage (β) equal to 0 (which implies a full coverage (actuarially fair) premium P = E(L)(1+β) =$25 x (1+0) = $25) and β = 0.6 (which implies an actuarially unfair premium of $40, the analytic solution for which is presented on p. 8 in the Insurance Economics lecture note).

Let’s use Solver to determine what the optimal coinsurance rate (\alpha ) is for U = W.5 when β = 0 (i.e., when insurance is actuarially fair and full coverage can be purchased for the actuarially fair premium of $25). To do this, open up the Bernoulli and Mossin Spreadsheet and invoke Solver by selecting “Data – Solver”; here’s how your screen will look at this point:

Solver1

This spreadsheet is based upon the so-called “power utility” function U = Wn, where is 0 < n < 1. Since we are interested in determining the optimal coinsurance rate for a consumer with U = W.5, we set cell B2 (labeled “exponent value”) equal to 0.500. With no insurance coverage (i.e., when \alpha = 0) , we find that E(W) = $95 and E(U(W)) = 9.334. Furthermore, the standard deviation (σ) of wealth is $43.30. This makes sense since the only source of risk in the model is the risk related to the potentially insurable loss.

Since we are interested in finding the value for \alpha which maximizes E(U(W)), Solver’s “Set Objective” field must be set to cell B1 (which is a cell in which the calculated value for E(U(W)) gets stored) and Solver’s “By Changing Variable Cells” field must be set to cell B3 (which is the cell in which the coinsurance rate \alpha gets stored). Since insurance is actuarially fair (β = 0 in cell B4), the Bernoulli Principle implies that the optimal value for \alpha is 1.0. You can confirm this by clicking on Solver’s “Solve” button:

Solver2

Not only is \alpha = 1.0, but we also find that E(W) = $95, σ = 0 and E(U(W)) = 9.75. Utility is higher because this risk averse individual receives the same expected value of wealth as before ($95) without having to bear any risk (since σ = 0 when \alpha = 1.0).

Next, let’s determine what the optimal coinsurance rate (\alpha ) is for U = W.5 when β = 0.60 (i.e., when insurance is actuarially unfair). As noted earlier, this implies that the insurance premium for a full coverage policy is $25(1.60) = $40. Furthermore, the insurance premium for partial coverage is $ \alpha 40. Reset \alpha ’s value in cell B3 back to 0, β’s value in cell B4 equal to 0.60, and invoke Solver once again:

Solver3

On p. 8 in the Insurance Economics lecture note, we showed analytically that the optimal coinsurance rate is 1/7, and this value for \alpha is indicated by clicking on the “Solve” button:

Solver4

Since \alpha = 1/7, this implies that the insurance premium is (1/7)40 = $5.71 and the uninsured loss in the Fire state is (6/7)(100) = $85.71. Thus in the Fire state, state contingent wealth is equal to initial wealth of $120 minus $5.71 for the insurance premium minus for $85.71 for the uninsured loss, or $28.57, and in the No Fire state, state contingent wealth is equal to initial wealth of $120 minus $5.71 for the insurance premium, or $114.29. Since insurance has become very expensive, this diminishes the benefit of insurance in a utility sense, so in this case only a very limited amount of coverage is demanded. Note that E(U(W)) = 9.354 compared with E(U(W)) = 9.334 if no insurance is purchased (as an exercise, try increasing β to 100%; you’ll find in that case that no insurance is demanded (i.e., \alpha = 0).

I highly recommend that students conduct sensitivity analysis by making the consumer poorer or richer (by reducing or increasing cell B5 from its initial value of 120) and more or less risk averse (by lowering or increasing cell B2 from its initial value of 0.500). Other obvious candidates for sensitivity analysis include changing the probability of Fire (note: I have coded the spreadsheet so that any changes in the probability of Fire are also automatically reflected by corresponding changes in the probability of No Fire) as well as experimenting with changes in loss severity (by changing cell C8).

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.