# Problem set 9 solution procedures and requirements

Problem Set 9 (due at the beginning of class on Thursday, April 25) is essentially a reparameterized version of the class problem that we will work on during tomorrow’s class meeting (also described in pp. 6-8 of the Credit Risk lecture note).

In order to fully comprehend the pricing of credit risk in the Black-Scholes-Merton framework, it is advised that students begin by solving the problem manually, followed by creating a spreadsheet model to validate their work.   The computation strategy for completing this problem set is best described as follows:

1. Calculate ${d_1}$ and ${d_2}$, where ${d_1} = \displaystyle\frac{{\ln (V(F)/B) + (r + .5{\sigma ^2})T}}{{\sigma \sqrt T }}$ and ${d_2} = {d_1} - \sigma \sqrt T$.  Since ${d_1}$ and ${d_2}$ represent critical values for the standard normal distribution, we calculate $N({d_1})$ and $N({d_2})$ accordingly.  Since $N({d_2})$ corresponds to the risk neutral probability that $F \ge B$ at date T, it follows that 1 –  $N({d_2})$ corresponds to the risk neutral probability that $F < B$ at date T; i.e., this is the risk neutral probability that the firm defaults on its promised debt payment.  Also, because of the symmetry of the standard normal distribution, 1 –  $N({d_2})$$N({-d_2})$.
2. Note that the value of risky debt, $V(D)$ corresponds to the value of safe debt ($B{e^{ - rT}}$) minus the value of the limited liability put option $V(Max[0,B - F])$, where F is the terminal value of risky assets, is the terminal (date T) value of a riskless zero coupon (also known as a “pure discount”) bond and $V(Max[0,B - F]) = B{e^{ - rT}}(N( - {d_2})) - V(F)(N( - {d_1}))$. Thus, the “fair market value for the bond” is determined by calculating $V(D) = B{e^{ - rT}} - [B{e^{ - rT}}(N( - {d_2})) - V(F)(N( - {d_1}))]$.   The dollar value of the limited liability put option is given by $V(Max[0,B - F]) = B{e^{ - rT}}(N( - {d_2})) - V(F)(N( - {d_1}))$, which also corresponds to the “fair premium” for credit insurance (cf. part 3 of Problem Set 9).
3. The class problem and Problem Set 9 also ask for the yield to maturity and credit risk premium.  The yield to maturity (YTM) for a period pure discount bond corresponds to the rate of interest which must be earned from date 0 to date T in order for the future value of $V(D)$ to be equal to B; i.e., $B = V(D){e^{YTM(T)}}$.  Solving for YTM in this equation, we find that $YTM = \ln (B/V(D))/T$.  The credit risk premium corresponds to the difference between the yield to maturity (YTM) and the riskless rate of interest r.  This risk premium compensates investors for bearing default risk costs.  Intuitively, it makes a lot of sense that there is a positive relationship between the risk of default and the credit risk premium.

# Formula sheet and hints for midterm 2

I have posted the formula sheet for midterm 2 at http://fin4335.garven.com/spring2019/formulas_part2.pdf; this formula sheet is also linked http://fin4335.garven.com/formula-sheets/ (see item 2).

A particularly important concept in finance (as well as on tomorrow’s exam) is the principle of riskless arbitrage.  Essentially, if one encounters two otherwise identical investments; i.e., same risks but different returns, then arbitrage profits may be earned by shorting the investment which has a lower return and using the proceeds of the short sale to fund the purchase of the investment which has a higher return.  This principle is at the heart of option pricing (particularly the delta hedging and replicating portfolio models) and even found its way in the portfolio/capital market theory topic (see the related problem set).  The other important idea on the midterm exam relates to understanding the consequences of asymmetric information and the formulation of strategies  for mitigating  moral hazard and adverse selection-related risks.  Anyway, these ideas are all well represented in the problem sets and various class problems on which we have worked since the first midterm exam.

See y’all tomorrow!

# Problem set 7 helpful hint

Here is a helpful hint for parts 1D and 2C of problem set 7.  As we showed last week in class, the weights for the minimum variance two-asset portfolio can be found by applying the following ratio in order to determine w1; upon calculating w1, then w2 = 1 – w1. (source: p. 7 of http://fin4335.garven.com/spring2019/lecture12.pdf):

${w_1} = \displaystyle\frac{{\sigma _2^2 - {\sigma _{12}}}}{{\sigma _1^2 + \sigma _2^2 - 2{\sigma _{12}}}}.$

Furthermore, in cases where correlation is equal to either  -1 or 1, the weighting scheme shown above guarantees that the minimum variance combination is riskless.  Therefore, if the expected return on a riskless two-asset portfolio is either greater or less than the expected return on a riskless asset, there’d be an arbitrage opportunity along the lines of what we showed in class on Tuesday pertaining to mispriced forward contracts, where we sold forward and bought the replicating portfolio when the forward was too expensive, and we bought forward and sold the replicating portfolio when the forward was too cheap.

# Problem Set 6 Hints and Spreadsheet

Problem Set 6 is due at the beginning of class on Thursday, March 7.

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/spring2019/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). # Information about tomorrow’s midterm exam in Finance 4335 Tomorrow afternoon’s midterm exam in Finance 4335 consists of four problems. Since your exam grade will be based upon the three highest scoring problems of these four, feel free to either work all four problems or just three of the four problems. Each problem is worth 32 points; thus three problems times 32 points each totals 96 points. I will add an additional 4 points on your exam if you also legibly write your name on the cover page in the space provided. Thus, the maximum number of points possible on this exam is 100 points. Furthermore, I have also posted the formula sheet for Midterm Exam 1, which will be included as part of the exam booklet. I recommend that y’all familiarize yourselves with this document sometime prior to tomorrow’s exam. I’d like to make an important point about the formulas provided on the formula sheet. What you’ll find there is not a complete census of all formulas used to date in Finance 4335. For example, I don’t include any insurance pricing formulas. By now, I assume that everyone knows that an actuarially fair price for an insurance policy is simply the expected value of the claim/indemnity under said policy. On the other hand, an actuarially “unfair” policy has a “premium loading” which represents a “markup” from the actuarially fair price. Also, I recommend when performing expected utility calculations, I recommend that you go out to no less than the 3rd digit to the right of the decimal point. Unfortunately, I cannot be at the exam tomorrow due to an important medical issue affecting a close member of my family. Professor Paul Anderson has graciously agreed to proctor tomorrow’s exam in my place. See y’all next week! Dr. Garven # Hints for solving problem set #4 Problem set #4 (which is due on Thursday, February 14) consists of two problems: 1) an optimal (expected utility maximizing) portfolio problem, and 2) a stochastic dominance problem. We’ll discuss stochastic dominance next Tuesday, but in the meantime allow me to provide you with some hints for setting up the first problem. The first problem involves determining how to (optimally) allocate initial wealth W0 =$100 to (risky) stock and (safe) bond investments for two investors who are identical in all respects except utility. Let $\alpha$ represent the allocation to stock; then the plan is to invest $100$\alpha$ in the stock and$100(1-$\alpha$) in the bond. The key here is to find the value for $\alpha$ which maximizes expected utility. The problem is based on the following facts:

• U(W) = W.5; for Investor A and U(W) = ln W for Investor B;
• W0 = \$100 for both investors;
• Current bond and stock prices are B0 and S0 respectively;
• End-of-period bond price is B1 = B0(1.05) with probability 1.0; and
• End-of-period stock price is S1 = S0(1.3) with probability .6 and S1 = S0(.7) with probability .4.

In order to compute expected utility of wealth for either investor, you must first determine state-contingent wealth (Ws). Since there is a 60% chance that the stock increases in value by 30%, a 40% chance that the stock decreases 30%, and a 100% chance that the bond increases in value by 5%, this implies the following:

• 60% of the time, Ws = $\alpha$W0(1.3) + (1-$\alpha$)W0(1.05) = $\alpha$100(1.30) + (1-$\alpha$)100(1.05) = $\alpha$130 + (1-$\alpha$)105 = 105 + 25$\alpha$.
• 40% of the time, Ws = $\alpha$W0(.7) + (1-$\alpha$)W0(1.05) = $\alpha$100(.7) + (1-$\alpha$)100(1.05) = $\alpha$70 + (1-$\alpha$)105 = 105 – 35$\alpha$.

Therefore, expected utility for Investor A is: E(U(W)) = .6(105 + 25$\alpha$).5 + .4(105 – 35$\alpha$).5, and expected utility for Investor B is E(U(W)) = .6ln (105 + 25$\alpha$) + .4ln(105 – 35$\alpha$). It is up to you to solve for the optimal value of $\alpha$ for each investor.  There are two ways to do this – via calculus or a spreadsheet model.  Actually, I would encourage y’all to work this problem both ways if time permits because doing so will help you develop an even better grasp of the underlying principles and concepts in Finance 4335.  However, at your option, you may rely solely on building your own spreadsheet model.  If you do this, in order to receive full credit, you need to email your spreadsheet model to risk@garven.com along with turning in the completed problem set.

# Problem Set 2 helpful hints

Problem Set 2 is now available from the course website at http://fin4335.garven.com/spring2019/ps2.pdf; its due date is Tuesday, January 29.

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 returns and standard deviations for 2-asset portfolios; see pp. 17-22 of the http://fin4335.garven.com/spring2019/lecture3.pdf lecture note for coverage of this topic. The second problem involves using the standard normal probability distribution to calculate the probabilities of earning various levels of return by investing in risky securities and portfolios.

# Visualizing Taylor polynomial approximations

In his video lesson entitled “Visualizing Taylor polynomial approximations“, Sal Kahn essentially replicates the tail end of last Thursday’s Finance 4335 class meeting in which we approximated y = eˣ with a Taylor polynomial centered at x=0.  Sal approximates y = eˣ with a Taylor polynomial centered at x=3 instead of x=0, but the same insight obtains in both cases, which is that one can approximate functions using Taylor polynomials, and the accuracy of the approximation increases as the order of the polynomial increases (see pp. 19-25 in my Mathematics Tutorial lecture note if you wish to review what we did in class on Thursday).

# Plans for next week’s Finance 4335 class meetings, along with a preview of future topics

Next week in Finance 4335 will be devoted to tutorials on probability and statistics. These tools are critically important in order to evaluate risk and develop appropriate risk management strategies for individuals and firms alike. Next Tuesday’s class meeting will be devoted to introducing discrete and continuous probability distributions, calculating parameters such as expected value, variance, standard deviation, covariance and correlation, and applying these concepts to measuring expected returns and risks for portfolios consisting of risky assets. Next Thursday will provide a deeper dive into discrete and continuous probability distributions, in which the binomial and normal distributions are showcased.

While I have your attention, let me briefly explain what the main “theme” will initially be in Finance 4335 (up to the first midterm exam, which is scheduled for Thursday, February 21). Starting on Tuesday, January 29, we will begin our discussion of decision theory. Decision theory addresses decision making under risk and uncertainty, and not surprisingly, risk management lies at the very heart of decision theory. Initially, we’ll focus attention on variance as our risk measure. Most basic finance models (e.g., portfolio theory, the capital asset pricing model (CAPM), and option pricing theory) implicitly or explicitly assume that risk = variance. We’ll learn that while this is not necessarily an unreasonable assumption, circumstances can arise where it is not an appropriate assumption. Furthermore, since individuals and firms are typically exposed to multiple sources of risk, we need to take into consideration the portfolio effects of risk. To the extent that risks are not perfectly positively correlated, this implies that risks often “manage” themselves by canceling each other out. Thus the risk of a portfolio is typically less than the sum of the individual risks which comprise the portfolio.

The decision theory provides a particularly useful framework for thinking about concepts such as risk aversion and risk tolerance. The calculus comes in handy by providing an analytic framework for determining how much risk to retain and how much risk to transfer to others. Such decisions occur regularly in daily life, encompassing practical problems such as deciding how to allocate assets in a 401-K or IRA account, determining the extent to which one insures health, life, and property risks, whether to work for a startup or an established business and so forth. There’s also quite a bit of ambiguity when we make decisions without complete information, but this course will at least help you think critically about costs, benefits, and trade-offs related to decision-making whenever you encounter risk and uncertainty.

After the first midterm, the remainder of the semester will be devoted to various other risk management topics, including the demand for insurance, asymmetric information, portfolio theory, capital market theory, option pricing theory, and corporate risk management.

# Problem Set 1 hint…

Problem Set 1 is due at the beginning of class on Tuesday, January 22. 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.