Category Archives: Helpful Hints

Preparation for tomorrow’s Final Exam review for Finance 4335

Here are some suggestions for preparing for tomorrow’s Final Exam review for Finance 4335:

1. Be sure to read and review my blog posting entitled “Hints about the final exam in Finance 4335… from earlier today.

2. I have posted the final exam formula sheet which will appear as part of the final exam booklet. Furthermore, the Standard Normal Distribution Function (“z”) Table will also appear as part of the final exam booklet.

3. Read and review the Finance 4335 Fall 2017 course synopsis.

4. For tomorrow’s review session, review problem sets 3-11 (solutions for which are available at http://risk.garven.com/?s=solutions+for+problem+set) and the Spring 2017 Final Exam Booklet and Solutions and come to class with any questions you may have concerning any of this material.

See y’all tomorrow!

Some hints for Problem Set 10

The classic capital budgeting model (such as you learned in Finance 3310) implicitly assumes that the firm has unlimited liability and faces linear taxes. When these assumptions hold, then the net present value (NPV) of a project is calculated by estimating expected values of future incremental after-tax cash flows and discounting them at an appropriate risk-adjusted discount rate. However, we showed during yesterday’s class meeting how limited liability and nonlinear taxes imply that the net present value of a project depends upon the manner in which incremental after-tax cash flows interact with cash flows from existing assets. Consequently, the after-tax value of equity is equal to the difference between the pre-tax value of equity and the value of the government’s tax claim (both of which we model as call options on the firm’s assets). Furthermore, project NPV corresponds to the difference in after-tax value of equity (assuming the project is undertaken), minus the after-tax value of equity (assuming the project is not undertaken).

Problem Set 10 provides an opportunity to apply these concepts.  Here are some hints for parts A through E of Problem Set 10 :

  1. In part A, apply the option pricing framework to determine the pre-tax value of equity (V(E), where V(E) = V(Max(0,F-B)), the value of debt (V(D), where V(D) = V(B – Max(0,B-F)), and the value of taxes (V(T), where V(T) = \tauV(Max(0,F-TS)), assuming that this investment is not undertaken.  Helpful hint: we performed these same calculations in class yesterday for the problem described on pp. 17-18 of the http://fin4335.garven.com/fall2017/risk_costly_chapter7.pdf teaching note.
  2. In order to determine whether the project should be undertaken, in part B you need to  after-tax equity value (i.e., V(E) – V(T)) which obtains under the assumption that the investment is undertaken. Once you obtain that result, the net present value (NPV) of the project is the difference between the after-tax value of equity (V(E) – V(T)) in part A (which you have already calculated) and the after-tax value of equity which obtains if the investment is undertaken. The decision to invest or not to invest depends upon whether the NPV of the investment is positive (in which case you undertake the project) or negative (in which case you do not undertake the project).
  3. An investment tax credit (ITC) is quite literally a check sent by the U.S. Department of the Treasury to the company; thus, the NPV when there is an ITC is equal to the NPV that you calculated in part B plus the value of the ITC.  At that point, whether you invest or don’t invest depends upon whether NPV is positive or negative (as in part B).  The ITC in this case increases project NPV by $1 million.
  4. In order to answer part D, you need to redo the calculation described in the first paragraph above using a 20% tax rate rather than a 35% tax rate.
  5. In order to answer part E, you can figure out the tax rate at which the firm is indifferent about making the investment by trial-and-error, or better yet, adapt the Tax Options spreadsheet located on the lectures notes page to the parameters upon problem set 10 is based and use Solver.

Clarification of expectations for and hints concerning Problem Set 9

A student asked me whether it’s okay to use an Excel spreadsheet to solve problem set 9.  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.

As I am sure you are already well aware, you can obtain most of the “correct” answers for problem set 9 by simply downloading and opening up the Credit Risk Spreadsheet in Excel and performing the following steps:

  1. For part A, open the Firm 1 worksheet, replace the “.4” in cell B3 with “.3”.  Then the answers for the three questions are in cells F4, F3, and F5 respectively.
  2. For part B, open the Firm 2 worksheet, replace the “.4” in cell B3 with “.5”.  Then the answers for the three questions are in cells F4, F3, and F5 respectively.
  3. For part C, assuming that  you are able to follow the logic presented in my On the economics of financial guarantees blog post from yesterday afternoon, the fair insurance premiums appear on both of the worksheets, and presumably you also understand from our study of financial derivatives that the expected return on a default-free bond is the riskless rate of interest.

The problem with simply plugging and chugging the spreadsheet template is that one can mechanically follow the steps outlined above without necessarily understanding the credit risk problem.  The key takeaway from our study of credit risk is that limited liability causes prices of bonds issued by risky (poor credit quality) firms to be lower than prices of bonds issued by safe (good credit quality) firms.  In the case of this problem set, firms 1 and 2 are identical in all respects expect for asset risk, and because of limited liability, this implies that in the absence of a financial guarantee, firm 2’s bonds are riskier than firm 1’s bonds. Thus, firm 2’s bonds have a lower market value (and a correspondingly higher yield, or expected return) than firm 1’s bonds, and firm 2 can expect to have to pay more than firm 1 for a financial guarantee which transfers the default risk from investors over to a financial guarantor.  In a competitive market, the fair premium for such a guarantee is given by the value of the limited liability put option.

By all means, make use of the Credit Risk Spreadsheet to validate your answers for the problem set.  But start out by devising you own coherent 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/fall2017/lecture16.pdf); this requires 1) calculating {d_1} and {d_2}, 2) using the Standard Normal Distribution Function (“z”) Table to find N({d_1}) and N({d_2}), and 3) inputting your N({d_1}) and N({d_2}) values into the BSM put equation, where the exercise price corresponds to the promised payment to creditors of $500,000, and the value of the underlying asset corresponds to the value of the firm, which is $1,000,000.  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)}}.

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/fall2017/lecture16.pdf and substitute the value of assets ($1,000,000) in place of S and the promised payment of $500,000 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) = $1,000,000) and V(E).  Upon calculating V(D) in this manner, then obtain these these bonds’ yields to maturity (YTM ) by solving for YTM in the following equation: V(D) = B{e^{ - YTM(T)}}.

On the economics of financial guarantees

On pp. 13-19 of the Derivatives Theory, part 3 lecture note and in Problem Set 9, we study how credit enhancement of risky debt works. Examples of credit enhancement in the real world include federal deposit insurance, public and private bond insurance, pension insurance, mortgage insurance, government loan guarantees, etc.; the list goes on.

Most credit enhancement schemes work in the fashion described below. Creditors loan money to “risky” borrowers who are at risk for defaulting on promised payments. Although borrowers promise to pay back $B at t=1, they may default (in whole or in part) and the shortfall to creditors resembles a put option with t=1 payoff of -Max[0, B-F]. Therefore, without credit enhancement, the value of risky debt at t=0 is

V(D) = B{e^{ - r}} - V(Max[0,B - F]).

However, when credit risk is intermediated by a guarantor (e.g., an insurance company or government agency), credit risk gets transferred to the guarantor, who receives an upfront “premium” worth V(Max[0,B - F]) at t=0 in exchange for having to cover a shortfall of Max[0,B - F] that may occur at t=1. If all credit risk is transferred to the guarantor (as shown in the graphic provided below), then from the creditors’ perspective it is as if the borrowers have issued riskless debt. Therefore, creditors charge borrowers the riskless rate of interest and are paid back what was promised from two sources: 1) borrowers pay D = B - Max[0,B - F], and 2) the guarantor pays Max[0,B - F].

economics_of_deposit_insurance

Credit risk teaching note and spreadsheet

I’d like to call your attention to my credit risk teaching note @ http://fin4335.garven.com/fall2017/creditrisk.pdf and my credit risk spreadsheet @ http://fin4335.garven.com/fall2017/creditrisk.xls. This teaching note provides a brief synopsis of today’s presentation of the credit risk topic, and the spreadsheet provides the code required in order to produce the table on page 8 of the teaching note.

Midterm exam 2 information…

Midterm 2 will be given during class on Thursday, November 2. 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 involve topics which we have covered since the first midterm exam. Topics covered include 1) demand for insurance, 2) moral hazard/adverse selection, 3) portfolio theory/capital market theory, and 4) financial derivatives (calls and puts specifically).

By the way, I have posted the formula sheet that I plan to use on the exam at the following location: http://fin4335.garven.com/fall2017/formulas_part2.pdf.

As I noted in my “Plans for next week in Finance 4335” blog posting, tomorrow’s class meeting will be devoted to a review session for 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.

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 last Thursday’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. For example, in my teaching note entitled “A Simple Model of a Financial Market”, I provide a numerical example where the interest rate is zero, there are two states of the world, a bond which pays off $1 in both states is worth $1 today, and a stock that pays off $2 in one state and $.50 in the other state is also worth one dollar. In that example, the replicating portfolio for a European call option with an exercise price of $1 consists of 2/3 of 1 share of stock (costing $0.66) and a margin balance consisting of a short position in 1/3 of a bond (which is worth -$0.33). Thus, the value of the call option is $0.66 – $0.33 = $0.33.
  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, you need to determine and price the components of the replicating portfolio; we will begin class on Tuesday, 10/24 by determining the the relative weightings (delta and beta) for the put’s replicating portfolio.

Problem Set 7 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}}}.