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.

On the economics of financial guarantees

In the Credit Risk 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, federally guaranteed student loans, public and private bond insurance, pension insurance, mortgage insurance, government loan guarantees, etc.; the list goes on.

Most credit enhancement schemes work in the following fashion. Creditors loan money to “risky” borrowers who are endowed with risky assets worth $V(F) today (at date = 0).  Borrowers are risky in the sense that they will default (in whole or in part) in the future (at date T) to the extent that $F < $B.  The shortfall to creditors resembles a put option with date payoff of -Max[0, B-F]. Therefore, without credit enhancement, the value of risky debt today (at = 0) is

V(D) = B{e^{ - rT}} - 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] which may occur at date T. 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].  Therefore, depositors get paid back B - Max[0,B - F] + Max[0,B - F] = B (note: in the picture below, = 1).


More exam 2 information…

I neglected to mention in my previous posting that Midterm 2 consists of 4 problems; you are only required to work 3 of the 4 problems, and each is worth a maximum of 32 points (4 points given for putting your name on the title page of the exam booklet).  If you work all 4 problems, the exam grade will be based upon the 3 highest scoring problems.

Formula sheet and hints for midterm 2

I have posted the formula sheet for midterm 2 at; this formula sheet is also linked (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!

Tomorrow’s meeting of Finance 4335 (review session for Midterm 2)

Tomorrow’s Finance 4335 class meeting will be devoted to a review session for Midterm 2, which is scheduled for next Tuesday, April 16, from 2-3:15 in Foster 314. I highly recommend reviewing problem sets 6-8 and the various class problems (moral hazard, adverse selection, option pricing). Solutions for all of these are available at Also, try solving the Fall 2018 Midterm 2 before tomorrow’s class if time permits. If you prepare this way, it will be a particularly productive class meeting indeed!

New extra credit opportunity

Here’s a new extra credit opportunity for Finance 4335. You can earn extra credit by attending and reporting on Dr. Tony Gill’s talk entitled “The Comparative Endurance & Efficiency of Religion: A Public Choice Approach”:

The Comparative Endurance & Efficiency of Religion: A Public Choice Approach

Wednesday, April 10, 2019 – 3:30 pm – 5:00 pm
Cox Lecture Hall
Armstrong Browning Library

If you decide to take advantage of this opportunity, I will use the grade you earn on your report to replace your lowest quiz grade in Finance 4335 (assuming that your grade on the extra credit is higher than your lowest quiz grade). The report should be in the form of a 1-2 page executive summary in which you provide a critical analysis of Dr. Gill’s lecture. In order to receive credit, the report must be submitted to me via email in either Word or PDF format by no later than Friday, April 12 at 5 p.m.

Finance 4335