Jun 16

FRM 2008 Episode #7: Credit A (Credit Risk Intro)

by David Harper, CFA, FRM, CIPM

FRM |

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About Episode #7 (Credit Risk: Into to Credit)

episode7_thumb

Please find links to earlier episodes (#1 to #6) at the end of this note. This episode is called Credit A. As you probably know, we follow the sequence of GARP's 2008 FRM Study Guide.

As usual I would like to offer a few tips in regard to this episode. This episode reviews: Saunders on individual loan risk and sovereign risk; de Servigny on credit ratings, default risk (PD) and loss given default (LGD); and Ong on expected loss (EL), unexpected loss (UL) and portfolio EL & UL.

Transition matrix

transition_matrix2

The 2008 FRM AIMs document forgets to introduce the credit rating transition matrix (see de Servigny Appendix 2A). About the transition matrix:

  • For a given beginning-of-period rating (left column), the transition matrix gives the probability of a migration to another rating before the period's end; we expect largest percentages in the diagonal as the most likely outcome is for the rating to persist.
  • The probabilities in a row are mutually exclusive and cumulatively exhaustive (MECE). Each row therefore characterizes an empirical distribution that must sum to 1.0 (100%).
  • If we allow ourselves to assume period-to-period independence (i.e., Markovian, which is empirically untrue but not the first rule we've broken!), we can extend the single period to multiple period (n-periods) by matrix multiplication; e.g., squaring the matrix gives a two-period transition matrix; for three periods, cube the matrix.

Through-the-cycle versus at-the-point approaches to ratings

altmans_arrow3

De Servigny writes about two conflicting approaches to credit ratings. This dichotomy is both timely and a useful lens for viewing credit risk models.

The diagram (from page 38 in the PPT deck) illustrates the continuum from point-in-time to through-the-cycle.

Note that the Merton-type model is nearly a pure point-in-time approach (why?), the rating agencies are nearer to through-the-cycle and Basel's IRB is in the middle (keep in mind as you approach Basel).

Probability of Default Implied by Term structure

impliedPD

Please work question #1 below. This gives further practice to a key idea in Tuckman: a term structure of spot rates implicitly contains a forward curve. You want to get comfortable with the following (see chart): the Corporate term structure is given by credit spreads above the Treasury (riskless) term structure. And both term structures embed two pieces of information:

  • The implied forward curves (riskless and corporate forwards)
  • Implied probabilities of default (both deduced via no arbitrage)

Merton Model

mertonkmv

You won't need to do the hard part of the Merton (solve for asset value and volatility). Our scope is the elegant logic of the Merton (see picture):

  • We expect asset value ($130 today) to grow to about $134 at end-of-period
  • That represents an end-of-period value that is 29% over the default threshold of $100; i.e., LN(134/100) = 29%. Where $100 is some function of debt; e.g., short-term + 50% of long-term
  • If the firm's (asset) volatility = 20%, then +29% is almost 1.5 standard deviations above the default threshold. The distance-to-default (DD) is therefore 1.46.
  • We can use an probability distribution to translate that into a probability of default (PD). For example, =NORMSDIST(-1.46) = 7.2% probability of default. But Moody KMV's will likely be higher as we already know normal underestimates the fat tails (and KMV does not use a parametric distribution to solve for implied PD/EDF).

Expected Loss (EL)

el_basel2

Note that the components in the formula for expected loss (EL = EAD * PD * LGD) will appear in Basel's internal rating-based (IRB) approach. See the essential similarity (diagram)? But we will see, when we examine the Basel IRB, that the expected loss is morphed into an unexpected loss by manipulation largely of the PD.

 

Economic Capital

credit_var2

On page 103 of the presentation deck, I have a chart that I hope you find helpful. This idea of economic capital is central. First, in mechanical terms, note the lack of a difference between economic and regulatory capital: both are cushion for unexpected losses. The difference, at a superficial level, is constituency: external regulators versus internally-driven (and therefore stakeholder or shareholder-driven). To illustrate their mechanical interchangeability, we could replace economic capital as the denominator in risk-adjusted return on capital (RAROC) with regulatory capital. About the other terms:

  • Note the distribution of this credit portfolio is deliberately non-normal with positive skew (the credit portfolio has limited upside, some expected losses, and skewed exposure on the downside)
  • Going from left to right (mounting losses), expected losses are priced into transactions ("business as usual"). Basel expects the banks to cover EL with reserves.
  • Unexpected loss (UL), as Ong's defines, is "simply the standard deviation of the unconditional value of the asset [exposure] at horizon"
  • See how UL is a special case of Value at Risk (a.k.a., capital at risk in this context)? The UL here is the worst loss we expect with a confidence that is low enough to warrant only one standard deviation.
  • Going further to the right is a matter of higher confidence. If, say, confidence is 99%, then economic capital is the difference between a worst expected loss at 99% and the expected loss.

My tip: before getting distracted by special terminology, consider that to the right of expected loss, it is only a matter of the (i) having a distribution and (ii) selecting a confidence level. A sufficiently low confidence level, at some solution, is the "one standard deviation loss" which happens to be the unexpected loss (UL). Going to the right, still at low confidence levels is typically called "capital at risk." Going further to the right, at higher confidence, is economic/regulatory capital. As a VaR concept, we understand none of these are the extreme worst cases scenario losses.

What is so great about economic capital? Some will say it is the common yardstick that can be used across the enterprise's major risk buckets. Distributions will surely vary (market versus credit versus operational risk) but armed with distributions, we can select a quantile (worst loss at 9X% confidence) and the losses can be expressed in a "common currency."

Portfolio Unexpected loss

ul_portfolio

The Ong readings culminates with an exercise for calculating portfolio EL and UL. I hope you get a chance to review the EditGrid/Excel spreadsheet the details these calcs.

Notice the essential similarity between portfolio UL and portfolio volatility under the traditional mean-variance framework (that we review in the Investment discipline).

 

New Learning Spreadsheets Added

learningxls

For this episode (Credit Risk A: Intro to Credit Risk), I uploaded the following all-new learning spreadsheets to the member page:

  • Gross Promised Loan Return (Saunders Ch. 11)
  • Altman's Z (Saunders Ch. 11)
  • Probability of Default (PD) Implied by Term Structure
  • Merton Model for PD (de Servigny Chapter 3)
  • Beta function (de Servigny Chapter 4)
  • Expected (EL) and unexpected loss (UL)
    (Ong Chapters 4 &5)
  • Portfolio expected & unexpected loss (Ong Chapter 6)

 

Because we now have uploaded over forty (40+) learning spreadsheets, I highlighted in yellow the more critical subset from an exam perspective. Yellow signifies an important or archetypal idea; yellow means: "I hope you review this spreadsheet." Non-yellow can be ignored, if your schedule does not allow.

editgrid

These "learning worksheets" can be accessed in three ways.

  1. Simply view in the browser,
  2. To open directly into Excel! Select File > Export As > Excel (.xls),or
  3. Most have a downloadable "native" Excel file (XLS) associated with the entry.

 

Screencast Tutorial

Paid member access the screencast in the member section. In addition to the viewable screencast:

  • You can downloadable the underlying Power Point slides (in PDF format). For this episode, there is a single 92 page deck.
  • An ipod format (.m4v)
  • A downloadable version of the screencast in a .zip file. (Save to new directory on local and launch the .html file.)

Non-members can sample the start of the screencast tutorial here.

 

Practice Questions

As before, I wrote the questions below to provoke your engagement in this episode. I have been asked, "do I need to work these questions?" The answer is "no, but I wish you would try." That's because I write them immediately after I record the screencast with the sole intent of giving you practice on high-density ideas in the screencast. They are not reused from the databank (I wrote these yesterday), I write them with the 2008 AIM list in hand. Some of them are surely more time-consuming than the average exam question. That's deliberate. For example, question #4 below is time-consuming; but working through the brilliantly elegant Merton model is to give hearty practice to at least three key ideas in the FRM (i.e., lognormal property of stock prices, probability distribution, structural approach to default prediction).

These questions do not replace/displace the upcoming core practice questions on the member page: These are "bonus questions," we will still produce large sets of flash-enabled practice questions linked to the new AIMs.

 

Question #1 (marginal PD implied by term structure)

Treasury spot rates are 4% and 5% at, respectively, one and two year maturities. The spot rates on a corporate bond are 6% and 7%, respectively, at one and two year maturities. Put another way, there is a +2% spread on the corporate bond at both the one- and two-year maturities.

(i) What is the implied probability of default (PD) over the one-year period?
(ii) What is the implied MARGINAL probability of default (marginal PD) in the second year?
(iii) What is the cumulative default probability (cumulative PD) over the two year period

Question #2 (Jorion Chapter 11)

Based on the "five key economic variables" that are useful in estimating the probability of rescheduling sovereign risk, what inference can we draw about the following (e.g., more likely to reschedule, less likely to reschedule)? Assuming all other things are equal (ceteris paribus):

(i) More exports
(ii) More imports
(iii) More net exports (exports - imports)
(iv) More foreign exchange reserves
(v) Higher Gross National Product (GNP)
(vi) Greater volatility of export revenues
(vii) Faster growth in domestic money supply

Question #3 (de servigny on credit ratings)

(i) In a September 2006 Special Comment, Moody's asserted that credit ratings face an inherent trade-off between accuracy and stability (i.e., ratings can be either accurate or stable but not simultaneously both). Can we relate this trade-off to Servigny's dichotomy between "at-the-point-in-time" versus "through-the-cycle" approaches to credit ratings?
(ii) In the assigned reading on subprime securitization (Understanding the securitization of Subprime Mortgage Credit by Ashcraft and Schuermann), the authors argue that "through-the-cycle ratings could amplify the housing cycle." Explain.
(iii) If we are given a credit ratings transition (migration) matrix, how can we use a structural credit model (e.g., Merton model) to estimate the rating of a firm? (or similarly, to estimate the equity needed to achieve a target rating?)

Question #4 (de Servigny on PD)

This is not easy but I really recommend this exercise because the Merton model elegantly employs several FRM building blocks. Once again, perceived complexity is due to the "mere combination" of basic ideas.

Assume a firm with current market value of $100. The firm's long-term debt is $60 and its short-term debt is $60. The expected return on the firm's assets is 12% with volatility of 10%.

(i) Under Moody's KMV approach, what is the firm's default threshold?
(ii) Since the asset's expected return is 12% per annum, with volatility of 10%, what is the asset's median (geometric) return over the five year period?
(iii) At the end of five years, what is the firm's distance to default (DD)?
(iv) Given the DD, assume normality to compute a Merton model probability of default?
(v) How is the KMV EDF likely to compare to this PD?

Question #5 (De Servigny on LGD)

(i) Unlike PD which can be estimated with a single figure, de Servigny says LGD should be captured by a distribution. Why? Which distribution is typical?
(ii) What are the key determinants of LGD/recovery according to de Servigny?
(iii) An unconditional model of expected loss (EL) might say EL = PD * LGD. Explain how collateral impacts this formula and how PD, LGD and collateral may be influenced by macroeconomic conditions.
(iv) According to de Servigny, which of the rating agencies incorporates LGD?
(v) How might we argue that Basel II is "procyclical" in regard to PD and/or LGD?

Question #6 (Ong on EL)

Assume the original commitment is $20 million, of which $10 million has been drawn by the borrower (i.e., $10 million is outstanding and $10 million is the unused commitment).

Assume usage given default (UGD) = 50%, probability of default (PD) = 1%, and loss given default (LGD) = 50%.

(i) What is the expected loss (EL)?
(ii) How does EL relate (vary with) PD and LGD (e.g., linear, nonlinearly)?
(iii) What is the meaning of UGD? What is the difference between UGD and LGD?
(iv) Bonus: we have everything we need to calculate unexpected loss (UL) except for which value. Which value do we require that's difficult?

Question #7 (Ong on UL)

Assume as above: $10 million outstanding plus $10 million unused commitment; usage given default (UGD) = 50%; probability of default (PD) = 1%; and loss given default (LGD) = 50%. Further, assume the standard deviation of LGD = 50%.

(i) What is the standard deviation of the PD/EDF?
(ii) What is the unexpected loss (UL)?

Question #8 (Ong on Portfolio EL & UL)

Assume a two-asset portfolio. The first exposure is as above ($10 million outstanding; $10 million unused commitment; usage given default (UGD) = 50%; probability of default (PD) = 1%; loss given default (LGD) = 50%; standard deviation of LGD = 50%). The second exposure is the same EXCEPT its LGD = 25% and the standard deviation of its LGD = 25%. Default correlation between the two exposures is 10%.

(i) What is the portfolio's expected loss (EL)?
(ii) What is the portfolio's unexpected loss (UL)?
(iii) What are the risk contributions, respectively, of each exposure?
(iv) What do the risk contributions sum to?
(v) What is the interpretation of the risk contribution? Why are the sum of unexpected losses (UL + UL) less than portfolio UL? Under what condition would (UL) + (UL) = portfolio UL?

My answers to these questions

 

Previous newsletters

Here are links to Episodes #1 through #6:

 

Thanks very much.

David Harper, CFA, FRM, CIPM
Founder
www.bionicturtle.com

David-Harper_100w

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