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# Edit Grid on Merton model's calculation of PD and LGD

#### liewpw05

##### New Member
Hi David,

I am writing to seek some clarification on the calculation of PD and LGD in the edit grid. I noticed that the expected rate of return of assets is used instead of the risk free rate.While on the other hand, i have observed in the subsequent spreadsheet on the calculation of the value of equity and debt using the Merton model, the risk free rate is used. Could you explain the rationale?

i have observed the same under the de servigny reading Chapter 3 on Default Risk-Quantitative Methodology

Thanks a lot

Regards
Peggy

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Peggy,

Nice observation. We could be clearer about that. (If you'd like a great backgrounder, here is a paper by Peter Crosbie @ KMV, who I consulted to before they were acquired by Moody's)

The Merton for credit has three major steps:

1. Estimate firm (asset) value and asset volatility; this is the hardest step. This step treats firm equity as a call option on firm assets and uses that relationship to solve for firm value and volatility (where firm value = stock price & firm volatility = volatility in the OPM). So, here is where you see risk free rate. Due to the challenging idea of risk-neutral valuation; i.e., we can use the risk-free rate to infer option value and it is true even for the real (risky) world. This idea (that we can use the riskless rate in the option pricing step) is easier to understand, IMO, by looking at the single step binomial, as the binomial is a general case of the Black-Scholes.

2. Okay, now we are really done with option pricing and the Merton insight. Calculate the distance-to-default (DD). Where DD is just the normalized distance from future expected firm value to default threshold. This is where you see expected return: to grow the firm asset value to its expected value in the future. Then measure standardized distance to default threshold. There is no option pricing here

3. Calculate PD/EDF. In the spreadsheet, parametrically by assuming normality (note the "dual error:" went from a lognormal distribution to a normal. Second, empirically skinny tails) and using =NORMSDIST(), which is just meant to be illustrative (as de Servigny says, too). KMV instead maps the DD to a "lookup table" based on actual (empirical) data. Put another way, in the XLS and illustrative example, the DD converts to a PD via a parametric distribution but the actual KMV maps to a PD via an empirical distribution (in the language of our other thread).

Thanks,
David

#### liewpw05

##### New Member
Hi David,

Thanks for the article attached. It's good reading.

Just to share with you some of the interesting things i read somewhere .The default probabilities calculated from the equity prices using the Merton 's model N(-d2) are risk-nuetral default probabilities, where d2 uses the risk free rate as an input. This text goes on to say " Moody's KMV provide a service that transform a default probability produced by Merton's model into a real-world default probability/EDF". This probably explains why the Distance to default formulae is the same formulae for d2 but uses the expected return of the asset instead risk free rate, since the EDF is derived from historical data as a basis.

By the way, i have two questions related to Loss Given Default:
1. What's the intuitive way to understand/remember the formulae for Loss Given Default in the edit grid?
2. On Basel II's Loss Given Default (LGD) in the credit risk weighted asset (RWA) under the the IRB approach:Is the stochastic nature of the LGD and the correlation with the PD considered in the Basel II credit RWA? I read from the FRM assigned reading on LGD that to avoid underestimating the expected loss and/or unexpected loss , both the above factors have to be taken into account.

Regards,
Peggy

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Peggy,

Thanks for sharing that snippet, that's great. It is a subtle distinction: we can price the (present) value of an option with the riskless rate, and it will transfer to the risky world, but this is not true of N(d2) which is the PD. The PD [i.e., N(d2)], unlike option value, is not indifferent to the riskless state; it is lower in the risky world due to the expected return of the asset. I would still suggest that the step #2 above is not the option pricing model. Or let me put another way: it seems to me more intuitive that we use the expected return in estimating distance to default. Perhaps that is the "easier" part. If you look at my practice question which applies the KMV (except for final step), it is pretty intuitively mechanical: grow the firm value, check distance to threshold in future, calculate probability. Does not require option pricing.

The harder part is: why is it that can we use the riskless rate when pricing the value of an option. This speaks to step #1; i.e., determining the value and volatility of the firm's assets. And I offer that we are using the riskless rate in only a very specific way: to compute the PV of the option, here being the equity of the firm.

1. I don't so much. FYI, it will not be tested. It is sort of a put option; i.e., if default, what is (debt) - (firm value), where this is LGD if firm value < debt. perhaps you can see the put in there? Then wraps the N() functions to probability adjust.

2. I attached PDF of a Basel reading that GARP dropped, which I wish they had retained as give helpful perspective. Note they cite this correlation as an hot area for research (i.e., LDG & PD correlation; they like copulas for this).

The short answer is: no, Basel does not do this. They avoid it instead by making the formula conservative: (i) requiring the LGD be a "downturn LGD;" a lower-than-expected LGD for bad economic times and (ii) by rolling up all correlations into the single factor (ASRF). In short, the PD is "used" (abused, really) as it is translated from an expected loss PD into higher unexpected loss PD. If you think about the credit loss distribution, they "neglect" an LGD-PD correlation (which may reach heights of model arrogance/difficulty anyhow) in favor of pushing the UL out further to the right with downturn LGD and the single factor asset correlation which is given an almost comically large job.

Finally, you will note this is not Ong's UL. His UL is one standard deviation. As i mentioned in the newsletter:
* Ong UL = 1 s.d.
* Internal capital = UL at internal confidence
* Regulatory capital = UL per Basel IRB

And Ong has shown that correlation impacts UL but makes no difference to EL as it does not enter either individual EL ( = PD*AE*LGD) or portfolio EL (sum of component ELs). Thanks for diving deep on this, many of your insights will defintely show up in the cram session, you've noticed some cool connections!

David

#### liewpw05

##### New Member
Hi David,

Yes, i appreciate the last point you made on the UL. I have found that confusing at first when going through the Ong reading so your screencast helps to clarify this matter.

I have a question on the recovery rates. In the De Servigny 's :Loss Given Default, there are 4 and 7 factors affecting the recovery rates of loans and traded bonds respectively. The 7 factors affecting the traded bonds seems to apply equally well to loans. Are these general factors affecting recovery rates irrespective whether it's a loan or traded bonds, while the 4 factors are specific to loan recoveries?

Regards,
Peggy

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Peggy,

My take on this is that the seven matter and the four not so much. First, the four are muddled (should incl. collateral) and undermined by his own empirical citation. Second, I agree with you, his seven are arguably a superset (e.g., bargaining power, by definition, includes both sides. So, this is the same). I think mainly, at the time he wrote this, the division is "I don't have too much data on loans" so i only have 4 factors, but "i have good data" on traded bonds, so i can cite 7. I think the seven is the good list, and especially as this reading is dated yet the exam questions are timely, I think the bond/loan distinction in de Servigny LGD is not heavy-handed. In both, key determinants are capital structure (seniority), collateral as primary and then the others. Interesting, since de Serigny wrote this, I'd argue macroeconomy has elevated in importance.

My only other thought on this is that, if there is a key difference (Bond/loan) vis-a-vis LGD, (this is just my thought here), it is found in Ong: that loans tend to be more complex with features than traded instruments. This is another way of saying, they contain more idiosyncratic features (e.g., instrument specific covenants). In the totally of the credit readings, I find this idiosyncratic versus systemic (e.g., economic state) a more interesting distinction here. But, re this reading, IMO the 7 overwhelm the 4. FYI, here is a great great piece on loan/bond and convergence (note how they start from equivalence)

David