# Merton model, a summary of the issues

#### RiskQuant

##### New Member
In addition, I want to ask you a question. Moody’s Analytics informs "100 largest financial institutions use KMV model". Then, with Merton model, is there a bank using Merton model to:
• estimate PD of customers (for calculating the capital requirement in Basel Accords) ?
Indeed, I only live in academic world, so that I don't know anything about real world !

Best.
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P.S. It is perfect If someone shows me the evidence of a specific bank using Merton model in Basel Accords. Thanks.

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#### Dr. Jayanthi Sankaran

##### Well-Known Member
Hi @RiskQuant,

The Merton model is used by credit analysts at brokerage firms and investors to understand how capable a company is at meeting financial obligations, servicing its debt and weighing the general possibility that the company will go into credit default.

Loan officers and stocks analysts also use the Merton model to analyze a corporation's risk of credit default.

Under the Advanced internal ratings based approach (Basel II), banks are allowed to estimate their own PD, EAD and LGD using internal models. In the case of exposure to public companies, default probabilities are commonly estimated using the Merton model.

This then is used to calculate the provisioning for Expected Losses and computing economic capital for Unexpected Losses. The goal is to define risk weights by determining the cut-off points between and within areas of the expected loss (EL) and the unexpected loss (UL), where the economic capital should be held, in the probability of default. Then, the risk weights for individual exposures are calculated based on the function provided by Basel II.

Hope that helps!

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
That's a great point @Dr. Jayanthi Sankaran : Basel's own internal ratings-based (IRB) approach is derived from the Merton model (i.e., it computes the default probability for an obligor based precisely on the above Merton model theory that an obligor's asset return has a normal/Gaussian distribution and will default if it falls below are certain quantile). But I would emphasis the "derived from" as Basel's IRB is not the same Merton as I outlined above. Above I outlined the classic Merton which estimates the PD for a single asset (and the associated KMV variation). Basel's IRB is certainly based on the same theory but extends (or really just begins from it) using a so-called ASRF framework (which makes huge assumptions about a portfolio of obligors for basically the sole purpose of achieving the convenience of portfolio invariance). In the way, the way I would put it is, Basel's IRB extends Merton's by utilizing a Gaussian copula (another convenience purchased at the cost of simplifying assumptions), where the asset's correlation to the systematic risk factor becomes a key assumption. In this portfolio-based model (i.e., Basel's IRB), under both the Foundation and Advanced IRB, the "average default probability" is estimated by the bank; technically, the IRB maps an average PD to a conditional PD. My point is: in Basel IRB, PD is an estimated input (just as Dr. Jayanthi says!) into a model that utilizes Merton's theory, compared to my description of the classic single-asset model which generates PD as the output.

That's one of the fascinating aspects, to me, of the IRB model. If you look at the IRB formula, it begins with N(N^-1(pd) .... if the asset correlation is zero (which is not allowed, there is a floor), then the numerator would start with this sort of dynamic: supply a PD of 2% and retrieve = N[N^-1(pd)] = NORM.S.DIST(NORM.S.INV(2%), TRUE) = 2.0%. Ha! The PD can be estimated any number of ways in order to generate the IRB input. Once it gets to the IRB, it's a Gaussian copula which is "merely using" the multivariate normal distribution to achieve the supreme convenience of relying all obligors on a single-factor (ie, correlation to systematic risk) so it's really about the copula more so at that point, and to an extent the multivariate normal distribution.

I am less qualified to say how many banks use a Merton-type model to derive these initial "average" PDs which become inputs in the PD. My understanding is that a pure Merton (specifically the lognormal asset value assumption) significantly underestimates the actual (and historical) defaults (as mentioned by Moody's above: "However, actual default experience departs significantly from the predictions of normally distributed DDs. For example, when a firm’s DD is greater than 4, a normal distribution predicts that default will occur 6 in 100,000 times. Given that the median DD of the entire sample of firms in the EDF dataset is not far from 4, this would lead to about one half of actual firms being essentially default risk-free. This is highly improbable"). So I would expect where Merton's theory is applied, it's applied in a way similar to Moody's mapping of DD to PD; or alternatively, a heavier-tail distribution can be assumed. I hope that's additive!

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#### Tania Pereira

##### Member
Subscriber
Hi, David!
In Step 2 (risk measurement): PD = N(-DD), How is calculate V(t) = ~ $13.34? • At the end of the period, firm will have an expected future value higher than today, due to positive drift. In this case, V(t) = ~$13.3
Thank you,

#### emilioalzamora1

##### Well-Known Member
Hi,

I am not the master and so I hope you can live with my answer as well.

Firm Asset (Value): 12.75
Future Value of Debt: 10
Expected return on its assets: 5%
Time (t) in years: 1
volatility of firm assets: 9.6%

In the Merton model we have a risk-free rate, but in this model we have an "expected growth on the value of the firm's assets, which is NOT riskless. Hence, we go from risk-neutral in option pricing to expected growth in a firm's assets in the Merton model.

Notice: the rf-rate in the Black Scholes equation is replaced with the expected rate of return (5%)

You have:
1. numerator (d2) + ln(12.75/10) + [0.05 - (0.096/2)^2 ]*1 = 0.288 (the expected future firm value is +28.8% standard deviations above the default threshold
2. denominator = 0.096 * sqrt(1) = 0.096

d2 (distance to default) is then simply:

0.288/0.096 = 3.00
which is the distance to default (measured in standard deviations, meaning that it would take a so called '3 sigma event' (which is very rare!) to put the firm into default mode: debt > firm assets. In words, the firm cannot fully pay its debts out of its current assets and have to file for bankruptcy proceedings.)

First Digression: Following the 08 crisis Alan Blinder has published one of the greatest books in the field of fin. crisis at all time "After the Music Stopped" about how it came about incl. various simple academic explanations (e.g. illiquidity vs. bankruptcy)
http://www.nytimes.com/2013/02/08/books/after-the-music-stopped-by-alan-s-blinder.html

Second Digression:
Funnily, if you look up what a 3-sigma event might be; you get: Hurricane Sandy is a 3-sigma type of event. And William Sharpe called Warren Buffett a three-sigma event as well (see "Tap Dancing to Work: Warren Buffett on practically everything" by C. Loomis.)

PD = N(-d2) = N(-3.00)

This is calculated either using
1.) NORMSDIST(-3.00) in Excel = 0.13% (prob. that the firm defaults is approx 0.13%)
2.) Reading off the inverse of 3.00 at the normal table (see below) which is (1-0.9987) = yielding approx. 0.0013 = 0.13%
http://www.bbm.colmmcguinness.org/live/pd-NormalTable.html

Expected Future Value of the Firm: 12.75 * exp[0.05 - (0.096/2)^2]*1 = 13.34

Finally notice the following:

The distance to default (DD) can either be expressed as:
2. returns normally: 3
1. levels (prices) lognormally: (13.34-10)/(13.34*0.096) = 2.61

The price-based lognormal DD of 2.61 is equivalent to the return-based normal DD of
3.0 (normal log returns --> lognormal prices).

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Subscriber
Tank you a lot!

I understood!

#### abey

##### New Member
Hi Mr. David,

Please can you help by providing a clear and deep explanation and steps in addressing this topic 'Assessing Merton-Implied Probabilities of Default Risk of Nigeria Public Companies'. This is my chosen topic for master's thesis and it's a new project, everything seems scatter right now. Please, I need your guide.

#### mm.ff

##### New Member
Hi,
I need to do some numerical example on Merton portfolio selection model.for this I choose some asset in my portfolio .Now I need μ and σ for this equation: dpi(t)=μipi(t)dt+σipi(t)dzi(t) ;i = 1,2,...,n. where like Merton paper μ and σ are instantaneous conditional expected percentage change in price per unit time and σ i is the instantaneous conditional standard deviation per unit time.I collected historical data for each asset.Now I don't now how to find μ and σ from historical data.
suppose historical data of 10 days for one of the asset in the portfolio are: {20, 21, 19, 22 , 21, 20, 21, 22, 22.5, 22.6}. How can I find the μ and σ ( instantaneous conditional expected percentage change in price per unit time and σ i is the instantaneous conditional standard deviation per unit time)?