Derivation Of The Probabilities Of Default Using Dd

Published Date: 02 Nov 2017

Estimate the current asset value (VA) and the volatility of asset return (σA);

Calculate the distance-to-default;

Derive the probabilities of credit default using the DD;

Estimation of the current asset value and the volatility of asset returns

First of all, we need to make some simplifying assumptions (Crouhy et al., 2000):

Firm’s assets market value is log-normally distributed. This means that log-asset returns are normally distributed;

Distribution of asset returns is steady during the time passes;

Capital structure is formed only of equity, short-term debt which is equivalent to cash, long-term debt which is perpetuity, and preferred shares.

After making these assumptions we are capable to find solutions for the value of firm’s equity and equity volatility:

(8) (9)

VE – Value of equity;

σE – Historical volatility of equity;

K – Leverage ratio in capital structure;

c – Average coupon payment on long-term debt;

rf – risk-free interest rate.

Since equity volatility is unstable, sensitive to the changes in asset value and only the value of equity can be directly observable, we cannot simultaneously resolve VE and σE for VA and σA. The only way is to pull out VA from VE and to set it like a function of equity value and the volatility of asset returns:

(10)

Calculation of the distance-to-default

Distance-to-default is represented by the number of standard deviations between the mean of the distribution of the asset value and critical threshold, also known as "default point", set at the par value of current liabilities including short term debt to be serviced over the time horizon, plus half the long-term debt (Crouhy et al., 2000). Distance-to-default is given by the following equation:

(11) (12)

DD – distance-to-default;

E(V1) – expected asset value in one year;

DPT – default point;

σA – standard deviation of future asset returns;

STD – short-term debt;

LTD – long term debt.

However, we have made an assumption that firm’s assets market value is log-normally distributed. This means that it is not enough just to set up the distance between the expected asset value in one year and the default point. In case of log-normality assumption distance-to-default is expressed by a unit of asset return standard deviation at time horizon T (Crouhy et al., 2000):

(13)

V0 – current market value of assets;

T – time horizon;

DPTT – default point at T;

µ – expected net assets return;

σ – annual volatility of assets.

Derivation of the probabilities of default using DD

This is the last stage of implementation of KMV Portfolio Manager framework. It consists of distance-to-default assignation to the actual probabilities of default which in KMV Portfolio Manager are called Expected Default Frequencies (EDFs).

This can be best shown by the example provided by Crouhy et al. (2000):

Table 3. EDF example assumptions

Assumptions

Current market value of assets (V0)

1000

Expected net annual growth of assets

20%

Expected asset value in one year (V0 (1.20))

1200

Annual asset volatility (σ)

100

Default point

800

Formed by author, source: Crouhy et al. (2000)

First we need to calculate distance-to-default using equation 13:

.

Next we do the assumption that among the population of, let’s say 5000 firms with a DD of 4 at one point in time, 20 of firms have defaulted one year later, then:

or 40 bp.

According to Moody’s the implied rating for this EDF would be Baa3 (see Apendix 2 for more EDF comparison to credit ratings).

CreditRisk+

As it was said before, then comparing to CreditMetrics and KMV Portfolio manager, CreditRisk+ is an actuarial model. CreditRisk+ is modeling default rate distributions of a portfolio. In this framework there are no assumptions made about the reasons of default. Any "kth" obligor can default with a probability Pk and it cannot default with a probability (1 - Pk). According to Crouhy et al. (2000) there are only two assumptions we need to make to start modeling credit risk with CreditRisk+ model, those assumptions are:

For a loan, the probability of default in a given period is the same for any other same period;

For a portfolio with a bigger number of obligors, the probability of default is small if obligors are considered separately, and the number of defaults of separate obligors is independent from one period to any other given period.

Then given these assumptions, the probability of realizing n default events in the portfolio in one year is given by Poisson distribution [1] (CSFB, 1997):

(14) (15)

P – probability;

µ – expected average number of default events in one year;

n – number of default events in one year;

PA – annual probability of default for obligor A.

According to CSFB (1997) the credit risk is modeled in two stages process, as it is shown in figure 4, it also requires following data inputs:

Obligor default rates

Credit exposures

Obligor default rate volatilities

Recovery rates.

Figure 4. Stages in the modeling process:

What is the FREQUENCY of defaults?

What is SEVERITY of the losses?

Distribution of default losses

Stage 1

Stage 2

Source: CSFB, 1997

Stage 1: Frequency of defaults

According to CSFB (1997) credit defaults occur as a sequence of events in such a way that it is neither possible to forecast the exact time of occurrence of any one default nor the exact the exact number of defaults. This was already presented by the Poisson distribution above (see equation 14).

Since the credit portfolio usually consists of number of different obligors which have different annual probabilities of default, the annual default probability can be determined by the credit ratings of obligors and mapping between default rates and credit ratings (look table 4).

Table 4. One-year default rates.

Source: Standard & Poor’s Global Fixed Income Research and S&P CreditPro®, 2011.

However default rates are not constant and have quite a high level of volatility over time and this volatility have to be included into the model. If volatility of default rate is not included the distribution of the number of default events becomes closely approximated by the Poisson distribution (CSFB, 1997). In Figure 5 below we can see how distribution of the default events changes if we include default rate volatility.

Figure 5. The distribution of default events.

Number of defaults

Probability

Volatility included

Volatility excluded

Source: CSFB, 1997.

Stage 1: Severity of the losses

CSFB (1997) states that in the event of a default of an obligor, a firm generally incurs a loss equal to the amount owed by the obligor less a recovery amount, which the firm recovers as a result of foreclosure, liquidation or restructuring of the defaulted obligor or the sale of the claim. For average recovery rates have a look at table 5. Recovery rates should take account of the seniority of the obligation and any collateral or security held (CSFB, 1997).

Table 5. Average recovery rates.

Source: Moody’s Global Corporate Finance, 2008.

According to Crouhy et al. (2000) the severity of each obligor, in CreditRisk+, is adjusted by the anticipated recovery rate, in order to calculate the loss given default.

Distribution of the default losses

In CreditRisk+ the losses are divided into bands in order to determine the loss distribution for the portfolio of loans and bonds. The level of exposure in each band is approximated by a single number (Crouhy et al., 2000). Each band is also seen like an independent portfolio consisting of loans and/or bonds. Expected loss in band is calculated as follows:

(16)

j – band; j = 1, 2, … , ;

εj – expected loss in band j in units of exposure;

µj – expected number of defaults in band j;

vj – common exposure in band j in units of exposure.

From that we can derive expected number of defaults:

(17)

If we calculate the expected loss for a obligor in units of exposure, i.e.,

(18) (19)

εA – expected loss of the obligor A;

λA – expected loss;

L – unit of exposure;

P – probability of default.

then the expected loss in a one year period in band j is:

(20)

So, it follows that the expected annual number of defaults is:

(21)

Yet, we have just found the way to derive the number of defaults in one band, to derive the distribution of losses through the entire portfolio we have to do following 3 steps:

Find probability generating function for each and every band:

(22)

Find the probability generating function for entire portfolio:

(23)

Find the loss distribution for whole portfolio:

(24)

That was the final step of modeling credit risk in portfolio level using CreditRisk+ framework.