Risk Management completo

Risk management class 1 – 14/09

Possible definitions of risk

• Oxford definition: Exposure to possibility of loss, a situation involving such a possibility.
• Business dictionary: A probability or threat of damage or any other negative occurrence that is caused by external or internal vulnerabilities, that may be avoided through preemptive action.

What to do to deal with it?

It is fundamental to talk about expected value and variability of value with respect to expectations. The difference between risk and uncertainty lies in the measurability: uncertainty surrounds an expected value, so we know distribution around that value. Risk management is very important in banks and insurance. In a probability distribution, we look at what happens in detail.

We account for factors we consider relevant to estimate risk in models. What is the condition to use a distribution? The probability of bad events is small in a normal distribution; we need a distribution with longer tails. For every return (negative or positive), there corresponds a probability. In reality, probabilities are not well-defined; to estimate a probability, we have to make some tests.

In the banking world, there is operational risk, which concerns low probability events that, if they happen, are devastating.

• Finance definition: The probability that an actual return on an investment will be lower than the expected return.
• Chance that an outcome is different from the expected outcome. Risk includes the possibility of losing some or all of the original investment.

Should I take into account only the probability of bad events or also the impact? In operational risk, for example, transactions between c/c, there are very frequent small errors, but the impact is not so dangerous. We are interested also in the impact.

How we might look at a financial risk

• Define the outcome “at risk”: what is the variable look at, e.g., expected return on investment.
• Define the risky event: it can be the possibility of negative return, lower than a threshold, or the default of the counterpart, or the interest rate risk.
• Try to estimate the probability of the event: for credit risk, it is the probability of default.
• Combine the probability with the impact: the impact on the portfolio of the event (LGD x EAD) (loss given defaults x exposure at default) (think to a bullet loan and the default of the counterpart. Take into consideration also collaterals, mortgages).

We will start with credit risk, but there exist many types of risks depending on the outcome you look at:

• Market risk: Risk on portfolio of assets.
• Operational risk: Risk of banks because of operations, computer system crash, and so on.
• Interest rate risk: When maturity mismatch with investment horizon, risk from change in interest rate.
• Liquidity risk: Risk to have no cash to afford the operating activity or to hold assets no more liquid in the market, e.g., there aren’t buyers for them.
• Reputational risk: Procedures, rules in the company that minimize reputational risk, for example, avoid corruption events.
• Model risk and solvency risk.

We need to talk about the dependence between risks above. Liquidity risk may be linked with market risk.

Managing risk

Look at banks: they need huge capitalization to absorb the losses that could derive from some of these risks, like credit risk. Banks by regulation have to put in place a model to know how much risk they can bear. One objective of risk management is to mitigate (decrease) the risk, and that can be done by reducing probability or decreasing impact. Remember that to have a decent return we have to take some risk. There is a trade-off between risk and return.

In summary, risk management regards several topics:

• First of all, risk measurement.
• Determine how much risk you can bear.
• Determine what can be done to mitigate risk, acting on probability or impact.
• Look at the interaction of risk with return.

Week 2 – Risk management

Credit Risk

Most important risk according to regulation also. It is used to identify a lot of risks in reality, so it is a generic risk of unexpected changes in the value of credit position or portfolio due to adverse movements of risk drivers related to the creditworthiness of counterparts. Basically, it is the risk that the counterpart is not able to fulfill its contractual obligations (repay a loan).

However, credit risk is not just pure default risk. Types of credit risk:

• Default risk: Risk that counterparty doesn’t fulfill its obligations.
• Migration risk: Risk of counterparty’s creditworthiness deterioration, involving a rating downgrade by rating agencies or perceived by bank’s credit analysts.
• Recovery risk: Risk linked to what happens after the default, that the realized recovery rate on defaulted exposures is less than estimated (even if default doesn’t occur yet).
• Exposure risk: Risk for the banks of sudden increases of counterparty’s exposure in the period preceding default (e.g., drawing on credit lines, exposure depends on the market value of a derivative).
• Spread risk: Risk that you have a credit outstanding, but the market asks a big spread for that kind of risk and it affects its market value. Risk of sudden increases of market risk premiums at the same rating terms.

Credit counterparty risk is the risk in the derivative credit market, such as Swap Interest Rate, which in the midterm can have a big positive value, because if I receive the fixed rate and in the meantime the interest rates go down...

Estimate credit counterparty risk

The default definition is not one:

Article 178 of EU regulation default definition establishes that a default shall be considered to have occurred when both of the two situations below take place:

• Unlikely to pay: More subjective, the institution considers that the obligor is unlikely to pay its credit obligations to the institution, the parent undertaking, or any of its subsidiaries in full, without recourse by the institution to actions such as realizing security.
• More than 90 days of delay in general: The obligor is past due more than 90 days on any material credit obligation to the institution, 180 days for exposure secured by residential or SME commercial real estate, as well as exposures to public sector entities.

Expected loss vs unexpected loss

How are expected losses composed?

Loss Expected over the period of receivables: EL = PD * LGD * EAD (exposure at default). These three components form the credit risk:

• PD: Probability an obligor isn’t able to fulfill his obligation towards the creditor.
• LGD: Quantify the loss relative to a receivable if a default occurs (1 – recovery rate).
• EAD: Is the amount of due receivables when a counterpart defaults.

The expected loss is not a risk. The risk is in the unexpected loss. The relation between risk and expected loss? There is not a link. The unexpected loss is the loss beyond the expected value of the loss distribution. It can be calculated by subtracting the expected loss from the loss level at the 99th percentile (or a high percentile in general). The expected loss thinking at CAPM comes from the systematic risk, while the unexpected loss is diversifiable, connected to specific risk. Unexpected loss is linked to the variance of the portfolio loss distribution. An unexpected loss depends on the covariance; the risk portion can be diversified through exposure choice. We need three risk measures to calculate the unexpected loss.

Instruments to measure the default risk

Ratings is an ordinal assessment of counterpart creditworthiness, measures probability of default. Ratings can be from a bank point of view or external if we see rating agencies. External rates exist only for listed companies and for sovereigns. Historical default rates can be used as a basis for determining the PD to be associated with the single rating categories. There is a new bond issue, the rating allows taking fast decisions for investors, if to invest or not. It is used a lot in risk management. From the moment you don’t pay an interest payment, no one will give you money in the bond market.

Understand when a rating is (probability of transitions between rating categories):

• Sollecited
• Unsollecited
• Analysis from financial statement of riskiness of a company, estimate probability that a counterpart defaults within a specific time period. In order to do it, we assume a binary target variable Y (default Y=1, not default Y=0) that will take value 1 whenever a latent variable Z takes values smaller than βX, where βX is a function of variables with explanatory power of the counterparty’s default risk.

21/09

Go on with credit risk

Recommended but not mandatory books:

• Credit risk analytics, Baesens, Rosch, Scheule
• Credit risk modeling using Excel and VBA, Posch
• Option, Futures and other derivatives (libro Asset Pricing)

Modelling of the basic credit risk measurement (continued)

Look at other two building blocks of credit risk:

• Loss given default: The estimate of the loss rate that should be valid at the moment of default. What we expect to lose in proportion to exposure, opposite to what we expect to recover. The LGD is a %. It is greater for a personal loan for buying a big TV compared to a mortgage loan guaranteed by a house (more in % terms).

The estimation methodologies can be divided into two families, depending on the available data:

• Market approach: We don’t have a comparable at default, estimation of LGD for a sector (that occurs in a given sector) looking at historical data. Neutralize the risk premium part.
• Workout LGD: Used for 99% of loans. This approach is based on the expected discounted value of future net cash flows (recoveries – costs). It takes into consideration past defaults and all the CFs (positive and negative) occurred after the default (incoming CFs). Consider costs that can be accounted to a specific loan and general costs to recover debt.

With this logic in mind, we can proceed in two ways, employ clusters (databases) built according to specific drivers, that cover entire loans that the bank has, and calculate the long run historical average for LGD. The second way regards employing econometric models that link LGD levels to explanatory variables (risk factors, e.g., credit segment, loan-to-value, geographic area, collateral, etc.).

To estimate Workout LGD we need:

• A historical dataset with recovery and cost cash flows of all closed defaults.
• A discounting rate including an adequate risk premium (CAPM, risk-free + risk premium).
• Exposure characteristics (size, guarantees..).

Recovery procedures length depends on the country they happen, how fast the courtrooms work. Higher LGD with longer process to judge. In Italy, it requires 6-7 years to recover on a mortgage. In the new regulation, the recovery value of non-secured debt will be 0.

• Exposure at default (EAD): Represents the exposure of a risky asset (on and off-balance-sheet) at the moment of default. It can be seen as the sum of:
• The amount of drawn credit facilities
• The amount of expected additional drawing

Keep in mind that banks have credit lines with corporates. Often EAD determination comes down to estimating a coefficient (Credit conversion factor), which is applied to the undrawn part of the credit, yields an estimate of expected additional drawings:

The K coefficient is typically estimated using historical data of defaulted committed credit facilities, on:

• A cluster base, clustering data according to specific drivers and calculating the LT historical average CCF factors for each cluster.
• Employing econometric models, that link drawn amounts to explanatory variables (e.g., activity sector, the geographical area or type of facility).

EAD now is different from EAD in 1 year because the assets bank holds now are different from assets in 1 year. These are basically the three main building blocks of credit risk (PD, LGD, and EAD).

PD model is based on a rating model but there is a further step. If you go to the bank and ask for a loan, the bank inputs your data into the model and PD is produced. It may be above or below a given threshold, in order to decide to give or not give you the loan. Ratings are used to decide if you get a loan or not. PD is used, in addition, to determine how much you have to pay for the loan. PD is important for pricing.

Week 3 – RM

Logistic Regression

It is a market practice for PD estimation, but you can use another to calculate PD according to regulatory (on the condition you explain why). Other approaches:

Returning to our logistic regression, it has the problem it cannot be estimated as a linear regression (not in Excel). Linear regression metrics:

y=XB+z = (X’X)ˆ *(X’y);-1→ β(cap)

Maximum likelihood estimation (MLE)

In order to gain intuition for MLE, let’s try to solve a simple coin flipping problem:

• On ten times flipping, we obtain 7 heads. What is the probability of heads, p, given the data? Probability of head given the data is 7/10.
• Now invert the question: What is the probability of the data – obtaining 7 heads in 10 coin tosses as a function of probability?

As a function to describe this situation, we can use a Bernoulli combination, that results in a binomial random variable with N=10 trials and unknown probability p:

P(7 heads) = (10;7) * p⁷ * (1-p)³

This function gives the likelihood (or probability) of the data as a function of the unknown parameter "p" likelihood function.

Now we have to maximize this likelihood function (L) → find the "punto di Massimo" della funzione (porre uguale a 0 la derivata)

• Take logarithm to eliminate exponent: log(L) = LL = log(10;7) + 7*log(p) + 3*log(1-p).
• Take derivative wrt p: -3d 7⟨/iquest; +0+dp p 1- p.
• Set the derivative equal to 0 and solve for p:

As a result of these calculations, we obtain our first result 7/10, to convince us this method works. Likelihood means as like as possible.

Next section:

Derive the logistic function, probability is not constant each time. Once I derive my PD from the likelihood, I don’t use it directly, but I apply some adjustments (calibration). Once we had the calibration, we have our PD model.

Logistics regression for PD estimation (continued)

We will have a look at how banks use these tools. Try to see what regulators say about it, how it numerically works. Risk measurement with likelihood function regression model goal is to know the PD for each client that asks e.g., a mortgage. Maximum likelihood method is just an estimation method.

Let’s start with a general approach, like a generalized linear model approach. A GLM model is defined by (properties):

• The probability distribution of the response variable Y.
• A linear predictor of Y, look at multivariate linear regression, the point is that you have some x’s that need to be linked linearly to our linear predictor: η = β + β X + X +… + β X.
• A link function, g(.), that describes how the mean of the response variable, E[Y ] = μ, depends on the linear predictor. Looks like a regression but we have not to put directly ‘Y’ on the right side because it is a function of expected value of Y g(μ) = η.

If we substitute for μ we obtain function of mean: μ = β[ ]( )= 0 + 1 +g E Y x …i 0 1 1 i

In the simple linear regression model y = β + x + ε, the responseβ β i 0 1 1 variable y is defined as a linear function of the explanatory variable x plus an error term, which as a basis for inference is typically assumed to be normally distributed with mean 0. GLM take into account more variables.

In the latter case, y will be distributed normally with: E(y ) = = + xμ β β i i i 0 1 1. -taking it as a linear predictor, it is easy to see the case g(μ) = μi i

The logistic regression as GLM

We know the linear model is not reliable to estimate the PD because it is not limited to 0 and 1. With the GLM we can define PD, something that needs to be between 0 and 1. The linear regression is clearly not a suitable model for a binomially distributed response variable. We can, though, apply GLM by choosing a suitable link function (that explains us how the mean of the binomially distributed response variable depends on the linear predictor). When y ~ B(n , p ) we model the proportion :i i i n i

[ ]y i =pE in i

Y can be either 0 or 1. n is the number of observations I have.i [ ]y i pE = i ni

Expected value of this proportion is equal to probability. I’m modeling an expected value and we have seen that in GLM we can use a link function to link the expected value to a linear predictor. We need a link function “g” that maps something between (0,1) into something that is an infinite interval, classic of linear combination that has no limits. We use the logit function as link function:

{ }[ ]yi E n iLn = = β +β X + X +.... + β X

logarithm of my expected value divided by one minus expected value and this is my "g" (remember g(μ) = η) and n(i) is the linear predictor.i i

Theoretically framework to estimate my PD. I substitute a logarithmic function to my infinite interval equation, keeping in mind the plot of a log, that goes between 0 and 1. In the log there is at the numerator the probability of default divided by the probability of non-default. We can have with logit continuous value between 0 and 1 and this makes sense. The regulator asks to calculate a PD, the minimum PD that can be assigned by regulation.