Predictive Modeling for Market Risk in the Banking Book

Predictive Modeling for Market Risk in the Banking Book

4EK614

Applied Operational Research in Business Consulting

With You Today


Tomáš SobotkaSenior Manager in Market Risk Team

[email protected]

Branislav Lovás

Senior Consultant in Market Risk Team

[email protected]

Market Risk & Regulatory Team

Quantitative Services

ALM

Treasury

Regulatory

Model Development

Model Validation

Securities Valuations

Derivatives Valuations

Data Analysis

Data Mining

Investment RoboAdvisors

Behavioral Modelling

Stress Testing

Forecasting

Regulatory Reporting

Methodological Review

Our projects

Our services

Eliška Kompanová

Senior Consultant in Market Risk Team

[email protected]

Adéla Nguyenová

Consultant in Market Risk Team

[email protected]

Housekeeping:

1. Basic understanding of statistical and mathematical concepts(mathematical modelling, time series)

2. Elementary knowledge of programming (Python, R, VBA)

Please see Annex to links with helpful sources.

Prerequisites

Day 1: Credit RiskDay 2: Market RiskClassroom: OnlineTime: 9:15am – 10:45am

11:00am – 12:30pm

Course Structure

Please note that you are not limited to methods presented today (and it is more than welcomed to new approaches)Non-Maturity Deposit Modelling1. Case study – you will obtain dataset for modelling balance of non-maturity deposits. Your task will be to analyze the dataset, split it to train set and test set. With train dataset, you will develop selected model for non-maturity deposit balances. Then you will take your test dataset and see how your model performs.2. Outputs – PPT presentation or PDF, summarizing the abovementioned outputs, and scripts in Python, R or VBA (including the spreadsheet) that were used. Please send these outputs to [email protected]. Submission deadline: 7 May 20213. Output presentation for EY (most important!)– short (10-15 minutes) presentation + Q&A about results of this assessment; the presentation will be delivered online and will take place during the week starting from 10 May 2021.

Course Assessment

1. PowerPoint slides, provided after the course

2. Jupyter notebook *.ipynb files, provided after the course

Study materials


mailto:[email protected]

Contents


1. Mathematical Crash Course

I. Somewhat Heuristic Introduction to Probability and

Stochastic Processes

II. Time Series Modelling

III. Linear Regression

2. Behavioral modelling in the Banking Book

I. Introduction to the Banking Book

II. Behavioral models

3 Non-Maturity Deposit modelling

I. Segmentation

II. Liquidity modeling of NMD

III. IR sensitivity models – β and optimal portfolio approach

IV. Other issues

4 Hands-on Case Study: Instructions for Module Assignments

Mathematical Crash Course

Somewhat Heuristic Introduction to Probability and Stochastic Processes

Time Series Modelling

Linear Regression

1

Questions 1: Basic Demographics


Go to www.menti.com and use the code 15 68 21 92


Probability Space

• Key object is a so-called probability space, which is a triplet (Ω, ℱ, 𝑃):

• Ω: Sample space (contains all possible outcomes)

• ℱ: 𝜎 –algebra on Ω, (contains all events we are interested in) i.e.:

• Ω ∈ ℱ

• If 𝐴1, 𝐴2… ∈ ℱ then ڂ𝑖=1+∞ 𝐴𝑖 ∈ ℱ

• If A ∈ ℱ then 𝐴𝐶 = Ω − 𝐴 ∈ ℱ

• 𝑃 – probability measure, i.e. a set function 𝑃:Ω ↦ [0, 1]:

• 𝑃 ∅ = 0; 𝑃 Ω = 1

• For pairwise disjoint sets Ai ∈ ℱ: 𝑃 𝑖=1ڂ+∞ 𝐴𝑖 = σ𝑖=1

+∞ 𝑃(𝐴𝑖)

• The above might sound theoretical, but the triplet is a cornerstone of any stochastic modelling

• Additionally, we might consider a filtered probability space (Ω, ℱ, ℱ𝑡 𝑡≥0, 𝑃) where ℱ𝑡 𝑡≥0 is an increasing set of sub-𝜎-

algebras ⊂ ℱ

Example:

• Rolling dice – what would be Ω?

• What is the smallest ℱ?

• What if we want to know what is the probability of casting an odd / even number?


Probability – Conditioning & Independence

• Assuming probability space Ω,ℱ, 𝑃 and two events 𝐴, 𝐵 ∈ ℱ

Conditional probability:

• 𝑃 𝐴 𝐵 =𝑃(𝐴 ∩ 𝐵)

𝑃(𝐵)≈ “probability of event A occurring if event B occurs”

Independence of two events:

• Two events 𝐴, 𝐵 ∈ ℱ are said to be independent if:

• 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 𝑃(𝐵)

Example:

• Rolling dice

• What is probability of 𝐴 = {1}conditional on B = {1,3,5}?

• What is a conditional probability of two independent events?

Questions 2: Roll a Dice




Random Variables

• Real-valued random variable is defined as 𝑋:Ω ↦ ℝ

• For math nerds: 𝑋 is a measurable function on Ω,ℱ, 𝑃

Technically:

• We denote,𝑋 = 𝑥 = 𝜔 ∈ Ω: 𝑋 𝜔 = 𝑥𝑋 < 𝑥 = 𝜔 ∈ Ω: 𝑋 𝜔 < 𝑥

𝑎 < 𝑋 < 𝑏 = 𝜔 ∈ Ω: 𝑎 < 𝑋 𝜔 < 𝑏

• Analogously, random variables 𝑋1, 𝑋2 are independent if for all 𝑥1, 𝑥2:

𝑃 𝑋1 = 𝑥1; 𝑋2 = 𝑥2 = 𝑃 𝑋1 = 𝑥1 𝑃(𝑋2 = 𝑥2)

Examples:

• Rolling dice

• 2 dice rolled at once, 𝑋 is the sum of casted values on dice

• Identity function

• Coin tossing bet

• Random walk of a drunk

Realization of 𝜔 ∈ Ω

𝜔 = H

𝜔 = T

$100

$0

Cumulative distribution function

• Let 𝑋 be a real-valued (continuous) random variable on Ω,ℱ, 𝑃 , then 𝐹 𝑥 : 𝐹 𝑥 = 𝑃 𝑋 ≤ 𝑥 =

∞−𝑥

𝑓𝑋 𝑠 𝑑𝑠, where

• 𝐹 𝑥 is the cumulative distribution function (CDF)

• 𝑓𝑋(𝑥) is the probability density function (PDF), (probability mass function for discrete random variables)

• The CDF:

• ranges from 0 to 1,

• is an increasing function,

• is right-continuous.


Useful Functions: CDF & PDF

Continuous random variableDiscrete random variable

• The normal distribution is bell-shaped, entirely defined by two parameters

• Mean - 𝜇

• Standard deviation - 𝜎

• The normal PDF can be analytically expressed:


Important Example: Normal Distribution

𝑓 𝑥 =1

𝜎 2𝜋𝑒–(𝑥–𝜇)2

2𝜎2

• Why is this important?

-0,3 -0,2 -0,1 0 0,1 0,2 0,3

Standard normal distribution

• The standard normal distribution has 𝜇 = 0 and 𝜎 = 1

Questions 3: Other probability distribution functions



Stochastic (random) process

• Let 𝜏 be a subset of [0, +∞], then a family of random variables indexed by 𝜏: 𝑋𝑡 𝑡∈𝜏 is a stochastic process. Two basic observations (for real-values stochastic processes):

• 𝑋𝑡 𝜔 : 𝜏 ↦ ℝ for a fixed 𝜔 is a sample path (or trajectory)

• 𝑋𝑡 𝜔 :Ω ↦ ℝ for a fixed 𝑡 is a random variable

Example – constructing a “Random Walk” model

• A simple random walk process 𝑋𝑛 𝑛∈ℕ0 can be constructed:

1. 𝑋0 = 0

2. Increment 𝑋𝑛+1 − 𝑋𝑛 is independent of all previous increments

3. Increment 𝑋𝑛+1 − 𝑋𝑛 has a coin toss distribution (binomial):

𝑃 𝑋𝑛+1 − 𝑋𝑛 = +1 = 𝑃 𝑋𝑛+1 − 𝑋𝑛 = −1 =1

2


Stochastic processes - Intro

• Stochastic time-series is a sequence of data collected at specified time points 𝑡1, 𝑡2, 𝑡3, …

• Trend, seasonality, cyclicity, irregularity

ARIMA(p,q): 𝑋𝑡 − 𝛼𝑋𝑡−1 −⋯− 𝛼𝑝𝑋𝑡−𝑝 = 𝜖𝑡 + 𝜃1𝜖𝑡−1 +⋯+ 𝜃𝑞𝜖𝑡−𝑞

• Auto regressive integrated moving average

• Three main parameters:

• p – number of autoregressive terms (AR) … current value as a linear combination of past values

• d – non-seasonal differences to achieve stationarity (I)

• q – number of lagged forecast errors in the prediction equation (MA) … current error as a linear combination of past errors

Generalized GBM

Stochastic time-series models


• Linear approach to modeling a relationship between one dependent variable (outcome) and multiple independent variables(inputs)

𝑦 = 𝑥𝑇𝛽

• Linearity

• Normality – errors have normal distribution

• Homoscedasticity – same variance

• Independence of observations

• Lack of perfect multicollinearity

• MLE:

𝑌 = (𝑋𝑇𝑋)−1𝑋𝑇𝑌

Linear Regression


y

x

Questions 4: Other models



Behavioral modelling in the Banking Book2


Key products offered by banks

Assets

Retail segmentPersonal loans

Mortgage (various types)

Debt consolidation

Overdrafts

Credit cards

SME segmentCommercial loans - secured/unsecured

Corporate segmentInvestment loans

Working capital loans

Liabilities

Across all segmentsCurrent accounts

Savings accounts

Term deposits (single/revolving)

Structured deposits (embedded options)

Bonds

Role of a Typical Commercial Bank


Deposits

Other Borrowing (Wholesale)

CapitalFixed Assets

Liquid Assets

Loans & Advances

Assets Liabilities

‘Typical’ Balance Sheet Role of Maturity Transformation

De

po

sit

Lo

an

He

dg

e

Ass

et

He

dg

e

Lia

bili

ty

Time

Banks play an important role as an intermediary in the financial system. They typically have two main functions:1. Take on deposits from the public (typically ‘short term’ in nature)

2. Lend out loans / mortgages (typically ‘long term’ in nature)

• Banking Book assets and liabilities are those which are intended to be held on the balance sheet until

maturity at amortized value.

• The banking book can also include those derivatives that are used to hedge exposures arising from the

banking book activity, including interest rate risk.

Banking Book


• In order to match assets and liabilities with the same interest

rate profile, banks need to know the maturity (re-price) date of

those assets and liabilities.

• The following assets and liabilities are often associated with

customer optionality and therefore the maturity is not clear:

Behavioral Models


Psychology

Decision-making

Finance / economics

Behavioral finance

Product Type Optionality

Fixed Term Loan Customers may have the option to re-pay the loan earlier than its maturity/reprice date

Fixed Term Deposit Customers may have the option to withdraw deposits earlier than the fixed term or to top-up deposits at the original fixed term rate

Non-Maturity Deposit There is no stated maturity but customers have the option to withdraw at any time

Pipeline Banks need to take a view on assets/liabilities which are contractual but have not hit the balance sheet yet.

Behavioral Models


Banks employ a number of techniques to assess the likely behaviour of these balances in order to match assets and liabilities of the same interest rate profile:

1. Fixed rate products with pre-payment / early withdrawal / top up optionality

2. Non-maturing deposits

Determined using

statistical modelling of behaviours

under different scenarios

• Market

• interbank and governments

• Macro-economic

• inflation, economic growth

• Behavioral

• reaction of customers to the market (withdrawing, saving, …)

• Personal

• Age, education, job, ..


Behavioral Models – input data types

What Risks are in the Banking Book?

Non-Maturity Deposit modelling3

Questions 5: Non-maturity deposits



• Liabilities in the Banking book

• Depositors can withdraw their funds at any time without any penalty

• Interest rates are unilaterally set up by the Bank

• One of the most stable sources of funding for banks; asset

• Contractual maturity – overnight (in reality significant effective duration)

• Retail savings, money-market accounts

• Modelling this early redemption option is very challenging

• NMD balances can be modelled under 3 different assumptions:

• Run-off

• Static balance-sheet

• Assuming inflow of new clients

• Factors affecting NMD:

• Market (interbank and governments)

• Macro-economic (inflation, economic growth)

• Behavioral (reaction of customers to the market)

► Main components discussed:

a) Liquidity runoff model / balances prediction

b) IR sensitivity model (replication portfolio)

Non-maturity deposits


Liquidity risk

management

Interest rate

risk / ALM

FTP

a) Liquidity runoff model

b) IR sensitivity model

c) Floor add-on

2.1 Modelling of NMD: Segmentation


► The first step in the deposit characterization process is to divide demand deposits into segments, which is criticalfor an accurate analysis.

► The main goal of product segmentation is to find homogeneous enough segments, while ensuring a minimumthreshold of materiality and simplicity.

► The segmentation process tries to reach a balance between three ideals:

► Typical segmentation criteria are:

1) Product type: Completely different products should not be examined together

2) Currency: Products with different currencies should not be pooled together.

► Segmentation process requires deep knowledge on the specifics, management strategy and historical behavior ofthe products.

Homogenity

SimplicityMateriality

2.2 Liquidity modelling of NMD: Overview


Overview

► A blend of quantifiable

statistical models coupled

with business adjustments

► Good segmentation of the

products is essential

Modelling approaches

► Segmentation based

on various factors, e.g.

currency, product time,

country, type of origination

(on-line / branch)

► Two types of statistical

models (in practice,

combination of both is used)

(next slides)

Model goal

► Assign a runoff profile

to deposit products

► The profiles should be

consistent with the historical

behaviour of the deposits

according to prevailing

market conditions

2.2 Liquidity modelling of NMD: Run-off profile


Stochastic time-series models

(ARIMA, generalized GBM)

► Modelling deposit balances evolution:

1. Runoff profile as an confidence interval of the fitted model (excl. new production)

2. Prediction of balances (including new production) expressed as expected value of the balance conditional on

current information – evaluated by:

I. Analytic expression (we know how to express this e.g. for GBM)

II. By Monte Carlo simulation

► Seasonality typically taken into account

► Statistical testing in place to validate appropriateness of the model

2.2 Liquidity modelling of NMD: Core volume and vintage analyses


► Balance volatility model determines what level of balances

will remain with the bank in the long term given a level of

confidence.

► As an output, balance volatility model divides the balances

into two portions:

► Core balances which remain with the bank under almost

all market conditions

► Non-core balances which fluctuate over time due to

market and idiosyncratic reasons and are typically

characterized as short term.

► The analysis of core vs. non-core balances is performed

through a growth based regression model.

► Balance volatility model fits the product balances to trend

models (exponential, linear or logarithmic) and find the

variation of actual balances around the trend line.

7 000

8 000

9 000

10 000

11 000

12 000

13 000

14 000

15 000

I-06 VII-06 I-07 VII-07 I-08 VII-08 I-09 VII-09 I-10 VII-10

Bala

nce

Months

Core balance Balance development Linear fit

Step 1: Fit the trend line

Step 2: Measure variation from the trend

Step 3: Determine the core portion

-

20

40

60

80

100

- 10 20 30 40 50 60

Rem

ain

ing

bala

nce

in %

Months

Product run-off profile

Jan 2006 Feb 2006 Mar 2006 Apr 2006

2.2 Liquidity modelling of NMD: Challenges and Market practices highlights


Challenges

► To distinguish between seasonal outflows and surge outflows (due to environmental

changes)

► Lack of historical balance data to support the granular segmentation analytics

► Difficult to have statistically sound models of type 1. for all products

► One might need to switch to a simpler expert model instead

► Good model governance is key here

Market practices highlights

► Hypothesis testing for confidence interval reliability

► Concentration risk

► Average account balance vs # accounts

► Seasonality & surge balance adjustments

2.3 IR sensitivity models – β and portfolio optimization: Overview


Overview

► As previously, a blend of

quantifiable statistical models and

expert adjustments is put in place

► Adjustments should align for

forward looking expectations on

rates and for business rationale

► Main output of the model –

replication portfolio

Modelling

approaches

► Two main concepts

are utilized,

sometimes they are

combined together

(i.e. via cointegration

analysis)

Model goal

► To stabilize NII and

to ensure correct

inputs for FTP

mechanisms

2.3 Modelling of NMD: Rate sensitivity analysis


► Characterization of administered rates for deposit products describes the significance of market rate changes on the rates paid by bank to thecustomer.

► By modeling the behavior of interest rates paid relative to market rates, a better understanding of repricing behavior and the sensitivity ofinterest expenses to market rates can be gained.

► The interest rate paid to indeterminate maturity products’ depositors is typically a combination of one or more market rates, and fixedcomponent.

► In order to determine which of market ratesexplain the changes in the rates paid by thebank, a multiple regressions are made.

► Coefficients of this regression equation tellwhat percentage of the portfolio reprices withthe associated market rate.

Δproduct rate = 40% Δ 1M PRIBOR

► This repricing behavior indicates that 40% ofthe portfolio reprices with the 1M PRIBOR,while the remaining 60% of the portfolio is rateinsensitive.

0,5%

1,0%

1,5%

2,0%

2,5%

3,0%

3,5%

4,0%

4,5%

5,0%

Inte

rest

ra

te

Months

Interest rate development

Product rate 1M PRIBOR 3M PRIBOR

6M PRIBOR 12M PRIBOR

2. β – rate sensitivity models

► Provide assumptions on how big change in interest rates one can pass to specific

client categories

► Beta (β) represents a deposit elasticity to changes in interest rates

► Retail NMD typically exhibit lower β compared to Wholesale deposits and savings

accounts

► Short-term/Long-term β and Linear /Dynamic β choices typically considered

2.3 IR sensitivity models – β and portfolio optimization: Modelling approaches


Typical utility functions

Criterion Optimisation goal

Volatility of the margin Minimize

Correlation between

customer and market ratesMaximize

Sharpe ratio Maximize

Typical β models

Model Description

Linear rate modelChange in rates proportional to the

benchmark

Mean reverting equilibrium

modelChanges due to short- and long-term effects

Direction dependent rate

model

Different sensitivity to increases and

decreases of the benchmark

Level dependent rate modelDifferent sensitivities when benchmark rates

are low / high

1. Portfolio optimization

► Typically connected with liquidity runoff (as

an upper bound or target) and regulatory

requirements

► One tries to find a portfolio of market

instruments that would have led to optimal

utility function value over relevant historical

time frame

► One might consider more than one criterion

(Sharpe, margin stability, duration, impact

under stress scenario)

2.3 IR sensitivity models – β and portfolio optimization: Modelling approaches


Short term β Long term β

► Influenced by tactical

pricing expectations, line of

business Judgment, and

acute historical experience

► Influenced by statistically determined

historical experience (long term), and

business judgement.

► Reflects deposit rate paid in equilibrium

► Constant recalibration and

back-testing

► Less frequent back-testing

Dynamic β Linear β

► Reflects the changing sensitivity of

customer rates depending on the level

of interest rates as well as the relative

change in interest rate across the life

of the deposit

► A constant sensitivity of changes in

customer rates to changes in

interest rates

► Provides a better convexity measure

to the deposit profile

► Captures the asymmetrical behavior

as well as beta response to changes

in the level of rates

► Operationally easier

► Requires a less frequent update to

the hedging strategy

► Challenging operationally

► Difficult to apply to FTP

► Can be difficult to implement micro-

level hedging strategy

► Requires frequent beta assumption

review. Most common review

frequency range from 6-12 months

► Could be significantly off from the

spot beta at a given time, resulting

in disincentives

► Reduces transparency for the

Business

0,5%

1,0%

1,5%

2,0%

2,5%

3,0%

3,5%

4,0%

4,5%

5,0%

Inte

res

t ra

te

Interest rate development

Product rate

1M PRIBOR

3M PRIBOR

6M PRIBOR

12M PRIBOR

2.3 IR sensitivity models – β and portfolio optimization: Challenges and Market practices highlights


Challenges

► What historical time-frame to use (e.g. short-term β vs long term

β), also applicable for portfolio optimization

► Manage operational complexity, especially for replicating

portfolio and dynamic β approaches

► Finding a correct segmentation of NMD volumes

Market practices highlights

► Portfolio optimization can be done by local optimizer with initial guess equal to previous

weight, or discrete steps

► Embedded floors taken into account:

a) Combination of β-Level and direction dependent rate models

b) via add-ons (incorporating value of IR floors)

c) or ignored

2.4 Other issues


• Negative IR policy (NIRP) and embedded optionality

• FTP rate

• An all-in FTP rate for a deposit product is the sum of base rate, term liquidity premium, and a contingent liquidity

cost.

• New benchmark rates

Questions 6: Your balance


0

500

1000

1500

1 4 7 10 13 16 19 22 25 28 31

Balance 1

0

200

400

600

800

1000

1 4 7 10 13 16 19 22 25 28 31

Balance 2

95

100

105

110

115

1 4 7 10 13 16 19 22 25 28 31

Balance 3

0

500

1000

1500

1 4 7 10 13 16 19 22 25 28 31

Balance 4


Questions 7: Modelling approaches



Hands-on Case Study: Instructions for Module Assignments

4

Practical Part – Jupyter Notebook


Freely available as part of Anaconda Distribution at: https://www.anaconda.com/distribution/

We use Python 3.7 version.

https://www.anaconda.com/distribution/

Course Assessment


• Please note that you are not limited to methods presented today (and it is more than

welcomed to new approaches)

• Non-Maturity Deposit Modelling

• 1. Case study – you will obtain dataset for modelling balance of non-maturity deposits. Your

task will be to analyze the dataset, split it to train set and test set. With train dataset, you will

develop selected model for non-maturity deposit balances. Then you will take your test dataset

and see how your model performs.

• 2. Outputs – PPT presentation or PDF, summarizing the abovementioned outputs, and scripts

in Python, R or VBA (including the spreadsheet) that were used. Please send these outputs to

[email protected]. Submission deadline: 7 May 2021

• 3. Output presentation to the Bank Board – short (10-15 minutes) presentation about results of

this assessment; the presentation will be delivered online. Presentations will take place during

week starting from 10 May 2021.

mailto:[email protected]

Course Assessment – structure of the presentation


1. Data description and model selection

2. Brief model description

3. How your model performs (make prediction for the fourth year and compare the prediction

with the real values)

4. Summarize your results

Course Assessment – evaluation


• You will present your results to the Bank Board (four of us)

• What will be important for us?

1. The quality of your presentation and deliverables

2. The quality of your model (assumptions, errors, ..)

3. The innovation

Annex + Q&A5

• The mathematical optimization (also known as mathematical programming) forms a cornerstone of quantitativeportfolio optimization; it can be described as a process of looking for the extreme of a function (maximum orminimum).

• Apart from asset allocation, optimization has many uses in quantitative finance – model calibration, financialderivates pricing or asset-liability management.

• Optimization problem can be summarized by three components:

and can be formally written as:

Optimization (1)


Optimization problem component Asset allocation application

Objective function 𝑓(𝑥)Portfolio risk as a function of portfolio

weights, e.g. portfolio volatility 𝑤𝑇Σ𝑤

Vector of control variables 𝑥 Vector of portfolio weights 𝑤

Constraints set ℂ Portfolio weights must add up to 100%

min𝑤∈ℂ

𝑓(𝑥)

𝑠. 𝑡. ℂ: 𝑔𝑖 𝒙 ≤ 0 𝑖 = 1,… , 𝐼 (inequality constraints)ℎ𝑗 𝒙 = 0 𝑗 = 1,… , 𝐽 (equality constraints)

(25)

Useful optimization property: max𝑓 𝑥 = min−𝑓 𝑥

• There are two types of function extremes:

• Local (maxima/minima) – extreme of the function in the neighborhood offeasible solutions.

• Global (maxima/minima) – extreme of the function in the set of all feasiblesolutions.

• The characteristics of the optimization problem (also referred to asmathematical programs) components of the determine its type, e.g.linear objective function with linear constrains form a linearprogram, a quadratic objective function with linear constraints leadsto quadratic program. Individual types of optimization problemshave their properties which can be usually exploited – e.g. in case ofconvex programs all local extremes are also global ones.

• Some of the optimization problems can be solved analytically withthe help of matrix algebra (e.g. portfolio variance minimization withequality constraints) but majority of problems require numerical,iterative algorithms, for example Newton-type methods such assteepest descent.

Optimization (2)


Global maximum

Local maxima

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Predictive Modeling for Market Risk in the Banking Book

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