Public

January 23, 2017

Dr. Grzegorz Goryl PRM

Value of credit risk for financial instruments Risky bond and cds pricing

UBS BSC Krakow, Risk Methodology

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Learning objectives

• Understand the importance of counterparty credit risk.

• Understand the application of Monte Carlo simulation for pricing risky assets.

• Understand the role of credit risk dependency for the valuation of financial instruments.

What I want you to take home ...

• Should we be sure of obtaining contractual cashflows from position in the financial instrument?

• Is there a way to model riskiness of our counterparty?

• …

Questions you may have …

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Table of contents

Learning objectives

Section 1 Introduction UBS 3

Section 2 Why do banks need quants? 9 Section 3 Bonds 13 Section 4 Credit Default Swaps 18

Section 5 Correlation 20 Section 6 Bond with CDS portfolio simulation 24

Section 7 Concluding remarks 27 Section 8 Q&A 29

Introduction UBS Section 1

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Get to know us

We are well-established and thoroughly global

We’re:

• a client-focused financial services firm with a 150-year history

• headquartered in Switzerland

• over 60,000 people in more than 50 countries

• committed to our wealth management businesses and our universal bank in Switzerland, along with our asset management and investment banking businesses.

We are a premiere global financial services company.

Switzerland 35%

Americas 35%

Asia Pacific 13%

Rest of Europe/ Middle East/Africa

8%

United Kingdom 9%

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What we do

Meeting our clients’ diverse needs

Wealth Management and Wealth Management Americas Financial services for wealthy individuals

Asset Management Asset management for institutional and other investors

Investment Bank Client-centered investment banking services

Personal & Corporate Banking for retail, corporate and institutional clients in Switzerland

Corporate Center From operations, legal, IT and risk to HR, finance, communications and branding, Corporate Center functions provide governance and support to all business divisions

We aim to be the world’s leading wealth manager and the premier universal bank in Switzerland, enhanced by an asset manager and investment bank which are world-class in their chosen areas of focus.

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UBS BSC Krakow and Wroclaw

UBS BSCs – planned growth and development

UBS BSC Nashville

UBS BSC Shanghai

UBS BSC Pune

Global Asset Management

Investment Bank

Business Solution Centers (BSCs)

Wealth Mngnt Americas

UBS Wealth Management

UBS Switzerland

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The world’s leading wealth manager

Source: Euromoney Private Banking and Wealth Management Survey 2015

Source: Financial Times Group

USD ~1 trillion of invested assets

USD ~1 billion of adjusted* pre-tax profit

USD ~1 million in revenues per financial advisor

The most productive financial advisors in the industry

Source: Morgan Stanley 2015 European Financials Conference

*See the last page of this document for a description of numbers described as "adjusted"

Asia Pacific

No.1 In 2015, UBS was named "Best Private Bank in Asia" for the third consecutive year.

Americas Globally

CHF ~ 2 trillion of invested assets; more than any other firm

Source: Scorpio Partnership 2015

1 in 2 billionaires has a relationship with UBS

Best Global Private Bank 2013, 2014 and 2015

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The premier universal bank in Switzerland

120k

private

2.5 million

150k wealth management

corporate

We serve Our clients

1/3 households in Switzerland

1/3 wealthy individuals in Switzerland

>90% of the 250 largest Swiss corporates

UBS in Switzerland

biggest private employer

employees branches

Swiss savings

1/6 CHF saved is managed by UBS

Third >21k 300

Why do banks need quants? Section 2

* A quant is a graduate with a quantitative background like mathematics, statistic, physics, economics, …

Why mathematics matters in Banking

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• PD • LGD • Collateral

Valuation

Client On-boarding

• Risk Authority

• CA vs. CO

Credit Decision

• RWA • CoC • EL • Provisions

Capital calculation

• Rating tools

• LGD

Pricing

• Stress figures

• Changes in Risk profile

Monitoring

Several statistical models are used along the full chain of steps in the lending business…

In a simplified word, lending business can be divided in five steps:

As all major Banks, also UBS has a dedicated team (Risk Methodology) of risk modellers with quantitative background like mathematics, statistics, physics, econometrics, economics, …

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Risk Methodology - Mission Statement

Mandate

Build and confirm the models which measure our credit, market, country and operational risks at an individual and aggregated level. In particular, we develop, validate and maintain the measures and models that:

• Support the control of risk taking of UBS

• Provide input to the calculation of UBS’s regulatory capital

Clients Divisional CROs and Credit Officers, Client Advisors, Regulators, Audit (internal and external), Quantitative Risk Controls, Banking Products, ...

Challenges Regulatory requirements, industrialization / automation, Big Data, internal data completeness and quality, IT constrains, …

Setup Existing teams in UBS in Zurich and LDN; Quant hub in UBS in Krakow

Job Profile Mathematicians, Physicists, Statisticians, Econometricians, etc.

Different Risk classes and different models…

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…and two big players

Quantitative Risk Models

Probability of default

models

Loss-given default models

Collateral valuation models

Market Risk VAR models

Operational risk models

Liquidity / Funding models

Stress models

Exposure models

UBS Risk Methodology (RM): • Credit

Methodology Retail & Corporates

• Credit Methodology Lombard

• Credit Methodology FI/Corporates

• Risk Aggregation Frameworks & Analysis

• Market Risk Methodology & Backtesting

UBS Model Risk Management & Control (MRMC): • MRMC Risk

Models

• Valuation Models

• Client Portfolio Models

• IB Risk Control

• Credit Exposure & Portfolio Valuations

• Model Oversight

Bonds Section 3

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Definition

A bond is a debt investment in which an investor loans money to an entity (typically corporate or governmental) which borrows the funds for a defined period of time at a variable or fixed interest rate.

• Zero-coupon bond

• Coupon bond

• Bond value

𝑉𝑉𝑏𝑏(𝑡𝑡) = 𝐶𝐶�𝐷𝐷(𝑡𝑡𝑖𝑖)𝑇𝑇𝑚𝑚

𝑡𝑡𝑖𝑖

+ 𝑁𝑁𝐷𝐷(𝑇𝑇𝑚𝑚)

P

N

Tm 0

P

N

Tm 0

C C C C C

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• Formula for risky bond valuation has to include default time

𝑉𝑉�𝑏𝑏 𝑡𝑡 = 𝜠𝜠 𝐶𝐶 � 𝐷𝐷(𝑡𝑡𝑖𝑖)𝑡𝑡𝑖𝑖≤𝑇𝑇𝑚𝑚

+ 𝑁𝑁𝐷𝐷(𝑇𝑇𝑚𝑚) 𝟏𝟏𝜏𝜏𝑏𝑏>𝑇𝑇𝑚𝑚 + 𝐶𝐶 � 𝐷𝐷(𝑡𝑡𝑖𝑖)𝑡𝑡𝑖𝑖≤𝜏𝜏𝑏𝑏

+ 𝑅𝑅𝑁𝑁𝐷𝐷 𝜏𝜏𝑏𝑏 𝟏𝟏𝜏𝜏𝑏𝑏≤𝑇𝑇𝑚𝑚 ℱ𝒕𝒕

• For simple cases one can apply analytical approach, however, when one wants to value multi-bonds portfolio, esp. with internal correlation, another approach is required – Monte Carlo simulation.

Risky Bond If a bond issuer defaults coupons are paid only until default. After default only part, recovery 𝑅𝑅 of notional is paid.

0

RN

τb

C C C C C C C C C C C C

N

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• Default time 𝜏𝜏 could be described as a random variable

• As such has a cumulated distribution probability:

𝐹𝐹 𝑡𝑡 = 𝑃𝑃 𝜏𝜏 < 𝑡𝑡

• Usually it is defined in pair with survival probability 𝑄𝑄(𝑡𝑡) – probability that company survive to time 𝜏𝜏:

𝐹𝐹 𝑡𝑡 = 𝑃𝑃 𝜏𝜏 < 𝑡𝑡 = 1 − 𝑄𝑄(𝑡𝑡)

• One may describe survival probability with so called constant hazard rate model:

𝑄𝑄 𝑡𝑡 = 𝑒𝑒−ℎ𝑡𝑡

Default time and survival probability

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• We know that if we have random variable 𝑋𝑋 with cumulated distribution probability function 𝐹𝐹 𝑥𝑥 , then random variable defined as 𝐹𝐹 𝑋𝑋 has the uniform distribution in the 0,1 interval.

• When we can draw uniformly distributed random variable 𝑈𝑈, like with the usage RAND() Excel function, we can draw values distributed according to 𝐹𝐹 𝑥𝑥 :

𝑋𝑋 = 𝐹𝐹−1 𝑈𝑈

• How the formula for default time in the constant hazard rate model would look like?

• Monte Carlo valuation:

1. Generate uniformly distributed random value 𝑢𝑢

2. Calculate default time 𝜏𝜏 = 𝐹𝐹−1 𝑢𝑢

3. Calculate sum of PV of cashflows untill min (𝑇𝑇𝑇𝑇, 𝜏𝜏)

4. Repeat 1-3 points several times and take average of values from point 3

Default time simulation Default time could be simulated from uniformly distributed random variable.

Credit Default Swaps Section 4

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𝑉𝑉𝐶𝐶𝐶𝐶𝐶𝐶(𝑡𝑡) = 𝜠𝜠 (1 − 𝑅𝑅)𝐷𝐷(𝜏𝜏𝑏𝑏) − 𝑠𝑠𝐶𝐶𝐶𝐶𝐶𝐶 � 𝐷𝐷(𝑡𝑡𝑖𝑖)𝑡𝑡𝑖𝑖≤𝑇𝑇𝑚𝑚

𝟏𝟏𝜏𝜏𝑏𝑏≤𝑇𝑇𝑚𝑚 − 𝑠𝑠𝐶𝐶𝐶𝐶𝐶𝐶 � 𝐷𝐷(𝑡𝑡𝑖𝑖)𝑡𝑡𝑖𝑖≤𝑇𝑇𝑚𝑚

𝟏𝟏𝜏𝜏𝑏𝑏>𝑇𝑇𝑚𝑚 ℱ𝑡𝑡

Hedging against credit risk with CDS A credit default swap is a particular type of swap designed to transfer the credit exposure between two parties. In a credit default swap, the buyer of the swap makes payments to the swap’s seller up until the maturity date of a contract. Payments are agreed as percentage of notional. In return, the seller agrees that, in the event that the debt issuer defaults or experiences another credit event, the seller will pay the buyer lost percentage of debt during the default.

1-R

Tm 0 τ s s s s s s s s s s s s

Correlation Section 5

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Risky bond and risky CDS In the real world market participant depends on each other. CDS issuers are not risk-free, and even could be correlated with the reference entity.

• If bond issuer defaults coupons are paid only until default. After default only part, recovery 𝑅𝑅, of notional is paid

• In case of cds issuer's default payment of premium is stopped and cds buyer loses protection

Tm 0

RN

τb

C C C C C C C C C C C C

N

1-R

Tm 0 τb τCDS s s s s s s s s s s s s

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• Marginal cumulated distribution probability for bond issuer default time 𝜏𝜏𝑏𝑏 𝐹𝐹𝑏𝑏 𝑡𝑡 = 𝑃𝑃 𝜏𝜏𝑏𝑏 < 𝑡𝑡 = 1 − 𝑒𝑒−ℎ𝑏𝑏𝑡𝑡

• Marginal cumulated distribution probability for bond issuer default time 𝜏𝜏𝐶𝐶𝐶𝐶𝐶𝐶 𝐹𝐹𝐶𝐶𝐶𝐶𝐶𝐶 𝑡𝑡 = 𝑃𝑃 𝜏𝜏𝐶𝐶𝐶𝐶𝐶𝐶 < 𝑡𝑡 = 1 − 𝑒𝑒−ℎ𝐶𝐶𝐶𝐶𝐶𝐶𝑡𝑡

• A copula is a multivariate probability distribution for which the marginal probability distribution of each variable is uniform

𝐶𝐶 𝑢𝑢1,𝑢𝑢2 = 𝑃𝑃 𝑈𝑈1 ≤ 𝑢𝑢1,𝑈𝑈2 ≤ 𝑢𝑢2

• Given univariate marginal distribution functions 𝐹𝐹1 𝑡𝑡1 , 𝐹𝐹2 𝑡𝑡2 the function any multivariate distribution function 𝐹𝐹 can be written in the form of copula function 𝐶𝐶

𝐹𝐹 𝑡𝑡1, 𝑡𝑡2 = 𝐶𝐶 𝐹𝐹1 𝑡𝑡1 ,𝐹𝐹2 𝑡𝑡2

• With usage of univariate normal distribution function Φ and bivariate normal distribution Φ𝜌𝜌 with correlation 𝜌𝜌 one may define

𝐶𝐶 𝑢𝑢1, 𝑢𝑢2 = Φ𝜌𝜌 Φ−1(𝑢𝑢1),Φ−1(𝑢𝑢2)

Joint distribution and Gaussian copula

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• Specifying 𝑢𝑢1 = 𝐹𝐹𝑏𝑏 𝑡𝑡1 and 𝑢𝑢2 = 𝐹𝐹𝐶𝐶𝐶𝐶𝐶𝐶 𝑡𝑡2 one gets bivariate distribution function 𝐹𝐹 with marginal 𝐹𝐹𝑏𝑏 and 𝐹𝐹𝐶𝐶𝐶𝐶𝐶𝐶

𝐹𝐹 𝑡𝑡1, 𝑡𝑡2 = Φ𝜌𝜌 Φ−1 𝐹𝐹𝑏𝑏 𝑡𝑡1 ,Φ−1 𝐹𝐹𝐶𝐶𝐶𝐶𝐶𝐶 𝑡𝑡2

• Having in this way specified joint distribution for default times 𝜏𝜏𝑏𝑏 and 𝜏𝜏𝐶𝐶𝐶𝐶𝐶𝐶 one can easily generate correlated defaults

• Let 𝑍𝑍𝑏𝑏 and 𝑍𝑍𝐶𝐶𝐶𝐶𝐶𝐶 be normal random variables with joint normal distribution Φ𝜌𝜌 and 𝑍𝑍𝑏𝑏 = Φ−1 𝐹𝐹𝑏𝑏 𝜏𝜏𝑏𝑏 with 𝑍𝑍𝐶𝐶𝐶𝐶𝐶𝐶 = Φ−1 𝐹𝐹𝐶𝐶𝐶𝐶𝐶𝐶 𝜏𝜏𝐶𝐶𝐶𝐶𝐶𝐶

• By definition of 𝐹𝐹 𝑡𝑡1, 𝑡𝑡2 , 𝜏𝜏𝑏𝑏 and 𝜏𝜏𝐶𝐶𝐶𝐶𝐶𝐶 have joint distribution 𝐹𝐹 with marginal distribution functions 𝐹𝐹𝑏𝑏 and 𝐹𝐹𝐶𝐶𝐶𝐶𝐶𝐶

• Therefore, in order to simulate correlated default times one need to calculate

𝜏𝜏𝑏𝑏 = 𝐹𝐹𝑏𝑏−1 Φ 𝑍𝑍𝑏𝑏 = −𝑙𝑙𝑙𝑙 1 −Φ 𝑍𝑍𝑏𝑏

ℎ𝑏𝑏

𝜏𝜏𝐶𝐶𝐶𝐶𝐶𝐶 = 𝐹𝐹𝐶𝐶𝐶𝐶𝐶𝐶−1 Φ 𝑍𝑍𝐶𝐶𝐶𝐶𝐶𝐶 = −𝑙𝑙𝑙𝑙 1 − Φ 𝑍𝑍𝐶𝐶𝐶𝐶𝐶𝐶

ℎ𝐶𝐶𝐶𝐶𝐶𝐶

Gaussian copula

Bond with CDS portfolio simulation Section 6

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• Simulations inputs

– Time to maturity 𝑇𝑇𝑚𝑚 = 10

– Risk free interests rate 𝑟𝑟 = 2%

– Hazard rate for the bond issuer ℎ𝑏𝑏 = 100𝑏𝑏𝑏𝑏𝑠𝑠 – Recovery rate for the bond and CDS issuers 𝑅𝑅𝑏𝑏 = 𝑅𝑅𝐶𝐶𝐶𝐶𝐶𝐶 = 0

– 106 Monte Carlo simulations

– CDS spread and bond coupons 𝑠𝑠𝑏𝑏 = 𝑠𝑠𝐶𝐶𝐶𝐶𝐶𝐶 = ℎ𝑏𝑏

• Simulations variables 𝜌𝜌 and ℎ𝐶𝐶𝐶𝐶𝐶𝐶

• Values from inputs

– Risk-free bond 𝑉𝑉𝑏𝑏(0) = 0.9094 = 0.0906 + 0.8187

– Risk free CDS 𝑉𝑉𝐶𝐶𝐶𝐶𝐶𝐶(0) = 0

• Calculated from simulations risky values 𝑉𝑉�𝑏𝑏 0 , 𝑉𝑉�𝐶𝐶𝐶𝐶𝐶𝐶 0 with portfolio 1% quantile

Simulation parameters Portfolio of bond and cds has been simulated with the assumption of correlation between default time.

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Results Taking into account the correlation effect results in a non-trivial behavior of value of portfolio even consisting of simple financial instruments.

Concluding remarks Section 7

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Conclusion

1. Even for a simple financial instrument like bond, there exists risk of defaulting our counterparty.

2. One way to hedge against of counterparty default is to buy protection via CDS contract.

3. Default time of our counterparty could be described as a random variable.

4. One may model credit risk in financial instrument using for example constant hazard rate model.

5. Pricing of value of credit risk could be done analytically (for simple cases) or using Monte Carlo simulations (esp. when one wants dependency between counterparties to be considered).

6. Dependency in credit risk world could be modelled using application of copulas concepts.

7. Correlation change the risk profile for a portfolio of financial instruments.

Q&A Section 8

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Questions? Asking is the key to a successful career

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Contact information

Dr. Grzegorz Goryl PRM Risk Methodology Tel. +48 12334 6528 grzegorz.goryl@ubs.com ubs.com/polandcareers