cppi dr. jörg kienitz treasury otc derivatives – head of quantitative analysis march 2007

CPPI

Dr. Jörg KienitzTreasury OTC Derivatives – Head of Quantitative Analysis

March 2007

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Mathfinance Workshop March 2007Verfasser: Dr. Jörg Kienitz, Treasury OTC Derivate

March 2007Fassung: 1.0

CPPI – The Basics

CPPI is the abbrevation for Constant Proportion Portfolio Insurance

The CPPI mechanism is a rules-based trading strategy. It seeks to maximise returns by way of leveraged exposure to a (portfolio) of risky asset(s) and providing a principal protection.

This takes place in certain risk thresholds. The risks are known as gap risk.

The are many modifications of the basic CPPI rules around.

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CPPI – Vocabulary

Bond Floor

The value of a Zerobond with the same time to maturity as the CPPI strategy. Could also be a coupon bearing bond.

Cushion

The cushion is the difference of the Bond Floor and the current value of the CPPI insured portfolio

Leverage Factor (Multiplier)

The LF is the factor multiplied with the cushion to give the possible amount to be invested in the risky assets. It represents the overnight risk inherent in the risky assets.

Protection Level

This is the amount of principal which should be protected. In classical CPPI the PL = 100%

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CPPI – Vocabulary

Maximum ExposureThe ME of the CPPI is the maximum level to which the capital is invested into the risky assets

Minimum ExposureThe ME of the CPPI is the minimum level to which the capital is invested into the risky asset. For classical CPPI the ME = 0.

Lock InThe Lock-In mechanism allows to lock in an upside already achieved during the lifetime of the CPPI

Deleverage

Deleverage is the event occuring if Cushion = 0. Then the portfolio is only worth the BF

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CPPI – CPPI Mechanics

Principal Protection Reserve

Cushion

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CPPI – CPPI Full Deleverage

Time of Deleverage

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CPPI – Risky Assets

Lock In Event

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CPPI – With Coupons

To achieve periodic payments the basic CPPI strategy could be modified to pay (half-) yearly coupons linked to LIBOR, e.g. LIBOR + 50 bp

Increases the risk of deleveraging, since one takes money out which decreases the cushion periodically!

The coupon is not guaranteed, e.g. would only be paid if the strategy would not deleverage.

A new risk arises, namely coupon shortfall

In our CPPI setting we examine Deleverage Probability, Coupon Shortfall Probability and Return

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CPPI – Risky Assets

The CPPI technique has been successfully applied to numerous asset classes.

• Credit

• Equity

• Funds

• Fixed Income

The observed distributions for each asset class can be considerably different, as for example

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CPPI – Baskets of Risky Assets

For real applications we consider a basket of risky assets, e.g. funds.

This allows to use correlation effects to increase the overall return and to reduce deleverage as well as coupon shorfall probability.

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CPPI – Our Approach

To Do

• Determine the universe of risky assets (mainly qualitative)

• Analysis of the universe using methods from time series analysis (mean, volatility, skew, kurtosis, correlation, etc.)

• Asset Allocation Approach to determine the efficient frontier

• Simulate the CPPI Mechanism for „optimal“ portfolios

The Simplest Setting

• Assume a Gaussian world and determine the mean vector and the covariance matrix

• Compute the Markowitz efficient frontier

• Run a one-factor simulation along the efficient frontier using

Mean basket = sum basket const

Variance basket = sum cov(basket const, basket const)

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CPPI – The Process

Simulate Stochastic Process (FX, Rate, Fund)

Compute Market Effects

Compute CPPI

Summarise Distribution

Deleverage Probability

Coupon ShortFall Probability

Output Statistics

CPPI „optimal“ portfolio

Time Series Returns

-0,0600

-0,0400

-0,0200

0,0000

0,0200

0,0400

0,0600

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

Time

Ret

urn

Asset 1 Asset 2 Asset 3 Asset 4 Asset 5

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CPPI – An Extended Approach

Extend the Models to cover a bright range of asset classes

Since there are many asset classes involved the Gaussian hypothesis is too restrictive

-> Use complex processes (e.g. NIG (Normal Inverse Gaussian) or VG ( Variance Gamma))

Compute the efficient frontier

-> Optimization is complex

Therefore

• We need a method to compute relevant figures from time series data

• We need a method to compute the efficient frontier

• We need a method to simulate fairly complex multidimensional processes

For creating optimization data

For simulating the CPPI strategy

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Tools

• Time Series Analysis

• Methods to determine figures from given historic data

• Optimization

• What is the best suited characterisation of risk?

• Simulation

• Flexible, robust Monte Carlo Engine

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Time Series Analysis

To use more complex stochastic processes we must be able to extract the relevant data to determine the processes parameters out of data.

For Geometric Brownian motion this can be done by computing the mean and the covariance structure using either time series data (for example if no quoted option data is available) or quoted prices (if available).

Therefore, we have to investigate for methods to compute the necessary parameters. Our findings suggest that it is possible for classes of Lévy processes such as variance gamma or normal inverse gaussian.

Use [P 04] as starting point

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Time Series Analysis

ANWWX

We consider the class of generalized hyperbolic distributions, i.e. processes which can be written as

With N a standard multivariate normal, W is a positive random variable independent of N and A is a dxk matrix. and are vectors.

In fact one could show

Both models allow the derivation of parameters out of time series data as well as quoted option prices.

WWNWX ,~|

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Monte Carlo Implementation

Once having the necessary data at hand we need to model the evolution of paths in the considered models. To this end we1 have developed a bunch of loosely coupled classes for simluation purposes:

• Random Number Generator

• Distributions

• Finite Difference Schemes for SDE Discretization

• PayOffs (The CPPI can be modelled as a path-dependent PayOff

• Risk Figures

• Regression

• …

1 joined work with Daniel Duffy from Datasim

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Monte Carlo Implementation – Big Picture

MC DirectorMC Output

MC Path Generator(Mediator)

MC Random(Template)

MC Option

MC PayOff(Factory)

FDM(Visitor)

MC Stats

SDE

MC Mesher

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Monte Carlo Implementation – Path Generation

MC Path Generator(Mediator)

FDM(Visitor)

SDE

MC Mesher

L/L

Numerical Scheme

Euler

…

Predictor Corrector

Milstein

Exact

Equidistant

Dates specific

…

L/NL NL/L

…

M1

…

M1

…

M1

…

M1

NL/NL

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Monte Carlo Implementation – Random Numbers

MC Random(Template)

Distribution

Random Generator

Normal Poisson Multi Normal …

Ran3

Mersenne Twister

Sobol

…

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Monte Carlo Implementation - PayOffs

MC Option

MC PayOff(Factory)

One AssetNon Path Dependent

Multi AssetPath Dependent

Multi AssetNon Path Dependent

One AssetPath Dependent

Call Barrier Spread Altiplano

AsianLookback Spread

Lookback

……

Quanto

BasketCPPI

……

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Monte Carlo Implementation – Design Patterns

We want to achieve efficiency, performance, accuracy, maintainability and interoperability.

Therefore we use C++ and several design patterns among them are:2

• Factory

Define an interface for creating an object, but let derived classes decide which class to instantiate.

The pattern is may better known as virtual construction because it allows to eliminate the need to bind specific classes into the code

• Property Pattern

Creating C++ classes means also to declare data and member functions that operate on data which in general is private. This implies that set/get functionality must be implemented for each object and hard coded data remain a compile-time phenomenom.

Taking this into account we model such data as property sets or idioms which are template classes

with named member data. The member data can in general be heterogenous. This means for example double, vector, classes, etc. 2 For a description of many more design patterns see [D 04] and [D 05]. For the design patterns used for Monte Carlo see [DK 07]

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Monte Carlo Implementation – Example: PayOff Factory

The CPPI can be seen as strategy depending on the taken path of the underlying.

Therefore, we will model it as a (path dependent) PayOff.

We will describe our setting in which the CPPI can be handled.

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Monte Carlo Implementation – Example: PayOff The Base Class

template<class D, class T> class PayOff

{

//Base class from which all payoff classes inherit

//Any instance has to define operator(), clone and the destructor

public:

PayOff(){}; // Constructor

virtual double operator()(D Discount, T Spot) const=0; // Price operators given discounts and Spots

virtual PayOff<D, T>* clone() const=0; // clone

virtual ~PayOff(){} // Destructor

private:

};

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Monte Carlo Implementation – PayOff Classes In Detail

class PayOffCPPI : public PayOff<double, std::vector<double> >{public:

PayOffCPPI();PayOffCPPI(const SimplePropertySet<string, double>& pset);

virtual double operator()(double Discount, std::vector<double> Spot) const;virtual double PayOff_Value(double Discount, std::vector<double> Spot) const;

virtual PayOff<double, std::vector<double> >* clone() const;

virtual ~PayOffCPPI(){}

SimplePropertySet<string, double> PayOffCPPI_Properties;

private:};

All CPPI parameters are stored as properties gathered into PayOffCPPI_Properties set

Construction with given Property Set

CPPI for one asset; For multiple assets just use your favourite matrix class!

For one realized path the value from CPPI strategy is computed

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Monte Carlo Implementation – Example: The Factory Class

template<class D, class T> class PayOffFactory

{

public:

typedef PayOff<D, T>* (*CreatePayOffFunction)(SimplePropertySet<string,double>);

static PayOffFactory<D, T>& Instance();

void RegisterPayOff(std::string, CreatePayOffFunction);

PayOff<D, T>* CreatePayOff(std::string PayOffId, SimplePropertySet<string, double> pset);

~PayOffFactory(){};

private:

std::map<std::string, CreatePayOffFunction> TheCreatorFunctions;

PayOffFactory(){}

PayOffFactory(const PayOffFactory<D, T>&){}

PayOffFactory& operator=(const PayOffFactory<D, T>&){ return *this;}

};

Used to represent spot value(s)

Used to represent discount factors

At runtime make the payoff known

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Monte Carlo Implementation – Example: Implementation of the Factory

template<class D, class T>PayOff<D, T>* PayOffFactory<D, T>::CreatePayOff(string PayOffId, SimplePropertySet<string, double> pset){ map<string, CreatePayOffFunction>::const_iterator i = TheCreatorFunctions.find(PayOffId);

if (i == TheCreatorFunctions.end()) {

std::cout << PayOffId << " is an unknown payoff" << std::endl;return NULL;

}

return (i->second)(pset);}

template<class D, class T>PayOffFactory<D, T>& PayOffFactory<D, T>::Instance(){ static PayOffFactory<D, T> theFactory; return theFactory;}

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Monte Carlo Implementation – Example: PayOff Construction using Factory

class PayOffConstructor{public:

PayOffConstructor(std::string);static PayOff<D, T>* Create(SimplePropertySet<string, double>);

};

template <class D, class T, class P>PayOffConstructor<D,T,P>::PayOffConstructor(std::string id){

PayOffFactory<D, T>& thePayOffFactory = PayOffFactory<D, T>::Instance();thePayOffFactory.RegisterPayOff(id,PayOffConstructor<D,T,P>::Create);

}

template <class D, class T, class P>PayOff<D, T>* PayOffConstructor<D, T, P>::Create(SimplePropertySet<string, double> pset){

return new P(pset);}

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Monte Carlo Implementation – Defining Property Sets

//Define a Simple Property SetSimplePropertySet<string, double> CPPI_Set;

//Define Properties to be collected into CPPI_SetProperty<string, double> Prop_CouponSpread;Property<string, double> Prop_InvestmentLevel;Property<string, double> Prop_Multiplier;

//Initalisation of PropertiesProp_CouponSpread = Property<string, double> („CouponSpread", 0.0085);Prop_Barrier = Property<string, double> („InvestmentLevel", 1);Prop_Barrier = Property<string, double> („Multiplier", 10);

//Adding properties to the setCPPI_Set(Prop_CouponSpread);CPPI_Set(Prop_InvestmentLevel);CPPI_Set(Prop_Multiplier);

Heterogenous properties are possible such as vectors, matrices, etc.

see [D 06] for details

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Monte Carlo Implementation – How to use Property Sets in Coding

//Using the Properties for coding the CPPI strategy

double CouponSpread = PayOffCPPI_Properties.value(„Coupon Spread“);

double MaximumExposure = PayOffCPPI_Properties.value(„MaxExposure“);

Therefore, coding the payoff uses essentially the names of the properties which can be used to model any derivative using payoff description languages and therefore extends the flexibility.

The usage of properties allows flexibility as we see it is easy to model any payoff (time dependent, multi factor, etc.). All that is needed can be put into the Property Set. E.g. if we want to have a barrier option not with one but two barriers simply extend the Property Set.

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Beyond the Basics

Let us come back to the initial problem of creating a „CPPI optimal“ basket

We are now able to use advanced models for simulation. But the optimization step is still missing.

To this end we have to investigate for a method replacing the Markowitz model in the Gaussian setting. The optimal approach would be a distribution free ansatz using time series or simulated time series data directly.

-> Enables us to use short time series

To this end we use CVAR (Expected Shortfall, … ) as a coherent risk measure. The efficient frontier can then be computed by minimizing:

This approach allows for using complicated dynamics, e.g. NIG, VG, Stochastic Vol, Stochastic Vol with jumps for CPPI purposes. See [RU 99] for details.

0;1

)1(

1

1

1;

i

N

ii

N

ii

Tx

xx

yxN

Min Quantile

yield

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Input to Optimization

6 Asset Time Series with 5000 computed returns

Constraints, such that each weight should not exceed 30% and each weight is positive

Constraint, such that the sum of all weights is 1 and weight2 + weight3 < 50%

Views can be added, such as the portfolio managers sees lower / higher returns for the period to be considered

For real applications we use a universe of up to 30 assets!

Asset Returns 100 for each

-0,20

-0,10

0,00

0,10

0,20

0,30

0,40

0,50

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Szenario

Ret

urn

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Output of Optimization

0,3%

0,8%

1,3%

1,8%

2,3%

2,8%

3,3%

3,8%

4,3%

0,0% 1,0% 2,0% 3,0% 4,0% 5,0%

5 % VaR and CVaR

Exp

ecte

d R

eturn

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Output of Optimization

Portfolio Weights

0,000%

5,000%

10,000%

15,000%

20,000%

25,000%

30,000%

35,000%0

,27

8%

0,4

95

%

0,7

12

%

0,9

29

%

1,1

46

%

1,3

63

%

1,5

80

%

1,7

97

%

2,0

14

%

2,2

31

%

2,4

48

%

2,6

65

%

2,8

82

%

3,0

99

%

3,3

16

%

3,5

33

%

3,7

50

%

Return

We

igh

t %

Asset 1 Asset 2 Asset 3 Asset 4 Asset 5 Asset 6

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Output of Monte Carlo Simulation

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115

0,00%

10,00%

20,00%

30,00%

40,00%

50,00%

60,00%

70,00%

80,00%

90,00%

100,00%

Fund Basket Bond Floor Cushion Fund Basket CPPI Allocation

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Output of Monte Carlo Simulation

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13%

IRR CPPI SSD IRR Fund

Coupon Default Probability

0,00% 2,00% 4,00% 6,00% 8,00% 10,00%

1

3

5

7

9

J ahr

Coupon Default

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Literature

[D 04] Duffy, D., „Financial Instrument Pricing Using C++“, Wiley 2004

[D 06] Duffy, D., „Introduction to C++ for Financial Engineers“, Wiley 2006

[DK 06] Duffy, D., Kienitz, J., Monte Carlo Methods in Finance Generic and Efficient MC Solver in C++, Wilmott November 2005

[DK 07] Duffy, D., Kienitz, J., Software Frameworks in Quantitative Finance, Part I

[DK 07] Duffy, D., Kienitz, J., „Monte Carlo Methods in C++“, Wiley forthcoming

[K 07] Kienitz, J., Stochastic Processes in Finance Part I, forthcoming[LS 05] Luciano, E., Schoutens, W., Multivariate Variance Gamma Modelling with Applications in Equity and Credit Risk Derivatives Pricing, ULM, Financial Modelling Workshop, September 2005[P 04] Protassov, R. „EM-based maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions with fixed . Statistics and Computing. Vol 14, issue 1.[RU 99] Rockafellar, T., Urysaev, S., „Optimization of Conditional Value at Risk“, Research report 99-4, Center for Applied Optimization, University of Florida.

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Disclaimer

The usual disclaimers apply. The views expressed in this presentation are solely that of the author and do not of Deutsche Postbank AG.

The author is responsible for any inaccuracies, omissions or errors.

cppi dr. jörg kienitz treasury otc derivatives – head of quantitative analysis march 2007

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