Risk and financial portfolio analytics - A technical Introduction

Risk and Financial Portfolio Analytics: A Technical Introduction

Oleksandr Romanko, Ph.D. Senior Research Analyst, Risk Analytics – Business Analytics, IBM Adjunct Professor, University of Toronto

Toronto SMAC Meetup January 15, 2015

© 2015 IBM Corporation 2

Please note:

IBM Risk Analytics statements regarding its plans, directions, and intent are subject to

change or withdrawal without notice at IBM’s sole discretion.

Information regarding potential future products is intended to outline our general product

direction and it should not be relied on in making a purchasing decision.

The information mentioned regarding potential future products is not a commitment,

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potential future products may not be incorporated into any contract. The development,

release, and timing of any future features or functionality described for our products

remains at our sole discretion.

Performance is based on measurements and projections using standard IBM benchmarks

in a controlled environment. The actual throughput or performance that any user will

experience will vary depending upon many factors, including considerations such as the

amount of multiprogramming in the user's job stream, the I/O configuration, the storage

configuration, and the workload processed. Therefore, no assurance can be given that an

individual user will achieve results similar to those stated here.


About me

Dr. Oleksandr Romanko

Senior Research Analyst, Quantitative Research at Risk Analytics, Business Analytics, IBM, with the company since 2010

Ph.D. in Computer Science from McMaster University

Author of over 10 papers and reports

Adjunct professor at University of Toronto and lecturer at McMaster University

Research areas:

business analytics, operational research, optimization, finance

portfolio optimization, multi-objective optimization

market and credit risk modeling and optimization

numerical methods for risk management

design of numerical algorithms and their software implementation


Making the world work better – pioneering the science

2008

1973 1969

1981


IBM Centennial: 100 Years of Innovation

© 2015 IBM Corporation

Analytics Jobs

Created by: Dennis Buttera


Data science

9


Business Analytics


Predictive Analytics What will happen?

Descriptive Analytics What has happened?

Prescriptive Analytics What should we do?

What is analytics?

Data Insight Action

Decide Analyze

Business Value

11

Analytics is the scientific process of deriving insights from

data in order to make decisions


History of analytics


Movies


Applications of big data analytics

Homeland Security

Finance Smarter Healthcare Multi-channel sales

Telecom

Manufacturing

Traffic Control

Trading Analytics Fraud and Risk

Log Analysis

Search Quality

Retail: Churn, NBO


Cloud


Bluemix

www.bluemix.net


Bluemix


Applied Statistics


What kind of data are we dealing with?

Types of data

• Quantitative

• Categorical (ordered, unordered)

Data collection

• Independent observations (one observation per subject)

• Dependent observations (repeated observation of the same subject, relationships

within groups, relationships over time or space)

Type of data drives the direction of your analysis

• How to plot

• How to summarize

• How to draw inferences and conclusions

• How to issue predictions

19


Quantitative data

Examples: financial return, temperature, age, income

Quick check: “Does it makes sense to calculate an average?”

Appropriate summary statistics:

– Mean and Median

– Standard Deviation

– Percentiles

More advanced predictive methods: Regression, Time Series Analysis, …

Plot your data!

20


Summarizing quantitative data

One-number summaries

– Mean

Average, obtained by summing all observations and dividing by the number of obs.

– Median

The center value, below and above which you will find 50% of the observations.

Summarizing your data with one number may not tell the whole story:

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Median = 19.8 Median = 19.8 Median = 10.5


Flaw of averages

“Plans based on average assumptions are wrong on average”

Average depth 3 ft


“Most observations fall within ±2 standard deviations of the mean.”

Standard deviation

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If the data is normally distributed

95 % of observations

Standard Deviation = 4.2

~95% of observations between 11.4 and 28.2


Descriptive statistics - example

Random sample of 5000 customers of a credit card company

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Amount spent on

primary card last

month

Debt to income

ratio (x100)

N Valid 5000 5000

Missing 0 0

Mean 1683.7340 9.9578

Median 1690.0670 8.8000

Std. Deviation 210.26680 6.42317

Minimum .00 .00

Maximum 2482.72 43.10


Percentiles

Generalizations of the median (50th percentile).

The pth is the data point below which p percent of the observations fall.

Often used to compare a single observation to a general population.

Examples:

– Standardized test scores

If you scored in the 93th percentile, your score was higher than that of 93% of test

takers.

– Finance and risk management

If your portfolio value-at-risk 95% is $10M, your portfolio loss will not exceed $10M

with probability 95%.

25


Percentiles - example

Percentiles can be another way of describing how spread out data values are.

Example: 5-Number Summary

Minimum – 25th percentile – Median – 50th percentile - Maximum

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Amount spent on

primary card last

month

Debt to income

ratio (x100)

Minimum .00 .00

Percentiles

25 1567.4658 5.1250

50 1690.0670 8.8000

75 1814.5430 13.5000

Maximum 2482.72 43.10


Distributions: Normal distribution

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Distributions

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Distributions

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Distributions

Estimate of the probability distribution of global mean temperature resulting

from a doubling of CO2 relative to its pre-industrial value, made from

100000 simulations


Simulation Modeling


Sums of random variables

For any random variable and a constant

Expectation of the sum of two random variables is equal to the sum of

expectations

and, therefore

Example: expected return of a portfolio

For the variance



How to compute the

probability distribution of the

sum of random variables?

We cannot add PDFs or

PMFs

The formula involves non-

trivial integration and is

known as convolution:

Use simulation to evaluate

such complex integrals




Simulation modeling – example 1

We want to invest $1000 in the US stock market for 1 year:

Invest into the S&P 500 market index (index fund)

Value of investment at the end of year 1:

Market return over the time period [0,1) is

Generate scenarios for the market return over the year and compute

decide on the number of scenarios and the set of scenarios for

generate scenarios

use historic scenarios

draw randomly from historic scenarios (bootstrapping)

draw random numbers from the assumed distribution (Monte Carlo)

visualize and analyze the approximate probability distribution of

In our example we assume that the return of the market over the next year

follow Normal distribution



Between 1977 and 2007, S&P 500 returned 8.79% per year on average with a

standard deviation of 14.65%

Generate 100 scenarios for the market return over the next year (draw

100 random numbers from a Normal distribution with mean 8.79% and standard

deviation of 14.65%):

Compute and plot

Number of values 100

Mean $ 1,087.90

Std Deviation $ 146.15

Skewness 0.0034442

Kurtosis 2.871695

Mode $ 1,118.96

5% Perc $ 837.40

95% Perc $ 1,324.00

Minimum $ 708.81

Maximum $ 1,458.52


Simulation modeling – example 1 in Matlab

600 700 800 900 1000 1100 1200 1300 1400 15000

5

10

15

20

25

Value at time 1

Fre

quency

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1600

700

800

900

1000

1100

1200

1300

1400

1500

Time

Valu

e

Simulated Value Paths

v0 = 1000; % initial capital

Ns = 100; % number of scenarios

% Generate Normal random variables

r01 = normrnd(0.0879, 0.1465, Ns, 1);

% Distribution of value at the end of year 1

v1 = (1 + r01) * v0;

% Plot a histogram of the distribution of outcomes for v1

[frequencyCounts, binLocations] = hist(v1, 10);

bar(binLocations, frequencyCounts);

% Plot simulated paths over time

time = 0:1:1;

plot(time,[v0*ones(100,1) v1],'Linewidth',2);


Why use simulation?

Example 1 illustrates very basic Monte Carlo simulation system

Simulation allows us to evaluate (approximately) a function of a random variable

in example 1 the function is simple

given distribution of , in some cases we can compute distribution of in closed

form, e.g., if followed a Normal distribution, then also follows a Normal

distribution with mean and standard deviation

if was not Normally distributed, or if the output variable were a more complex

function of the input variable , it would be difficult and practically impossible to

derive the probability distribution of from the probability distribution of

Other advantages of simulation:

simulation enables visualizing probability distribution resulting from compounding

probability distributions of multiple input variables (example 2)

simulation allows incorporating correlations between input variables (example 3)

simulation is a low-cost tool for checking the effect of changing a strategy on an output

variable of interest (example 4)

Next, we extend example 1 to illustrate such situations



You are planning for retirement and decide to invest in the market for the next

30 years (instead of only the next year as in example 1). Your initial capital is

still

Assume that every year your investment returns from investing into the

S&P 500 will follow a Normal distribution with the mean and standard deviation

as in example 1.

Value of investment after 30 years:

The return over 30 years will depend on the realization of 30 random variables

Observations:

sum of Normal random variables is Normal

here we have multiplication of Normal random variables, is it Normal?



Between 1977 and 2007, S&P 500 returned 8.79% per year on average with a

standard deviation of 14.65%

Simulate 30 columns of 100 observations each of single period returns:

Compute and plot


Mean $ 12,587.62

Std Deviation $ 10,948.39

Skewness 3.349066

Kurtosis 28.24214

Mode $ 4,458.97

5% Perc $ 2,655.55

95% Perc $ 32,481.38

Minimum $ 609.75

Maximum $194,355.00



0 1 2 3 4 5 6

x 104

0

5

10

15

20

25

30

35

40

Value after 30 years

Fre

quency

0 5 10 15 20 25 300

1

2

3

4

5

6x 10

4

Time

Valu

e





r_speriod30 = normrnd(0.0879, 0.1465, Ns, 30);


v30 = v0 * prod(1 + r_speriod30, 2);


[frequencyCounts, binLocations] = hist(v30, 10); bar(binLocations, frequencyCounts);


time = 0:1:30; v_t = v0*ones(Ns,1);

for(t=1:30) v_t = [v_t v0 * prod(1 + r_speriod30(:,1:t), 2)]; end

plot(time,v_t,'Linewidth',2);



0 2 4 6 8 10 12 14

x 104

0

50

100

150

200

250

300

350

400

450

500


Fre

quency

0 5 10 15 20 25 300

2

4

6

8

10

12

14x 10

4

Time

Valu

e





r_speriod30 = normrnd(0.0879, 0.1465, Ns, 30);


v30 = v0 * prod(1 + r_speriod30, 2);


[frequencyCounts, binLocations] = hist(v30, 100); bar(binLocations, frequencyCounts);


time = 0:1:30; v_t = v0*ones(Ns,1);

for(t=1:30) v_t = [v_t v0 * prod(1 + r_speriod30(:,1:t), 2)]; end

plot(time,v_t,'Linewidth',2);



You are planning for retirement and decide to invest in the market for the next

30 years. Your initial capital is

You have an opportunity to invest in stocks and Treasury bonds:

allocate 50% of your capital to the stock market (S&P 500 index fund) today

allocate 50% of your capital to bonds today

Assume that every year your investment returns from investing into the

S&P 500 and Treasury bonds will follow a Normal distribution with the mean

and standard deviation as in example 2 (for S&P 500), mean 4% and standard

deviation 7% for bonds. Assume correlation -0.2 between the stock market and

the Treasury bond market.

Covariance matrix:

Value of investment after 30 years:



Simulate 30 years of 100 observations each of single period correlated returns:

Compute and plot


Mean $ 7,892.80


Skewness 2.921482

Kurtosis 20.48869

Mode $ 5,050.96

5% Perc $ 2,951.82

95% Perc $17,457.43

Minimum $ 1,408.63

Maximum $79,729.34



0 1 2 3 4 5 6 7 8

x 104

0

200

400

600

800

1000

1200


Fre

quency

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8x 10

4

Time

Valu

e




mu = [0.0879; 0.04]; % expected return

sigma = [0.1465^2, -0.0021; -0.0021, 0.07^2]; % covariance matrix

% Generate correlated Normal random variables

stockRet = ones(Ns,1);

bondsRet = ones(Ns,1);

for iYear = 1:30

scenarios = mvnrnd(mu, sigma, Ns);

stockRet = stockRet .* (1 + scenarios(:,1));

bondsRet = bondsRet .* (1 + scenarios(:,2));

end


v30 = 0.5*v0*stockRet + 0.5*v0*bondsRet;



Using scenario generation procedure from example 3 for decision-making

Compare portfolios:

50-50 portfolio allocation in stocks and bonds (Strategy A)

30-70 portfolio allocation in stocks and bonds (Strategy B)

Compute and plot


Mean $ 1,865.13


Skewness 3.506451

Kurtosis 40.18968

Mode $ 687.75

5% Perc $ -254.41

95% Perc $ 6,027.23

Minimum $-1,829.78

Maximum $45,972.08


Mean-Variance Portfolio Selection


Measuring risk and portfolio selection

Consider n assets with random returns:

proportion invested in asset i

exp. return and standard dev. of

the return of asset i

variance-covariance matrix

Portfolio expected return and variance:

Set of admissible portfolios:

Portfolio Return ( )

Pro

ba

bili

ty d

en

sity

0

Variance

(standard deviation)

Mean

return

Portfolio return distribution ( ) is assumed to be Gaussian (Normal)

Consider n assets with random returns:

proportion of total funds invested in asset i

expected return and standard deviation of

the return of asset i

correlation coefficient of i’s and j’s returns

vector of expected returns

variance-covariance matrix (PSD)

Expected return and variance of the resulting portfolio:

Set of admissible portfolios:

Portfolio selection

49


Portfolio selection

A feasible portfolio x is efficient if it has:

maximal expected return among all portfolios with the same variance,

minimum variance among all portfolios with the same expected return.

Mean-variance optimization (Markowitz, 1952):

Alternative formulations

Solving for all the values of V, R, or gives efficient portfolios:


Portfolio selection

Portfolio optimization problem – efficient frontier:


Portfolio selection

Extensions of mean-variance model: introduce transaction costs

Mean-variance portfolio optimization problem – two

objectives:


Portfolio selection

Mean-variance portfolio optimization problem – efficient

frontier and portfolio composition:


Questions


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Risk and financial portfolio analytics - A technical Introduction

Data & Analytics