introduction to "the checklist" - ten steps for advanced risk and portfolio management

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Introduction to:

The “Checklist”

Ten Steps for Advanced Risk and Portfolio Management

The “Checklist” – Introduction

Last update: 31 July 2016

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Advanced Risk and Portfolio Management

The “Checklist”

Advanced Risk and Portfolio Management

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The “Checklist”

Summary

The Checklist is a holistic ten-step approach to risk and portfolio management that applies i) across all asset classes; ii) to Asset Management, Banking and Insurance; iii) at portfolio and at enterprise level

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The Checklist: general framework

The “Checklist”

General framework

current time investment horizon

Data P&L

Goal: manage risk and optimize performance of a portfolio between the current time and a future investment horizon . To perform our tasks, we have access to data, cumulated over time up to

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1a - Quest for Invariance: Risk drivers (Example: two stocks)

i) Log-values follow (approximately) random walks over one-day steps ii) Log-values determine the P&L of the two stocks

The risk drivers for stocks are the log- (adjusted) values

The “Checklist”

Step 1a – Quest for invariance: risk drivers (E)

time series of daily log-values time series of daily values

Input Output

Goal: Determine risk drivers for the two stocks under consideration

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time series of daily comp. returns time series of daily log-values

1b - Quest for Invariance: Invariants (Example: two stocks)

i) Since the log-values follow (approximately) a random walk, their increments, i.e. the compounded returns, are (approximately) i.i.d. across time

ii) The compounded returns determine the evolution of the log-values (risk drivers)

The invariants for stocks are the daily compounded returns:

The “Checklist”

Step 1b – Quest for invariance: invariants (E)

Goal: Identify the invariants (i.i.d. variables) from the time series analysis of the risk drivers (log-values)

Input Output

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2 – Estimation (Example: two stocks)

The “Checklist”

Step 2 – Estimation (E)

Let us assume that the distribution of the invariants is bivariate normal: Since the compounded returns are invariants, and thus i.i.d., we can apply the Law of Large Numbers, and the expectation and covariance matrix can be estimated from the invariants realizations by the sample mean and sample covariance matrix

Goal: Estimate the joint distribution of the daily compounded returns of the two stocks

time series of daily comp. returns estimated normal distrib. for daily comp. returns

Input Output

invariants: compounded

returns

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By applying the random walk recovery function recursively, we can express the risk drivers at the investment horizon as

From the distribution of the invariants, we obtain the distribution of the risk drivers at the investment horizon, which is jointly normal

where (days).

Goal: Compute the distribution of the risk drivers (log-values) at the horizon with days.

distribution of the risk drivers at the horizon

comp returns distr. random walk current log-value

3 – Projection to the horizon (Example: two stocks)

The “Checklist”

Step 3 – Projection to the horizon (E)

Input Output

risk drivers: log-values

invariants: comp. returns

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4 – Pricing at the horizon (Example: two stocks)

The “Checklist”

Step 4 – Pricing at the horizon (E)

distribution of the risk drivers at the horizon

Goal: Compute the distribution of the ex-ante P&L’s of the stocks

distribution of the ex-ante P&L’s (1st order Taylor approx)

The values of the stocks at the horizon (with days) can be written as

The P&L’s read

Starting from the joint normal distribution of the risk drivers we obtain that the joint distribution of the P&L’s is normal with

first order Taylor approx

Input Output

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5 – Aggregation (Example: two stocks)

The “Checklist”

Step 5 – Aggregation (E)

distribution of the ex-ante P&L’s

holdings

distribution of the portfolio ex-ante P&L

Goal: Compute the distribution of the portfolio ex-ante P&L

Given the holdings (number of shares) in the two stocks: the portfolio ex-ante P&L reads Starting from the joint normal distribution of the stocks P&L’s we obtain that the portfolio ex-ante P&L is normal with

Input Output

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6 – Ex-ante Evaluation (Example: two stocks)

The “Checklist”

Step 6 – Ex-ante Evaluation (E)

distribution of the portfolio ex-ante P&L

Goal: Evaluate ex-ante the portfolio, by computing its ex-ante volatility

Let us assume that, as investors, we evaluate allocations based solely on volatility, represented by the standard deviation, without any concern for the expected returns. The satisfaction is the opposite of volatility of the ex-ante portfolio P&L distribution:

satisfaction/risk associated to the portfolio

Input Output

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7 – Ex-ante Attribution (Example: two stocks)

The “Checklist”

Step 7 – Ex-ante Attribution (E)

Goal: i) Linearly attribute the portfolio ex-ante P&L to the S&P500 + a residual; ii) Additively attribute the volatility of the portfolio’s P&L to S&P500 and residual

Factor: return of the S&P500

Exposure:

Residual:

Risk attribution:

joint distribution of ex-ante P&L and factors

exposures joint distribution: risk contributions

Input Output

contribution from the S&P500:

contribution from the residual:

Attribution model:

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8 – Construction (Example: two stocks)

The “Checklist”

Step 8 – Construction (E)

Goal: find optimal portfolio to hedge second stock

optimal allocation P&L’s distribution satisfaction/risk constraints

Input Output

first order condition

We compute the minimum-variance (hence, minimum volatility) portfolio on the efficient frontier that is long one share of the second stock ( )

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9 – Execution (Example: two stocks)

The “Checklist”

Step 9 – Execution (E)

We apply the simplest execution algorithm, namely "trading at all costs". This approach disregards any information on the market or the portfolio and delivers immediately the desired final allocation by depleting the cash reserve.

Goal: Achieve the optimal allocation by rebalancing the current allocation

current allocation optimal allocation

market order amount, trade price

Input Output

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10 – Dynamic allocation (Example: two stocks)

The “Checklist”

Step 10 –Dynamic allocation (E)

If the investment went up over the last period, we re-invest the proceeds in the same allocation; if the investment lost value, we liquidate 20% of our portfolio and keep the proceeds in cash.

Goal: Decide policy to rebalance stocks week after week

Input Output

one-period allocation decision and execution

process: Steps 1- 9

dynamic policy

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The Checklist: General case and videos

The “Checklist”

The Checklist: general case and videos

The ten steps of the Checklist appear trivial in the over-simplified two stocks example discussed so far. However, each of them is actually complex and fraught with pitfalls, and needs a deep discussion. An overview of the general key concepts for each step is given in the following. Furthermore, we point toward multiple advanced approaches to address the non-trivial practical problems of real-life risk modeling, with the support of a few videos based on applications to real data.

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Risk drivers are random variables that: i) follow a homogeneous pattern across time ( random walk)

ii) determine the joint P&L generated by the instruments

1a - Quest for Invariance: Risk drivers (General case)

The “Checklist”

Step 1a – Quest for invariance: risk drivers (Gen. case)

risk drivers path

information available at time t pricing

function

P&L of the n-th instrument

past time series of the risk drivers raw data

Input Output

Goal: Determine risk drivers for all the financial instrument under consideration

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The “Checklist”

Step 1a – Quest for invariance: risk drivers (Gen. case)

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1b - Quest for Invariance: Invariants (General case)

past time series of the risk drivers past time series of the invariants

The “Checklist”

Step 1b – Quest for invariance: invariants (Gen. case)

Goal: Identify the invariants for the risk drivers from the time series

Input Output

current information “next-step” function

The invariants are random variables that

i) are independent and identically distributed (i.i.d.) across different time steps

ii) determine the evolution of the risk drivers

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1b - Quest for Invariance: Invariants (General case)

past time series of the risk drivers past time series of the invariants

The “Checklist”

Step 1b – Quest for invariance: invariants (Gen. case)

Input Output

Goal: Identify the invariants for the risk drivers from the time series

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Invariance test on stock daily compounded returns

[Play clip on youtube]

1b - Quest for Invariance: Invariants (Video)

The “Checklist”

Step 1b – Quest for invariance: invariants (V)

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https://youtu.be/WA_a5_K3528





Simple estimation approaches fit a distribution to the past realizations of the invariants

These approaches can be improved by using Flexible Probabilities (FP), i.e. by associating

specific weights with the past realizations of the invariants .

Flexible Probabilities can be specified via - Time conditioning (window/exponential decay) - State conditioning

2 – Estimation (General case)

Goal: Estimate the joint distribution of the invariants

The “Checklist”

Step 2 – Estimation (Gen. case)

joint distribution of the invariants

Input Output

invariants time series, Flexible Probabilities, views

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2 – Estimation (General case)

The “Checklist”

Step 2 – Estimation (Gen. case)

joint distribution of the invariants

Input Output

More advanced techniques also process other sources of information (“views” )

Goal: Estimate the joint distribution of the invariants

time series of invariants, Flexible Probabilites, views

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2 – Estimation (Video)

The “Checklist”

Step 2 – Estimation (V)

Flexible Probabilities: blending time conditioning (exponential decay) with state conditioning (market indicator obtained by smoothing and scoring VIX log-returns) [Play clip on youtube]

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https://youtu.be/OxuF8Ye9wCI

https://youtu.be/tEIrE3Nv1Wc

https://youtu.be/tEIrE3Nv1Wc

https://youtu.be/OxuF8Ye9wCI




The “Checklist”


Bayesian estimation: Normal-inverse-Wishart posterior shrinks towards the sample distribution (large dataset) or towards the prior distribution (high confidence) [Play clip on youtube]

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https://youtu.be/ZiHKmDHnyx0






The “Checklist”


Random Matrix Theory describes the steepening of the spectrum of the sample covariance due to estimation [Play clip on youtube]

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https://youtu.be/F0GkreFgZao

https://youtu.be/F0GkreFgZao





The “Checklist”


Maximum likelihood estimation with Flexible Probabilities for time series of different length [Play clip on youtube]

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https://youtu.be/uXbRtIBN_8I

https://youtu.be/uXbRtIBN_8I



Time series analysis (Step 1,2)

Goal: Compute the joint distribution of the projected path of the risk drivers

estimation interval

Projection (Step 3)

distribution of the invariants, current information, projection function

path of the risk drivers “projection” function

(iterated “next-step” function)

path of the invariants

3 – Projection to the horizon (General case)

The “Checklist”

Step 3 – Projection to the horizon (Gen. case)

distribution of the projected path of the risk drivers

current information

Input Output

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3 – Projection to the horizon (General case)

The “Checklist”

Step 3 – Projection to the horizon (Gen. case)

Goal: Compute the joint distribution of the projected path of the risk drivers

distribution of the invariants, current information, recovery function


Input Output

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3 – Projection to the horizon (Video)

The “Checklist”

Step 3 – Projection to the horizon (V)

[Play clip on youtube] Projection of a Brownian motion

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https://youtu.be/lMFMqLrfFAI

https://youtu.be/lMFMqLrfFAI



3 – Projection to the horizon (Video)

The “Checklist”

Step 3 – Projection to the horizon (V)

[Play clip on youtube] Projection of a Cauchy process

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https://youtu.be/eK9rQfMc4Vg

https://youtu.be/eK9rQfMc4Vg



4 – Pricing at the horizon (General case)

The “Checklist”

Step 4 – Pricing at the horizon (Gen. case)


Goal: Obtain the distribution of the ex-ante P&L’s of the instruments

distribution of the ex-ante P&L’s

As seen in Step 1a, each P&L is a deterministic function of the paths of the risk drivers and of the current information (terms and conditions, current market quotes...)

Input Output

Given the distribution of the paths , we obtain the joint P&L’s distribution as follows

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P&L of a stock at the horizon with Taylor first and second order approximations superimposed (the log-value follows a Brownian motion)

[Play clip on youtube]

4 – Pricing at the horizon (Video)

The “Checklist”

Step 4 – Pricing at the horizon (V)

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AL???

https://youtu.be/Ha__QZXs81k

https://youtu.be/Ha__QZXs81k

AL???



The “Checklist”

Step 5 – Aggregation (Gen. case)

5 – Aggregation (General case)

Goal: Compute the distribution of the portfolio ex-ante performance

Input Output

P&L distrib. holdings (benchmark holdings )

We consider a portfolio with holdings (units)

First, we compute the current value of the portfolio

Counterparty valuation adjustments and liquidity adjustments may be required.

Next, we compute the distribution of the ex-ante performance

instruments ex-ante P&L’s

Standardized holdings (portfolio weights or relative weights): affine functions of

portfolio and benchmark holdings

Mkt/credit/oper. P&L distrib.

Ex-ante performance distrib.

Portfolio value

The operational P&L

can be modeled with the same

techniques as credit, and it is

assumed independent of the

market and credit P&L.

Oerational component The market and credit ex-ante performance is the excess

P&L or the excess return with respect to a benchmark,

and can be written as

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The “Checklist”

Step 5 – Aggregation (Gen. case)

5 – Aggregation (General case)

Enterprise risk management relies on the same tools:

P&L distrib. holdings (benchmark holdings )

Input Output

Given we compute the distribution of the market and credit ex-ante performance

Mkt/credit/oper. P&L distrib.

Ex-ante performance distrib.

Portfolio value

Bank, Insurer, Asset management company

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The “Checklist”

Step 5 – Aggregation (V)

5 – Aggregation (Video)

Portfolio of options: P&L distribution via the Historical with Flexible Probability approach [Play clip on youtube]

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https://youtu.be/K2CbTvnijns

https://youtu.be/K2CbTvnijns



Portfolio’s P&L under the credit simplified regulatory framework [Play clip on youtube]

The “Checklist”

Step 5 – Aggregation (V)

5 – Aggregation (Video)

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https://youtu.be/9SpSFjXqPF4

https://youtu.be/9SpSFjXqPF4



6 – Ex-ante Evaluation (General case)

The “Checklist”

Step 6 – Ex-ante Evaluation (Gen. case)

Ex-ante performance distribution

Goal: Assess the portfolio performance by evaluating its summary risk statistics

To assess the goodness of the portfolio we summarize the corresponding ex-ante performance distribution with an index of satisfaction

or, equivalently, of risk:

Given the distribution of the ex-ante performance we can compute

satisfaction/risk associated to the portfolio

Input Output

expected utility/certainty equivalent: mean-variance, higher moments, prospect theory

spectral/distortion: VaR (economic capital), CVaR, Wang

non-dimensional ratios: Sharpe, Sortino, Omega and Kappa ratios

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6 – Ex-ante Evaluation (General case)

The “Checklist”

Step 6 – Ex-ante Evaluation (Gen. case)

Classification of risk measures ( )

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7 – Ex-ante Attribution (General case)

The “Checklist”

Step 7 – Ex-ante Attribution (Gen. case)

joint distribution of ex-ante performance and factors

exposures joint distribution risk contributions

Goal: i) Linearly attribute the portfolio ex-ante performance to risk factors + a residual; ii) Additively attribute the risk/satisfaction index to the factors and the residual

Attribution model:

portfolio-specific exposures factors

residual

The exposures (and residual) can be obtained - bottom up: aggregating factor models for the single instruments - top down (Factors on demand): tailoring the attribution model to the portfolio

Satisfaction/risk attribution: contributions from factors

and residual (k=0)

Input Output

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8 – Construction (General case)

The “Checklist”

Step 8 – Construction (Gen. case)

P&L’s distribution satisfaction/risk constraints

Goal: Find optimal holdings that maximize satisfaction, subject to investment constraints

optimal allocation

Optimization problem:

Investment constraints on allocation, budget, leverage, etc.

Quasi optimal solution can be obtain via a 2 step mean-variance approach

1) Efficient mean-variance frontier

2) Satisfaction maximization

optimal allocation

Input Output

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9 – Execution (General case)

The “Checklist”

Step 9 – Execution (Gen. case)

Goal: Achieve the optimal allocation by rebalancing the current allocation

current allocation optimal allocation

orders’ amounts, trades times/prices

Input Output

To optimize the execution strategy and achieve the optimal allocation , the following steps are applied recursively: 1. Order scheduling: market impact model is chosen and the trading P&L optimized. At

time t, the “parent” order is split into “child” orders with expected execution times.

2. Order placement: the first child order is executed by processing real time order book

information and market signals (trade autocorrelation, order imbalance, volume clustering,...)

3. Order routing [optional]: limit and market orders are split across different trading venues

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9 – Execution (Video)

The “Checklist”

Step 9 – Execution (V)

Volume clustering signal [Play clip on youtube]

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https://youtu.be/PQ9vd1Lx9qM

https://youtu.be/PQ9vd1Lx9qM



10 – Dynamic allocation (General case)

The “Checklist”

Step 10 – Dynamic allocation (Gen. case)

Goal: Sequence one-period target allocations and respective executions, according to a dynamic policy

dynamic policy one-period allocation decision and execution

process: Steps 1- 9

Input Output

A dynamic allocation is a sequence of portfolio allocations defined in terms of the one-period holdings which are held constant over the period . The key to implement a dynamic allocation is the existence of an underlying allocation policy, i.e. a function of the information available at time , which defines the respective one-period allocation Examples of dynamic allocations: - systematic strategies (based on signals) - portfolio insurance (based on heuristics or on option pricing theory)

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Characteristic portfolio strategy based on reversal signals [Play clip on youtube]

10 – Dynamic allocation (Video)

The “Checklist”

Step 10 – Dynamic allocation (V)

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https://youtu.be/RHHrAgrM23c

https://youtu.be/RHHrAgrM23c



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introduction to "the checklist" - ten steps for advanced risk and portfolio management

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