the "checklist" - executive summary
TRANSCRIPT
Advanced Risk and Portfolio Management
’ Advanced Risk and Portfolio Management
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
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
’ The “Checklist” > Executive Summary
1- Risk drivers identification (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
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
2 - 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:
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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3 – Estimation (Example: two stocks)
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
4 – Projection to the horizon (Example: two stocks)
Input Output
risk drivers: log-values
invariants: comp. returns
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5 – Pricing at the horizon (Example: two stocks)
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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6 – Aggregation (Example: two stocks)
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
’ The “Checklist” > Executive Summary
7 – Ex-ante Evaluation (Example: two stocks)
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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8 – Ex-ante Attribution (Example: two stocks)
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:
’ The “Checklist” > Executive Summary
9a – Construction: Portfolio optimization (Example: two stocks)
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 ( )
’ The “Checklist” > Executive Summary
9b-9c – Construction: dynamic allocation (Example: two stocks)
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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10 – Execution (Example: two stocks)
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
’ The “Checklist” > Executive Summary
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
1 - Risk drivers identification (General 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” > Executive Summary
2 - Quest for Invariance: Invariants (General case)
past time series of the risk drivers past time series of the invariants
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
’ The “Checklist” > Executive Summary
2 - Quest for Invariance: Invariants (General case)
past time series of the risk drivers past time series of the invariants
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]
2 - Quest for Invariance: Invariants (Video)
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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
3 – Estimation (General case)
Goal: Estimate the joint distribution of the invariants
joint distribution of the invariants
Input Output
invariants time series, Flexible Probabilities, views
’ The “Checklist” > Executive Summary
3 – Estimation (General 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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3 – Estimation (Video)
Flexible Probabilities: blending time conditioning (exponential decay) with state conditioning (market indicator obtained by smoothing and scoring VIX log-returns) [Play clip]
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3 – Estimation (Video)
Bayesian estimation: Normal-inverse-Wishart posterior shrinks towards the sample distribution (large dataset) or towards the prior distribution (high confidence) [Play clip]
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3 – Estimation (Video)
Random Matrix Theory describes the steepening of the spectrum of the sample covariance due to estimation [Play clip]
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3 – Estimation (Video)
Maximum likelihood estimation with Flexible Probabilities for time series of different length [Play clip]
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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
4 – Projection to the horizon (General case)
distribution of the projected path of the risk drivers
current information
Input Output
’ The “Checklist” > Executive Summary
4 – Projection to the horizon (General case)
Goal: Compute the joint distribution of the projected path of the risk drivers
distribution of the invariants, current information, recovery function
distribution of the projected path of the risk drivers
Input Output
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4 – Projection to the horizon (Video)
[Play clip] Projection of a Brownian motion
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4 – Projection to the horizon (Video)
[Play clip] Projection of a Cauchy process
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5 – Pricing at the horizon (General case)
distribution of the projected path of the risk drivers
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
’ The “Checklist” > Executive Summary
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]
5 – Pricing at the horizon (Video)
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6 – 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.
Operational 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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6 – 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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6 – Aggregation (Video)
Portfolio of options: P&L distribution via the Historical with Flexible Probability approach [Play clip]
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Portfolio’s P&L under the credit simplified regulatory framework [Play clip]
6 – Aggregation (Video)
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7 – Ex-ante Evaluation (General 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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7 – Ex-ante Evaluation (General case)
Classification of risk measures ( )
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8 – Ex-ante Attribution (General 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
’ The “Checklist” > Executive Summary
9a – Construction: Portfolio optimization (General 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
’ The “Checklist” > Executive Summary
9b-9c – Dynamic allocation (General 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) -> Step 9b - portfolio insurance (based on heuristics or on option pricing theory) -> Step 9c
’ The “Checklist” > Executive Summary
Characteristic portfolio strategy based on reversal signals [Play clip]
9b – Dynamic allocation (Video)
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10 – Execution (General 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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10 – Execution (Video)
Volume clustering signal [Play clip]
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