Risk under parameter uncertainty and price movement

Risk under
Parameter Uncertainty &
Price Movement
FRAGILITY–LATENT RISKS & UNSTABLE HEDGES–MADE VISIBLE
ANISH R. SHAH, CFA ANISHRS@INVESTMENTGRADEMODELING.COM
MAR 8, 2022
ANISHRS@INVESTMENTGRADEMODELING.COM
© 2022 ANISH SHAH

Two slide overview (1/2)
1. Risk decomposition is a tool to understand portfolio’s net risk
2. Net risk = what remains after hedges cancel
For example, in market neutral, longs – shorts = hedged market effect of 0 + other risks
3. Hedged risks are often bigger (before canceling) than the net
Imagine the amount of market effect in each side of long/short
4. Numbers are inexact
→ ‘hedged’ risks aren’t so contribute to the net
© 2022 ANISH SHAH

Two slide overview (2/2)
5. How can numbers be off? Two ways
Parameter uncertainty from estimation error
Dynamic portfolio composition over the horizon of interest
Price movement alters position weights
Hedges hold for an instant, fluctuate as weights shift
6. What to do?
Regard everything as uncertain
Work from estimates of mean and variance instead of fixed values
Parameters – center and spread, e.g., as generated during inference/forecasting
Weights – distribution of future portfolio weights over horizon
7. Risk decomposition reports expected value ± standard deviation
Reveals latent fragility
© 2022 ANISH SHAH

Especially valuable for
1. Leverage
2. Concentrated positions
3. Alternative/illiquid/hazily-modeled assets
4. Optimized portfolios
5. Infrequently rebalanced portfolios
© 2022 ANISH SHAH

Mostly conceptual presentation
Math is in the paper
Shah, A. (2020). Portfolio risks under estimation uncertainty and price movement.
http://ssrn.com/abstract=3774239
Criticism and questions: AnishRS@InvestmentGradeModeling.com
© 2022 ANISH SHAH

Risk decomposition
S&P500 volatility by factors
Factor Exposure Risk contribution
Beta 1.03 22.77
Mkt Cap 2.76 3.15
Relative Strength 0.46 0.92
Price Volatility -0.38 0.74
⁞
Dividend Yield 0.18 -0.08
Stock Specific 0.25
28.74
© 2022 ANISH SHAH
S&P500 volatility by sector buckets
Sector Wt Risk contribution
Information Technology 32.0 9.89
Consumer Discretionary 16.3 5.16
Financials 14.8 4.10
Health Care 10.9 2.70
Consumer Staples 7.8 1.88
⁞
Telecom 2.4 0.43
100 28.74
Breaks portfolio’s volatility (in variance or std dev) into sources

Risk contributions geometrically
Menchero and Davis (2011). Risk
contribution is exposure times
volatility times correlation
Risks = directions (like north-south,
east-west)
Standard deviation = length
An investment (travel leg) is a
combination of risks
Portfolio = sum of investments
Risk contribution = length in the
direction of the portfolio
Movement in other directions
nets to zero
© 2022 ANISH SHAH
ATL
PHX
DEN
BOS

What can possibly go wrong?
Risk contributions are calculated from
◦ Covariance between securities
◦ Portfolio weights
Ingredients are inexact
◦ Risks that appear neutralized aren’t to some degree
Moreover, the conventional view considers only the net
◦ The magnitude of components involved doesn’t matter
e.g., Long beta=1 stock, short another beta=1 and
Long beta=3 stock, short another beta=3 appear the same, but…
◦ the second has much more error and instability, contains large hidden risks
© 2022 ANISH SHAH
← estimated with error
← fluctuate with prices

How things go wrong in pictures
Imagine two directions of risk – market and
chicken feed – and investing long-short
◦ Beta hedged → no market in net portfolio →
no market in risk contributions
What if beta – estimated, not known – isn’t
perfectly hedged?
◦ The large market components from each side,
mismatched, drive the risk of the net
© 2022 ANISH SHAH
market
cf long
short
net
market
cf long
short
net
Perceived-as-hedged directions vanish regardless of sensitivity to underlying offsetting risks
◦ Imagine looking at the equity of a levered company without separately seeing the debt
◦ Hidden unstable risks at both the individual investment and portfolio levels
Optimization finds perceived hedges
market
cf long
short
net

Effect of price movement
Stability of hedges – hedged today doesn’t mean hedged tomorrow
Stylized example on two beta=1 portfolios:
Portfolio A: 50% beta=1 stock + 50% another beta=1 stock
◦ Portfolio maintains beta=1 regardless of price changes
Portfolio B: 50% cash + 50% beta=2 stock
◦ Market rises 10%. Portfolio is 45% cash + 55% stock, has beta=1.1
◦ Market falls 10%. Portfolio is 55% cash + 45% stock, has beta=0.9
◦ Beta exposure is constantly changing, unstable
Can quantify stability by assessing risk across the future distribution of portfolio weights
© 2022 ANISH SHAH

What do you do?
Problems
◦ Parameter estimates are unavoidably imperfect
◦ Portfolio composition changes with price movement
Solution
◦ Evaluate across the range of what’s possible
◦ Use parameters’ mean and covariance
◦ Model distribution of portfolio weights over the horizon of interest
◦ ± range accompanies numbers in risk decomposition
© 2022 ANISH SHAH

Risk contributions – usual calculations
Portfolio = sum of items (defined according to how we want to cut risk)
◦ e.g., factor exposures, holdings by industry, holdings of individual securities
let σP
2 = var(portfolio), σk
2 = var(item k)
sk = standard deviation contribution from item k
= movement in direction of portfolio
= σk × corr(k, portfolio) = σk × cov(k, portfolio) / (σk σP)
= cov(k, portfolio) / σP
vk = variance contribution
= cov(k, portfolio)
Contributions sum to portfolio volatility: ∑k vk = σP
2 ∑k sk = σP
© 2022 ANISH SHAH
Differ only by σP

Risk contributions with uncertainty
Suppose interested in contributions by security
◦ let w = portfolio weights, C = covariance between securities
vk = variance contribution from security k
= cov(holdings in k, portfolio) = wk ∑j Ckj wj
Say parameters are uncertain and weights shift
◦ vk is random
◦ E[vk] = E[wk ∑j Ckj wj] = ∑j E[wk Ckj wj]
◦ C, w independent → = ∑j E[Ckj]E[wk wj] = ∑j E[Ckj] (E[wk] E[wj] + cov[wk, wj])
◦ Cov[vk,vj], after distributional assumptions, involves means and covariances of w and C
◦ Note: Mean and covariance of standard deviation contributions, sk = vk / σP , can’t be easily computed
Instead, work with a notional center E[vk] / sqrt(E[σP
2]) and approximate covariance from linearization
© 2022 ANISH SHAH
Need to be modeled

Modeling uncertainty
w: Comes from a model of price dynamics over the horizon of interest, e.g., overnight or until
the next rebalance
C: One way is an uncertain covariance model as in
◦ Shah, A. (2015). Uncertain covariance models. http://ssrn.com/abstract=2616109
◦ http://www.slideshare.net/AnishShah23/uncertain-covariance-62814740
© 2022 ANISH SHAH

Factor-modeled covariance in pictures
© 2022 ANISH SHAH
GOOG
Market
Risk Factors
Exposures
Stock-Specific
Effect
AAPL
GE
Growth-Value
Spread
Together these constitute a
model of how securities move –
jointly (winds and sails) and
independently (motors)

© 2022 ANISH SHAH
GOOG
Uncertain Risk Factors
Uncertain
Exposures
Uncertain
Stock-Specific
Effect
Factor variances have a mean and
covariance
Exposures have a mean and
covariance
Stock-specific effects have a
mean and variance
Calculated quantities – e.g., risk
contributions, portfolio variance
– have a mean and variance
Uncertain factor-modeled covariance

Example of (ad hoc) creating a non-
Bayesian uncertain model
13 observations of Northfield 2-Week Near Horizon USA Fundamental Model
◦ Feb 2021 to Feb 2022, 5th business day of each month. (Thanks Dan, Ghaz, Jennifer at Northfield)
Means - use final observation (Feb 2022) values
◦ Factor covariance F, exposures E, stock-specific risks ε2
Uncertainty covariance
◦ Let a parameter’s error at t = its value at t+1 – value at t
◦ Parameters’ covariance estimated as the 2nd moments of error
Minor points
◦ Assumptions on independence and on the distribution of E and w (but not F !)
◦ The level of ε2 is connected to F – e.g., to capture correlation tightening – via observations of its cross-
sectional median
© 2022 ANISH SHAH

40 stock long-short portfolio
In each of 10 sectors, sort companies by name – first two long, second two short
Starting portfolio = conventional minimum variance with 100% long, 100% short
Future weights = distributed under 2 week price movement, derived via conventional risk model
© 2022 ANISH SHAH
Variance ± Exposure ± Conv Risk Uncertain Risk ±
Total 7.55 7.68 4.31
Beta 618.2 240.2 -0.01 0.02 0.01 0.05 0.08
Mkt Cap 11.9 4.4 0.06 0.04 -0.01 0.00 0.00
Relative Strength 71.9 22.6 0.19 0.08 0.27 0.33 0.32
⁞
Stock Specific 1 0.25 6.39 6.33 5.03

40 stock long-short portfolio (cont.)
Can break the uncertain risk’s mean into sources
◦ Parameter means (akin to conventional)
◦ Price movement
◦ Exposure uncertainty
© 2022 ANISH SHAH
Conv Risk Uncertain Risk means w unc E unc
Total 7.55 7.68 7.42 0.07 0.18
Beta 0.01 0.05 0.01 0.02 0.03
Mkt Cap -0.01 0.00 -0.01 0.00 0.00
Relative Strength 0.27 0.33 0.27 0.00 0.06
⁞
Stock Specific 6.39 6.33 6.29 0.04
Why different?

Sidebar: As in life, big problem erases
small problem
Recall (from 9 slides ago)
◦ Item’s σ contribution = cov(item, portfolio) / std dev of portfolio
◦ Item = e.g., effect of stripped-of-market oil exposure
Add a new risk to the portfolio, e.g., a bunch of market to a long-short
◦ The item itself hasn’t changed and didn’t become hedged
What happens to
◦ cov(item, portfolio)?
◦ std dev of portfolio?
◦ item’s σ contribution?
In pictures: increasingly orthogonal to the portfolio, the counted direction of the item shrinks
© 2022 ANISH SHAH

40 stock long-short portfolio (cont.)
See portfolio cut by sector holdings instead of factor
© 2022 ANISH SHAH
Conventional Risk Uncertain Risk ±
Total 7.55 7.68 4.31
Consumer Staples 1.32 1.36 6.16
Utilities 1.09 1.12 4.37
Information Technology 0.90 0.90 0.24
⁞
Totals remain the same

SP500 under the same scenario
Exposure ± Conventional Risk Uncertain Risk ±
Total 28.74 28.74 5.30
Beta 1.03 0.01 22.77 22.77 8.71
Mkt Cap 2.76 0.01 3.15 3.15 1.09
Relative Strength 0.46 0.03 0.92 0.92 0.29
Price Volatility -0.38 0.02 0.74 0.74 0.17
Book/Price -0.49 0.01 0.64 0.64 0.20
Earnings Variability -0.84 0.03 0.31 0.31 0.07
⁞
Stock Specific 0.25 0.25 0.07
© 2022 ANISH SHAH

Scenarios - SP500 with different beliefs
about future
Previous slide toothless because uncertainty modeled from a calm period
Suppose factor covariance and stock specific variance levels are proportional to VIX2
◦ The future has a 90% chance of continuing, drawn from 7-Feb to 28-Feb weekly VIX (23, 28, 29, 30),
and a 10% chance of being extreme, VIX 60
◦ Even better would be a set of differently shaped factor covariance matrices, e.g., from history
◦ Note: no distributional assumptions on F or ε2
© 2022 ANISH SHAH
Conventional Risk Uncertain Risk ±
Total 40.43 40.44 16.64
Beta 32.03 32.03 26.04
Mkt Cap 4.43 4.43 3.60
Relative Strength 1.29 1.30 1.07
⁞

Summary
Risk decomposition is a canonical, useful tool to assess a portfolio’s risks
But perfect alignments (as arise from optimization) hide risks
… and ingredients are uncertain, so such alignments don’t really exist
… and price movement breaks them even if they did
The solution is to analyze with parameters and weights explicitly modeled as uncertain
Reveals latent risks and fragility
Tell your risk provider! (if you want this information on your portfolios)
Ideas are relevant to other risk applications
◦ For more stability, what about optimizing a portfolio’s expected future variance?
◦ or penalizing uncertainty during optimization
© 2022 ANISH SHAH

Risk under parameter uncertainty and price movement

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