Presentation on theme: "Presentation to the Swiss CFA Society Geneva, May 11, 2010 Zurich, May 19, 2010 Portfolio Design Renato Staub May 2010."— Presentation transcript:
1 Presentation to the Swiss CFA Society Geneva, May 11, 2010 Zurich, May 19, 2010 Portfolio DesignRenato StaubMay 2010
2 Objectives Here, we deal with the expectation based market allocation. ObjectivesHere, we deal with the expectation based market allocation.In classical portfolio management, this is referenced as ‘active’ allocation.We want to improve the asset allocation (AA) design process. In particular, we are:Recognizing that history may be a questionable guide to the future by using historical parameters to simulate the opportunityCalibrating the amount of risk taken, given the opportunity, in order to reasonably expect to achieve the risk budget over timeProviding a framework for checking the consistency across capabilitiesMaking the investment process more transparent
3 ConceptsGiven the value/price signals over time, we are looking for theAppropriate expectation based AA strategyAccording amount of riskComposition of this strategyWe combine the following concepts:Value/Price as based on discounted cash flow modelsRandom walksMean reversionInformation analysisWe integrate them in a single framework.
4 Nature of Asset Allocation Nature of Asset AllocationThere are concepts that do not entail imbalances, e.g. Black-Scholes.That is, the market goes up or down with equal probability.By contrast, AA assumes that marketsDeviate from their intrinsic valueRevert in the long run.Without mean reversion, AA cannot add value.Hence…
5 … we want to ride this wave! … we want to ride this wave!
6 Eye-Catcher We simulate Value/Price of a mean reverting market Eye-CatcherWe simulateValue/Price of a mean reverting marketMonthly over 1000 yearsThe market’s average return equals 8.5% (f1).The upper chart entails allUnder-valuations >20% and <30%The lower chart shows the subsequentAnn. 3-year returns (f2).We observe: f2 >> f1.Without mean reversion, f2 would equal f1.
7 Terminology We deal with two key inputs, i.e. A market’s price, P TerminologyWe deal with two key inputs, i.e.A market’s price, PIts fundamental value, VP can be observed for liquid markets.V cannot be observed. It must be estimated by a concept.We model in log space, i.e. v = log(V) and p = log(P).The value-price relationship is defined as follows:vp = log(V/P) = log(V) – log(P) = v-pAssuming that p reverts to v over the duration d, this impliesxr = (v-p) / dxr is the expected return component due to price correction.
8 Numerical Example The following parameters are given: Numerical ExampleThe following parameters are given:V = 100 (by definition)P = 80d = 3 yearsV/P = 100/80 = 1.25vp = log(V/P) = v-p = log(100/80) =xr = (v-p) / d = / 3 = 7.44%That is, we expect an additional log. return of 7.44% p.a. due to price correction.However, this is a raw expectation.Later on, there will be a further modification to xr.
9 Plan We simulate the vp evolution over a long time. PlanWe simulate the vp evolution over a long time.We calibrate the simulation such that the resultingvp spanvp volatilityReversion timeare in line with practical experience.We infer the information embedded in this process.We translate this information into a portfolio.We investigate the portfolio, in particularIts performanceOther important properties
10 Process - p We assume that the price follows a random walk. Process - pWe assume that the price follows a random walk.The shocks are based on our forward looking covariance matrix.The market price is shocked proportionally to its size, that ispt+1 = pt + However, in order to avoid infinite dispersion, we must ensure mean reversion.To that end, we adjust the equation as followspt+1 = pt (1-pp) + ppp is p’s gap sensitivity of mean reversion.From time series analysis, we know < 0 The process is stationary >= 0 The process is non-stationary
11 Process – v Nobody knows v, and this is why we estimate it. Process – vNobody knows v, and this is why we estimate it.We assume it to fluctuate around its (unknown) ‘true’ intrinsic value, i.e.vt+1 = vt (1-vv) + vvv is v’s gap sensitivity of mean reversion.But in practice, we often review a model in case of a large vp discrepancy.This applies in particular to markets of low model confidence.Technically, it means a gap sensitivity of v vs. p, that isvt+1 = vt (1-vv) + (vt-pt) (1-vp) + v = vt+1 = vt (2-vv-vp) - pt (2-vp) + vWe reference this effect as ‘chasing’: the (perceived) value chases the price.This means a narrowing of vp, i.e. the perceived opportunity becomes smaller.
12 Process – Combining v and p Process – Combining v and pCombining v and p, we getvpt+1 = vt (2-vv-vp) - pt (2-pp -vp) + v + pThe next chart is a vp simulation based on monthly data over 1000 years.Notably, vp is confined to a certain bandwidth - because of mean reversion.The breadth of the band depends on the input parameters.
13 Process - InformationAs a result of mean reversion, there is a positive correlation between theExpected return from correction (i.e. the signal)Observed subsequent return from correctionHence, we are interested in the correlation between the two, that isR = corr(Ei[xr(i,i+d)], xr(i,i+1))where xr(i,i+d) is the return from correction between time i and time i+d.Because of the substantial noise components, this correlation is small.In other words, our signal, xr, is far from perfect.We defineR = IC = Information Coefficient
14 Process - Property Shape Invariance Theorem: Process - PropertyShape Invariance Theorem:vp evolutions with identical b‘s entail identical information.As an example, the following charts portray two markets. TheBlue market has shock components of 2s.Red market has shock components of s.Mean reversion parameters of both markets are identical.As expected, both ICs inferred from simulation equal
15 Process - Calibration The main calibration parameters are the Process - CalibrationThe main calibration parameters are theVolatilitiesMean reversion parametersThe chart to the right shows the percentilesof our simulation based vp distribution.As a reference, our valuation considered U.S. equity60% overvalued in (techno bubble)50% undervalued in (credit crisis)The according percentiles relative to the marketrisks are wider for equity than for bonds.This is because we set stronger mean reversionfor bonds than equity.
16 FLAM and PAR The Fundamental Law of Active Management (FLAM) infers: FLAM and PARThe Fundamental Law of Active Management (FLAM) infers:A portfolio’s information ratio (IR) equals the IC of the comprising markets (i.e. signal projection) times the square root of the number of independent market bets available. *)The Proportional Allocation Rule (PAR) concludes:To allocate efficiently, we should allocate proportional to the signal. **)WhileFLAM describes performancePAR tells us how to achieve it*) Assuming the various markets have identical ICs**) Assuming the signals are uncorrelated and have identical volatilities
17 PAR – Schematic Example PAR – Schematic ExampleAt time 0The price equals p and the value equals vAccording to PAR, we buy an amount of (v-p)Between time 0 and 1The price changes by Hence, the value price discrepancy changes to (v-p-)At time 1’According to PAR, we adjust the quantity by -Between time 1 and 2The price changes by -We have the quantity (v-p-) at price pThe net gain equals 2
18 PAR – Schematic Example PAR – Schematic ExampleThe same principle works for two up moves combinedwith two down moves.For all four combinations, i.e.up-up-down-downup-down-up-downdown-down-up-upDown-up-down-upPAR results in a net gain of 22 .Exposure Theorem:Following PAR at constant volatility, the extra return grows with time.On the other hand, the size of the extreme vp is irrelevant.
19 Slide in – Geometrical Representation of Risk Slide in – Geometrical Representation of RiskRisk can be depicted by the length of a vector.And correlation equals the cosine of the angle between two risks.In the following triangle, the labels meanRisk of Portfolio 1 (Pf1): s1Risk of Portfolio 2 (Pf2): s2Correlation between Pf1 and Pf2: cos(j)Relative Risk (Tracking Error)between Pf1 and Pf2: s1 - s2s1s1 - s2js2
20 Signal - InformationWe represent both the signal and the observation by vectors.The more the signal points in the direction of the observation, the better it is.Hence, a signal’s prediction quality equals itsProjection onto the observation axisThe projection equals theCorrelation between signal and observationCosine of angle jAgain, “correlation” and “cosine” are twodifferent labels for the same thing.SignaljObservationProjectedSignal
21 Signal - Modificationxri is the naïve expected extra return for market iBut only its projection will materialize statisticallyHence, we IC-correct itFurther, a return must always be put in relation to its distribution.Thus, we divide xri by the volatility of market i.Ultimately, the “true” substance of xri equalsxri
22 Signal - ModificationAssume an equity and a bond market have an si of equal size.Identical allocation to equity and bonds implies more portfolio risk from equity.Since PAR assumes identical risks, we must rescale one more time by the risk:Ultimately, the suggested allocation equalsThe portfolio risk is linear in f.
23 Allocation – Vector Approach Allocation – Vector ApproachAll bets are made vs. cash (only).That is, in case of cash, C, and two markets, A and B, the vector approachBets A vs. CBets B vs. CDoes not bet A vs. BHence, the set of all bets is a one-dimensional structure.This is why we call it “Vector Approach”.The vector approach is mainly a bet of cash vs. the entire market.
24 Allocation – Matrix Approach (First Scenario) Allocation – Matrix Approach (First Scenario)The matrix approach makes all possible bets, that is, also the bet A vs. B.The relative risk between A and B is marked in red.The triangle ABC is positioned such that its cornerstouch the corresponding iso-vp lines.As B is more undervalued than A, we go long B vs. A.Based on the vp differential and the riskgeometry, this bet will be of average size.That is, the bet A vs. B has mainly implicationsin terms of diversification.VP=20%VP=7%VP=0%BCA
25 Allocation – Matrix Approach (Second Scenario) Allocation – Matrix Approach (Second Scenario)The vp differential between A and B is unchanged.But A and B are correlated much stronger vs. the first scenario.Hence, we scale by a smaller risk distance.As a result, A vs. B is by far the strongest bet.That is, its implication goes beyond diversification only.Rather, it is supposed to beef up return.VP=20%VP=7%BAVP=0%C
26 Signal – Hosting the Matrix Approach Signal – Hosting the Matrix ApproachWe build the difference between two IC corrected extra returns.And the risk is the relative risk between the two markets.That is:wij is the position based on the relative bet between market i and market j.The above equation also serves the vector approach, in which case j labels cash.
30 Simulations – Valuation and Suggested Allocation Simulations – Valuation and Suggested Allocation
31 Allocation - Comparison Allocation - ComparisonVector approach:xri and wi have always identical signs.Cash is a special bucket, as all bets are made vs. cash.Hence, the cash dispersion is massively larger.Matrix approach:xri and wi do not necessarily have identical signs.Market i may be undervalued but shorted vs. most other markets.The reason is: it may be less undervalued than other markets.Cash is a bucket like any other.Hence, the cash dispersion is much smaller.
33 Performance - Comparison Performance - ComparisonResulting Information Ratios (IR), based on 20 market bets:Vector Approach: 0.93Matrix Approach: 1.48Overall, the level of the resulting IR seems (too) high.This is partially due to the stationarity underlying our framework.Realistically, that’s the best possible assumption.However, the high IR is also due to efficient use of information.Notably, the matrix approach performs much better.The reason is its better diversification.The portfolio can be scaled through f to any risk/return level at the given IR.
34 Summary and Conclusions Summary and ConclusionsWe develop a formal allocation process that isTransparentConsistent across marketsTo that end, weCalibrate and simulate a mean reverting value/price processExtract and translate the embedded signalsWe present two translation approachesVector approach: all market bets are made vs. cashMatrix approach: there are also bets between marketsThe matrix approach performs better, as it is better diversified.We may deviate from the suggested allocation; the according performance difference is attributed to non-fundamental factors.
35 Thank you very much for your Attention! Thank you very much for your Attention!
36 References Grinold, Richard C. and Ronald Kahn. “Active Portfolio Management.” Probus Publications, Chicago, 1995. Hull, Jon C., 1993, Options, Futures, and other Derivatives, Second Edition. Prentice Hall, Englewood Cliffs, NJ. Staub, Renato. “Signal Transformation and Portfolio Construction for Asset Allocation.” Working Paper, UBS Global Asset Management, Sept Staub, Renato. “Deploying Alpha: A Strategy to Capture and Leverage the Best Investment Ideas”. A Guide to 130/30 Strategies, Institutional Investor, Summer 2008. Staub, Renato. “Are you about to Handcuff your Information Ratio?” Journal of Asset Management, Vol. 7, No. 5, 2007. Staub, Renato. “Unlocking the Cage”. Journal of Wealth Management, Vol. 8, No. 5, 2006. Staub, Renato. “Deploying Alpha Potential”. UBS Global AssetManagement, White Paper, 2006.