Portfolio Construction and Risk Managementg

Consider the portfolio selection model:

wherewhere

Quantifying the inputs to the optimization is good discipline.1

Portfolio Construction and Risk Managementg

Risk ManagementS f f ( )Start with quantification of what you know (risk)Prepare for what is not knowable (uncertainty)

C fChallenge simplifying assumptions1) Only care about the mean and variance of the

b bilit di t ib ti f tf li tprobability distribution of portfolio returnsa) Ignore skewness (asymmetry) and kurtosis (fat tails)b) Event riskb) Event riskc) Market failured) Government Interventione) Investment horizon – liquidity demand versus

provision2

Portfolio Construction and Risk Managementg

2) Parameter Stationarity (or one period model)a) Regime Changea) Regime Changeb) Time variation characteristics (bonds)c) Cuspiness (securitized asset tranches)d) Dynamic interaction (liquidity->fundamentals-

>liquidity => downward spirale) Time-varying covariance structure (i.e., correlationse) Time varying covariance structure (i.e., correlations

increase during crisis periods)

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Portfolio Construction and Risk Managementg

Risk Management Approaches1) Multi-Dimentional Risk Analysis (calculate sensitivity exposures

to state variables) 2) Stress tests on each exposure3) Value-in-Stress (multiple scenario analysis)4) Value at Risk (VaR) is probabalistic

a) Requires covariance matrix estimation) qb) i.e., 95% VaR is L= –V x 1.645 x portfolio standard deviation .

There is a 5% chance of loosing more than L.c) Distributional assumption for tail riskd) Parameter estimation

i. Historical data (with or without forward looking adjustments)ii. Forward simulation

e) Model the tail of the distribution

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Portfolio Construction and Risk Managementg

Dealing with the unknowableDealing with the unknowable

1) Stop Loss (then what?)

2) Flight to quality hedges

3) Long/short strategies (no net market or macro-factor3) Long/short strategies (no net market or macro factor

exposures)

4) L k4) Lock-ups

5) Portfolio Insurance

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Portfolio Construction and Risk Managementg

General Valuation Equation

V0 =x1

(1+ R1)+

x2

(1+ R1)(1+ R2)+ ...+ xN

(1+ R1)...(1+ RN )

For bonds, the cash flows in the numerator are expected coupons and principal payment(s) in each period. For common stock N is infinity and the numerator containsFor common stock, N is infinity and the numerator contains expected cash flows to equity holders.The R' s are the per period expected required rates of return to compensate investors for the riskiness of their respective cashcompensate investors for the riskiness of their respective cash flows.

Risk Premi m is a f nction of priced risk factors and the sec rit ’sRisk Premium is a function of priced risk factors and the security’s sensitivity to those exposures.

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Portfolio Construction and Risk Managementg

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Portfolio Construction and Risk Managementg

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Pension Plan Asset-Liability Management

2006 Pension Protection Act

d f d d l d f d l lRequires underfunded plans to move toward a 100% funding level

Provisions come into effect on a rolling schedule over several years

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Pension Plan Asset-Liability ManagementProvisions combined with market performance trends are affecting asset 

allocations now

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Pension Plan Asset-Liability ManagementMajor Factors Affecting Pension Plan Funding Status 

Performance of assetsPerformance of assets

Company contributions 

Number of employees covered and cost per employee

Discount rates used to calculate the present value ofDiscount  rates used to calculate the present value of 

future obligations

AA rated Long dated corporate bondsAA rated Long‐dated corporate bonds

Average of AAA, AA, and A rated corporate bonds

Note that the discount factors have fallen sharply during 2009 =>Note that the discount factors have fallen sharply during 2009 => 

increase in PV of future obligations11

Pension Plan Asset-Liability ManagementAsset Allocation:  Select a portfolio that minimizes the 

standard deviation of the plan’s surplus

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Pension Plan Asset-Liability Management

Liabilities rate of return series:

Rate of change in the present value of future liabilities for the change in discount rates (AAA‐A corporate bond rates g ( pprescribed by the 2006 pension protection act). 

The term structure of liabilities in this example is that of theThe term structure of liabilities in this example is that of the average  pension plan.

Th i ki f h l ’ li bili i i h i i fThe riskiness of the plan’s liabilities in the time‐series of returns is defined by the time period selected.

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Pension Plan Asset-Liability ManagementAsset Allocation:  Select a portfolio that minimizes the 

standard deviation of the plan’s surplus

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