dynamic asset-liability management and decision support

PF economicsStochastic optimization model

Risk exposureDSP interfaces and implementation

Case study and numerical evidencesConclusions

Dynamic Asset-Liability management anddecision support for Pension Funds

Giorgio Consigli1

Vittorio Moriggia, Giacomo Landoni, Sebastiano Vitali, Elena Benincasa,Filomena Petronio

Angelo Uristani 2, Marina Zeyss, Federico Passerini

1University of Bergamo2Allianz Investment Management

UniBG June 12, 2015 – Pension Fund Managementworkshop

G. Consigli ALM 4PF via dynamic stochastic optimization




Outline

1 PF economics

2 Stochastic optimization model

3 Risk exposure

4 DSP interfaces and implementation

5 Case study and numerical evidences

6 Conclusions





Investment management

We focus on a PF portfolio management problem formulated as adynamic stochastic program over a 20 year horizon with annualpension payments.The PF manager is assumd to seek an optimal trade-off betweenshort, medium and long-term goals. An extended asset universe isconsidered including liquid and illiquid assets.The Pension Fund liability or DBO is computed as discounted valueof inflation adjusted pension payments through an appropriatereplicating portfolio of discount bonds and caps and floors.We assume constant contribution rates and employee-sponsorssharing rule during the accumulation phase.





PF key variables





PF net pension payment





Liability pricing model

The PF liability - specifically its Defined Benefit Obligation (DBO) - iscomputed as discounted values of inflation adjusted pensionpayments.Th interest rate dynamics are assumed to be consistent with Black’spricing model for interest-rate sensitive instruments.

M∑T=t+1

(ZCB(t, T )− CAPLET (t, T ) + FLOORLET (t, T ))





Liability pricing model: numerical exampleWe show that such portfolio actually matches the value of futurenominal liabilities discounted at the real interest rates, computed bysubtracting inflation caps or floors from the prevailing nominal interestrates.Consider the liability cash flows expected at the end of years 1 and 2,evaluated at t = 0, with compounded infation for payments to occur atT . In t = 0 we have:

Year Infl ZCB Cap Floor Rpl. ptf valueT=1 1,05 113,86 2,48 0,40 111,77T=2 1,04 195,28 4,75 0,38 190,92

DBO 302,699748Disc. value 303,15061





Liability pricing model: volatility estimation

Volatility fitting procedure

As for caps and floors, starting with two different matrices of optionpremiums (one for cap-contract prices and one for floor-contractprices), an optimal volatility surface for different tenors and strikes iscomputed that minimize the distance between the input data and thereplicating portfolio.





Notation and model variables

For t ∈ T , a discrete and finite time space, n ∈ Nt are nodal labelswith children n+ and parent node n−. We denote by a(n) thesequence of all ancestors for any final or intermediate node and withc(n), the subtree originating from n. tn is the time of node n

Variables at the intermediate stages:

Income returns on fixed income assets,Expiring fixed income securities, Annualpension payments, Cash imbalances.

Variables at the decision stages:

Income returns, capital gains, UGL on theextended asset universe, Expiring fixedincome securities, Corporate sponsors’contributions, Net pension payments, Interestson cash imbalances, Selling decisions andnew investments.






For t ∈ T , n ∈ Nt

Ωn liquidity gap: net pension payments minus portfolio incomereturn;

K1n ALM Risk: interest rate fluctuations times the duration gap;

Λn defined benefit obligation (DBO): present value of theaccrued pension obligation;

At,s fair value of plan assets: change in fair value of plan assetsdetermined by the total portfolio return;

Kfn Investment Risk Capital;

Πcumn total cumulated realized portfolio return: unexpected

additional contributions motivated by increasing funding gap.







LNETn net pension payments

Π1,INVn portfolio income return

xi,m,n are portfolio holdings of asset i, node n, bought inm ∈ a(n). X−n =

∑i

∑m x−i,m,n and X+

n =∑

i x+i,n are sellings

and buyings. Xi,n =∑

m∈a(n) xi,m,n and Xn =∑

iXi,n.

For t ∈ Tint, n ∈ Nt

LZn cash imbalances over intermediate stages

LZn = Xi,n− · ξi,n +Xexp

i,n −LNETtj ,s





Short, medium and long-term objective variables


Liquidity gap plus ALM risk: Y1,n = Ωn +K1n + Ψn−

Plan sponsors’ unexpected contributions: Y2,n < γ · Φu,+n

with Φu,+n − Φu,−

n = Λn −An where:

γ plan sponsor contribution tolerance;Φu,+

n , Φu,−n positive and negative part of Φu

n = Y2,n;

Return on plan assets per unit tail risk: Y3,n =Πcum

n

Kfn

Net defined benefit obligation (DBO): Y4,n = Λn−An

The four targets are assumed to be set at the 1, 3, 10, 20 yearhorizon respectively.





Objective functionThe optimization problem is represented by:

maxx∈X

(1− α) ·∑j

λjE(Yn|Fj)− α ·∑j

λjE(Yj − Yn|Fj)

s.t for all n ∈ Nt, t ∈ T

Xn = Xn−(1 + ρn) +X+n −X−n

zn = X−n −X+n + Φu

n + Φdn + Π1,INV − LNET

n + Lzn(1 + πn) + zn−(1 + rn)

Πfn = Xn−ξn +

∑m∈a(n)

X−m,nγm,n + Πfn−

X+n +X−n = ϑXn−(1 + ρn)

Kfn ≤ µ(Xn − Λn −Kt

n)





Asset return model

The asset universe includes a set of investment opportunities, whoseprice ρ and cash returns ξ must be specified and input to theoptimization problem:

Barcap Treasury 1-5, 5-10, 10-20, 20-30 and 30+ year maturity

10 year Securitised, IG and SG Corporate Bond indices

10, 20 and 20+ year inflation-linked fixed income indices

MSCI EMU (Public) Equity index and Private equity index

Indirect GPR Europe real estate index

Infrastructure Cyclical and Defensive investments (100 years)

Renewable energy and Floaters





Investment risk capital

The risk capital consumption is given by:

Kfn = Kf1

n +Kf2n (1)

Kf1n =

(∆A

n− −∆Λn−)drn(tn − tn−) +Kf1

n− (2)

Kf2n =

√∑i∈A

∑j∈A

Xi,n−Xj,n−kij(tn − tn−) +Kf2n− (3)

where ∆An =

∑i

∑m∈a(n) xi,m,n∆i,n and ∆Λ

n =∑

T>tnΛn,T × (t− tn)

are the asset and liability durations in node n. Kf1 and Kf2 define theinterest rate and market risk exposure of the portfolio.





Implementation framework





Case study

We present a set of results focusing on:

the liability pension payment modul based on an underlying interest rateprocess consistent with the lognormal assumptions at the grounds ofBlack’s pricing model for interest-rate sensitive instruments;

the generation and solution time for stochastic programs of increasingdimension under the different solution approaches;

the scenario analysis of the obtained optimal solution.





Liability Pricing Model

The benefit indexation scheme is described by a replicating portfolioincluding a cap and a floor set of rates: the inflation adjustmentpensions are revalued according to the occurred inflation but with amaximum possible revaluation and also with a protection of minimumrevaluation.





Dynamic Global PF Asset Allocation





Scenario Analysis





Portfolio asset composition





Investment risk capital exposure





Surplus





Conclusions

Pension Funds ALM is becoming an extremely relevant R&Darea also in relatibvely conservative pension systemscharactrized by a majority of state-controlled tier 1 frameworks,

We have analyzed the key elements of a DB ALM systemdeveloped for a large corporation, where a very simple liabilitymodel has been adopted,

In a discrete model, large scale stochastic optimization providesan efficient methodology to tackle in a very realistic way severalregulatory and financial issues affecting Pension Fundsnow-a-day.


dynamic asset-liability management and decision support

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