Empirics on Portfolio Insurance Design Risk - the case of ...In Brief Portfolio Insurance CPPI...

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Empirics on Portfolio Insurance Design Riskthe case of CPPIs

Raquel M. Gaspar

ISEG Universidade de Lisboa

10 September 2018

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Motivation

1 Mathematical finance models are useful toprice and hedge financial derivatives, but alsodesign and develop new financial products/strategies.

2 An important example are the so-called portfolio insurance(PI) strategies sold by financial institutions – the issuers – toinvestors.

3 CPPI real life case.

4 Need to evaluate risks embedded in existent portfolioinsurance products, both

managed by issuers, andtaken by investors.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Risk Management Proposals

1 Always benchmark against naive (trivial) alternatives.

2 Access existence of possible design risk.

3 Evaluate the need of a model and model risk implications.

4 Focus on real life empirical data risk measures.

5 Take both issuers and investors prespective.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Outline

1 Portfolio Insurance2 CPPI Results3 Empirics4 Design Risk5 Open Issues

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Portfolio Insurance

DefinitionGiven an investment horizon T , an inception date 0, and initialwealth V0, a dynamic portfolio insurance strategy is aself-financing strategy that, for any 0 ≤ t ≤ T ,

allocates wealth, Vt , according to a pre-specified allocationmechanism between

risky asset (or set of risky assets)Examples: stock, index, bond, portfolios managed by thirdentities, ...riskless asset

and is able to guarantee a fixed percentage η ≤ 1, of theinitial investment at maturity. I.e. VT ≥ ηV0.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

PI Notation

Let S and B be the price processes of the risky asset and theriskless assets, respectively.By definition, the value of a PI strategy, (Vt)t∈[0,T ], is

Vt = νBt Bt + νS

t St ⇔ νBt = Vt − νS

t StBt

.

The pre-specified allocation mechanism,{νS

t

}t∈[0,T )

, is aset of rules established at the inception date 0.F the running value of the future guarantee and is known asthe floor

Ft = EQt

[e−∫ T

t rsdsηV0

].

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

(Unrealistic) Assumptions

Assumption 1: Re-allocation is feasible in continuous timeand without costs. There are no fees.

In real lifeDaily or weekly allocation mechanisms => introduces gap riskTransaction costs and/or fees => introduce deduction risk

Assumption 2: The riskless asset exists and has positiveexpected returns.

In real lifeno risk-free asset => introduces guarantee risknegative interest rates => introduces stronger, lower than 1,bounds on η.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Portfolio Insurance Strategies

Investment in PI strategies may make sense when:Investors believe the underlying risk asset will increase untilthe maturity = > they wish to participate in the upsidepotential.But still, they are aware they may be wrong and want to limitlosses, because they are risk averse.

All of us know naive PI strategies:Only invest in risky assets what one accepts to loose =>safety first strategyInvest all wealth in risky assets but “get out” as soon as youalready lost the most you are willing to loose => stop lossstrategy

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

PI – illustration

V = 100, η = 90%, r = 4%

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

SLPI - Stop Loss Portfolio Insurance

MechanismEntirely invest the initial wealth into the risky asset.If and when the strategy value falls below the pre-establishedfloor, fully disinvest from the risky asset and place the strategyvalue in the risk-free asset to provide the future guarantee.

NaiveOnly one possible re-allocation dateDefine τ = inf{t > 0 : V SLPI

t = Ft}, the first time the valueof the strategy hits the floor.The allocation mechanism

{νS

t

}SLPI

t∈[0,T )is

νSt =

{VtSt

t < τ

0 , t ≥ τ

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

SLPI - Stop Loss Portfolio InsuranceIllustration of two possible paths for the underlying:

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Safety first Strategy (CPPI 1)

MechanismInvest, at inception, the present value of the future guarantee inthe risk-free asset and the remaining in the risky-asset.

NaiveNo re-allocation at allThe allocation mechanism

{νS

t

}CPPI 1

t∈[0,T )is trivial

νSt = νS

0 = V0 − F0S0

⇒ νBt = Ft

Bt.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Safety first Strategy (CPPI 1)Illustration:

ST = 140

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

OBPI - Option Based Portfolio Insurance

Term SheetAt maturity, the strategy is worth

VT = FT︸︷︷︸guarantee

+ q ×max(ST − S0; 0)︸ ︷︷ ︸participation in the upside of risky asset

MechanismAt time 0, invest the present value of the future guarantee in therisk-free asset and use the remaining to buy as many as possible(q) at-the-money call options. Wait until maturity.

The participation rate q is fixed at time 0 and equalsq = V0−F0

C0, where C0 is the arbitrage-free price of the ATM

call on the risky asset.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

OBPI - Option Based Portfolio Insurance

Can we always find ATM call options on the market ?NO.

There may be no options traded on particular underlying weare interested in.Hard to find options with long to very long maturities.

Can we derive the allocation mechanism{νS

t

}OBPI

t∈[0,T )?

Is the OBPI strategy a well defined strategy ?

In general, NO.Only if:

We make an additional assumption – model assumption – tobe able to dynamically replicate the call component =>model risk.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Constant Proportion Portfolio Insurance

MechanismAt any time, invest in the risky asset the difference between yourwealth and the running value of the floor multiplied by apre-specified constant m > 1.The remaining is invested in the risky free asset.

Trivial model-free implementation.The allocation mechanism

{νS

t

}CPPI m

t∈[0,T )is

νSt St =

multiplier︷︸︸︷m × (Vt − Ft)︸ ︷︷ ︸

cushion

⇔ νSt = m ct

St

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Constant Proportion Portfolio Insurance

No model risk to the issuer.Relatively easy do explain to investors.Extremely popular in the industry.Real life products multipliers range from 3 to 7.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Constant Proportion Portfolio InsuranceIllustration

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Some Results

Let us consider a Black-Scholes world, and do Monte Carlosimulations for a variety of drift-volatility scenarios.How well do CPPIs in comparison to the other PI strategies?

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Some Results

CPPIs with m > 1 have high probability of terminal value extremelyclose to the guarantee, even in boom scenarios.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Some Results

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Some Results

In almost all scenarios the naive strategies – SLPI and CPPI1 – stochastically dominate non-naive CPPIs with m > 1, in2nd or 3rd order. They do not dominate OBPIs.The performance of CPPIs get worst the higher the maturityT and is extremely sensitive to the volatility, but not to thedrift parameter.In fact, the terminal value ST , has little impact on theperformance of CPPIs with m > 1.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Some Results

What if we still consider a BS world, but impose a terminalvalue of the underlying risky asset (much) higher than itsinitial value, using conditional Monte Carlo simulations?

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Some Results

We show CPPIs are extremely path-depend, which goes againstthe European nature of PI strategies.

η = 100%,T=5, r=4%, σ = 40%

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Some Results

Even in the cases when we consider that ST is 5 times higher thanits initial value, still CPPIs with m > 1 can easily end up very closeto (or at) the guarantee.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Some Results

Even in the cases when we consider that ST is 8 times higher ...

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Empirics

Of course we do not live in a BS world.

Does it help to consider empirical distributions? => NO!

The overall results do not change much.When simulating based upon empirical distributions, CPPIs,show even weaker performance.

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Empirics

Has there ever been a good moment for long maturity CPPIinvestments?

Considers fictitious investments in PI strategies starting at allpossible dates in history for a given risky underlying – e.g.S&P500, bond indices, ... – and compare the performance ofdifferent PI strategies.

Can you guess?

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Risks

What is the problem with CPPIs?

market risk?=> NO! All PI strategies have the same risky asset.model risk?=> NO! Only OBPI have it and that is born by issuers.gap risk?=> NO! All PI strategies have it and that is born by issuers.

Can we talk about a Design Risk?

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

CPPI - Design Risk

Proposal of Definition:

Design RiskThe risk that is:

unrelated with the underlying performance, andintroduced (by the issuer) via the add-hoc definition of anallocation mechanism

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Open Issues

How to formally define design risk?How to isolate/evaluate design risk existent in investmentstrategies or structured products?

In Brief Portfolio Insurance CPPI Results Empirics Design Risk Open Issues

Thank you!