Market Consistent and Sub-consistent Valuation · Introduction to Market Consistent Valuation...

Introduction to Market Consistent ValuationMarket Consistency and Sub-consistency in Incomplete Market

Hedging in incomplete marketsStudying cases of MV and CVaR

Empirical Assessment

Market Consistent and Sub-consistent Valuation

Hirbod AssaInstitute for Financial and Actuarial Mathematics

Institute for Risk and UncertaintyUniversity of Liverpool

(Joint with Nikolay Gospodinov FED Atlanta)

November, 2015WatRISQ Seminar, Waterloo

Hirbod Assa Institute for Financial and Actuarial MathematicsInstitute for Risk and UncertaintyUniversity of Liverpool (Joint with Nikolay Gospodinov FED Atlanta)Market Consistent and Sub-consistent Valuation




Outline

Introduction to Market Consistent Valuations.

Market Consistency and Sub-consistency in Incomplete Market.

Hedging in Incomplete Market

Studying cases of MV and CVaR.

Empirical Assessment.





Introduction

Solvency II, adopted by the European Commission in 2009 and modified in2014, is a framework for regulation and supervision of insurance and reinsurancecompanies in the European Union.Solvency II pillars

Pillar 1 quantitative requirements for valuation, capital requirmentPillar 2 requirements for risk management, governancePillar 3 addresses transparency, reporting to supervisory

One of the main requirements of Solvency II, that goes into effect as of January2016, is that, for supervisory purpose of determining the solvency position, allinsurers need to value their assets and liabilities in a market-consistent manner.

More specifically, the market-consistent valuation should be based on marketprices – as opposed to historical cost accounting – and fluctuate with changes inmarket conditions.

The objective of Solvency II is to quantify this risk in a market-consistentfashion so that the solvency capital held by the insurance companies properlyreflects the economic risk inherent in their balance sheet.





This is prompted ...

There are two main reason

FIRST: Introduction of new insurance financial oriented products insurancepayoffs e.g.,

1-equity-linked insurance contracts

2-equity linked pension

3- catastrophe and longevity bonds

4-CDS and investment insurance

5-etc

include both financial and non-financial risk which may not be directlyhedgeable.

SECOND: Insurance companies investment in financial assets.





Literature review

The literature on market consistent valuation is new and relatively small.

Wuthrich et al. (2010) develop a dynamic setting for a market consistent for lifeor non-life insurance.

Wuthrich and Merz (2013) introduce sound risk measurement practices thatform the basis of good risk management policies

Happ et al. (2015): in incomplete markets by utility indifference pricing

Pelsser and Stadje (2014) propose time-consistent and market-consistentvaluation: (i) cannot generate arbitrage (ii) fully hedged portfolios cannotimprove the actuarial evaluation

Barrieu and El Karoui (2008): by minimizing the risk of the unhedged part. Seealso Barrieu and El Karoui (2005), Malamud et al. (2008) and Knispel et al.(2011).





Risk Evaluator

Let Π : L∞ → R be a mapping.

P1. Π(0) = 0 (N);

P2. Π(λx) = λΠ(x), for all λ ≥ 0 and x ∈ L∞ (PH);

P3. Π(x + c) = Π(x) + c, for all x ∈ L∞ and c ∈ R (CI);

P4. Π(x) ≤ Π(y), for all x , y ∈ L∞ and x ≤ y (M);

P5. Π(x + y) ≤ Π(x) + Π(y), for all x , y ∈ L∞ (SA);

P6. Π(λx + (1− λy)) ≤ λΠ(x) + (1− λ)Π(y) (Convexity).

1,2 and 6⇒ ∃∆Π ⊆ L1

Π(X ) = supQ∈∆Π

EQ[−X ]





Different Possible Risk Evaluator

Let ρ be a coherent risk evaluator

Shortfall Risk Π(X ) = ρ((−X ) ∧ 0)

Coherent Risk Π(X ) = ρ(X )

Mean variance Π(X ) = −E(X ) + λσ(X )

Value at Risk Π(X ) = VaRα(X ) = −qα(X )





Pricing Rule

Let X be the set of all price-able (or fully hedged or hedge-able) positions. (CAN BESUB-SPACE< CONE< ETC)Let π : X ⊆ L∞ → R be a mapping.

P1. π(0) = 0 (N);

P2. π(λx) = λπ(x), for all λ ≥ 0 and x ∈ L∞ (PH);

P3. π(x + c) = π(x) + c, for all x ∈ L∞ and c ∈ R (CI);

P4. π(x) ≤ π(y), for all x , y ∈ L∞ and x ≤ y (M);

P5. π(x + y) ≤ π(x) + π(y), for all x , y ∈ L∞ (SA);

P6. π(λx + (1− λy)) ≤ λπ(x) + (1− λ)π(y) (Convexity).

1,2,6⇒ ∃R ⊆ L1

π(X ) = supz∈R

E [zx]





Examples of pricing Rules

Test assets x0, x1, ..., xN with prices p0, p1, ..., pN .

Stochastic discount factor z ∈ L1, we haveE (zxi ) = pi ,∀i = 0, 1, ...,N.

The set X ⊆

N∑i=0

aixi , ai ∈ R, i = 0, 1, ..,N

.

R ⊆ SDF.Consider Ft∈[0,T ]. A random process xt0≤t≤T is adapted of prices.

Financial market: of a N + 1: (x0,t , x1,t , ..., xN,t)0≤t≤T .X ⊆c +

N∑i=0

∫ T

0hi,tdxi,t , c ∈ R, hi,t is a pred. process, i = 0, 1, ...,N

.

The set R ⊆ SDF: equivalent martingale measures.For any z ∈ SDF, it is clear that

E

(z

(c +

N∑i=0

∫ T

0hi,tdxi,t

))= c .

Important implication π is linear on the set X .





Market Consistency Type I and II

So far we have

X , the set of hedgable positions

π : X → R, the market evaluation

Π : L∞ → R, the risk evaluation

Definition

Let Π be a risk evaluator and π be a pricing rule. Then, Π is market (sub-) consistentof type I if

Π(x)(≤) = π(x),∀x ∈ X . (1.1)

Also, we say that market (sub-) consistency of type II holds if

Π(x + y)(≤) = π(x) + Π(y),∀x ∈ X and ∀y ∈ L∞. (1.2)





Theorem

Theorem

Let π be a pricing rule on the space X = L∞(G) for a σ-filed G ⊆ F . If Π satisfiesconditions P1 and P6, then in a complete market (i.e, X is a linear space and π islinear), the following conditions are equivalent:

1 Consistency of type I holds;

2 Consistency of type II holds;

3 Π(x) = supz∈L1|EG (z)=1 E(zx)− c(z), for a penalty function

c : z ∈ L1|EG(z) = 1 → R≥0 ∪ +∞.





Incompletness

Any type of friction: (Jouini and Kallal:1995a), (Jouini and Kallal:1995b) and (Jouiniand Kallal:1999) argue that for a wide range of market imperfections such as

1-dynamic market incompleteness,

2-short selling costs and constraints,

3-borrowing costs and constraints,

4- proportional transaction costs,the pricing rule is sub-linear.





Digression to Convex Analysis

For any set X , we introduce

X⊥ = z|E(zx) ≤ 0,∀x ∈ X.

For any function f : L∞ → R ∪ +∞, we introduce

f ∗(z) = supx∈L1

E(zx)− f (x).

For a lower semi-continuous convex function f , we have

f = f ∗∗.

For any function f , we introduce

f (x) = f (x), x ∈ X and +∞, otherwise.

∆π = z|E(zx) ≤ π(x), ∀x ∈ X.





First result: Market sub-consistency

Theorem

If Π posses properties P1 and P6, in a market with sub-linear pricing rule π, thefollowing conditions are equivalent:

1 Π is market sub-consistent of type I;

2 Π is market sub-consistent of type II;

3 Π(x) = supz∈∆πE(zx)− c(z), for a penalty function c ≥ 0;

4 Π∗ <∞ ⊆ ∆π .

Furthermore, if Π is also positive homogeneous (property P2) then all conditions areequivalent to ∆Π + X⊥ ⊆ ∆π .





Second result: Market consistency type I

Theorem

If Π posses properties P1 and P6 in a market with sub-linear pricing rule π, thefollowing conditions are equivalent:

1 Π is market consistent of type I;

2 z ∈ L1| infz⊥∈A⊥ Π∗(z − z⊥) = 0 = ∆π .

Furthermore, if Π is also positive homogeneous (property P2) then all conditions areequivalent to ∆Π + X⊥ = ∆π .





Examples: Type I consistency

If X = L∞(G), for a pricing rule π, the following examples are typical examples ofmarket consistent evaluations of type I:

1 Π(x) = π (δEG(maxx − EG(x), 0p) + EG(x)) ,

2 Π(x) = π(δVaRGα(x − EG(x)) + EG(x)),

3 Π(x) = π(δCVaRGα(x − EG(x)) + EG(x)),

where VaRGα and CVaRGα are conditional VaR and CVaR.





Third result: Market consistency type II

Theorem

If Π posses properties P1 and P6 in a market with sub-linear pricing rule π, thefollowing are equivalent

1 Π is market consistent of type II;

2 ∀z ∈ Π∗ <∞ and x ∈ X , π(x) = E(zx).

Furthermore, if Π is positive homogeneous (property P2) then all conditions areequivalent to ∀(z, x) ∈ ∆Π ×X , π(x) = E(zx).





Market consistency type II: discussions

Corollary

If the evaluator Π posses properties P1 and P6, then market consistency of type IIholds if and only if the pricing rule is linear.

Corollary

If the type II market consistent evaluation Π possesses properties P1 and P6, then forany z ∈ Π∗ <∞, we have Π∗ <∞ ⊆ z + X⊥ ∩ (−X⊥). If Π also possessesproperty P2, we have ∆Π ⊆ z + X⊥ ∩ (−X⊥).

However, the typical examples satisfy these conditions.





Best Estimate and Hedging

The hedging strategy which we introduce is based on the idea of a best estimatefor actuarial valuation of an insurance position.

According to Article 76 in Solvency II, the value of technical provisions is equalto the sum of a best estimate and a risk margin.

The best estimate corresponds to the probability-weighted, expected presentvalue of future cash flows, using the relevant risk-free interest rate1.

The hedging strategy that we develop follows a similar logic by assuming thatany non perfectly hedgeable position can be divided into two parts: a part whichis fully hedged (the best estimate) and a part which is unhedged and hence risky(the risk margin). However, for reasons that are discussed bellow, we will extendthe concept of a best estimate along a new dimension.

1http://www.solvency-ii-association.com/Hirbod Assa Institute for Financial and Actuarial MathematicsInstitute for Risk and UncertaintyUniversity of Liverpool (Joint with Nikolay Gospodinov FED Atlanta)Market Consistent and Sub-consistent Valuation




Hedging

Example: Let us suppose the market M consists of n trad-able assets with the randompay-offs X1, . . .Xn.

Time horizon 0 and T

A random pay-off Y is replicable if ∃a1, . . . an either in R or R+ such that

Y = a1X1 + a2X2 + · · ·+ anXn.

We would also say that Y is fully hedged by X1, . . . ,Xn.

what if @a1, . . . , an such that

Y = a1X1 + a2X2 + · · ·+ anXn.





Some solutions

Intrinsic risk (mean square framework) , (Schweizer 1990,1991, 1992)Minimizes the mean square of the cost.INDEPENDENT FROM AGENT.Super-hedging (El Kaoui and Quenez 1995):Find a1, . . . an such that

a1X1 + a2X2 + · · ·+ anXn ≥ Y

and also some criteria is minimized (e.g the hedging price is minimized)TOO EXPENSIVE.Efficient-hedging (Follmer and Leukert. 2000), (Nakano 2004) and (Rudloff2007) :Find a1, a2, . . . , an such that the shortfall risk is minimized

Risk (Y − a1X1 − a2X2 − · · · − anXn) ∧ 0

subject to some criteria (e.g. the price falls below the super-hedging price).(ASSA AND MALLAHI 2013)No-Good-Deal hedging (Balbas, Balbas and Garrido 2010) and (Assa andBalbas 2011)

Risk(a1X1 + a2X2 + · · ·+ anXn − Y )

subject to some criteria (e.g. budget constraint).

Good Deal Price≤ Efficient Price ≤ Super-hedge PriceHirbod Assa Institute for Financial and Actuarial MathematicsInstitute for Risk and UncertaintyUniversity of Liverpool (Joint with Nikolay Gospodinov FED Atlanta)Market Consistent and Sub-consistent Valuation




Imposing the problem

Position to hedge and price Y .

The set of fully hedged positions X (e.g. = Span(X1, . . . ,Xn)).

The cost (of hedge) = Price.

The risk (of hedge)=Risk.





A General Idea

Find the closest fully hedged ( e.g. X = a1X1 + a2X2 + · · ·+ anXn) randompay-off mimicking Y .

“Project” Y on X ( e.g. Span(X1, . . . ,Xn) or Span(X1, . . . ,Xn)+)

Cost(aggregate)

@@@R

Implied hedging costX

Spread riskY − X





The hedging problem

So we have

minX∈X

Risk(X − Y )︸︷︷︸Risk margin

+ Price(X )︸︷︷︸Best estimate

where Y is a position to be priced and X is a set of fully hedged positions (e.g.Span(X1, . . . ,Xn)).





Hedging problem

Rewrite the hedging problem as

Ππ(Y ) = infX∈XΠ(X − Y ) + π(X ).

Ππ is a risk evaluation.





Market principles

Normality (N). Ππ(0) = 0.

No Good Deal Assumption (NGD). There is no financial position x such that

Π(−x) < 0 , π(x) ≤ 0.

(similarities to No Arbitrage)Compatibility (C). The hedging problem has a finite infimum.

The first principle simply recognizes that the price of zero is always zero. The secondprinciple states that any risk-free variable has a positive cost. The last principle pointsout that the hedging problem always yields a price.





First results

Theorem

Assume that Π and π satisfy properties P1–P4. Then,

(N)⇔ (NGD)⇔ (C).

Theorem

Assume that the risk evaluator Π and the pricing rule π posses properties P1, P2 andP6. Then, the following statements are equivalent:

1 The hedging problem HP is finite.

2 ∆Π ∩∆π 6= ∅.3 There is no Good Deal in the market.

In all cases, HP can be represented as

Ππ(y) = supz∈∆Π∩∆π

E(zy), ∀y ∈ L∞.





Literature on Good Deal

Cochrane and Saa-Requejo first introduce the notion of a Good Deal as afinancial position with particularly high Sharpe ratio.

The definition of a Good Deal has been extended in Cerny and Hodges. NODESIRABLE IS FREE !

Jaschke and Kuchler, first address the relation between Good Deals andcoherent risk evaluators.

Cherny extends the definition of Good Deals to positions with a highperformance ratio (a generalization of Sharpe ratio).

Assa and Balbas to study when Good Deals exist!

Typical example : (ASSA and MALLAHI 2012) CDS (Credit Default Swap).





Second results

Proposition

If π : X → R is a pricing rule and Π : L∞ → R is a market evaluator, then Ππ ,introduced in HP, is market sub-consistent of either type.

The following theorem states that market sub-consistent evaluation can be representedas HP.

Theorem

Let π : X → R be a pricing rule possessing property P1, and let Π : L∞ → R be a riskevaluator. Then, the evaluator Π is market sub-consistent of type II if and only ifΠ = Ππ . Furthermore, if Π possesses property P6, the same is true for marketconsistency of type I.





Maximal inferior sub-consistency

Definition

A market sub-consistent risk evaluator Πm is a maximal inferior risk evaluator if

1 Πm ≤ Π;

2 Πm(x) ≤ π(x),∀x ∈ X ;

3 For any other risk evaluator Π′ with properties 1 and 2, Π′ ≤ Πm.

Theorem

Assume that the risk evaluator Π and the pricing rule π possess properties P1, P2 andP6. If there is no Good Deal, then

Πm = Ππ .

Therefore, Πm can be represented as

Πm(y) = supz∈∆Π∩∆π

E(zy), ∀y ∈ L∞.





Information and law invariant risk evaluators

How to model information: by partition.

Let us suppose Σ is a finite partition of Ω and manager has access to itsinformation G = σ(Σ). Law invariant coherent risk evaluators on L∞ have thefollowing property

Π(E [X |G]) ≤ Π(X )

So lack of information will result in risk underestimation and consequentlyproducing Good Deals,

Π(E [X |G]) + π(X ) ≤ Π(X ) + π(X ).

In particular since Π(X ) ≤ Π(X + ε), ignoring any complexity results inproducing Good Deals.





Hedging problem for MV

MVλ(X ) = λσ(X )− E(X )π(X ) = E(X )

.

infX∈XMVλ(X − g) + E(X ).

Y = ΘX′ + e.

whereX = (1, X1, . . . , Xn) = (1,X1 − E(X1), . . . ,Xn − E(Xn))Y = Y − E(Y ).It is independent from λ





Hedging problem for CVaR

CVaRα(X ) = 1

α

∫ α0 VaRs(X )ds

π(X ) = E(X ).

infX∈XCVaRα(X − g) + E(X ).

VaRα(

ΘX′ − Y)

= 0.

whereX = (1, X1, . . . , Xn) = (1,X1 − E(X1), . . . ,Xn − E(Xn))Y = Y − E(Y ).






Economic Risk:

1 Inflation, monthly percentage changes in CPI, denoted by (INF)

2 Real Interest, the monthly return on a three-month T-bill, denoted by (RI)

3 Term Spread, the difference between a long- and a short-term government bond,denoted by (TS)

4 Default Spread, the difference between corporate bonds rated Baa by MoodysInvestor Service and Aaa corporate bonds, denoted by (DS)

5 Dividend Yield, The monthly dividend yield on the S&P 500, denoted by (DIV)

6 Consumption Growth, Monthly real per-capita consumption growth, denoted by(CG)

FRED2, BGFRS3, Shiller’s4

2Federal Reserve Economic Data3Board of Governors of the Federal Reserve System4http://www.econ.yale.edu/ shiller/data.htm






Securities:

1 Risk Free, three-month T-bill rate (as a proxy), denoted by(RF)

2 Market , Market Risk minus Risk Free, denoted by (RM-RF)

3 Size, Small Minus Big, denoted by (SMB)

4 Book-to-market value, High Minus Low, denoted by (HML)

5 Momentum, Up Minus Down, denoted by (UMD)

6 Term factor, the difference between a long-term government bond return andthe three-month T-bill rate, denoted by (TERM)

7 Default factor, the difference between the return on a portfolio of long-termcorporate bonds and a long-term government bond return, denoted by (DEF)

FRED, BGFRS, and Fama and French data library.





OLS and Heteroskedasticity

Most of the literature in economic hedging, factor pricing and also the one bySchweizer is based on OLS .

Statistically justifying quantile regression.

Quantile regression provides information on distribution(skewness, kurtosis)

Heteroskedasticity is significant issue for OLS.

OLS is no longer efficient in the presence of non-normal errorterms.

Two type of significance: 1-nonzero coefficients, 2- significancedifference from OLS.

Financially justifying quantile regression.

Using risk management based on coherent risk evaluators(CVaR), financially justify the use of quantile regression.





Heteroskedasticity

Breusch-Pagan test.

Inflation χ2 = 22.83, Prob. > χ2 = 0.0018

T-Bill χ2 = 64.72, Prob. > χ2 = 0.0000

Term Spread χ2 = 78.94, Prob. > χ2 = 0.0000

Default Spread χ2 = 48.42, Prob. > χ2 = 0.0000

Dividend χ2 = 96.30, Prob. > χ2 = 0.0000

Consumption Growth χ2 = 6.60, Prob. > χ2 = 0.4715





Figure: Normal-quantile plots−

2−

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0.2

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−.4 −.2 0 .2 .4Inverse Normal

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G

−2 −1 0 1 2Inverse Normal





Inflation

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050.

100.

15H

ML

0 .2 .4 .6 .8 1Quantile

−3.

00−2.

00−1.

000.

001.

00R

F

0 .2 .4 .6 .8 1Quantile

−0.

050.

000.

05U

MD

0 .2 .4 .6 .8 1Quantile

−0.

20−

0.10

0.00

0.10

TE

RM

0 .2 .4 .6 .8 1Quantile

−0.

200.

000.

200.

400.

60D

EF

0 .2 .4 .6 .8 1Quantile





Stylized Facts

Except for Consumption Growth, there is a direct relation between the level ofrisk aversion and the share of DEF in the hedging portfolios.In hedging Inflation there is a direct relation between the level of risk aversionand the shares of RM-RF.Hedging Interest Rate implies a reverse relation between the level of riskaversion and the shares of RF as well as UMD and RM-RF. Moreover, inhedging Interest rate there are significant differences between quantile-hedgingand the mean-variance hedging of RF and DEF.In hedging Term Spread, there is a reverse relation between TERM and the levelof risk aversion. In addition there is a significant difference betweenquantile-hedging and the mean-variance hedging of TERM and DEF.There is a reverse relation in hedging Default Spread between the level of riskaversion and shares of TERM.There is significant differences for hedging Dividend Yield, betweenquantile-hedging and mean-variance hedging for shares of RF, UMD and DEF.As one can see in Figure 6, there is no significant difference in quantile andmean-variance hedging. Indeed, we expected that from the Breusch and Pagantest in regressing Consumption Growth. Moreover, there is not a particularmonotonicity relation between the level of risk aversion and the factor shares inquantile-hedging.





Summary and Highlights

Studied market consistency and sub consistency in incomplete markets.

Type I and II are absolutely different for consistency while for sub-consistencythey look alike.

Introduced hedging and maximal inferior sub-consistency.

Characterizing market consistency and sub-consistency with hedging.

Characterizing maximal inferior market sub-consistent with hedging.


Market Consistent and Sub-consistent Valuation · Introduction to Market Consistent Valuation...

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Transcript of Market Consistent and Sub-consistent Valuation · Introduction to Market Consistent Valuation...