The Theory and (Best) Practices of

1
Frontiers in Financial Market Mathematics
Bologna, September 13th, 2006

• The Theory and (Best) Practices of
• Liability-Driven Investment (LDI)

Lionel Martellini EDHEC Risk and Asset
Management Research Center lionel.martellini_at_edhe
c.edu www.edhec-risk.com
2
Outline

• Introduction
• A Brief History of ALM
• An Academic Perspective on LDI Solutions
• Dynamic Allocation Strategies in ALM
• A Numerical Illustration
• Non-Linear Payoffs in ALM
• Mathematical Appendix

3

• Introduction
• A Brief History of ALM
• An Academic Perspective on LDI Solutions
• Dynamic Allocation Strategies in ALM
• A Numerical Illustration
• Non-Linear Payoffs in ALM
• Mathematical Appendix

4
Introduction Pension Fund Crisis in the US

• By some estimates (), the under-funding of
  private-sector pension plans is estimated to be
• 450 billion in the United States
• By the same estimates (), the under-funding
  among SP 500 companies alone is 243 billion
  this represents 40 of the estimated profits for
  SP 500 firms for the year 2004.
• For some company, pension deficit is (much)
  larger than market cap consider for example
  United Airlines
• Pension fund deficit 9.8 billion ()
• Market cap 73.45 million

() Richard Berner and Trevor Harris, EBRI/ERF
Policy Forum 55, Morgan Stanley, 6 May 2004.
() Declared by PBGC on 22 Jun. 2005, on a
hearing held by the U.S. House Subcommittee on
Aviation.
5
Introduction Pension Fund Crisis in the UK
FRS17 () deficit for FTSE 100 companies ( bn)
from June 2002
Ratio of assets to FRS17 () liabilities () on
50 companies of the FTSE 100
Source Accounting for Pensions Survey 2005,
2004, 2003, Lane, Clark and Peacock Actuaries
Consultants. () Under new accounting standards
FRS17, assets are valued at market value and
liabilities are discounted with an AA corporate
bond discount rate the surplus or deficit of the
scheme will appear in the balance sheet of the
sponsor company. Before FRS17, the liabilities
were discounted at a fixed rate (transition
period 2001-2005).
6
Introduction Continental Europe is also in
Trouble

• By some estimates (), the overall deficit for
  the European companies in the Dow Jones STOXX 50
  is 116 billion by the end of 2004.
• Germany has the highest deficit with 4.34
  billion.

() Source Annual survey conducted by LCP
(Lane, Clark Peacock Actuaries Consultants).
7
Introduction A Dramatic Shift from Surplus to
Deficit - US

• SP 500 DB Pension Plans
• Net surplus over 200 billion ? Net deficit over
  200 billion in about 3 years

Source SP 500 Pension Status Report, 2005
8
Introduction Origin of the Crisis Actuarial
Aspects

• The increase in life expectancy leads to an
  increase in (present value of) pension costs.
• Life Expectancy for Men and Women at 65
• This however does not explain the fast and
  unexpected shift from surplus to deficit.

9
Introduction Origins of the Crisis Market
Conditions

• Two primary causes
• Perfect storms of adverse market conditions
• Decline in equity markets ? decrease in pension
  plan assets
• Decrease in interest rates ? increase in pension
  liabilities
• Weakness of risk management and asset allocation
  practices
• Could the crisis have been avoided by better
  asset allocation decisions?

10
Introduction Perfect Storm of Adverse Market
Conditions
Base rate in US, Europe and UK
Evolution of SP 500 index
11
Introduction Asset Allocation Risk Management
(Mis)Practices

• Investment in equity
• By 1992, holdings in equities by pension funds
  were 75 in the UK, 47 in the US, 18 in the
  Netherlands and 13 in Switzerland.
• As a result of domination of equities, the
  increase in liability value that followed
  decrease in interest rates was only partially
  offset by the parallel increase in the value of
  the bond portfolio.
• Was a too high investment in equity before the
  start of the bear market the real problem?
• In 2001, midway through the bear market, pension
  funds had 64 of their total assets in equities
  in the UK, 60 in the US, 50 in the Netherlands,
  and 39 in Switzerland.
• Better asset allocation and risk management
  practices would have avoided the pension fund
  crisis, and might help contribute in finding a
  solution to the problem.

12

• Introduction
• A Brief History of ALM
• An Academic Perspective on LDI Solutions
• Dynamic Allocation Strategies in ALM
• A Numerical Illustration
• Non-Linear Payoffs in ALM
• Mathematical Appendix

13
A (Very) Brief History of ALM Different
Approaches to ALM

• Cash-flow matching immunization
• Cash-flow matching involves a perfect match
  between the cash flows from the portfolio of
  assets and the commitments in the liabilities
  inflation-linked instruments are often used in
  that perspective.
• Immunization to the extent that perfect matching
  is not possible, this technique allows the
  residual interest rate risk created by the
  imperfect match between the assets and
  liabilities to be managed in an optimal way.
• AM counterpart investing in risk-free asset.
• Surplus optimization
• In a concern to improve the profitability of the
  assets, and therefore to reduce the level of
  contributions, it is necessary to introduce into
  the strategic allocation asset classes (stocks,
  nominal bonds) that are not perfectly correlated
  with the liabilities.
• AM counterpart investing in optimal risky
  portfolio.

14
A (Very) Brief History of ALM Looking Forward

• Missing ingredients?
• Two-fund separation theorem in AM, the optimal
  portfolio involves a combination of the risk-free
  asset and the risky portfolio with best
  risk-return trade-off.
• Portfolio insurance these strategies generate a
  limited exposure to the risky portfolio when it
  performs well, and a downside risk protection
  when it performs poorly.
• Quid of LDI solutions?

15

• Introduction
• A Brief History of ALM
• An Academic Perspective on LDI Solutions
• Dynamic Allocation Strategies in ALM
• A Numerical Illustration
• Non-Linear Payoffs in ALM
• Mathematical Appendix

16
An Academic Perspective on LDI Solutions Asset
Prices

• We consider n risky assets, the prices of which
  are given by
• A risk-free asset, the 0th asset, is also traded
  in the economy the return on that asset,
  typically a default free bond, is given by
• (W)(W1,,Wn) is a standard n-dimensional
  Brownian motion.

17
An Academic Perspective on LDI Solutions Risk
Premium Process

• Under these assumptions, the market is complete
  and arbitrage-free and there exists a unique
  equivalent martingale measure Q.
• Define
• where ? is the risk premium process
• Then, Z is a martingale, and Q is the measure
  with a Radon-Nikodym density Z with respect to
  the historical probability measure P.
• By Girsanov theorem, the following process is a
  Q-martingale

18
An Academic Perspective on LDI Solutions
Liability Process

• Introduce a separate independent process for
  specific liability risk
• Because of the independence between systematic
  risk exposure and specific liability risk, we
  have that
• with

19
An Academic Perspective on LDI Solutions EMMs
in the P0-Market

• In the presence of liability risk that is not
  spanned by existing securities, the set of all
  measures under which discounted prices are
  martingales, where the risk-free asset is used a
  numeraire, is given by
• with
• where the risk premium for pure liability risk
  is (for sL,e ? 0)

20
An Academic Perspective on LDI Solutions EMMs in
the L-Market

• The liability portfolio is the natural numeraire
  portfolio in this economy the value of assets
  relative to liabilities is the funding ratio, the
  key variable in ALM.
• The dynamics of relative prices is
• which can also be written as
• where we have defined the following processes
  (independent Brownian motions under QL- see next
  slide)

21
An Academic Perspective on LDI Solutions EMMs
in the L-Market (Cont)

• In the L-market, the set of all measures under
  which discounted prices are martingales, where
  the liability portfolio is used a numeraire, is
  given by
• with
• In the L-market, the risk premium are
• For market price
• For pure liability risk (for sL,e ? 0)

22
An Academic Perspective on LDI Solutions
Objective

• Objective
• where the investment policy is a (column)
  predictable process vector representing
  allocations to risky assets, with the reminder
  invested in the risk-free asset.
• Different from standard AM problem (Merton (1969,
  1971))
• In ALM, what matters is not asset value per se,
  but asset value relative to liability value
  (a.k.a. funding ratio), which dynamics is given
  by

23
An Academic Perspective on LDI Solutions
Solution

• Optimal terminal funding ratio value
• Value function
• with

24
An Academic Perspective on LDI Solutions
Solution Cont

• Optimal portfolio strategy
• We thus obtain a two (three) funds separation
  theorem
• The first portfolio is the standard log-optimal
  efficient portfolio.
• Amount invested is inversely proportional to the
  investors Arrow-Pratt coefficient of
  risk-aversion.
• The second portfolio is a liability-hedging
  portfolio it can be shown to have the highest
  correlation with the liabilities alternatively,
  it is a portfolio that minimizes the local
  volatility of the funding ratio.

25
An Academic Perspective on LDI Solutions LDI
Solutions

• Building block 1
• This is cash-flow or duration matching use
  derivatives (interest rates and inflation swaps)
  along with traditional fixed-income instruments
  to manage the interest rate risk
  (immunization/dedication).
• Leverage is usually employed in this leg.
• This part is customized with respect to the
  liability structure of the client.
• Building block 2
• There is a need for risk management, and not only
  asset management, in performance seeking
  portfolio.
• This legitimates the use of absolute return
  strategies and non-linear payoffs hedge funds
  and/or derivatives/structured products.

26
An Academic Perspective on LDI Solutions Sketch
of the Proof Martingale Approach

• The optimization program reads
• 
  such that
• Lagrangian
• FOCs
• 
  

(1)
(2)
27
An Academic Perspective on LDI Solutions Sketch
of the Proof Cont

• From (1), we obtain
• 
  
• Substituting (A-3) into (A-2), we solve for l ,
  which we plug back into (3) to get
• We then obtain the indirect utility function
• and standard calculation of expectation of an
  exponential of a Gaussian variable give the
  announced result

(3)
28
An Academic Perspective on LDI Solutions Sketch
of the Proof Cont

• To obtain optimal portfolio weights, consider
• 
  
• Use Itos lemma to find the dynamics of this
  funding ratio process
• and identify the volatility terms

29
An Academic Perspective on LDI Solutions
Dynamic Programming

• Using the dynamic programming approach, we check
  that
• with
• and f a solution to the non linear Cauchy
  problem
• which is separable in F and can be written as
• with g as before.

30

• Introduction
• A Brief History of ALM
• An Academic Perspective on LDI Solutions
• Dynamic Allocation Strategies in ALM
• A Numerical Illustration
• Non-Linear Payoffs in ALM
• Mathematical Appendix

31
Dynamic Allocation Strategies in ALM Funding
Ratio Constraints

• In the United States, the Pension Benefit
  Guaranty Corporation (PBGC), which provides a
  partial insurance of pensions, charges a higher
  premium to funds reporting a funding level of
  under 90 of "current" liabilities.
• In GB, there was a Minimum Funding Requirement
  (MFR) that came into effect in 1995 the 2004
  Pensions Bill has replaced the MFR with a
  scheme-specific statutory funding objective to be
  determined by the sponsoring firm and fund
  trustees.
• In Germany, Pensionskassen and Pensionsfonds must
  be fully funded at all times to the extent of the
  guarantees they have given.
• In Switzerland, the minimum funding level is
  100, with an incentive to conservative
  management (investment in equities, for example,
  is limited to 30 of total assets for funds with
  less than 110 coverage ratio).
• In the Netherlands, the minimum funding level is
  105 plus additional buffers for investment risks.

32
Dynamic Allocation Strategies in ALM Minimum
Funding Requirement (MFR)

• We now add constraints on the funding ratio, in
  complete market setup.
• Formally, there can be two types of constraints,
  explicit or implicit.
• In a program with explicit constraints, marginal
  indirect utility from wealth discontinuously
  jumps to infinity
• 
  such that
• In a program with implicit constraints, marginal
  utility goes smoothly to infinity at the MFR

33
Dynamic Allocation Strategies in ALM Implicit
Constraints Solution (Complete Markets)

• Consider the following constraint ALM problem
• Optimal terminal funding ratio value
• Value function

34
Dynamic Allocation Strategies in ALM Implicit
Constraints Solution (Cont)

• Optimal portfolio strategy
• We now have a state-dependent, as opposed to
  static, allocation to the two funds.
• The first portfolio is the standard mean-variance
  efficient portfolio.
• Consider the fraction of wealth At allocated to
  the optimal growth portfolio
• It is given by

35
Dynamic Allocation Strategies in ALM Implicit
Constraints Interpretation

• It appears that the fraction of wealth At
  allocated to the optimal growth portfolio is
  equal to a constant multiple m of the cushion,
  i.e., the difference between the asset value and
  the floor defined as At - kLt.
• This is reminiscent of CPPI strategies, which the
  present setup extends to a relative risk
  management context.
• While CPPI strategies are designed to prevent
  final terminal wealth to fall below a specific
  threshold, extended CPPI strategies (a.k.a.
  contingent immunization strategies) are designed
  to protect asset value not to fall below a
  pre-specified fraction of some benchmark value,
  here liability value.

36
Dynamic Allocation Strategies in ALM Explicit
Constraints Solution (Complete Markets)

• Consider the following constraint ALM problem
• 
  such that
• Optimal terminal funding ratio value
• Value function

37
Dynamic Allocation Strategies in ALM Explicit
Constraints Solution (Cont)

• Optimal portfolio strategy
• with
• The fraction of wealth At allocated to the
  optimal growth portfolio

38
Dynamic Allocation Strategies in ALM Explicit
Constraints Interpretation

• Note that
• The replicating portfolio for the payoff kLT
  consists of investing in the liability hedging
  portfolio, which is perfectly correlated to the
  liability portfolio in the complete market
  setting its initial value is kL0.
• Therefore the optimal strategy consists in
  allocating the initial wealth A0 so as to invest
  kL0 in the liability-replicating portfolio and
  invest the remaining wealth, A0 kL0, in an
  option that will deliver the surplus, if any, of
  the value of the unconstrained payoff, i.e., the
  payoff resulting from following the optimal
  strategy from the unconstrained solution.
• This is reminiscent of OBPI strategies, which the
  present setup extends to a relative risk
  management context (exchange option).

39
Dynamic Allocation Strategies in ALM MRP
Requirement Revisited

• The previous framework can be extended.
• Formally
• such
  that
• This setting conveniently nests the previous
  cases
• When a0, we recover the fully constrained case
• When a1, we recover the unconstrained case.
• Alternative one can introduce a condition that
  penalizes both a high probability of a (relative)
  loss, and a high expected (relative) loss when it
  occurs

40
Dynamic Allocation Strategies in ALM Examples of
Implementation

• Approaches similar in spirit have been introduced
  by
• State Street Global Advisors (SSgA) under the
  form of Dynamic Risk Allocation Model (DRAM).
• AXA-IM under the form of Dynamic Contingent
  Immunisation strategies.

DCI in a negative market scenario
Evolution of funding ratio static vs dynamic
Source State Street Global Investor dec 2004
Supplement
Source Pension fund strategy research report
3, AXA IM and BH
41

• Introduction
• A Brief History of ALM
• An Academic Perspective on LDI Solutions
• Dynamic Allocation Strategies in ALM
• A Numerical Illustration
• Non-Linear Payoffs in ALM
• Mathematical Appendix

42
A Numerical Illustration Generating Scenarios

• There are 3 main risk factors affecting asset and
  liability values
• Interest rate risk(s)
• Inflation risk
• Stock price risk
• Example of a standard model ()
• Model calibrated so as to be consistent with
  long-term parameter estimates ()

() Modeling of Economic Series Coordinated with
Interest Rate Scenarios (2004), by Ahlgrim,
D'Arcy and Gorvett ( research project sponsored
by the Society of Actuaries).
43
A Numerical IllustrationStylized Pension Fund
Problem

• Stylized pension fund problem
• We consider a stream of inflation-protected fixed
  payments (normalized at 100) for the next 20
  years.
• To achieve this goal, some initial contribution
  is required.
• One natural solution consists of buying equal
  amounts of zero-coupon TIPS with maturities
  ranging from 1 year to 20 years (liability
  matching portfolio).
• The performance, however, is poor.
• Find present value of liability-matching
  portfolio L(0) we obtain L(0) 1777.15 (rather
  close to 20x100, very expensive!)
• Distribution of surplus at date 20 trivial (no
  possible deficit).
• So as to save on the necessary contribution, it
  is reasonable to add risky asset classes to spice
  up the return.
• There is a (deficit) risk involved, however.

44
A Numerical IllustrationResults ()
all values are given as present values at
starting date (initial investment as of 1777.15
that is the present value of liabilities)
losses relative to L(0) in parentheses
45
A Numerical IllustrationDistribution of Final
Surplus/Deficit
Distribution of final surplus/deficit
46
A Numerical IllustrationALM versus AM
A portfolio efficient in an AM sense is not
necessarily efficient in an ALM sense, and
vice-versa.
47
A Numerical IllustrationDynamic Portfolio
Strategies

• We now turn to dynamic portfolio strategies.
• We consider 4 different implementations of the
  extended contingent immunization method
• Consider stocks as performance portfolio and take
• Case 1 k90, and m4
• Case 2 k95, and m4
• Consider stock-bond portfolio with highest Sharpe
  ratio as performance portfolio
• Case 1 k90, and m4
• Case 2 k95, and m4
• We also test the case of extended OBPI (in which
  case we only consider the stock-bond portfolio
  with highest Sharpe ratio as performance
  portfolio).
• Results show very significant risk management
  benefits.

48
A Numerical Illustration Static versus Dynamic
Strategies
49
A Numerical Illustration Extended OBPI
50

• Introduction
• A Brief History of ALM
• An Academic Perspective on LDI Solutions
• Dynamic Allocation Strategies in ALM
• A Numerical Illustration
• Non-Linear Payoffs in ALM
• Mathematical Appendix

51
Non Linear Payoffs in ALMRisk Management in ALM

• We have argued that an optimal ALM policy
  involves investing in two funds
• A liability hedging portfolio, for risk
  management purposes
• The standard growth optimal portfolio, for asset
  management purposes
• In the presence of funding ratio constraints, the
  optimal solution involve a dynamic trading
  strategy with the standard efficient portfolio
  and the liability hedging portfolio.
• These strategies are reminiscent of CPPI or OBPI
  portfolio insurance strategies which they extend
  to a relative (w.r.t. liabilities) risk context.

52
Non-Linear Payoffs in ALM Hedge Funds and
Structured Products in ALM

• These dynamic asset allocation strategies are
  attractive because they allow for significant
  risk management benefits.
• On the other hand, they involve time-changing
  asset allocations decisions, which can generate a
  possible source of confusion between the ALM and
  AM processes.
• There exists an alternative that consists in a
  static investment in
• Leveraged position in liability-matching
  portfolio
• Performance portfolio containing assets with
  non-linear payoffs.
• Two natural candidates are
• Hedge funds active investment vehicles
• Derivatives and structured products passive
  investment vehicles

53
Non-Linear Payoffs in ALM Using Swaps for
Leveraged Positions in ALM

• Consider a liability with benefits that are
  predominantly inflation-linked.
• Therefore inflation-linked assets would be an
  appropriate match.
• Inflation-linked bonds are cash instruments with
  lowest liability risk.
• An alternative asset and liability match would be
  to
• Buy a risky portfolio of straight bonds or invest
  in absolute return strategies (more on this
  later).
• Use the swaps market to convert the bond
  cashflows to the precise inflation-linked
  cashflows required to pay the schemes projected
  benefit payments.
• Pension scheme pays fixed flows (extracted from
  the portfolio)
• Scheme receives inflation linked cash flows
  tailored to meet the projected liabilities.
• Relative to index-linked gilts this solution
  would
• Provide more precise management of inflation
  risk.
• Introduce leverage, and hence generate a higher
  expected return, due to additional performance of
  risky portfolio but additional risk (in case
  portfolio underperforms the fixed swap rate).

54
Non-Linear Payoffs in ALM Example of an
Inflation Swap

• Buy a diverse portfolio of bonds or invest in an
  absolute return performance portfolio
• Swap the fixed cash flows generated from the
  portfolio for Retail Price Index (RPI) linked
  returns.

RPI-linked cashflows to match pension
payments Notional x (CPPIT/CPPI0)-1)
RPI-linked pension payments
Bank
Fund
Fixed cash-flows extracted from return on bond
portfolio or performance portfolio. Notional x
((1fixed rate)T-1)
55
Non-Linear Payoffs in ALM Illustration

• Overview of a stylized fund (British pension fund)

56
Non-Linear Payoffs in ALM Illustration Cont

• Cash Flow Matching strategy Mechanics of risk
  hedging

Inflation M35.61
_at_ 2.5 inflation
_at_ 5 Interest rate
2026 Cash Flow M34.44
Non inflated M55.76
How to immunize the PV to inflation and interest
rate changes
2026
PV in 2006 (34.44(55.7635.61)/(15)20)
57
Non-Linear Payoffs in ALM Illustration Cont
Pays a single ZC breakeven inflation rate () of
2.9
Swap Counterpart
Pension Fund
Inflation
Infl.Breakeven M43.01
M 98.77
Zero Coupon Bond _at_4.51 M40.88
Non inflated M55.76
Non inflated M55.76
Fund buys in 2006 a ZC (4.51 is current yield on
20Y ZC bond 40.8898.77/(14.51)20)
Fund pays in 2026
Fund must pay in 2026
() Means that yield on 20 years nominal bonds is
290 BPs higher (1.61) than yeild on 20 years
real bond (4.51). This means that inflation
would have to average more than 2.8 per year
until the maturity of the bond for the
inflation-linked bond to do as well as nominal
bond of similar term. Investors do not
necessarily expect inflation to be as high as
2.8, since they do not know what the future will
bring they are willing to sacrifice some current
yield for inflation protection on the principal.
58
Non-Linear Payoffs in ALM Illustration Cont

• Cash Flow Matching strategy with inflation swap
  investor holds ZC swap
• Pension fund is now immunized against interest
  rates and inflation changes PV of future
  obligations is LOCKED
• Unfavorable Inflation increases to 3.5 a year
  and rates fall to 4

MtM Swap M 5.56
M 55.76 inflated _at_3.5 p.a PV M 50.64

M 50.64
ASSETS
Zero Coupon worth _at_4 M45.08
LIABILITIES
Zero Coupon redempts at M 98.77 in 2026
Zero Coupon worth _at_4 M45.08
Fund bought in 2006 a ZC New price is
45.0898.77//(14)20)
5.5655.76(13.5)-55.76(12.9)/(14)20
50.6455.76(13.5)/(14)20
59
Non-Linear Payoffs in ALM Illustration Cont

• Cash Flow Matching strategy
• Favorable Inflation decreases to 2 a year and
  rates increase to 5

ASSETS
LIABILITIES
Zero Coupon redempts at M 98.77 in 2026
Zero Coupon worth _at_5 M 37.23
Zero Coupon worth _at_5 M 37.23
M 55.76 inflated _at_2 PV M 31.23
M 31.23
MtM Swap M -6.00
Fund bought in 2006 a ZC
60
Non-Linear Payoffs in ALM Illustration Cont

• It should be noted that the fund is hedged
  against inflation and interest rate risk but not
  against mortality risk.
• ? high impact on the pension fund cash flow
  profile

61
Non-Linear Payoffs in ALM Another Illustration

• Example Retail Price Inflation (RPI)-linked
  annuities
• Match with inflation-linked assets but only an
  imprecise cashflow match possible with
  index-linked bonds and supply of inflation-linked
  corporate bonds insufficient.

62
Non-Linear Payoffs in ALM Inflation Swap Overlay
for CF Matching

• An inflation swap can be used to exchange
  cash-flows generated by a bond portfolio for
  RPI-linked cash-flows to match the precise nature
  and timing of annuity payments.
• This gives a more precise inflation match than
  with index-linked gilts and allows freedom to
  invest in a wide range of underlying assets.

63
Non-Linear Payoffs in ALM Linear versus non
Linear Payoffs in A(L)M
Linear Exposure / Symmetric Payoff / Static
Allocation Strategies
Active Mutual Funds
ETFs, Passive Mutual Funds
Active Strategies
Passive Strategies
Structured Products
Hedge Funds
Non Linear Exposure / Dissymmetric Payoff /
Dynamic Allocation Strategies
64
Non-Linear Payoffs in ALM Hedge Funds in ALM

• Hedge funds are not needed ingredients in the
  liability matching portfolio cash and
  derivatives fixed-income products (such as
  inflation and interest rate swaps) are already
  doing a fine job.
• On the other hand, they can be very useful
  ingredient of the performance seeking portfolio.
• Because hedge funds have both alpha and beta
  benefits, they can be used both as satellites and
  in the core of the performance-seeking portfolio
• Hedge funds as satellites because they evolve in
  a less regulated environment, they can maximize
  the potential for alpha generation for a given
  level of skill.
• Hedge funds in the core be cause hedge funds are
  not only exposed to traditional risk factors, but
  also to alternative risk factors, suitably
  selected strategies can be used for return
  enhancement and/or risk reduction purposes.
• Moreover, their focus on absolute return and
  their non-symmetric payoffs make them natural
  candidates in the performance-seeking portfolio
  when leverage is used in implementing the
  liability-matching portfolio, making cash-rate
  the new liability-driven benchmark.

65
Non-Linear Payoffs in ALM Active Non Linear
Exposures
With traditional asset classes, negative returns
are (at least) as numerous and large as positive
returns on the other hand, hedge funds returns
offer a non-linear payoff
Active portfolio management with an absolute
return focus implies opposing a risk management
process on the downside to an idea generation
process of the upside
Data from 01/97 to 12/2004. MSCI world index is
used as a proxy for equity Lehman Global
Treasury is used as a proxy for bonds Edhec HF
indices are used as proxies for HFs.
66
Non-Linear Payoffs in ALM Structured Products in
ALM

• In the same vein, structured products with convex
  payoffs also offer asymmetric risk management
  benefits.
• While the range of such products can be rather
  wide, popular example are CPPI, OBPI based on
  standard options, and OBPI based on exotic
  options (e.g., option on the maximum value of the
  underlying asset).
• These structured products can be implemented
  using dynamic asset allocation decisions and/or
  static investment in options (OBPI and E-OBPI).
• These are passive investment strategies the
  focus is on providing the investor with (limited)
  access to the risk premium associated with
  investing in risky assets such as stocks, without
  all the associated risks.

67
Non-Linear Payoffs in ALM Structured Products in
Risk Management

• Generally speaking, two different approaches to
  risk management can be followed.
• Risk diversification, i.e., reducing risk by
  optimal asset allocation techniques on the basis
  of imperfectly correlated assets.
• Risk hedging, i.e., reducing risk by using some
  form of insurance contract (derivative
  instrument) on a given underlying asset, the
  packaging of both the underlying asset and the
  insurance contract being known as a structured
  product.
• Given that allocation and structuration are two
  different, and perhaps competing, forms of risk
  management, whether the benefits of these two
  approaches can be added and combined so as to
  generate even greater risk reduction benefits
  appears to be a rather non-trivial question.
• Structured products, which allow their user to
  achieve a non-linear option-like exposure with
  respect to the return on traditional asset
  classes, are natural investment vehicles for
  institutional investors, who have a particularly
  strong preference for non-linear payoffs because
  of the non-linear nature of the liability
  constraints.
• What is the optimal static allocation to such
  dynamic strategies (structured products)?

68
Non-Linear Payoffs in ALM Methodology
A Focus on Extreme Risks We assume the
investor minimises CVaR (Conditional
Value-at-Risk) for a given level of returns, i.e.
he has a focus on managing extreme risks.
Modelling Financial Markets We account for (a)
Mean reversion of stock and bond returns (b)
Stochastic interest rate and assume the
investor has access to the following asset
classes (i) Stock Market Index (ii) Bond
Index Zero-coupon bond with constant
time-to-maturity. (iii) Guaranteed Structured
Product (GSP), with path dependent payoff

• Example of VaR et CVaR level
• We take an investment with an expected 10-year
  loss of
• maximum 10 in 99 of cases ? VaR(99)10
• on average 12 in 1 of other cases ?
  CVaR(99)12

69
Non-Linear Payoffs in ALM Passive non Linear
Exposures
The graph shows 2500 return scenarios for the GSP
based on paths generated from our model for asset
price dynamics
70
The structured product improves the efficient
frontier considerably. Investors with strong
risk aversion (points 1-3) replace stocks and
bonds by the GSP. Risk-seeking investors
(points 5-9) replace the bonds in their portfolio
with the GSP. Only the most risk-seeking
investors (point 10) would have a zero allocation
to the GSP.
71
Non-Linear Payoffs in ALM Accounting for Weight
Constraints
In practice, institutional investors may not be
willing or allowed to invest a dominant part of
their portfolio in structured products. Adding
even a limited fraction of the overall allocation
to structured products allows for significant
benefits in terms risk adjusted performance
(increase in the ratio of return/CVaR).
Changes in the optimal allocation and the Risk
return tradeoff when the allocation to the GSP is
constrained.
72

• Introduction
• A Brief History of ALM
• An Academic Perspective on LDI Solutions
• Dynamic Allocation Strategies in ALM
• A Numerical Illustration
• Non-Linear Payoffs in ALM
• Mathematical Appendix

73
Mathematical AppendixReferences, Proofs and
Mathematical Precisions

• In what follows are presented more detailed
  references and mathematical results.
• This material illustrates the power of the change
  of numeraire technique, often used in derivatives
  pricing (quanto options, exchange option,
  fixed-income derivatives, etc.) in portfolio
  optimization problems.
• By recognizing that the liability portfolio is
  the natural numeraire in this economy, and by
  considering portfolio value dynamics under the
  associated equivalent martingale measure, we can
  use convex duality techniques to solve the
  optimal allocation problem, with and without MFR
  constraints.

74
Related Literature CT ALM Models

• Merton (chapter 21, 1990)
• Allocation decision of a university endowment
  fund
• Intertemporal hedging demand due to liability
  risk
• Rudolf and Ziemba (JEDC, 1995)
• Allocation decision of a pension fund
• Time-varying opportunity set with currency rates
  as state variables
• Sundaresan and Zapatero (RFS, 1997)
• Asset allocation of a pension fund as well as
  retirement decisions
• Also involves a fund separation theorem

75
Related Literature Other (CT) Related Papers

• Papers on portfolio decision with benchmarking
• Browne (MS, 2000) complete market setting
• Tepla (JEDC, 2001) include constraints on
  relative performance
• Papers on portfolio decision with inflation risk
• Brennan and Xia (JF, 2002) incomplete market
  setting
• Papers on portfolio decisions with minimum target
  terminal wealth
• Grossman and Vila (JB, 1989) rationalize the
  demand for put options
• Cox and Huang (JET, 1989) positive terminal
  wealth constraints
• Basak (RFS, 1995) discuss the equilibrium
  implications
• Grossman and Zhou (JF, 1996) agents only consume
  at terminal date

76
The Model Asset Prices

• We consider n risky assets, the prices of which
  are given by
• A risk-free asset, the 0th asset, is also traded
  in the economy the return on that asset,
  typically a default free bond, is given by
• We assume that the scalar r, the (1xn) (column)
  vector m (mi)i1,,n and the (nxn) matrix s
  (sij)i,j1,,n are progressively-measurable and
  uniformly bounded processes, and that s is a non
  singular matrix that is also progressively-measura
  ble and bounded uniformly.
• (W)(W1,,Wn) is a standard n-dimensional
  Brownian motion.

77
The Model Risk Premium Process

• Under these assumptions, the market is complete
  and arbitrage-free and there exists a unique
  equivalent martingale measure Q.
• Define
• where ? is the risk premium process
• Then, Z is a martingale, and Q is the measure
  with a Radon-Nikodym density Z with respect to
  the historical probability measure P.
• By Girsanov theorem, the following process is a
  Q-martingale

78
The Model Liability Process

• Introduce a separate independent process for
  specific liability risk
• Because of the independence between systematic
  risk exposure and specific liability risk, we
  have that
• with

79
The Model EMMs in the P0-Market

• In the presence of liability risk that is not
  spanned by existing securities, the set of all
  measures under which discounted prices are
  martingales, where the risk-free asset is used a
  numeraire, is given by
• with
• where the risk premium for pure liability risk
  is (for sL,e ? 0)

80
Liabilities as a Numeraire Portfolio EMMs in the
L-Market

• The liability portfolio is the natural numeraire
  portfolio in this economy the value of assets
  relative to liabilities is the funding ratio, the
  key variable in ALM.
• The dynamics of relative prices is
• which can also be written as
• where we have defined the following processes
  (independent Brownian motions under QL- see next
  slide)

81
Liabilities as a Numeraire Portfolio EMMs in the
L-Market (Cont)

• In the L-market, the set of all measures under
  which discounted prices are martingales, where
  the liability portfolio is used a numeraire, is
  given by
• with
• In the L-market, the risk premium are
• For market price
• For pure liability risk (for sL,e ? 0)

82
The Unconstrained Problem Objective

• Objective
• where the investment policy is a (column)
  predictable process vector representing
  allocations to risky assets, with the reminder
  invested in the risk-free asset
• Define by Fw the funding ratio process, i.e.,
  assets at time t relative to the liabilities, for
  an investor following the strategy w starting
  with initial wealth A0 and given initial
  liability value L0

83
The Unconstrained Problem Solution

• Optimal terminal funding ratio value
• Value function
• with

84
The Unconstrained Problem Solution Cont

• Optimal portfolio strategy
• We thus obtain a two (three) funds separation
  theorem
• The first portfolio is the standard log-optimal
  efficient portfolio.
• Amount invested is inversely proportional to the
  investors Arrow-Pratt coefficient of
  risk-aversion.
• The second portfolio is a liability-hedging
  portfolio it can be shown to have the highest
  correlation with the liabilities alternatively,
  it is a portfolio that minimizes the local
  volatility of the funding ratio.

85
The Unconstrained Problem Sketch of the Proof
Martingale Approach

• The optimization program reads
• 
  such that
• Lagrangian
• FOCs
• 
  

(1)
(2)
86
The Unconstrained Problem Sketch of the Proof
Cont

• From (1), we obtain
• 
  
• Substituting (A-3) into (A-2), we solve for l ,
  which we plug back into (3) to get
• We then obtain the indirect utility function
• and standard calculation of expectation of an
  exponential of a Gaussian variable give the
  announced result

(3)
87
The Unconstrained Problem Sketch of the Proof
Cont

• To obtain optimal portfolio weights, consider
• 
  
• Use Itos lemma to find the dynamics of this
  funding ratio process
• and identify the volatility terms

88
The Unconstrained Problem Dynamic Programming

• Using the dynamic programming approach, we check
  that
• with
• and f a solution to the non linear Cauchy
  problem
• which is separable in F and can be written as
• with g as before.

89
The Unconstrained Problem Welfare Gains from
Market Completion

• In the past few years, investment banks have
  started issuing dedicated liability-matching OTC
  derivatives.
• These dedicated derivatives solutions allows for
  a better hedging of investors liabilities
• In practice, they allow for quasi-perfect hedging
  of financial risks, but not of other risks
  (actuarial risks), unless a specific re-insurance
  solution is put into place
• We can estimate the increase in investors
  welfare that would emanate from completing the
  market as
• This measure can be used in an empirical
  exercise.

90
The Constrained SolutionMinimum Funding
Requirement (MFR)

• We now add constraints on the funding ratio, in
  complete market setup.
• There can be two types of constraints, explicit
  or implicit.
• In a program with explicit constraints, marginal
  indirect utility from wealth discontinuously
  jumps to infinity
• 
  such that
• In a program with implicit constraints, marginal
  utility goes smoothly to infinity at the MFR

91
The Constrained Solution Implicit Constraints
Solution (Complete Markets)

• Consider the following constraint ALM problem
• Optimal terminal funding ratio value
• Value function

92
The Constrained Solution Implicit Constraints
Solution Cont

• Optimal portfolio strategy
• We now have a state-dependent, as opposed to
  static, allocation to the two funds.
• The first portfolio is the standard mean-variance
  efficient portfolio.
• Consider the fraction of wealth At allocated to
  the optimal growth portfolio
• It is given by

93
The Constrained Solution Implicit Constraints
Interpretation

• It appears that the fraction of wealth At
  allocated to the optimal growth portfolio is
  equal to a constant multiple m of the cushion,
  i.e., the difference between the asset value and
  the floor defined as At - kLt.
• This is reminiscent of CPPI strategies, which the
  present setup extends to a relative risk
  management context.
• While CPPI strategies are designed to prevent
  final terminal wealth to fall below a specific
  threshold, extended CPPI strategies are designed
  to protect asset value not to fall below a
  pre-specified fraction of some benchmark value.

94
The Constrained Problem Sketch of the Proof

• The Lagrangian reads
• 
  
• FOC
• with l from the budget constraint
• 
  

95
The Constrained Problem Sketch of the Proof
Cont

• To obtain optimal portfolio weights, consider
• 
  
• Use Itos lemma to find the dynamics of this
  funding ratio process
• and identify the volatility terms with

96
The Constrained Soution Explicit Constraints
Solution (Complete Markets)

• Consider the following constraint ALM problem
• 
  such that
• Optimal terminal funding ratio value
• Value function

97
The Constrained Solution Explicit Constraints
Solution Cont

• Optimal portfolio strategy
• with
• The fraction of wealth At allocated to the
  optimal growth portfolio

98
The Constrained Solution Explicit Constraints
Interpretation

• Note that
• The replicating portfolio for the payoff kLT
  consists of investing in the liability hedging
  portfolio, which is perfectly correlated to the
  liability portfolio in the complete market
  setting its initial value is kL0.
• Therefore the optimal strategy consists in
  allocating the initial wealth A0 so as to invest
  kL0 in the liability-replicating portfolio and
  invest the remaining wealth, A0 kL0, in an
  option that will deliver the surplus, if any, of
  the value of the unconstrained payoff, i.e., the
  payoff resulting from following the optimal
  strategy from the unconstrained solution.
• This is reminiscent of OBPI strategies, which the
  present setup extends to a relative risk
  management context (exchange option).

99
The Constrained Problem Sketch of the Proof

• The Lagrangian reads
• 
  
• FOC
• Then
• 
  

100
The Constrained Problem Sketch of the Proof
Cont

• To obtain optimal portfolio weights, consider
• 
  
• Price the option component and use Itos lemma to
  find the dynamics of this funding ratio process
• and identify the volatility terms with