State-of-the-Art in Asset-Liability Management

EDHEC Research Day 2007
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7 juin 2007 - Pôle Universitaire Léonard de Vinci - Paris La Défense
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Research Day 2007
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State-of-the-Art in Asset-Liability
Management
Lionel Martellini, Professor of Finance and Scientific Director of the EDHEC Risk
and Asset Management Research Centre
[email protected]
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Overview
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Introduction
• From AM to ALM
• Fund Separation Theorem in AM
ALM in Institutional Money Management
• A (Very) Brief History of ALM
• Fund Separation Theorem in ALM & LDI Solutions
ALM in Private Wealth Management
• ALM Decisions in Household Finance
• Managing Money with a Goal
ALM in Sovereign State Risk Management
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• From AM to LM
• Optimal Sovereign Debt Management
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Introduction
The Investor’s Standpoint: From AM to ALM
• Typical asset allocation problem (with no consumption):
Max E ( u(AT ))
( wt )t≥0
See Markowitz (1952) for static setting, Hakansson (1969) & Samuelson (1969) for dynamic setting in discrete-time, Merton (1969, 1971) for dynamic setting in continuous-time.
• Asset-liability management models extend the analysis to account for
the presence of liability constraints, so that the focus is not on terminal
wealth, but on terminal wealth relative to liabilities (either in terms of
surplus or funding ratio).
A
Max E u( T L )
Max E ( u(AT − L T ))
(w )
(w )
T
t t≥0
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t t≥0
(
)
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Introduction
Pension Fund Crisis
• What matters is not asset value, but the value of assets compared to
liabilities.
- After the pension fund crisis at the turn of the millennium, the situation has improved, but remains problematic.
- By some estimates (*), the aggregate pension deficit decreased by $14.8 billion during 2005, leaving an aggregate pension
deficit of $96 billion for the 100 companies involved in Milliman pension fund survey 2006.
- 21 out of the 100 companies were in a surplus position at the end of 2005, up from 19 in 2004, 17 in 2003, and only 11 in 2002, although still significantly less than the 90 that had surplus assets
at the end of 1999.
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Introduction
Fund Separation Theorem in AM
• Key ingredients of optimal decisions in AM
- Two-fund separation theorem: in AM, the optimal portfolio involves a combination of the risk-free asset and the risky portfolio with the best risk-return trade-off.
- Optimal AM portfolio strategy:
w* =
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1
(σσ ')−1 ( μ − r1)
γ
- Quid of ALM?
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ALM in Institutional Money Management
A (Very) Brief History of 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 assets.
• Surplus optimization
- 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.
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- AM counterpart: investing in optimal risky portfolio.
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ALM in Institutional Money Management
Fund Separation Theorem in ALM & LDI Solutions
• Optimal portfolio strategy
w* =
⎛
1
1⎞
(σσ ')−1 ( μ − r1) + ⎜ 1 − ⎟ (σ ')−1 σ L
γ
γ⎠
⎝
• We thus obtain a two (three) funds separation theorem
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- The first portfolio is the standard log-optimal efficient portfolio.
- Amount invested is inversely proportional to the investor’s 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.
- This is strongly reminiscent of newly-introduced LDI strategies.
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ALM in Institutional Money Management
Surplus Optimization versus LDI Solutions
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ALM in Institutional Money Management
ALM Decisions in Household Finance
• Most private bankers implicitly promote an ALM approach to wealth
management.
- In particular, they claim to account for the client’s goals and constraints.
- The technical tools involved, however, are often non-existent or
ill-adapted.
• Current practice in addressing clients’ specific goals and constraints is
relatively limited.
- While the client is routinely asked all kinds of questions regarding current situation, goals, preferences, constraints, etc., the resulting
service and product-offering mostly boil down to a rather basic
classification in terms of risk profiles.
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- Private bankers’ real added-value should lie in the ability to account for the client’s specific goals and constraints.
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ALM in Institutional Money Management
ALM Decisions in Household Finance
An absolute return performance, often perceived as a natural choice in
the context of private wealth management, shall not be a satisfactory
response to the needs of a household facing long-term inflation
risk, where the concern is capital preservation in real, as opposed to
nominal, terms.
Similarly, a household whose objective would be related to the
acquisition of a property would accept low and even negative returns
in situations when real estate prices significantly decrease, but will not
be satisfied with relatively high returns if such high returns are not
sufficient to meet a dramatic increase in real estate prices.
In such circumstances, a long-term investment in stocks and bonds
with a performance weakly correlated with real estate prices would not
be the right investment solution.
ALM techniques are best suited for designing investment portfolios
that allow investors to reach specific goals.
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ALM in Institutional Money Management
Real Estate Price Risk
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ALM in Institutional Money Management
Managing with a Goal: Real Estate Acquisition
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Sovereign State Liability Management
The Issuer’s Standpoint: From AM to LM
• Pure asset management problem consists of finding the optimal
allocation to various asset classes so as to generate the optimal
risk-return trade-off.
• From a liability management perspective, the question is to find the
optimal allocation to the various classes of liabilities so as to generate
the optimal trade-off between cost and risk of the debt portfolio.
• Formally, assume a short position of initial size L(0); the goal is to
minimize the terminal value of the debt portfolio for a given level of
uncertainty.
• This objective can be equivalently obtained by:
Max E ( u(−L T ))
(θ t )t≥0
U is some concave utility function
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Min E ( v(L T ))
(θ t )t≥0
V is some convex loss function
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Sovereign State Liability Management
From LM to (a different kind of ) ALM
• The problem can be extended to account for the dynamics of
revenues and expenses generated by the assets to be financed.
• The initial value of the asset is A0, and the terminal value is AT.
• The program is the counterpart of an ALM problem, and reads:
Max E ( u(AT − L T ))
(θ t )t≥0
• More generally
Max E ( u(AT − L T ))
( wt ,θ t )t≥0
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Sovereign State Liability Management
Optimal Debt Management
More specifically, consider an institution (here a sovereign state) that
needs to raise an amount Lo at the initial date so as to finance its initial
deficit.
Subsequently, the state needs to find the optimal allocation to various
classes of liabilities so as finance future primary deficits as well as
servicing of debt, until time-horizon T (assumed to be finite so far), at
which point the institution will have to pay back the residual amount LT.
Obviously, the institution will hope that LT will be as small as possible;
also the risk-averse institution perceives uncertainty around LT as
undesirable.
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Sovereign State Liability Management
Short Versus Long-Term Debt
• If, for example, we think of these liabilities as being debt instruments
such as standard bonds, then the goal is to issue the needed amount of
debt at the initial date while aiming at minimizing its face value.
• In particular, if the debt allocation decision is between (roll-over of)
short-term bonds versus issuing a long-term bond, the risk-cost trade-off
is as follows:
- The yield on long term-bonds is typically higher so that LT will be higher on average with long-term bonds
- On the other hand the long-term bond is safe while roll-over, which is cheaper on average, is risky because of uncertainty over future short-term rates.
• The question is to derive:
- The optimal allocation to the long-term bond and to the roll-over of short-term debt so as to generate the best cost-return trade-off.
- Trade-off between nominal versus real debt needs also to be analyzed.
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Sovereign State Liability Management
Nominal versus Real Debt
• Increasing the share of inflation-indexed bonds leads to a reduction of
the cost of debt since the state is selling insurance against inflation and
receives the associated premium.
• On the other hand, it increases the risk in the cost of debt financing
(uncertainty in coupon payments); this is the standard risk-return trade-off.
• However, this increase in risk effect should be mitigated by the fact that
increases in inflation also lead to increases in the state fiscal revenues.
• In other words, issuing inflation-indexed bonds leads to an increase in
risk from a pure liability perspective but not necessarily from a combined
liability-asset management perspective.
• In this context, it might happen that inflation-indexed debt dominates
nominal debt both in terms of risk and return, and the optimal
composition of a government debt portfolio will be affected accordingly.
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