Market Risk Management of Investment Portfolios

Riccardo Rebonato

QUARC - Royal Bank of Scotland

Oxford University

December 2002

Focus on asset allocation choice for investment portfolios

• What questions are we trying answer?• How can one get a fair measure of the risk/reward profile

associated with a large portfolio of assets (eg, Mortgage-Backed Securities) held in an investment portfolio?

What is a reasonable measure to monitor that the risk in a portfolio is consistent with an institution’s tolerance for ‘losses’?

What does one mean by ‘loss’ in the context of an investment portfolio?

My goal

I shall describe in some detail a particular application, but the implications are much wider: they go at the heart of the applicability of a probabilistic approach to risk management for certain important types of portfolio

The Context

The regulators are becoming increasingly interested in measuring and controlling market risk outside the trading books, eg IR exposure in investment portfolios (insurance risk, balance sheet risk), structural IR risk, PE, pension funds, insurance risk…

The ‘default reaction’ has been to embrace the VaR methodology

Limits of the VaR methodology

The VaR methodology and the measures (ES) in its family have theoretical and practical intrinsic limitations:

• Estimation problems with long holding periods• Implicit assumption of market efficiency: the

MTM change only has a privileged status if it truly reflects changes in the ‘fundamentals’

• This is no longer true if supply and demand affect prices (rates)

Further Shortcomings of VaR-based Measures

An important distinction:• Discretionary trading

– The size of trading book positions are at the discretion of the trading desk, subject to limits on the maximum size

• Investment Portfolio– An investment portfolio must invest the

available funds in some form of assets (perhaps cash)

A direct consequence

• Absolute risk limits, based on a monetary loss, are appropriate for trading portfolios

• For investment portfolios, relative risk limits make more sense (‘What would the risk/return trade-off have been if the portfolio had been invested in a reference portfolio?’) – Similar in spirit to the Sharpe ratio.

What are the requisites for a more meaningful analysis?

The approach must explicitly take into account the accrual-accounting nature of the portfolio

The approach must be able to tackle different parts of the balance sheet (deposits, loans, funding, insurance claims) on a consistent basis – what if we cannot MTM one side of the balance sheet?

The proposed methodology must account for IR-induced behaviour (eg, pre-payments) in a realistic way

The (unlikely) possibility of a forced or unplanned unwinding of the portfolio must be taken into account, but should not drive the analysis.

What are the benefits of this analysis?

The ability to see what fluctuations in NII from the whole balance sheet one can expect over a reasonably long time horizon

Consistent treatment of different parts of a balance sheet Gaining a better idea of the effectiveness of an investment

portfolio made up of a given asset class in hedging the other parts of the balance sheet (eg, MBS and deposits)

Comparing the risk/reward profile arising from investing at the margin in MBS, Corporate Bonds, etc.

A framework to establish a control structure which is attuned to the needs and practice of an investment (as opposed to MTM) portfolio: incentives and controls of the portfolio manager are aligned

Preamble

• I do not think that any of the following is of great use in answering these questions:

Mark-to-market-based limits Mark-to-market-only valuation Overly rigid control structures that might force

liquidation at inopportune times VaR-based limits

Why is MTM not appropriate?

• One important difference between holding the same security on a trading book or in an investment portfolio is the liquidity price component.

• It hits directly the trading-book trader (with short-term, VaR-based limits).

• It can be monitored, managed, avoided by the portfolio manager with a sufficiently long time horizon who is not constrained by a MTM-based limit structure.

• Example of Danish MBSs

• If the MTM is not appropriate, the actual cashflows (the NII) become the natural ‘yardstick’.

• This approach also provides the natural connection with other parts of the balance sheet – e.g. deposits – where a MTM measure is virtually impossible.

• However, the NII measure can depend crucially on the chosen time horizon.

• The crucial problem becomes: how should one choose the holding period over which the NII is evaluated in order to get meaningful answers?

[skip]The MTM and the NII measures are intimately linked:

the single number MTM is just ‘some’ average of the NII over all the possible future path. If the NII were calculated all the way to the maturity of the last security in a portfolio, and averaged along the paths, one would obtain an ‘actuarial’ value.

This actuarial value will normally be higher than its MTM value. The extra value has two components:

the compensation for risk (roughly the same for the MTM trader and the portfolio manager)

the compensation for liquidity (only required by the MTM trader).

Caveat: The importance of an appropriate choice

of the reference portfolio• If the reference portfolio is chosen to match

the liabilities (in the limit perfectly), the overall risk and extra reward go to zero keeping the ratio a finite, non-zero number

• However, if the overall portfolio variance is too small, the comparison with a real mismatched portfolio is not very meaningful

An appealing approach (bird’s eye view):

Construct a series of IR paths Along each path construct the NII Observe the distribution of these paths at the final

maturity Exactly as in the VaR case, express our aversion to

losses by specifying a value of cumulative NII + change in MTM that we should only under-perform in X% of the cases.

The ‘irrelevant’ liquidity does not enter the analysis.

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But, will the measure be dominated by the NII (the more relevant information) or by the MTM component (that combines useful information about future NIIs but also contains the ‘noise’ from the liquidity component)?

The answer lies in the choice of time horizon

A pragmatic solutionThe shorter the horizon, the more the measure is dominated by the

MTM changes• The longer the horizon, the greater the impact of what we want

to capture (i.e. the NII)• If the horizon is too long, it does not constitute an effective risk

control• If the horizon is too short, the liquidity element (that we do not

want) can be large.

A pragmatic solution: HORIZON =

DURATION OF THE ASSETS’ PORTFOLIO.

The methodology in more detail

• Step 1: Constructing a series of IR paths

This step must produce future IR paths that incorporate as accurately as possible the information from historical correlated moves of yield curves, collected over a long period of time.

Do we need accurate paths?

These simulations must be accurate because they will be used

to make comparison between different asset classes to gauge risk/reward profiles to establish reasonable bands of variation for the NII

The usual solution

• Do a PCA of the (relative) changes

• Retain m eigenvectors

• Rescale the eigenvalues

• Evolve the yield curve by drawing iid random draws (one for each factor and time step)

Observation 1

• The procedure is valid only if the underlying joint process is a diffusion. I can always orthogonalize a covariance matrix, but I will only get back the original process if it was a joint diffusion to start with.

Observation 2

• By drawing iid random draws that are independent both across factors and serially one is making avery strong assumption about the lack of serial dependence in the data

• I shall show that this assumption is strongly unwarranted

ALL OF THIS MIGHT NOT BE ENOUGH!

QUARC has put together a yield curve simulator that:

• Recovers all the eigenvectors/eigenvalues correctly

• Recovers the unconditional variance of the rates correctly

• Recovers the autocorrelation of rates of different maturity correctly

• Recovers the correct distribution of curvatures across the yield curve

• I will show that even all of this might not be enough for a probabilistic measure of risk

NII (incl. MTM) for a portfolio consisting of a single new Pass-Through valued at $200m, reinvesting into Pass-Throughs at the prevailing mortgage rate with 50bp servicing fee. The Curve Generator uses

historical vols and reversion levels.

Pass-Through NII+MTM distribution, reinvestment in Pass-Through

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Pass-Through

Same NII distribution, expressed as annualised yield

NII(+MTM) Yield for a Pass-Through, Reinvesting in Pass-Throughs

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Net cash distributions in each year for the same simulation

Distributions of net interest flows for single Pass-Through, reinvesting in Pass-Throughs

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Net cash yield distribution in each year for the same simulation

Distribution of net interest yields for single Pass-Through, reinvesting in Pass-Throughs

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NII (incl. MTM) for a portfolio consisting of a new Pass-Through valued at $100m and an at-the-money 5-year treasury of $100m principal, reinvesting into Pass-Throughs at the prevailing mortgage rate with a 50bp servicing fee.

The Curve Generator uses historical vols and reversion levels.

NII(+MTM) for portfolio consisting of a Pass-Through and a 5-year ATM Treasury of $100m value each, reinvesting in Pass-Throughs

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PT+bond

Same NII distribution, expressed as annualised yield

Annualised 5-year yield (incl MTM) for a portfolio consisting of a Pass-Through and a 5-year Treasury of $100m value each, reinvesting in Pass-Throughs

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Net cash distributions in each year for the same simulation

Distributions of ne t inte rest flows for Pass-Through and 5-year Bulle t Bond, re investing in Pass-Through

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Net cash yield distribution in each year for the same simulation

Net cash yie ld for Pass-Through and 5-year bulle t Bond, re investing in Pass-Throughs

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Everything looks fine…

… but does it pass some simple sanity checks?

Problems: Comparing Different Bonds

Par bonds: real word evolution

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Crucial dependence of the results on:

• Reversion levels

• Implied/actual volatilities

How robust are the results to these inputs?

The impact of volatility

• Direct impact on – Option positions– Mortgage-Backed Securities

• Indirect impact at the long end on the curvature of the yield curve

This is the return analysis for various par bonds with risk-neutral reversion levels and unscaled vols.

Par bonds: approximately risk neutral evolution

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Return analysis for various par bonds with rescaled vols.

Par bond Rescaled vols: approximately risk neutral evolution

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The importance of the Reversion Levels: The Race

• Choose a funding strategy• Buy an asset, funded as above• Select an IR path• Record the net cash flows• By asset maturity look at the money left• Current yield curves and the race between spot

rates and forward rates• The speed of the spot rates crucially depends on

the reversion levels

When does the simulation make money?

• The race between the forward rates and the spot rates

• The interplay between reversion speed, reversion level and ‘risk’

• The contribution of risk aversion

How risky is a bond?

• Should I recover with the real-world simulation the market price of a bond? No! Apart from liquidity, there is a risk component (variance of the asset+liability portfolio) that demands compensation.

• But: Whose risk? Can I speak of the risk of a portfolio neglecting its funding? I will recover the market yield compensation (ranking of different bonds) only if my funding is the same as the market’s!

• Portfolio: 5.5-year bond funded with

– 6-month paper

– 2-year notes

– 5-year bonds

• Pension portfolio managers and the price of 30-year Gilts (low yield but, given their liabilities, very low risk in the portfolio)

Possible Applications of the Approach

If a trader truly believes in• the quality of the data• the accuracy of the statistical analysis• the ‘relevance’ of the datathe approach (because it misprices current bonds) can give

useful information about the relative desirability of trading (investment) strategies (relative-value trading)

But, is the approach really appropriate for risk management purposes?

Does it Matter for Risk Management?

Perhaps the exact location of the centres of the distribution is crucial for the trader, but not so important for the risk manager? Not so…

Different time scales than ‘usual’ VaR applications: • if the simulation period is not long, the results are

not meaningful (MBS portfolio)• if the simulation period is long, small differences

in the reversion levels can give rise to very different results

Which distribution is riskier?

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The link between ‘risk’ and reversion level

• The relative location of the two distributions is very sensitive to the difficult-to-estimate (and perhaps non-stationary) reversion levels

• Because of ‘the race’, small changes in reversion levels can create a vastly different relative positioning of the two distributions a very different ‘risk’.

Conclusions So Far

• For an investment portfolio, – the MTM measure is not the most appropriate, and an NII

measure conveys more information– relative rather than absolute risk management makes

more sense

• For the NII measure to be meaningful the holding horizon must be long

• If the holding horizon is long the statistical problems become very large – especially in the ‘return’ component of the equation (cfr. equity premium)

Is there an alternative to the statistical-based approached?

• More and more risk management applications embrace the probabilistic approach

• The IR paths present in the real world and in the risk-adjusted world only differ in their statistical weight

• The problems with the approach above is not that the paths were ‘wrong’, but that their probability of occurrence strongly depended on difficult-to-estimate parameters

A possible alternative

• Specify one or more reference investment alternatives (example of MBS)

• Generate (either using the simulation engine, or by choice, or by combining the two approaches) a series of scenarios which are particularly problematic for the actual portfolio

• Analyse the results (and, possibly, stipulate that the real portfolio should not under-perform the worst of the reference portfolios by more than a certain amount.)

A possible alternative

First scenario - "go down, then go up 4% parallel"

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Fundamental issues

• Results obtained with 10 years’ worth of data – it still might be atypical

• ‘The future looks like the past’• The probabilistic approach with long horizons

suffers from intrinsic data problems• Is a probabilistic approach the best?• The uncertainty principle of risk management

Senior Management and Risk Appetite

• The role of SM is to specify a ‘risk appetite’: a trade-off between possible downside and expected reward

• VaR-based approaches reduce this decision to a percentile choice for a large portfolio in isolation lack of strategic dimension

• A scenario-based analysis can be extended in a coherent manner to different parts of the balance sheet, different businesses in a large Group, different strategic alternatives

• It brings Risk Management at the core of the strategic decisions taken by an institution

• An alternative to the Economic Capital approach