**DELPHI DURATION**

*by
Warren Edwardes*

*This article is based on an article published
as:*

**"Why
Delphi has the Answers; Delphi Duration, an A/LM technique",
Apr. 92, Balance Sheet Magazine, London**

__Introduction__

Nearly every Asset and Liability conference
seems to include a passionate discussion between the proponents
of gap analysis, duration, simulation and risk point systems.
This paper describes a technique I call *"Delphi Duration".
*It encompasses the essential features of the various methodologies.

The bank or building society(S&L) treasurer is often placed in an uncomfortable dilemma. He is expected to perform under a variety of conflicting objectives ranging from the enhancement of net interest income to mark to market value maximisation. This lack of consistency in clearly defining the treasurer's mandate has led to the disparate ALM methods listed above. Perhaps the Delphi Duration system could serve the requirements of all monitoring parties.

Marcia Stigum's general definition of interest
rate exposure in *"Managing Bank Assets & Liabilities",
*1983 is worth recalling: "Interest rate exposure is the
uncertainty introduced into a bank's earnings by possible changes
in interest rates." As in her book, I have used *"he"*
to mean person so as to avoid awkward English.

Complete immunization to the perils of fluctuating interest rates is seldom carried out. Most banks, to varying degrees, will take on some interest rate risk so as to enhance profits. However, it is not clear that shareholders believe that banks are proficient at forecasting interest rates and then successfully taking advantage of such forecasts.

Banks have access to detailed data on their own corporate and credit-card loan portfolios. It could be argued that through an analysis of his own book and a continues presence in the money markets a bank Treasurer should be able to consistently outperform the yield-curve.

A loss through an unknown or inadvertent exposure to interest rates is, however, unforgivable. A profit under such circumstances would be regarded as a very low quality earning and not be highly valued by shareholders.

Off balance sheet instruments are now often used to meet customer demand for fixed rate mortgages and fixed rate retail deposit products. The very rapid maturation of the interest rate swaps market is clear from Dr Stigum's 1983 comment: "... if a depositor wants to place 3-year money with a bank at an attractive rate, a bank that is determined to run a matched book will be able to accept that deposit only if it uncovers an acceptable 3-year asset at a fixed rate."

A prudent treasurer should first, subject to the adequacy of his information system, aim to fully hedge his book (on paper at least). Any desired interest rate bet should only then be carried out with the benefit of information on what the bank stands to lose. With the capital requirements attaching to swaps, the treasurer would not, in practice, enter into swaps to hedge his book and then simultaneously reverse them in accordance with his interest rate views. He would effect swaps representing his net requirements. They should, however, be deliberate.

The interest rate gap is the principal amount by which a bank's assets that reprice in a particular time bucket exceed its liabilities that reprice in that same period. It is the next interest rate refixing date on a floating rate note that counts and not the maturity date. The latter is a determinant of liquidity and is to be covered in a separate paper on "Composite Liquidity Scores".

Gap analysts create time buckets for grouping assets and liabilities. These tend to be monthly upto one year and then quarterly, annually, bi-annually and quinquennially. With the low power/price ratio of modern computers weekly or even daily buckets could be used. This would, however, lead to a loss of clarity. Gaps are calculated for each time horizon and then cumulated across time buckets to produce a triangular matrix report.

Essentially, a gap analyst that forecasts a rise in interest rates will seek to have his assets repricing before his liabilities. A positive gap will be created for near term buckets of the bank's asset/liability structure.

Whilst gap management can incorporate predictions of rate turning points, (if such forecasts are plausible), it does not take any notice of current or predicted yield curves. Detailed reports on changes in net interest income or net present value as a result of shocks in interest rates are not provided. Most importantly, gap analysis ignores reinvestment risk and the time value of money.

Gap analysis is useful but in itself not an adequate ALM tool.

Gap analysis provides an overview of the volatility of net interest income. It does this through a series of statistics over various time buckets and groups of time buckets. Duration analysis attempts to encapsulate the sensitivity of the present value of a bank's book to changes in interest rates. It does so in a single statistic.

Duration has its roots in actuarial science
and is commonly used in fixed income asset management. The duration
of a bond is its __weighted average__ life to its next __repricing
date__. It differs from its term to maturity in that in addition
to the term to the principal repayment, the term to the fixed
coupons are taken into account. A 5-year floating rate CD linked
to 6-month LIBOR has a maturity of 5-years but a life for duration
purposes of 6 months. The weights used for duration calculation
purposes are the present values, positive or negative, of each
cash flow. Duration is therefore: "the present-value weighted
average life".

Under Macaulay's duration present values are
calculated using the __same__ average discount rate for all
flows. Duration can also be calculated using discount rates reflecting
the current nominal money market nominal yield to maturity curve.
This can be further improved upon by discounting by the zero coupon
or spot rates pertaining to each cash flow date.

It will be observed that the zero curve tends to be steeper than the nominal curve following the stripping out of the interim cash flows. The Duration(D) of a series of cash flows c(t), can be modified to represent the elasticity of price(P) with respect to changes in interest rates(i).

The price of the flows c(t) is determined by:

**P = tS [c(t) / (1+i)t]**

**D = {tS [c(t) * t / (1+i)t]} / {tS [c(t) / (1+i)t]}**

Differentiating P with respect to i we get:

**dP/di = tS [-c(t) * t / (1+i)(t+1)]**

**= -D * P / (1+i)**

Macaulay's Duration assumes a parallel shift in a flat yield curve. Even assuming the use of the zero coupon yield curve to discount rather than average or nominal yields, Duration is still less than realistic. Yield curves can and do change in slope. Long term rates are usually less volatile than short rates.

Duration calculates the value of a small change in price with respect to a small change in interest rates. Convexity, which represents the second differential of P with respect to i, can also be usedto indicate stability. Both provide objective measurements of interest rate sensitivity.

Simulation has a number of advantages and disadvantages with respect to gap analysis and duration. As the name describes, projections are made of the balance sheet and profit and loss account under various interest rate scenarios: "most likely", declining, flat and rising.

There are a number of simulation models available in the market. Some include Econometric linear regression forecasting techniques. They have near unlimited freedom to project future inflows and outflows of current accounts, savings accounts overdrafts and mortgages. A variety of changes in interest rates are assumed around the bank's forecasts and these are used to project not only the flows themselves but also changes in net interest income and perhaps present values.

Strategies such as the launch of 5-year fixed rate mortgages or option-based savings schemes could be tested using simulation models.

The use of an all-singing, all-dancing simulation model could lead to dangerous over-confidence. The treasurer must guard against spurious accuracy. The system is heavily dependent on the ability to accurately forecast interest rates. Full simulation can provide a useful but very subjective view of the health of the bank's balance sheet.

There are a number of risk point methodologies from the simple to the probabilistic.

As with gap analysis and simulation, assets and liabilities are inserted into time buckets according to their next interest rate repricing date.

A risk point system is an index measure that assigns weights for each time bucket. The weights are then applied to the gaps (denominated in currency) in each bucket to provide a composite gap or risk point statistic.

A simple system assigns weights for each monthly time bucket in the first year from 1 to 12. This assumes that a gap in the twelfth month is twelve times as volatile as a gap in the first month. Even the simplest form of duration suggests that this is incorrect.At the other extreme, risk units are based on the historical volatility of particular instruments such as 3-month LIBOR or 5-year gilts.

The use of historical volatility assumes that it is a good predictor of future volatility. The options market suggests otherwise. Historical volatility is often quite different from implied volatility from traded options. The latter could be used as a market based predictor. Options are not, however, available for every term.

Other than the subjectivity used in determining the volatility calculation period, Risk Points represent a fairly mechanical although objective method of ALM. By the same token it is somewhat rigid and based on debatable assumptions.

Delphi duration was designed to encapsulate the essential features of gap analysis, duration, simulation and risk point measurement. It was also designed to bolt onto existing systems, thereby minimising costs and interaction difficulties.

As with risk points and gap analysis, time buckets are used. Furthermore, Delphi Duration incorporates the calculation of present values. Moreover, a flat yield curve is not used since full information is available about the current yield curve. The current zero coupon curve is calculated from the yield curve and used for discounting. Unlike duration, however, parallel shifts in the curve are not assumed. A range of duration measures is calculated given a variety of shocks to the curve. As with Delphic economic forecasting, each member of ALCO opines on the probability of each new yield curve scenario with the ALCO chairman determining a composite Delphi Duration.

The Delphi is an index measure with the index set at a base of 1 for, say, changes in the time bucket for a term of 1 year. For the base scenario, the present value of a 1% rise in interest rates across the curve is calculated. The present value changes for each monthly time bucket are then converted into an index by dividing by the value of the change in the present value of a flow in the 1-year bucket. This gives a duration measure of 1 for a 1-year flow, less than 1 upto one year and greater than 1 over a year.

Likewise, a constant 1% fall in rates is assumed and Delphis calculated.

Changes in the slope of the curve can be assumed for the other scenarios e.g. a 1% change in the 1-year accompanied by a 2% or 0.5% change in the 1-month rate.

The ALCO members each assign probabilities to the various scenarios and the Chairman, in turn, weights the views of his committee based on their track records. The Delphi Duration weights need not be changed more frequently than the monthly ALCO meeting. Some risk point systems recommend continuous fine-tuning.

The Delphi Duration system could be seen as a simple manageable system that balances objective mathematical purity with the subjective judgment of experienced treasury managers. It avoids excessive forecasting of interest rates. Subjectivity is limited to the question: "If 1-year rates rise (fall) by 1% tomorrow how would you expect the rest of the yield curve to move?" The Treasurer does not have to operate a "black-box" with secret formulae. Simplicity is the key to understanding. Furthermore, the ALCO members are obliged to formulate a view and are actively involved in the decision making process.

A somewhat cynical view would be that the appropriate A/L management system would depend on the performance measurement system in force for the bank in its shareholders' eyes and for the treasurer in terms of his bonus and redundancy trigger point: accrued interest income or mark-to-market.

Issues such as the use of risk-free rates or margin-inclusive rates for discounting have not been covered here. Into which time bucket should one place low grade Commercial Paper, VRNs or overdrafts? Systems tend to analyse the principal amounts being rolled over. They should also allow for the flows of interest.

Last year, a regular eminent speaker on the subject pointed out that a view on rates is insufficient. He stressed the importance of the view in relation to the ruling yield curve. A paying punter asked him if rates were expected to fall what should he do? The questioner replied to the guru saying that the yield curve was flat - short rates were identical to medium term rates. Embarrassingly, the expert ignored the difference between nominal rates and annualised rates - the compounding effect. He recommended a lengthening of investment maturities!

Positive action does not demand accurate forecasting of interest rates. It does not depend on a view in relation to current rates. It does, however, require a judgment of rates in relation to market-implied forward/forward rates, FRAs or futures. Notwithstanding the above, I hope I have not over-pontificated lest I fall into a trap similar to that of the eminent guru.

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