FINANCIAL INVESTMENTS Faculty:Bernard DUMAS

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FINANCIAL INVESTMENTSFaculty Bernard
DUMAS Hedging and Overlays or Risk
Management session 2-2
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Overview

• Hedging defined
• How is it done?
• Statistical hedging
• Hedge ratio
• Example of hedging with stock index futures

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Hedging defined
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Hedging defined

• A hedger is a person with a pre-existing, given
  position (not to be modified)
• who uses financial instrument (e.g., futures
  contract) to reduce or eliminate a dimension of
  risk in the position
• To hedge to enter transactions that will
  protect against loss through a compensatory price
  movement, Random House dictionary.

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Hedging or risk management

• Pre-existing position may be
• Holding of security (portfolio investor)
• Indirect holding of factor risk (portfolio
  investor)
• Holding of fixed (non traded) asset (corporate
  firm)
• Person could
• Sell off the position
• But suppose that position contains several
  dimensions of risk, some of which are to be kept
• Need to add a security (such as a futures or
  derivative contract) specifically to offset the
  unwanted dimension of risk
• Hedge is accompaniment overlaid on original
  holding
• Hedging is an afterthought
• In portfolio context, logically, hedging
  instrument should have been included in the
  portfolio choice problem in the first place
• Why this would have been better
• In the afterthought approach, the purpose is to
  reduce risk cost of hedging seen as a minor
  consideration
• Would make more sense in the corporate context

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How is it done?
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Payoff profile of a forward contract
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The use of a forward contract to hedge
Payoff
Spot exchange rate /
Forward exchange rate
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Statistical hedging
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Statistical hedging

• To hedge the exact asset underlying the futures
  contract is straightforward
• the optimal hedge ratio for a hedger is to sell
  one futures contract for each present unit of the
  underlying asset that he/she owns
• The purpose of statistical hedging is to discuss
  the optimal hedging policy when the asset you
  want to hedge does not have a hedging instrument
  directly written on it
• Hedge is going to be approximate
• Hedge ratio obtained by statistical procedure
• There will remain a residual risk or basis risk
• For instance hedging instrument has maturity
  shorter than the anticipated holding of the asset

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Statistical hedging

• variance-minimizing hedge ratio
• General idea choose the way you run the
  regression to accommodate your situation
• Regress
• left-hand side Return on asset that you are
  currently holding and that you want to hedge
  (i.e., remove)
• on
• right-hand side Return of the instrument(s)
  used for hedging purposes
• Slope coefficient is hedge ratio
• Obtain futures in an amount equal to the opposite
  of the hedge ratio
• In this way, cancel off the risk

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Statistical hedging

• Hedge ratios come in two forms
• (equations below for situation in which you own
  the spot and use the futures for hedging)
• Absolute
• Dollar price changes ?S ST - S0 ?F FT - F0
• Note LHS should really also include dividend or
  interest return
• S0 initial spot price of asset to be hedged F0
  initial futures price
• ST uncertain asset price at time T FT
  uncertain futures price at time T
• h absolute hedge ratio (number of contracts)
• ? risky residual risk or basis risk or
  nonhedgable risk
• After hedging you will hold
• Relative
• Here ? relative (or ) hedge ratio ( dollar
  amount of contracts per dollar of asset to be
  hedged)

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Hedging with several futures

• OLS regression coefficients provide the
  minimum-variance hedge ratios, so you run the
  multiple regression
• This is a hedge-design tool.
• Include all available instruments in the
  right-hand side
• You implement the hedge only in the dimensions
  you want to hedge
• The right-hand side need not include all the
  causes of fluctuation of the left-hand side.
  There may not be an instrument available to hedge
  all of these causes even if you wanted to.
• Instead, could make use of factor loadings (see
  next lecture).
• Add futures to the portfolio to get total loading
  to become zero.

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Summary on statistical hedging

• Residual risk ? risk that will remain after
  hedging
• When you hedge, you get rid of price risk and you
  keep basis or residual risk.
• A hedge is fully effective only if the futures
  price changes and asset price changes are
  perfectly correlated (zero basis risk)
• Hedging effectiveness is measured by the adjusted
  R-squared from the regression of asset price
  changes on futures price changes

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Estimating hedge ratios

• Price-change interval must be selected (e.g.
  daily, weekly, monthly, etc. price changes)
• Higher frequency implies more information but
  also more noise
• Prices of asset and futures must be simultaneous
• Prices may have seasonalities

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Illustration using SP 500 futures data during a
year

• Suppose we want to hedge 50 million invested in
  the SP 500 portfolio, using SP500 futures
• The two are not perfectly correlated only because
  of dividends and interest rate
• Say that a year has 252 trading days, which
  creates
• 251 daily price changes,
• Or 51 weekly price changes,
• Or 25 bi-weekly price changes
• We use the price changes of the nearby futures
  contract.
• When switching futures contracts, care must be
  taken to splice the price change series correctly

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Illustration using SP 500

• Splicing the futures price series
• daily, weekly and biweekly regressions of
  portfolio return on SP 500 futures data during
  year
• Highly correlated in this example because the
  only difference between the SP500 index and the
  futures written on it comes from uncertainty on
  interim dividends and interest rate

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Illustration using SP 500

• Consider optimal hedge ratio using weekly
  regression 0.9914
• SP 500 index level is 1100 at the beginning of
  the year,
• so number of units of index to be hedged is
  50000000/1100 45454
• Each futures contract is for 500 times the index
  level,
• so number of futures to sell assuming one-to-one
  hedge is 45454 /500 90.91
• Optimal hedge is to sell 90.91 ? 0.9914 90.13
  contracts
• 95 confidence interval can be constructed from
  regression slope estimate confidence interval

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Conclusion on hedging

• Sensitivity measured either as slope of
  regression
• hedging canceling the sensitivity to a source
  of risk