FINANCIAL INVESTMENTS Faculty:Bernard DUMAS

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Université de Lausanne Master of Science Spring
2008
FINANCIAL INVESTMENTSFaculty Bernard
DUMAS Hedging and Overlays or Risk
Management session 6-2
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Overview

• Statistical hedging
• Hedge ratio
• Example of hedging with stock index futures
• Functional hedging (delta hedging dynamic asset
  allocation)
• Options in portfolio management
• Portfolio insurance, risk management and
  guaranteed products

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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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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
  are left with 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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Functional hedging
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Options in portfolio management Ability to
replicate a derivative security

• In the absence of transactions costs and
  extraneous risk, it is possible to replicate any
  derivative profile with a portfolio made up of
• Some amount of riskless investment (possibly
  negative)
• plus some amount of investment in the underlying
  or primitive security
• Consider example of a call.

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Ability to replicate a derivative security
(contd)

• Consider the succession of two points in time

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The Black-Scholes formula

• C value of a call S value of underlying K
  strike price
• N() EXCEL function NORMSDIST

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The Black-Scholes formula
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Price sensitivities

• Delta change in option
  price w.r.t. change in asset price.
• Amount of underlying to be held to replicate
  the option.
• Difference between C and ? ? S is amount B to be
  borrowed.
• Note Gamma change in delta w.r.t. change in
  asset price

Underlying S
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Risk management functional hedging

• Dynamic Asset Allocation, Guaranteed products and
  Portfolio insurance
• The Protective put policy creates a floor or
  guarantee
• Keep the portfolio of assets and
• Buy a put on a basket (you have to sell a bit of
  each asset to finance the option)
• Or replicate the put (i.e., construct it
  yourself)
• Recall that a put option is equivalent to (i.e.,
  can be replicated by) holding a negative variable
  amount of the stock and holding the short-term
  riskless asset
• Or sell off assets, invest most of the money in
  riskless bond and
• Buy a call on a basket
• Or replicate a call on a basket (i.e., construct
  it yourself)
• The two forms of replication lead to identical
  holdings

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Protective put
100
Floor
0
Value of underlying
0
100
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Protective put

• The capital you should have to start with is
• the price of one share (of basket) the price of
  a put on one share,
• (which is equal to the present value of the
  exercise price the price of a call on one
  share)
• You allocate that capital in the following way
• hold ? share(s) of stock basket (this is the ?
  of the call or one plus the ? of the put)
• remainder of the capital in the risk-less asset
• Take note the strategy is self financing.

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Months
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Portfolio insurance

• no initial cost
• floor K
• rate of participation upward ? lt 1

?
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Conclusion on hedging

• Statistical and functional hedging are similar
  concepts. In both cases,
• sensitivity measured either as slope of
  regression or as derivative ??/?S
• hedging canceling the sensitivity to a source
  of risk
• The two types of sensitivities can be combined
  (as in chain rule of calculus)
• Statistical hedge ratio ? delta

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Appendix
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Problems of implementation of portfolio insurance

• Advantages of futures-based replication over
  options
• more liquid
• longer maturities
• rollover is easier
• most options are American type
• desired exercise price may not be available
• If there is a basis risk (e.g., the portfolio to
  be insured is not a market index portfolio),
• use beta of the portfolio to be insured relative
  to the index that underlies the futures contract
  beta?delta
• Problems in either case extraneous risks
• variable rate of interest
• variable dividend yield on the index
• time-varying volatility or jumps

• The trouble with option-based strategies they
  involve a horizon date
• that may or may not suit your needs.
• if it does not, you must roll over the hedge
• Strategy may be implemented
• with options (equity index options)
• or by replication
• in that case, it is most convenient to replicate
  not with spot securities but with index futures.