Evaluating Hedging Performance

1
Evaluating Hedging Performance

• Donald Lien
• University of Texas San Antonio
• October 7, 2005

2
Background Papers

• International Review of Financial Analysis, 2004
• Journal of Futures Markets, 2005
• International Review of Economics and Finance,
  forthcoming
• Working papers under review

3
Futures Hedging

• Use futures contracts to reduce the risk in a
  given spot position
• Hedge ratio is defined as the ratio of the
  futures position to the spot position
• Risk is measured by variance, extended Gini-mean
  difference, Value at Risk, expected shortfall,
  lower partial moment (i.e., general downside risk)

4
Calculating Expectations

• Predictable and unpredictable components
• Information set
• Conditional versus unconditional expectation
• Conditional versus unconditional variance

5
Optimal Hedge Ratio

• Naïve one against one
• OLS simple regression method
• VECM generalized hedge ratio
• Random Coefficient Model
• GARCH Model
• Stochastic Volatility Model

6
Hedging Effectiveness

• The percentage reduction of hedged portfolio
  variance from the spot variance
• Equivalence between hedging effectiveness and R2
• Some replace variance by standard deviation
• General definition replaces variance with
  alternative risk measures

7
The Use of Hedging Effectiveness

• It measures the usefulness of the futures
  contract in risk reduction
• It is applied to choose the better hedge strategy
• Ex post comparison sometimes relies upon R2
  instead of actual calculation
• Ex ante comparison updates hedge ratio with a
  window sliding method

8
Frustration with Empirical Findings

• Sample calculations replace population results
• Ex post analysis indicates a better R2 but not
  better hedging effectiveness
• Ex ante comparison supports the dominance of the
  OLS hedge ratio in hedging effectiveness

9
A Diversion Forecasting

• Yt a b Yt-1 ut
• Unconditional forecast Yfu m a/(1-b)
• Conditional forecast Yfc a b Yf-1
• E(Yf Yfu)2 s2/(1-b2) where s2 is the
  variance of ut
• E(Yf Yfc)2 s2 lt E(Yf Yfu)2

10
Some Assumptions

• When the size of the estimation sample is
  sufficiently large, sample estimates of m, a, and
  b are reliable
• When the size of the comparison sample is
  sufficiently large, sample estimate of the MSE is
  reliable
• Assume the model is correctly specified and there
  is no structural change from the estimation
  sample to the comparison sample

11
Forecast Comparison

• The conditional forecast outperforms the
  unconditional forecast in terms of mean squared
  errors
• Mean squared errors are actually conditional and
  unconditional variances

12
Hedging Scenario (IRFA Paper)

• pt a b pt-1 ut, where pt is the change in
  spot prices, i.e., spot return
• ft c dft-1 vt , where ft is the change in
  futures prices, i.e., futures return
• Previous spot and futures prices are included in
  the information set of the hedger

13
Notation

• h is a hedge ratio
• hu is the OLS (i.e., unconditional) hedge ratio
• hc is the conditional hedge ratio
• Su is the variance of ut
• Sv is the variance of vt
• Cuv is the covariance between ut and vt
• Var(.) is the variance
• Cvar(.) is the conditional variance

14
Variance Calculations

• Cvar(pt - hft) Su -2hCuv h2Sv
• Var(pt - hft) Su/(1-b2) -2hCuv/(1-bd)
  h2Sv /(1-d2)
• The following is known
• Var(pt - hft) ECvar(pt - hft) the
  unconditional variance of the conditional mean

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Optimal Hedge Ratios

• hc minimizes Cvar(pt - hft) hc Cuv/Sv
• hu minimizes Var(pt - hft) hu kCuv/Sv,
• where k 1-d2/1-bd
• hc hu when d 0, i.e., the futures price
  follows a random walk
• hc gt hu when b 0, i.e., the spot price follows
  a random walk
• In general, hc gt hu when d(d-b) gt 0

16
Comparison Results

• Conditional hedge ratio is smaller than the
  unconditional hedge ratio only when (i) Both spot
  and futures markets exhibit persistence with
  stronger persistence prevailing in the spot
  market (i.e., b gt d gt 0) (ii) Both spot and
  futures markets exhibit mean reversion with
  stronger reversion prevailing in the spot market
  (i.e., 0 gt d gt b)

17
Hedging Performance

• Var(pt - huft) lt Var(pt - hcft)
• Cvar(pt hcft) lt Cvar(pt huft). That is,
  Var(pt a bpt-1 hcft hcc hcdft-1)
• lt Var(pt a bpt-1 huft huc hudft-1)
• Hedging effectiveness is based upon the sample
  unconditional variance
• We expect hu to outperform hc in both estimation
  and comparison samples

18
Cointegration Framework (JFM paper)

• pt a b pt-1 gpet-1 ut
• ft c dft-1 gf et-1 vt
• et is the difference between the spot and futures
  prices, i.e., the basis
• Higher lag orders of spot and futures returns can
  be included in the spot and futures return
  equations

19
Analytical Results

• Cointegration (ECM) hedge ratio tends to be
  larger than the OLS hedge ratio (JFM, 1996)
• Improvement in hedging performance (using R2) is
  small (QREF 2004)
• The OLS hedge ratio outperforms the ECM hedge
  ratio in post sample unless there is a major
  structural change

20
Empirical Results (IREF paper)

• Examination of 24 commodity futures markets
  validates the analytical predictions
• MV hedge ratio underperforms ECM or naive hedge
  ratios in more than a half of the 24 cases.
  However, a Chow test indicates structural changes
  in these cases

21
Not-yet-published Results

• Unless there is a major structural change, on
  average the OLS hedge ratio would outperform
  GARCH hedge ratios in post sample comparisons

22
Abuse of Hedging Effectiveness

• Except the OLS hedge ratio, most hedge ratios are
  derived with an attempt to minimize conditional
  risk measures (such as conditional variance)
• Hedging effectiveness relies upon unconditional
  risk measures
• It is therefore inappropriate to compare
  different hedge ratios in terms of hedging
  effectiveness

23
Whats Next?

• There is only one method to calculate
  unconditional risk
• Calculations of conditional risk are model based
• How to evaluate conditional risk in the post
  sample?
• Are risk managers considering conditional risk or
  unconditional risk?

24
Possibly Disturbing Results

• The conventionally estimated hedging
  effectiveness is a biased estimator of the true
  hedging effectiveness
• Consequently, the comparisons of the hedging
  effectiveness from various strategies may be
  meaningless
• For the 24 commodity markets, fortunately the
  bias tends to be small although the standard
  deviation is large for some cases

25
Estimation Bias

• Regardless of the hedge strategies, the estimated
  hedging effectiveness incurs a downward bias
• The same result applies when one adopts an
  alternative Sharpe-type performance measure

26
Thank you very much

• Comments and suggestions are most welcome.
  Please contact me at don.lien_at_utsa.edu.