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Hedging Strategies Using Futures

Long & Short Hedges

  • Long futures hedge, you know you will purchase the asset and wish to lock the price
  • Short futures hedge, you know you will sell an asset in the future and wish to lock the price
  • In favor of hedging, Companies should
    • Focus on the main business they are in
    • Take steps to minimize risks from
      • interest rates
      • exchange rates
      • other market variables
  • Against hedging,
    • Shareholders usually are well diversified, can make their own hedging decisions
    • May increase risk to hedge when competitors don't
    • But a situation where
      • loss on hedge
      • gain on the underlying, is difficult to explain

Basis Risk

  • \(\text{Basis} = \text{Spot Price}-\text{Futures Price}\)
    • Sometimes on a financial asset, it is \(\text{Futures Price}- \text{Spot Price}\)
  • Basis risk \(\impliedby\) uncertainty about basis when hedge is closed out

Long Hedge for Purchase of an Asset

  • \(F_{1}:\) Futures price at hedge-setup time
  • \(F_{2}:\) Futures price at asset-purchase time
  • \(S_{2}:\) Asset price at purchase time
  • \(b_{2}:\) Basis at time of purchase
  • Cost of asset = \(S_{2}\). Gain on futures \(F_{2}-F_{1}\)
  • So net amount paid = Cost - Gain = \(S_{2} - (F_{2} - F_{1}) = F_{1} + b_{2}\)
    • \(b_{2} = S_{2} - F_{2}\) (positive basis, profit)

Basis a/o a date means, how much is the spot more than the futures price then. Are you at an advantage or disadvantage, if you want to purchase the asset?

Example

  • June 8
    • Will need to purchase 20,000 barrels of crude oil in Oct or Nov.
    • Futures contract for delivery every month.
      • Contract size = 1000 barrels
    • Takes long position in 20 Dec Futures contracts.
      • \(F_{1} = \$48\) per barrel
  • Nov 10
    • Company ready to purchase crude oil and closed out its futures contract
    • \(S_{2} = \$50\) per barrel
    • \(F_{2} = \$49.1\) per barrel
    • Gain on futures contract: \(49.1 - 48 = 1.1\) per barrel
    • Basis on Nov 10 = \(50 - 49.1 = 0.9\)
    • Effective price paid per barrel \(= 50 - 1.1 = 48.9\)
    • Or \(F_{1} + b_{2} = 48 + 0.9 = 48.9\)
    • Total price paid = \(48.9 \times 20000 = 978,000\)

Think: First paid 48 per barrel. Then the value of that 48 increased to 49.1, so that extra 1.1 per barrel is your gain. So, now instead of an extra $2 per barrel compared to June 8, you only need to pay $0.9 per barrel, which is your basis. So, totally you spent, \(48 + 0.9 = 48.9\), saving $1.1 per barrel (hedged this loss).

Short Hedge for Sale of an Asset

  • Same definitions
Example
  • March 1
    • Expected to receive 50M Jap Yen
    • Yen futures contracts delivery: March, June, Sep, Dec
    • 12.5 M Yen per contract
    • MNC shorts four Sep futures contracts today (march 1)
    • Closes when it receives Yen at July End.
    • \(F_{1} = 1.0800\) cents per yen
  • July End
    • \(S_{2} = 1.0200\) per yen
    • \(F_{2} = 1.0250\)
    • Gain = \(F_{2} - F_{1} = 0.0550\) cents per yen
    • \(b_{2} = S_{2} - F_{2} = - 0.0050\) cents per yen
    • Effective price = \(S_{2} - \text{Gain}\) = \(F_{1} + b_{2}\) cents per yen
      • \(= 1.0800 + (-0.0050) = 1.0750\)
    • Total amount received (because we are in short position)
      • \(= 1.0750 \times 50M = \$537,500\)

Choice of Contract

  • Choose delivery month \(\geq\) end of life of hedge, but closest to
  • If no futures contract on asset being hedged, choose contract whose futures price most highly correlated with asset price.

Optimal Hedge Ratio

  • \(h^* = \rho \dfrac{\sigma_{S}}{\sigma_{F}}\) \(\implies\) Amount of exposure that should be hedged
  • \(\sigma_{S}:\) Std dev of \(\Delta S\), change in the spot price during the hedging period
  • \(\sigma_{F}:\) Std dev of \(\Delta F\), change in the Futures price during the hedging period
  • \(\rho:\) correlation between \(\Delta S\) and \(\Delta F\).

Optimal Number of Contracts

  • \(N^* = \dfrac{h^*Q_{A}}{Q_{F}}\)
  • \(Q_{F}:\) Size of one futures contract.
  • \(Q_{A}:\) Size of position being hedged.

Note, when optimal number of contracts without taking into account daily settlements, the quantities of stocks are taken. But when considering it, we take the value of the stocks. Equity also is a daily settlement case… so we take the value of the portfolio.

Daily Settlement

  • \(N^* = \hat{h} \dfrac{V_{A}}{V_{F}}\)
  • \(V_{A}:\) Value of the position being hedged \(=SQ_{A}\), \(S\) is today's spot.
  • \(V_{A}:\) Value of the Futures price times size of one contract \(=FQ_{F}\), \(S\) is today's spot.
  • \(\hat{\sigma}_{S}:\) Std dev of \(\Delta S\), daily change in the spot price during the hedging period
  • \(\hat{\sigma}_{F}:\) Std dev of \(\Delta F\), daily change in the Futures price during the hedging period
  • \(\hat{\rho}:\) correlation between daily \(\Delta S\) and \(\Delta F\).
  • TAILING THE HEDGE = \(\dfrac{N^*}{(1+i)}\) the interest, \(i\) earned over the remaining life of the hedge. When settlements are made daily
    • Gains from futures position can be reinvested to earn interest
    • Losses must be funded, incurring an interest cost.
    • Timing of cashflows is not captured in the standard model.
    • Multiply \(N^*\) by \(\dfrac{S_{0}}{F_{0}}\)

Portfolio Hedging

V_F 252500
V_A 5050000
h 30 = \(\beta\dfrac{ V_{A}}{V_{F}}\)
After three months
S_2 900
F_2 902
Gain from short futures position $810,000.00 = \(h \times (F_{2}-F_{1}) \times \dfrac{\text{\#shares}}{\text{Contract}}\)
Loss on index -10% \(= \dfrac{S_{2}- S_{1}}{S_{1}}\)
Dividend on index 0.0025 \(= 1\% pa = 0.25\%\) per 3 months
Net return on index over 3 months -9.75% \(= -10\% + .25\%\)
Risk free interest rate 0.01 \(= 4\% pa = 1\% pm\)
Expected return of portfolio -15.13% (CAPM formula)
Expected value of portfolio $4,286,187.50
Expected value of hedgers position $5,096,187.50 (Expected value + Gain from futures)

Changing \(\beta\)

\[ (\text{target } \beta - \text{current } \beta) \times \dfrac{V_{A}}{V_{F}} \]
  • Negative means short
  • Positive means long

Why people hedge equity returns?

  • You want to be out of the market for a while, you have to sell the entire portfolio, then when you want back in, repurchase it. But that causes a lot of transaction costs. Thus, hedging will help avoid that.
  • For fixing your returns at \(\alpha +r_{f}\), if the average \(\beta\) of stocks is 1.0, in both good and bad times.

Stack & Roll

  • Roll future contracts forward to hedge future exposures
  • Initially enter to hedge exposures up to a time horizon
  • Just before maturity, close and replace with new contract reflecting new exposure

Liquidity Issues

  • Danger: losses realized on hedge, while gains on underlying exposures are unrealized. (Low liquidity of underlying exposures)
  • Liquidity Crisis: MGRM spent millions to maintain its hedge, without realizing any of its future gains on paper.