.. slideconf:: :slide_classes: appear ============================================================================== Hedging with Futures ============================================================================== Long and Short Hedges ============================================================================== A hedge is an investment that limits the risk of another risky investment. It typically consists of an offsetting position. .. raw:: - Futures are used to hedge against price fluctuations of an asset that must be bought or sold at a future date. .. raw:: - A long hedge is used when you will purchase an asset in the future. .. raw:: - A short hedge is used when you will sell an asset in the future. Short Hedge Example ============================================================================== Suppose it is May 15th and an oil producer has negotiated to sell 1 million barrels of crude oil on Aug 15. :math:`\smash{S_0 = \$80}` and :math:`\smash{F_0 = \$79}`. .. raw:: - The producer will gain/lose \$10,000 for each 1 cent increase/decrease in the spot price. .. raw:: - A hedge would consist of shorting 1000 futures contracts (for 1000 barrels each) with expiration as close to Aug 15th as possible. .. raw:: - Suppose :math:`\smash{S_T = \$75}` on Aug 15 - what are total profits for the producer? .. raw:: - Suppose :math:`\smash{S_T = \$85}` on Aug 15 - what are total profits for the producer? .. raw:: Note that the producer will not want to deliver the barrels for the futures contract, but will close out early. Long Hedge Example ============================================================================== Suppose it is Jan 15th and a copper producer has negotiated to buy 100,000 pounds of copper on May 15. :math:`\smash{S_0 = 340}` cents and :math:`\smash{F_0 = 320}` cents. .. raw:: - The producer will gain/lose \$1000 for each 1 cent decrease/increase in the spot price. .. raw:: - A hedge would consist of buying 4 CME Group futures contracts (for 25,000 pounds each) with expiration as close to May 15th as possible. .. raw:: - Suppose :math:`\smash{S_T = 320}` cents on May 15 - what is the total cost for the producer? .. raw:: - Suppose :math:`\smash{S_T = 305}` cents on May 15 - what is the total cost for the producer? .. raw:: Note that the producer will not want to take delivery for the futures contract, but will close out early. Imperfect Hedging ============================================================================== It is typically difficult for an investor to exactly hedge a position. .. raw:: 1. A futures contract may not exist for the exact asset to be hedged. - E.g. VIX futures. .. raw:: 2. The expiration of the futures may not coincide with the end of the termination of the position in interest. .. raw:: 3. The termination date of the underlying position may be unknown. Basis ============================================================================== For traders, the basis is defined as the difference between the futures and spot prices. .. raw:: - Traditional definition: :math:`\smash{b_t = S_t - F_t}`. .. raw:: - For financial assets: :math:`\smash{b_t = F_t - S_t}`. .. raw:: - Recall .. math:: \begin{align*} F_t & = e^{c(T-t)} S_t \\ \Rightarrow b_t & = S_t - F_t = \left(1-e^{c(T-t)}\right)S_t. \end{align*} .. raw:: - Note that the basis can be positive or negative. Basis Fluctuation ============================================================================== The basis can fluctuate through time. .. raw:: - The futures price should converge to the spot price at expiry. This is a deterministic change in the basis related to shortening time window :math:`\smash{T-t}`. .. raw:: - The basis may also fluctuate due to random variation in the cost of carry, :math:`\smash{c}`. - This is caused by random fluctuations in :math:`\smash{r}`, :math:`\smash{r_f}`, dividends, storage costs, etc. Stylized Basis Fluctuation ============================================================================== .. image:: Hedging/basis.png :width: 7.5in :align: center Actual Basis Fluctuation ============================================================================== .. image:: Hedging/basisVariation.png :width: 7.5in :align: center Hedging and Basis ============================================================================== Consider an arbitrary asset with the following spot and futures prices at :math:`\smash{t_1}` and :math:`\smash{t_2}`: :math:`\smash{S_1 = \$2.50}`, :math:`\smash{F_1 = \$2.20}`, :math:`\smash{S_2 = \$2.00}` and :math:`\smash{F_2 = \$1.90}`. .. raw:: - :math:`\smash{b_1 = \$0.30}` and :math:`\smash{b_2 = \$0.10}`. .. raw:: - If you hold the asset and plan to sell at :math:`\smash{t_2}`, how can you hedge? .. raw:: - Profit: :math:`\smash{S_2 + F_1 - F_2 = F_1 + b_2 = \$2.30}`. .. raw:: - If you need to purchase the asset at :math:`\smash{t_2}`, how can you hedge? .. raw:: - Cost: :math:`\smash{S_2 + F_1 - F_2 = F_1 + b_2 = \$2.30}`. Basis Risk ============================================================================== Note that the profit/cost of the hedging strategies above is :math:`\smash{F_1 + b_2}`. .. raw:: - If :math:`\smash{b_2}` is known at :math:`\smash{t_1}`, then a perfect hedge could be designed. .. raw:: - Basis change due to :math:`\smash{t_2-t_1}` is perfectly foreseeable. .. raw:: - Basis fluctuation due to random variations in :math:`\smash{c}` is not perfectly foreseeable. Contract Choice ============================================================================== Perfect hedges typically don't exist. Important decisions for the hedge include: .. raw:: - An asset with a futures contract that closely approximates the asset to be hedged. .. raw:: - Expiry close to the necessary terminal date of the hedge. .. raw:: - Typically, expiry is chosen to be a month following hedge termination so that delivery doesn't occur and rolling is unnecessary. Example: Hedging Yen ============================================================================== Suppose it's March 1st and you will receive 50 million Yen at end of July. Yen futures contracts are for delivery of 12.5 million Yen in Mar/Jun/Sep/Dec. .. raw:: - How can you hedge? .. raw:: Assume the following spot and (Sep) futures prices (cents/Yen): :math:`\smash{F_1 = 0.9800}`, :math:`\smash{S_2 = 0.9200}` and :math:`\smash{F_2 = 0.9250}`. .. raw:: - What is the price you pay for Yen? .. raw:: - :math:`\smash{F_1 + b_2 = 0.9800 + (0.9200 - 0.9250) = 0.9750}` per Yen. .. raw:: - For 50 million Yen: :math:`\smash{50,000,000 \times 0.9750 = 48,750,000}` cents or :math:`\smash{\$487,500}`. Cross Hedging ============================================================================== A *cross hedge* occurs when the asset being hedged is different from the asset underlying a futures contract. .. raw:: - Since the two assets may not be perfectly correlated, an adjustment must be made to determine the optimal number of futures contracts to hold. .. raw:: - The optimal number of futures is determined via the *hedge ratio*. Hedge Ratio ============================================================================== The optimal hedge ratio is defined as .. math:: \begin{align} h^* & = \rho \frac{\sigma_S}{\sigma_F} \\ \sigma_S & = Sd(\Delta S) \\ \sigma_F & = Sd(\Delta F). \end{align} .. raw:: - :math:`\smash{h^*}` is the slope of a regression of :math:`\smash{\Delta S}` on :math:`\smash{\Delta F}`. .. raw:: - The hedge ratio is typically computed by estimating :math:`\smash{\rho}`, :math:`\smash{\sigma_S}` and :math:`\smash{\sigma_F}` with historical data. Hedge Ratio with Returns ============================================================================== Note that .. math:: \begin{align} \sigma_{r_S} & = Sd(\Delta S/S) = Sd(\Delta S)/S \\ \sigma_{r_F} & = Sd(\Delta F/F) = Sd(\Delta F)/F \end{align} where :math:`\smash{r_S}` and :math:`\smash{r_F}` are returns (not price changes). .. raw:: - An alternate definition of the hedge ratio is: .. math:: \begin{gather} \tilde{h} = \rho \frac{\sigma_{r_S}}{\sigma_{r_F}} = \rho \frac{\sigma_S/S}{\sigma_F/F} = h^* \frac{F}{S} \\ \Rightarrow h^* = \tilde{h} \frac{S}{F}. \end{gather} Hedge Ratio Regression ============================================================================== .. image:: Hedging/hedgeRatioReg.png :width: 7.5in :align: center Hedge Ratio ============================================================================== Note that if the asset underlying the futures is identical to the asset being hedged: .. raw:: - :math:`\smash{\rho = 1}`, :math:`\smash{\sigma_S = \sigma_F}` and :math:`\smash{h^* = 1}`. .. raw:: If :math:`\smash{\rho = 1}` and :math:`\smash{\sigma_S = 2\sigma_F}`, you need to hedge with two futures. .. raw:: - The assets are perfectly correlated, but price swings in the futures are only half as large as those for the asset being hedged. Optimal Number of Contracts ============================================================================== The optimal number of contracts to purchase for a cross hedge is .. math:: \begin{align} N^* & = h^* \frac{Q_S}{Q_F} = \tilde{h} \frac{S Q_S}{F Q_F} = \tilde{h} \frac{V_S}{V_F}. \end{align} where :math:`\smash{Q_S}` is the size of the position being hedged, :math:`\smash{Q_F}` is the size of a futures contract and :math:`\smash{V_S}` and :math:`\smash{V_F}` are their total valuations (shares times price). Hedge Ratio Example ============================================================================== An airline company needs to purchase 2 millions gallons of jet fuel in 1 month. .. raw:: - CME Group heating oil futures are the best contract to use as a hedge. .. raw:: - One contract is for delivery of 42,000 gallons of heating oil. .. raw:: - The table on the following slide has historical data to estimate the optimal hedge ratio. Hedge Ratio Example ============================================================================== .. image:: Hedging/hedgeRatioData.png :width: 7.5in :align: center Hedge Ratio Example ============================================================================== Using the data: - :math:`\smash{\hat{\rho} = 0.928}`, :math:`\smash{\hat{\sigma}_S = 0.0263}` and :math:`\smash{\hat{\sigma}_F = 0.0313}`. .. raw:: .. math:: \begin{align} \hat{h}^* & = \hat{\rho} \frac{\hat{\sigma_S}}{\hat{\sigma_F}} = 0.928 \frac{0.0263}{0.0313} = 0.78 \end{align} .. raw:: .. math:: \begin{align} \hat{N}^* & = \hat{h}^* \frac{Q_S}{Q_F} = 0.78 \frac{2,000,000}{42,000} \approx 37. \end{align} Hedging an Equity Portfolio ============================================================================== Suppose you want to hedge an equity portfolio which has some sensitivity to the market, :math:`\smash{\beta}`: .. math:: \begin{align} r_p & = r_f + \beta (r_I - r_f) + \epsilon. \end{align} .. raw:: - :math:`\smash{r_p}`, :math:`\smash{r_I}` and :math:`\smash{r_f}` are the portfolio, market index (S\&P 500) and risk-free returns, respectively. .. raw:: - Since :math:`\smash{\beta}` is the slope of the regression of excess returns, it is the hedge ratio, :math:`\smash{\tilde{h}}`. Hedging Equity Portfolio Example ============================================================================== Suppose: - :math:`\smash{P_{SP500} = 1000}`. - :math:`\smash{P_{ES} = 1010}`. - Portfolio Value :math:`\smash{\$5,050,000}`. - :math:`\smash{r_f = 0.04}` per annum. - Index dividend yield :math:`\smash{0.01}` per annum. - :math:`\smash{\beta = 1.5}`. Hedging Equity Portfolio Example ============================================================================== .. image:: Hedging/equityIndexHedge.png :width: 7.5in :align: center Why Hedge? ============================================================================== Note that the index hedge results in a portfolio that earns grows at the risk-free rate. .. raw:: - So why hedge? .. raw:: - Perhaps you think your portfolio will earn positive *non-market* return (alpha) but don't want to be exposed to the market. .. raw:: - Perhaps you want to hold the portfolio for a long period of time, but need a brief reduction of risk exposure. Changing Beta ============================================================================== A complete hedge (as above) makes the effective beta zero. .. raw:: - Suppose you simply want to change the beta of your portfolio to a new value, :math:`\smash{\beta^*}`? .. raw:: .. math:: \begin{align} N^* = (\beta - \beta^*) \frac{V_P}{V_F}. \end{align}