Hedging Strategies Using Futures 3 Hedging Strategies Using Futures 指導教授：吳銘政 95320005 蔡雅蘋 95320022 鐘于媜 95320008 林佩儀 95320025 林彥伶 95320009 賴枚汮 95320029 羅婉瑜 95320021 陳佩絨 95320030 蔡佳君

目錄 3.1 Basic principles 3.2 Arguments for and against hedging 3.3 Basis risk 3.4 Cross hedging 3.5 Stock index futures 3.6 Rolling the hedge forward Summary

3.1 Basic principles (基本原則) 當個人或公司選擇期貨來避險時，通常目標是盡可能在適當的位置抵銷風險。 Ex:公司考慮到如果將商品價格3個月內全部增加1分，將會有$10,000的收益;但如果將商品價格在3個月內全部減少1分，會有$10,000的損失。困難點在於，公司的財務主管應該使用設計好的空頭避險來抵銷風險。

為了規避現貨價格下跌的風險，在期貨市場建立空頭部位（賣出期貨）的避險策略。 Short Hedges 空頭避險 為了規避現貨價格下跌的風險，在期貨市場建立空頭部位（賣出期貨）的避險策略。 空頭避險又稱為賣出避險(short hedge)，是指由於持有現貨商品，但擔心將來因價格下跌而遭受損失，所以在期貨市場賣出期貨(sell futures)，而由期貨的收益來規避因為現貨價格下跌的風險。空頭避險也可能因為未來會有現貨的長部位，卻擔心未來現貨會下跌，而預先賣出期貨

由於避險者在現貨市場賣空現貨或未來會買入現貨部位，但是擔心如果未來現貨價格上漲將有損失，因此在期貨市場買進期貨多頭避險 Long Hedges多頭避險 多頭避險又稱為買進避險(long hedge)，規避現貨價格上漲，造成未來購買現貨之成本增加的風險，而在期貨市場建立多頭部位（買進期貨）的避險方式。 由於避險者在現貨市場賣空現貨或未來會買入現貨部位，但是擔心如果未來現貨價格上漲將有損失，因此在期貨市場買進期貨多頭避險

3.2 Arguments for and against hedging (支持與反對避險的論點) Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables

Arguments against Hedging Shareholders are usually well diversified and can make their own hedging decisions It may increase risk to hedge when competitors do not Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult

3.3 Basis risk (基差風險) 實務上，基於下列這三點原因: 1.在未來某特定日，該特定資產的市價不一定等於期貨契約價格 2.投資人(風險規避者)買入或賣出該特定資產的日期具有不確定性 3.投資人可能在期貨契約交割日以前必須結清使得投資人利用期貨契約買入或賣出該特定資產仍然存在著風險。因為該風險來自於該特定資產的現貨價格和期貨契約價格的差額--(基差)，且基差不是維持不變的，而是會隨時改變，所以產生了”基差風險”。

Bais=Spot price of asset to be hedged –Futures price of contract used The Basis 基差 基差是期貨價格與現貨價格的差距 Bais=Spot price of asset to be hedged –Futures price of contract used →基差(B)=現貨價格(S)-期貨契約價格(F)

S1 :spot price at time t1 (在時點t1的現貨價格) 基差變化對多頭與空頭避險效果的影響 S1 :spot price at time t1 (在時點t1的現貨價格) S2 :spot price at time t2 (在時點t2的現貨價格) （Final Asset Price ） F1 :futures price at time t1 (在時點t1的期貨價格) （Initial Futures Price） F2 :futures price at time t2 (在時點t2的期貨價格) (Final Futures Price） b1 :basis at time t1 (在時點t1期貨之基差) →b1= S1- F1 b2 :basis at time t2 (在時點t2期貨之基差) →b2 =S2- F2

若基差轉強（例如基差由3變5）-- (b2- b1 )>0 則避險者在時點t2 購入現貨的淨成本將上升，對多頭避險不利 基差變化對多頭避險效果的影響 某一避險者預期將在時點t2買入現貨，那他可以用F1 的價格購入期貨來避險，在時點t2 的期貨部位利潤為 (F2 –F1)，而現貨的買入價格為S2 ，所以避險後在時點t2 購入一單位現貨的淨成本為： →淨成本= S2 – (F2 –F1) = F1 + b2 若基差轉強（例如基差由3變5）-- (b2- b1 )>0 則避險者在時點t2 購入現貨的淨成本將上升，對多頭避險不利 若基差轉弱（例如基差由5變3）-- (b2- b1 )<0 則避險者在時點t2 購入現貨的淨成本將下降，對多頭避險有利

基差變化對空頭避險效果的影響 某一避險者在時點t1擁有現貨並預期在時點t2會將它出售，那他可以用F1 的價格賣出期貨來避險，在時點t2 的期貨部位利潤為 (F1–F2 )，而現貨的出售價格為S2 ，所以避險後在時點t2 出售一單位現貨的淨收益為： →淨收益= S2+ (F1–F2) = F1+ b2 若基差轉強（例如基差由3變5）-- (b2- b1 )>0 則避險者在時點t2 出售現貨的淨收益將上升，對空頭避險有利 若基差轉弱（例如基差由5變3）-- (b2- b1 )<0 則避險者在時點t2 出售現貨的淨收益將下降，對空頭避險不利 結論： 基差轉強→對空頭避險有利 基差轉弱→對多頭避險有利 (P52)

Choice of Contract 使用期貨契約避險之契約選擇 避險者以期貨契約從事避險的目的，主要是想以較小的基差風險來取代較大的現貨價格風險，進而達成減少其所面對的未來不確定風險的目的。 避險者使用期貨契約避險，其效果好壞決定於期貨價格與現貨價格的相關性高低。因此避險者選擇期貨契約需考慮兩個因素: 1.期貨契約標的資產的選擇 2.期貨契約交割月份的選擇

3.4 Cross hedging (交叉避險) 用不同的商品（當然這兩樣商品具有正相關的關係）且到期日不同的商品來進行避險稱之。 當我們發現市場上沒有類似現貨所發行的期貨來避險時，而找另一個現貨價格有正相關，或者是同質性的產品來避開價格上的風險。 舉例來說：首先，假設我們在市 場上沒有12月燕麥的期貨商品。一個種植燕麥的商人，想要找一個能與燕麥價格有正相關的商品來避開風險，這時候我們燕麥商發現了小麥與燕麥的價格有著正相關的關係，就用一買一賣的方式來鎖住價差，避開燕麥風險。這就叫交叉避險。

Calculating the minimum variance hedge ratio ΔS : Change in spot price (現貨價格的改變), S , during a period of time equal to the life of the hedge ΔF : Change in futures price (期貨價格的改變), F , during a period of time equal to the life of the hedg σs : Standard deviation of ΔS (ΔS 的標準差) σF : Standard deviation of ΔF (ΔF的標準差) ρ : Coefficient of correlation between ΔS andΔF (ΔS 和ΔF的相關係數) h＊ : Hedge ratio that minimizes the variance of the hedger‘s position (最小的變動避險比率) h＊=ρσs/σF … (3.1) 如果ρ=1和σs=σF則避險比率h＊=1，表示期貨價格完全反映了現貨價格

Optimal number of contracts QA : Size of position being hedged (units) - 避險部位的多寡 QF : Size of one futures contract (units) - 期貨合約的多寡 N* : Optimal number of futures contracts fot hedging (避險期貨合約的最適數量) N*= h ＊ QA QF … (3.2)

Tailing the hedge N*= h＊ VA VF

3.5 Stock index futures we now move on to consider stock index futures and how they are used to hedge or manage exposures to equity prices. A stock index tracks changes in the value of a hypothetical portfolio of stocks. the weight of a stock in the portfolio equals the proportion of the portfolio invested in the stock. Dividends are usually not included in the calculation so that the index tracks the capital gain/loss from investing in the portfolio. Some indices are constructed from a hypothetical portfolio consisting of one of each of a number of stocks. Other indices are constructed so that weights are proportional to market capitalization

stock indices (股票指數) 一日合同 =250元 * 價值標準普爾 期貨股價指數是現金的結算， 而不是由交付的標的資產。

hedging an equity portfolio stock index futures can be used to hedge a well-diversified equity portfolio. P：current value of the portfolio F：current value of the stocks underlying one futures contract if the portfolio mirrors the index , the optimal hedge ratio, h*, equals 1.0 and equation(3.2) shows that the number of futures contracts that should be shorted is N*=P/A

when b=1, the return on the portfolio tends to mirror the return on the market; when b=2, the excess return on the portfolio tends to be twice as great as the excess return on the market; when b=0.5, it tends to be half as great;and so on. in general, h* = β, so that equation(3.2)gives N*=β (P/F)

Value of index in three months: 900 Futures price of index today: 1,010 Futures price of index in three months: 902 Gain on futures position: 810,000 Return on market: -9.750% Expected return on portfolio: -15.125% Expected portfolio value in three months (including dividends): 4,243,750 Total expected value of position in three months: 5,053,750

Reasons for Hedging an Equity Portfolio Table 3.4 shows that the hedging scheme results in a value for the hedger’s position at the end of the 3-month period being about 1%higher than at the beginning of the 3-month period. There is no surprise here. It’s natural to ask why the hedger should go to the trouble of using futures contracts.

Hedging can be justified if the hedger feels that the stocks in the portfolio have been chosen well. Another reason for hedging may be that the hedger is planning to hold a portfolio for a long period of time and requires short-term protection in an uncertain market situation.

Changing the Beta of a portfolio In the example in table 3.4, the beta of the hedger's portfolio is reduced to zero. sometimes futures contracts are used to change the beta of a portfolio to some value other than zero.

In general, to change the beta of the portfolio from β to β In general, to change the beta of the portfolio from β to β *, where β > β *, a short position in (β - β *)P/F contracts is required. when β > β *, a long position in (β* - β)P/F contracts is required.

Exposure to the Price of an Indidvidual Stock hedging an exposure to the price of an individual stock using index futures contracts is similar to hedging a well-diversified stock portfolio. he hedge provides protection only against the risk arising from marked movements, and this risk is a relatively small proportion of the total risk in the price movements of individual stocks. it can also be used by an investment bank that has underwritten a new issue of the stock and wants protection against moves in the market as a whole.

3.6 Rolling the hedge forward Sometimes the expiration date of the hedge is later than the delivery dates of all the futures contracts that can be used. The hedger must then roll the hedge forward by closed out one futures contract and taking the same position in a futures contract with a later delivery date. Hedges can be rolled forward many times.

Time t1： Shore futures contract 1 Time t2： Close out futures contract 1 Short futures contract 2 Time t3： Close out futures contract 2 Short futures contract 3 ˙ Time tn： Close out futures contract n-1 Short futures contract n Time T： Close out futures contract n

Table 3.5 Data for the example on rolling oil hedge forward. Date Apr. 2007 Sept. 2007 Feb. 2008 June 2008 Oct. 2007 futures price 68.20 67.40 Mar. 2008 futures price 67.00 66.50 July 2008 futures price 66.30 65.90 Spot price 69.00 66.00

Suppose that in April 2007 a company realizes that it will have 100,000 barrels of oil to sell in June 2008 and decides to hedge its risk with a hedge ratio of 1.0. (In this example,we do not make the “tailing” adjustment described in Section 3.4.) The current spot price is $69.

The company therefore shorts 100 October 2007 contracts. In September 2007 it rolls the hedge forward into the March 2008 contracts . In February 2008 it rolls the hedge forward again into the July 2008 contract.

The October 2007 contract is $68.20 $67.40 ( per barrel) (profit of $0.80) The March 2008 contract is $67.00 $66.50 ( per barrel) (profit of $0.50)

The July 2008 contract is $66.30 $65.90 ( per barrel) (profit of $0.40) The final spot price is $66.

The dollar gain per barrel of oil from the short futures contracts is (68. 20-67.40)+(67.00-66.50)+(66.30- 65.90)=1.70

The oil price declined from $69 to $66. Receiving only $1 The oil price declined from $69 to $66. Receiving only $1.70 per barrel compensation for a price decline of $3.00 may appear unsatisfactory. However, we cannot expect total compensation for a price decline when futures prices are below spot price. The best we can hope for is to lock in the futures price that would apply to a June 2008 contract if it were actively traded.

The daily settlement of futures contracts can cause a mismatch between the timing of the cash flows on hedge and the timing of the cash flows from the position being hedged. In situations where the hedge is rolled forward so that it lasts a long time this can lead to serious problems.

Summary

THE END THANK YOU~