homework 4

苏教译林四年级英语上册教案Unit2 Let’s make a fruit salad 教学目标：1.能初步听懂、会读、会拼写单词a banana、 a grape、a mango、a pineapple；复习水果类单词an orange, a peach,a pear,an apple。

2.能听懂、会说、会读、会运用句型Do you have…?并且会用Yes, I do.和No，I don’t 来回答。

3.初步理解some和any的使用方法。

4.了解西方国家的饮食文化。

5.充分利用学生已有的知识和生活经验，创设生活化的真实情境，引导学生在运用语言中学习语言，在学习新的语言知识后创造性地运用语言。

教学重点：能初步听懂、会读、会拼写单词a banana、 a grape、a mango、a pineapple；复习水果类单词an orange, a peach,a pear,an apple。

2.能听懂、会说、会读、会运用句型Do you have…?并且会用Yes, I do.和No，I don’t 来回答。

3.充分利用学生已有的知识和生活经验，创设生活化的真实情境，引导学生在运用语言中学习语言，在学习新的语言知识后创造性地运用语言。

教学难点：1. 初步理解some和any的使用方法。

教学疑点学生能否理解some和any的使用方法，渗透的语法知识能否完全掌握？教学准备：挂图，卡片，多媒体（PPT），学生自带水果，老师做的水果沙拉教学过程：Step 1 Greeting and warm up1.GreetingT: Hello, boys and girls. How are you ?S: I’m fine, thank you. And you?T: I’m well.2.Warm upPlay a game----Fast response游戏规则：快速闪现问题，抢答例如：What can you see?I like monkeys, What do you like ?Do you like monkeys ?What colour is the …?Step 2 Presentation1.Play a game----Magic bag游戏规则：收口的口袋里装些玩具和水果。

Unit2 Let’s make a fruit salad教学目标：1.能初步听懂、会读、会拼写单词a banana、 a grape、a mango、 a pineapple；复习水果类单词an orange, a peach,a pear,an apple。

2.能听懂、会说、会读、会运用句型Do you have…?并且会用Yes, I do.和No，I don’t 来回答。

3.初步理解some和any的使用方法。

4.了解西方国家的饮食文化。

5.充分利用学生已有的知识和生活经验，创设生活化的真实情境，引导学生在运用语言中学习语言，在学习新的语言知识后创造性地运用语言。

教学重点：能初步听懂、会读、会拼写单词a banana、 a grape、a mango、 a pineapple；复习水果类单词an orange, a peach,a pear,an apple。

2.能听懂、会说、会读、会运用句型Do you have…?并且会用Yes, I do.和No，I don’t 来回答。

3.充分利用学生已有的知识和生活经验，创设生活化的真实情境，引导学生在运用语言中学习语言，在学习新的语言知识后创造性地运用语言。

教学难点：1. 初步理解some和any的使用方法。

教学疑点学生能否理解some和any的使用方法，渗透的语法知识能否完全掌握？教学准备：挂图，卡片，多媒体（PPT），学生自带水果，老师做的水果沙拉教学过程：Step 1 Greeting and warm up1.GreetingT: Hello, boys and girls. How are you ?S: I’m fine, thank you. And you?T: I’m well.2.Warm upPlay a game----Fast response游戏规则：快速闪现问题，抢答例如：What can you see?I like monkeys, What do you like ?Do you like monkeys ?What colour is the …?Step 2 Presentation1.Play a game----Magic bag游戏规则：收口的口袋里装些玩具和水果。

能听懂、会说、会读、会运用句型.can...Ca.you....Yes..can.No..can’t.
能够正确读出辅音字母f的发音, 并自己归纳一些词。
能唱歌曲I can skate and I can jump
2.情感态度目标：能够运用Nice.Cool.Great!Hav..try.Don’.b.sad.等词句。
Step 2 Presentation
1.读一读黑板上的表示动作的单词及赞美性的语言。
2.请同学运用句型Ca.you....Yes..can.No..can’t.来自由问答。
S1.Hello.....Ca.you..... S2.Yes..can..S1.Nice!.Cool./Great./Wonderful!
S.H.ca.pla.football. (出示词组pla.football)
T: What can he do?
S: He can play basketball.(通过预习, 应该有同学能说出来)
T: Wonderful! (出示词组 play basketball, 教学朗读)
观察单词football 足球, foot是脚, ball球, 用脚踢的球, 谓之足球。
T.(做动作..ca.swim.Ca.yo.swim.S.Yes..can..No..can’t.
T..ca.skate.Wha.ca.yo.do.S..can...
T.Wha.abou.you.. S..can....
(老师在与学生对话的过程中, 遇到学生说的已学的动作类的单词, 可以请同学拼一拼, 并板书, 老师针对学生的回答, 可以用Nice! Cool! Great! Wonderful!等语言来进行表达, 同时也为本课的内容服务, 老师可以出示准备好的这些单词的卡片, 直接贴在黑板上。)

Solutions to Homework4FM5021Mathematical Theory Applied to Finance4.15Use the rates in Problem4.14(given in the table below)to value an FRA where you will pay5%for the third year on$1million.Maturity(years)Rate(%per annum)1 2.02 3.03 3.74 4.25 4.5The forward interest rate for the third year with continuous compounding is0.037(3)−0.03(2)3−2=0.051,i.e.5.1%,which measured with annual compounding is equal toe0.051−1=0.05232or5.232%.Since the3-year interest rate is3.7%with continuous compounding,the value of the FRA is1,000,000×(0.05232−0.05)×1×e−0.037×3=$2,078.85.4.16A10-year8%coupon bond currently sells for$90.A10-year4%coupon bond currently sells for$80.What is the10-year zero rate?(Hint:Consider taking a long position in two of the4%coupon bonds and a short position in one of the8%coupon bonds.)A long position in two4%coupon bonds and a short position in one8%coupon bond lead to a cashflow of$90−2×$80=−$70in year0and a cashflow of$200−$100=$100 in year10.Since the coupons cancel out,$100in10years is equivalent to$70today. Therefore,the10-year rate with continuous compounding satisfies100=70e10R.Hence,R=110ln10070=0.0357,i.e.3.57%per annum.5.2What is the difference between the forward price and the value of a forward contract?The forward price of an asset is the price at which you would agree to buy or sell the asset at a future time.The value of a forward contract is zero at the time when it isfirst entered1into.As time passes,the underlying asset price changes,and the value of the contract may become positive or negative.Consider,for example,a forward contract on an investment asset with price S0that provides no income.The forward price F0is given byF0=S0e rT,where r is the risk-free rate,and T is the time to maturity.If K is the delivery price for a contract that was negotiated some time ago,the value of the forward contract f today isf=(F0−K)e−rT=S0−Ke−rT.5.3Suppose that you enter into a6-month forward contract on a non-dividend-paying stock when the stock price is$30and the risk-free interest rate(with continuous compounding)is12%per annum.What is the forward price?The forward price is$30e0.12×0.5=$31.86.5.5Explain carefully why the futures price of gold can be calculated from the spot price and other observable variables whereas the futures price of copper cannot.Gold is an investment asset.Its futures price as a function of the spot price of gold S0,the risk-free rate r,and time to maturity T isF0=S0e−rT.If the futures price is relatively high(F0>S0e−rT),investors will buy gold and short fu-tures contracts.If the futures price is too low(F0<S0e−rT),they will sell or short gold and long futures contracts.Copper is a consumption asset.If the futures price is too high,buying copper and shorting futures contracts is profitable.However,because investors do not in general hold the asset, the strategy of selling copper and buying futures is not available to them.Therefore,there is an upper bound,but not a lower bound,to the futures price.5.6Explain carefully the meaning of the terms convenience yield and cost of carry.What is the relationship between futures price,spot price,convenience yield,and cost of carry?Ownership of the physical commodity may provide benefits that are not obtained by holders of futures contracts.These benefits from owing the physical asset are referred to as the convenience yield provided by the commodity.The cost of carry is the interest that is paid tofinance the asset plus the storage cost less the income earned on the asset.The relationship between the futures price(F0),the spot price(S0),the convenience yield (y),and the cost of carry(c)isF0=S0e(c−y)T,2where T is the time to maturity of the futures contract.5.14The2-month interest rates in Switzerland and the United States are, respectively,3%and8%per annum with continuous compounding.The spot price of the Swiss franc is$0.6500.The futures price for a contract deliverable in2months is$0.6600.What arbitrage opportunities does this create?The theoretical futures price for a contract deliverable in two months is0.6500e(0.08−0.03)×212=$0.6554.The actual futures price is,therefore,too high.An arbitrageur would buy Swiss francs and short Swiss francs futures.5.16Suppose that F1and F2are two futures contracts on the same commod-ity with times to maturity,t1and t2,where t2>t1.Prove thatF2≤F1e r(t2−t1),where r is the interest rate(assumed constant)and there are no storage costs. For the purposes of this problem,assume that a futures contract is the same as a forward contract.Assume thatF2>F1e r(t2−t1).It is easily seen that this situation gives rise to arbitrage opportunitties.An investor can make a riskless profit by1.Taking a long position in a futures contract which matures at time t1.2.Taking a short position in a futures contract which matures at time t2.When thefirst futures contract matures,the asset is purchased for F1using funds borrowed at rate r.It is then held until time t2at which point it is exchanged for F2under the second contract.The costs of the funds borrowed and accumulated interest at time t2is F1e r(t2−t1).A positive profit ofF2−F1e r(t2−t1)is then realized the time t2.This arbitrage opportunity would not exist for a long time. Therefore,F2≤F1e r(t2−t1).5.20Show that equation(5.3)is true by considering an investment in the asset combined with a short position in a futures contract.Assume that all income from the asset is reinvested in the e an argument similar to that in footnotes2and4and explain in detail what an arbitrageur would do if equation (5.3)did not hold.Suppose we buy N units of the asset and the income from the asset is reinvested in the asset.The income from the asset causes the holding of the asset to grow at a continuously compounded rate q.By time T the holding has grown to Ne qT units of the asset.We,3therefore,buy N units of the asset at time zero at a cost of S0per unit,and enter into a forward contract to sell Ne qT units for F0per unit at time T.The cashflows from this strategy areTime0:−NS0Time T:NF0e qTBecause there is no uncertainty about these cashflows,the present value of the time T inflow must equal the time zero outflow when we discount at the risk-free rate.NS0=(NF0e qT)e−rT,i.e.F0=S0e(r−q)T,which is exactly equation(5.3).If F0>S0e(r−q)T,an arbitrageur would borrow money at rate r and buy N units of the asset.At the same time the arbitrageur would enter into a forward contract to sell Ne qT units of the asset at time T.As income is received,it is reinvested in the asset.At time T the loan is repaid and the arbitrageur makes a profit of N(F0e qT−S0e rT).If F0<S0e(r−q)T,an arbitrageur would short N units of the asset investing the proceeds at rate r.At the same time the arbitrageur would enter into a forward contract to buy Ne qT units of the asset at time T.When income is paid on the asset,the arbitrageur owes money on the short position.The investor meets this obligation from the cash proceeds of shorting further units.The result is that the number of units shorted grows at rate q to Ne qT.The cummulative short position is closed out at time T and the arbitrageur makes a profit of N(S0e rT−F0e qT).4。