衍生品Ch04翻译.pdf

Chapter 4 第四章第四章 Interest Rates 利率利率 1 Types of Rates 利率种类利率种类 Treasury rates 国债利率 LIBOR rates 伦敦银行同业拆出利率 Repo rates 再回购利率 2 Treasury Rates 国债利率国债利率 Rates on instruments issued by a government in its own currency 国债利率是政府发行的以自身货币为计价单位的金融 产品的利率 短期国库券 T Bills 中期国库票据 T Notes 长期 国库债券 T Bonds 目前美国联邦政府发行的国债总额约为10万亿美元 其中约30 由外国政府持有 3 三个月美国国债利率三个月美国国债利率 4 10年期美国国债利率年期美国国债利率 5 Treasury Rates 国债利率国债利率 6 LIBOR and LIBID 英国银行同业拆出 入 利率英国银行同业拆出 入 利率 LIBOR is the rate of interest at which a bank is prepared to deposit money with another bank The second bank must typically have a AA rating LIBOR指一家银行准备给将资金存入另一家银行时的 利率 第二家银行必须有AA评级 LIBOR is compiled once a day by the British Bankers Association on all major currencies for maturities up to 12 months 每天有英国银行家协会都会编制基于主要货币的到期 时间至多12个月的LIBOR利率 7 LIBOR and LIBID 英国银行同业拆出 入 利率英国银行同业拆出 入 利率 LIBID is the rate which a AA bank is prepared to pay on deposits from anther bank LIBID是AA评级银行向其它银行存款支付的利 率 欧洲市场利率 不容易受某个国家操纵 8 Repo Rates 再回购利率再回购利率 Repurchase agreement is an agreement where a financial institution that owns securities agrees to sell them today for X and buy them bank in the future for a slightly higher price Y 再回购协议指持有证券的金融机构同意今天以X价格将 证券出售给银行 并且以一个稍高的价格Y将证券买回 The financial institution obtains a loan 金融机构获得贷款 The rate of interest is calculated from the difference between X and Y and is known as the repo rate 由证券卖出与买回差价计算的利率称为再回购利率 9 联邦基金利率联邦基金利率 Federal Fund Rate 联邦基金利率是指美国同业拆借市场的美国同业拆借市场的利率利率 其最主 要的隔夜拆借隔夜拆借利率利率 美国联邦储备系统 联储 各会员银行为调整准备金 头寸和日常票据交换轧差而相互拆放联邦基金的利 率 联邦基金的借贷以日拆为主 利率水平由市场资金供 求决定 成为美国金融市场上最重要的短期利率 是反映货币 市场银根松紧最为敏感的指示器 联邦公开市场委员会的成员每年约每七个星期进行例 会议定联邦基金目标利率联邦基金目标利率 10 隔夜指数互换利率隔夜指数互换利率 OIS Overnight Indexed Swaps Rates 是一种将隔夜利率交换成为若干固定利率的利率互换 OIS 的期限通常不会超过一年 也有一个月 一周甚 至几天 在正常的市场情况下 隔夜指数掉期往往低于 Libor 英国银行间同业拆借利率 Libor OIS利差反应 信贷市场困难程度 2008年10月 11 The Risk Free Rate 无风险利率无风险利率 The short term risk free rate traditionally used by derivatives practitioners is LIBOR 传统上 LIBOR利率被衍生品从业者视作短期无风险 利率 The Treasury rate is considered to be artificially low for a number of reasons See Business Snapshot 4 1 由于一些原因 国债利率被认为是人为地低 短期国债被高估 安全性 流动性 监管需求 税务 优惠 甚至有可能出现负利率 12 The Risk Free Rate 无风险利率无风险利率 As will be explained in later chapters 以后解释 Eurodollar futures and swaps are used to extend the LIBOR yield curve beyond one year 欧洲美元期货和互换可以用来扩展LIBOR收益率曲线至一年 以上 The overnight indexed swap rate is increasingly being used instead of LIBOR as the risk free rate 隔夜指数互换利率 OIS 相对于LIBOR而言日益多被用作无风险 利率 13 Measuring Interest Rates利率的计量利率的计量 The compounding frequency used for an interest rate is the unit of measurement 计量的单位是利率的复利频率 The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers 一年复利4次和一年复利1次的差异类似于英里 和千米的差异 14 Impact of Compounding复利的影响复利的影响 When we compound m times per year at rate R an amount A grows to A 1 R m m in one year 如果利率R每年复利m次 A在一年里增长为A 1 R m m 15 Compounding frequency Value of 100 in one year at 10 Annual m 1 110 00 Semiannual m 2 110 25 Quarterly m 4 110 38 Monthly m 12 110 47 Weekly m 52 110 51 Daily m 365 110 52 Continuous Compounding连续复利连续复利 In the limit as we compound more and more frequently we obtain continuously compounded interest rates 复利频率无限大我们得到了连续复利利率 100 grows to 100eRT when invested at a continuously compounded rate R for time T 如过按连续复利利率R计算 100 在期限T内增长为 100e RT 100 received at time T discounts to 100e RT at time zero when the continuously compounded discount rate is R 如果按连续复利折现率R折算 T期获得的 100折现到0期 为 100e RT 16 Conversion Formulas 转换公式转换公式 Define 定义 Rc continuously compounded rate 连续复利利率 Rm same rate with compounding m times per year 每年复利m次利率 17 Rm R m Rm e c m m Rm c ln 1 1 Examples 例子例子 10 with semi annual compounding is equivalent to 2ln 1 05 9 758 with continuous compounding 每半年复利的10 年利率等价于连续复利的9 758 年利率 8 with continuous compounding is equivalent to 4 e0 08 4 1 8 08 with quarterly compounding 连续复利的8 年利率等价于每季度复利的8 08 年利率 Rates used in option pricing are nearly always expressed with continuous compounding 期权定价中的利率几乎总是用连续复利来表示 18 Zero Coupon Rates 零息利率零息利率 A zero rate or spot rate for maturity T is the rate of interest earned on an investment that provides a payoff only at time T T年期的零息利率 即期利率 是指在投资仅 在T期末获得支付的利率 19 Example Table 4 2 page 81 例子例子 20 Maturity years Zero rate cont comp 0 5 5 0 1 0 5 8 1 5 6 4 2 0 6 8 Bond Pricing 债券定价债券定价 To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate 按某一合适零息利率贴现每期现金流来计算债券的价格 In our example the theoretical price of a two year bond providing a 6 coupon semiannually is 在我们例子 息票率6 每半年付息一次的2年期的债 券的理论价格为 21 333 1039839 0 05 0 50 058 1 00 064 15 0 068 2 0 eee e Bond Yield 债券收益率债券收益率 The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond 债券的收益率是使债券现金流的现值等于其市 场价格的贴现率 Suppose that the market price of the bond in our example equals its theoretical price of 98 39 假设在我们例子中债券的市场价格等于它的理论价格 22 Bond Yield 债券收益率债券收益率 The bond yield continuously compounded is given by solving 债券收益率 连续复利 由下式给出 to get y 0 0676 or 6 76 23 3331039839 0 510152 0 eeee yyyy Par Yield 平价收益率平价收益率 The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value 对于具有一定期限的债券平价收益率是使得债券价格 价格等于其面值的券息率券息率 In our example we solve 在我们的两年债券例子中 我们解下面 24 g compoundin semiannual with 876 get to 100 2 100 222 020680 51064001058050050 c e c e c e c e c Par Yield continued 平价收益率平价收益率 一般 如果每年息票支付m次 1债券到期支付的现值为d 1息票日支付年金的现值为A 在我们例子中 m 2 d 0 87284 A 3 70027 25 Data to Determine Zero Curve 确定零息利率的数据确定零息利率的数据 26 Bond Principal Time to Maturity yrs Coupon per year Bond price 100 0 25 0 97 5 100 0 50 0 94 9 100 1 00 0 90 0 100 1 50 8 96 0 100 2 00 12 101 6 息票每半年支付一次 前三个月零息利率 10 127 前半年10 649 第一年10 536 The Bootstrap Method 息票剥离法息票剥离法 An amount 2 5 can be earned on 97 5 during 3 months 三个月可以从97 5元中赚取2 5元 Because 100 97 5e0 10127 0 25 the 3 month rate is 10 127 with continuous compounding 因为100 97 5e0 10127 0 25 连续复利下 三个月利率 为10 127 Similarly the 6 month and 1 year rates are 10 469 and 10 536 with continuous compounding 同样 连续复利下 6个月期和一年期利率分 别为10 469 和10 536 27 The Bootstrap Method continued 息息票剥离法 续 票剥离法 续 To calculate the 1 5 year rate we solve 为了计算1 5年期利率 我们解 to get R 0 10681 or 10 681 得到R 0 10681 或 10 681 Similarly the two year rate is 10 808 同样 两年期利率为10 808 28 9610444 5 10 110536 05 010469 0 R eee Zero Curve Calculated from the Data 数据计算的零息利率曲线数据计算的零息利率曲线 Figure 4 1 page 84 29 9 10 11 12 00 511 522 5 Zero Rate Maturity yrs 10 127 10 469 10 536 10 681 10 808 Forward Rates 远期利率远期利率 30 The forward rate is the future zero rate implied by today s term structure of interest rates 远期利率是由今天利率期限结构所蕴含的未来 的零息利率 确定的 Forward Rates 远期利率远期利率 31 第一年零息利率3 前两年零息4 存两年 先存第一年 再存第二年 108 33 那么这两个零息利率隐含了合理的第二年的零 息利率是多少 从现在看 Formula for Forward Rates 远期利率公式远期利率公式 Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded 假设T1和T2时的零息利率 连续复利 为R1和R2 The forward rate for the period between times T1 and T2 is T1到T2时期的远期利率为 This formula is only approximately true when rates are not expressed with continuous compounding 如果不是连续复利 上面公式仅是近似正确 32 R TR T TT 2211 21 Formula for Forward Rates 远期利率公式远期利率公式 公式推导 33 222111 100100 T TTRRTR eee F Application of the Formula 公式应用公式应用 34 Year n Zero rate for n year investment per annum Forward rate for nth year per annum 1 3 0 2 4 0 5 0 3 4 6 5 8 4 5 0 6 2 5 5 5 6 5 Instantaneous Forward Rate 瞬时远期利率瞬时远期利率 The instantaneous forward rate for a maturity T is the forward rate that applies for a very short time period starting at T It is 期限为T的瞬时远期利率是从T期开始的一个很短期限 的远期利率 where R is the T year rate 这里R是T年期收益率 35 RT R T Upward vs Downward Sloping Yield Curve 向上向上vs向下倾斜收益率曲线向下倾斜收益率曲线 For an upward sloping yield curve Fwd Rate Zero Rate Par Yield 向上倾斜的收益率曲线 远期利率 零息利率 平价利率 收益率 For a downward sloping yield curve Par Yield Zero Rate Fwd Rate 向下倾斜的收益率曲线 平价利率 收益率 零息利率 远期利率 36 Forward Rate Agreement 远期利率协议远期利率协议 37 A forward rate agreement FRA is an OTC agreement that a certain rate will apply to a certain principal during a certain future time period 远期利率协议是一种场外交易 这种交易约 定在将来某一时间以某一利率借入或借出某一 数量的本金 Forward Rate Agreement Key Results 远期利率协议 重要结果远期利率协议 重要结果 An FRA is equivalent to an agreement where interest at a predetermined rate RK is exchanged for interest at the market rate 利率互换协议等价于一个用事先确定的利率RK交换市场 利率RM的协议 An FRA can be valued by assuming that the forward LIBOR interest rate RF is certain expected to be realized 通过假定远期LIBOR利率RF 能够实现来给FRA定价 38 Forward Rate Agreement Key Results 远期利率协议 重要结果远期利率协议 重要结果 This means that the value of an FRA is the present value of the difference between the interest that would be paid at interest at rate RF and the interest that would be paid at rate RK 这意味着FRA的价值是支付利率分别为RF 和 RK 的利息差的现值 39 Valuation Formulas 定价公式定价公式 If the period to which an FRA applies lasts from T1 to T2 we assume that RF and RK are expressed with a compounding frequency corresponding to the length of the period between T1 and T2 如果FRA从T1 持续到T2 我们假设RF和RK为由与T1 到 T2 期限长度对应的的复利频率计算所得 With an interest rate of RK the interest cash flow is RK T2 T1 at time T2 利率为RK 在T2 时利息现金流为RK T2 T1 With an interest rate of RF the interest cash flow is RF T2 T1 at time T2 利率为RF 在T2 时利息现金流为RF T2 T1 40 Valuation Formulas 定价公式定价公式 When the rate RK will be received on a principal of L the value of the FRA is the present value of 如果收到本金为L的利率RK 则FRA的价值为如下现值 received at timeT2 When the rate RK will be paid on a principal of L the value of the FRA is the present value of 如果支付本金为L的利率RK 则FRA的价值为如下现值 received at time T2 41 12 TTRR FK 12 TTRR KF Example 例子例子 An FRA entered into some time ago ensures that a company will receive 4 s a on 100 million for six months starting in 1 year 一个FRA合约使得一家公司在一年后收入6个月期限 固定利 率4 本金为1亿美元的现金流 Forward LIBOR for the period is 5 s a 同期的LIBOR远期利率为5 The 1 5 year rate is 4 5 with continuous compounding 1 5年期连续复利利率为4 5 The value of the FRA in millions is FRA的价值 百万美元 是 42 467050050040100 510450 e Duration 久久期期 Duration of a bond that provides cash flow ci at time ti is 在ti 提供现金流ci 债券久期是 where B is its price and y is its yield continuously compounded B 是价格 y 是收益率 连续复利 43 B ec tD i yt i n i i 1 Duration 久久期期 44 Key Duration Relationship 重要久期公式重要久期公式 Duration is important because it leads to the following key relationship between the change in the yield on the bond and the change in its price 久期很重要因为它导致了下面关于收益率变化和价格 变化之间重要的公式 久期衡量债券价格对利率的敏感度 45 yD B B Key Duration Relationship 重要久期公式重要久期公式 46 Key Duration Relationship continued 重要久期公式重要久期公式 续 续 When the yield y is expressed with compounding m times per year 如果y是一年复利m次的收益率 The expression 表达式 is referred to as the modified duration 被称作为 修正久期 47 my yBD B 1 D y m1 Key Duration Relationship continued 重要久期公式重要久期公式 续 续 48 Bond Portfolios 债券债券组合组合 The duration for a bond portfolio is the weighted average duration of the bonds in the portfolio with weights proportional to prices 债券组合的久期是构成债券组合每一个债券的久期的 加权平 均 其权重与相应债券价格成比例 The key duration relationship for a bond portfolio describes the effect of small parallel shifts in the yield curve 债券组合久期关系描述的是收益率曲线一个小的平行移动的影响 What exposures remain if duration of a portfolio of assets equals the duration of a portfolio of liabilities 如果资产组合和负债组合久期相同 还有什么风险暴露 49 Convexity 凸性凸性 The convexity C of a bond is defined as 凸性的定义如下 This leads to a more accurate relationship 这可以导出一个更为准确的关于收益率变化和价格变 化之间的关系 50 B etc y B B C n i yt ii i 1 2 2 2 1 2 2 1 yCyD B B Convexity 凸性凸性 When used for bond portfolios it allows larger shifts in the yield curve to be considered but the shifts still have to be parallel 当用于债券组合时 允许大一点的收益率曲线移动 但 是移动也必须是平行 51 Theories of the Term Structure 利率期限结构利率期限结构理论理论 Expectations Theory forward rates equal expected future zero rates 预期理论 远期利率等于预期未来即期利率 Market Segmentation short medium and long rates determined independently of each other 市场分割理论 短期 中期 长期利率独立决定 Liquidity Preference Theory forward rates higher than expected future zero rates 流动性偏好理论 远期利率高于预期未来即期利率 52 Liquidity Preference Theory 流动性偏好理论流动性偏好理论 Suppose that the outlook for rates is flat and you have been offered the following choices 假设利率期限结构前景是平的 并且你有如下选择 Which would you choose as a depositor Which for your mortgage 你会选择哪一个作为一个储蓄者 按揭呢 53 Maturity Deposit rate Mortgage rate 1 year 3 6 5 year 3 6 Liquidity Preference Theory cont 流动性偏好理论流动性偏好理论 续续 To match the maturities of borrowers and lenders a bank has to increase long rates above expected future short rates 为了匹配借款人和贷款人的到期时间 银行必须使长期利率 高于预期未来短期利率 In our example the bank might offer 在我们的例子 银行可以提供 54 Maturity Deposit rate Mortgage rate 1 year 3 6 5 year 4 7 案例 加州案例 加州 Orange County破产案破产案 奥兰治县曾是美国加利福尼亚州一个富庶的地方 1994 年12 月6 日 奥兰治县由于其主要投资基金 在利率衍生品投资上的失误造成了16 亿美元的亏 损 该县政府不得不申请破产保护 价值一万亿美元的市政债券市场应声下跌1 加 州债券价格下跌3 4 55 案例 加州案例 加州 Orange County破产案破产案 奥兰治县曾是美国加利福尼亚州一个富庶的地方 1994 年12 月6 日 奥兰治县由于其主要投资基金 在利率衍生品投资上的失误造成了16 亿美元的亏 损 该县政府不得不申请