【正文】 $Y = face value = tyear discount factor = Value today of a dollar to be receive in tyears = $X/ $Y ( 0 , )( 0 , ) R t tD t e ???m periods per year pounding: Continuous pounding: ? ?1( 0 , )1 ( 0 , ) mtDtR t m ???(0, )Dt(0 , )$ $ 1 mtRtYXm???? ? ? ?????Future versus Spot Rates s1 = 8% r2 = 10% r3 = 11% r4 = 11% s1= 8% s2= % s3= % s4= % Do not use yield curve to price bonds ? They can not be priced by discounting cash flow with the same yield because of different structure of CF. ? The Principle of No Arbitrage or The Law of One Price ? Use spot rates (yield on zerocoupon Treasuries) instead! 例子：收益率比較 時期 1 2 價格 YTM 貼現(xiàn)率 % % A現(xiàn)金流 100 1100 % B現(xiàn)金流 0 1080 % Bond Yields and Spot Rates ? we can construct coupon bonds from portfolios of zeroes, and we can construct zeroes from portfolios of coupon bonds. ? This means that,in the absence of arbitrage,the prices of zeroes imply prices for coupon bonds and the prices of coupon bonds imply prices for zeroes. Continue ? Notice that the yield is a blend or a kind of average of the different zero rates associated with the cash flows. In other words, the yield must be between the highest and lowest spot rates. FORWARD RATE ? DEFINITION: the interest rate today that will be paid on money to be ? borrowed at some specific future date and ? to be repaid at a specific more distant future date ? 遠期利率是指從未來某個日期開始的遠期債務合約所要求的利率。 FORWARD RATE 1 假設某個 1年期零息債券的即期利率為 8%，而另一個兩年期的零息債券的即期利率則為 10%。 如果投資者投資 1元錢在一個兩年期的零息債券上 ， 兩年后將獲得 1（ ） 2元 。 那么 ， 從現(xiàn)在看 ， 第 2年（ 第 2年初到第 2年末 ） 的遠期利率是多少呢 ？ 第 2年的遠期利率可計算如下： [1 （ ） 2/]1=% FORWARD RATE 2 222,11 )1()1)(1( sfs ????f1,2 is the forward rate from year 1 to year 2 (1+r3)3=(1+r1)(1+f1,3)2= (1+r1)(1+f1,2)(1+f2,3) f1,3 is the annualized forward rate from year 1 to year 3 FORWARD RATE 3 ? More generally for the link between years t1 and t: tttttt sfs )1()1()1( ,111 ???? ???Continuous Compounding (simpler formula): 2 2 1 11221( ) ( ) ( , ) s t t s t tf t ttt? ? ????2 2 1 1 1 2 2 1( ) ( ) ( , ) ( )s t t s t t f t t t te e e? ? ? ???3 、 YIELD CURVES ? DEFINITION: a graph that shows the YTM for zerocoupon Treasury securities of various terms (maturities) on a particular date YIELD CURVES ? YIELD CURVES AND TERM STRUCTURE ? yield curve provides an estimate of ? the current TERM STRUCTURE OF INTEREST RATES ? yields change daily as YTM changes Yield Curve An upwardsloping yield curve indicates that Treasury Securities with longer maturities offer higher annual yields Yield % Time to Maturity Yield Curve Shapes Normal Level or Flat Inverted Yield Curves at Various Points in Time 0 5 10 15 20 25 30 17 16 15 14 13 12 1 1 10 9 8 7 6 5 2 3 4 February 17, 1982 January 2, 1985 October 22, 1996 September 18, 2022 August 2, 1989 October 15, 2022 Annualized Treasury Security Yields Number of Years to Maturity 我國收益率曲線 中美收益率曲線比較 中國和美國國債收益率曲線的比較01234563MO 1YR 3YR 5YR 7YR 9YR11YR 13YR 15YR 17YR 19YR 21YR 23YR 25YR 27YR 27YR期限收益率%美國國債收益率 中國國債理論收益率 中國國債實際收益率4 . THE TERM STRUCTURE OF INTEREST RATES ? Term structure theory deals with the effect that time has on interest rates. ? It seeks to answer the question of why bonds with different maturities should have different yields. TERM STRUCTURE THEORIES ? THE FOUR THEORIES 1. THE UNBIASED EXPECTATION THEORY 2. THE LIQUIDITY PREFERENCE THEORY 3. MARKET SEGMENTATION THEORY 4. PREFERRED HABITAT THEORY UNBIASED EXPECTATIONS ? Basic Theory: the forward rate represents the average opinion of the expected future spot rate for the period in question ? in other words, the forward rate is an unbiased estimate of the future spot rate. ? If this theory is correct, then the shape of the yield curve is also an accurate indicator of expected future spot rates. TERM STRUCTURE THEORY: Unbiased Expectations ? THEORY 1: UNBIASED EXPECTATIONS ? the expected future spot rate equals the forward rate ? in equilibrium es1,2 = f1,2 where es1,2 = the expected future spot f1,2 = the forward rate TERM STRUCTURE THEORY: Unbiased Expectations 222,11 )1()1)(1( sess ????222,11 )1()1)(1( sfs ????TERM STRUCTURE THEORY: Unbiased Expectations ? CHANGING SPOT RATES AND INFLATION ? Why do investors expect rates to rise or fall in the future? ? spot rates = nominal rates ? because we know that the nominal rate is the real rate plus the expected rate of inflation TERM STRUCTURE THEORY: Unbiased Expectations ? CHANGING SPOT RATES AND INFLATION ? Why do investors expect rates to rise or fall in the future? ? if either the spot or the nominal rate is expected to change in the future, the spot rate will change TERM STRUCTURE THEORY: Unbiased Expectations ? CHANGING SPOT RATES AND INFLATION ? Why do investors expect rates to rise or fall in the future? ? if either the spot or the nominal rate is expected to change in the future, the spot rate will change TERM STRUCTURE THEORY: Unbiased Expectations ? Current conditions influence the shape of the yield curve, such that ? if deflation expected, the term structure and yield curve are downward sloping ? if inflation expected, the term structure and yield curve are upward sloping TERM STRUCTURE THEORY: Unbiased Expectations ? PROBLEMS WITH THIS THEORY: ? upwardsloping yield curves occur more frequently ? Investors are not indifferent to risk ? CoxIngersollRoss have investigated this hypothesis and find that it is not consistent with an economic equilibrium. TERM STRUCTURE THE