TermStructure

# TermStructure - The Term Structure of Interest ...

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Unformatted text preview: The Term Structure of Interest  Rates  Term Structure of Interest Rates  •  Diﬀerent interest rates prevail in the market  for borrowing over diﬀerent =me horizons.  •  The term structure of interest rates describes  the rela=onship between the term of  borrowing and the rate for borrowing.  •  It can be expressed in many diﬀerent ways.  Term Structure of Interest Rates  •  A diﬀerent term structure can be derived for  each (type of) instrument.  •  We will focus on the market benchmark term  structure derived from the treasury market.  –  Treasury securi=es are default‐free, so this  represents a term structure that is free from  considera=ons of creditworthiness  –  The treasury market is the most liquid market, so  there are no illiquidity concerns  Types of Interest Rates  •  •  •  •  Yield‐to‐Maturity  Spot Rate  Forward Rate  Short Rate  Yield‐to‐Maturity  •  Widely used for bonds  •  Given by the constant interest rate that  equates the discounted value of the future  cash ﬂows under the bond and its current  market price.  •  Also called the internal rate of return.  Yield‐to‐Maturity  •  For a bullet bond with coupon payment c, n coupon payments, face value F and price P, the  yield (y) is the solu=on of the equa=on:  1 − (1 + y )− n P = cF + F (1 + y )− n y •  The yield curve gives the yield as a func=on of  bond maturity.  € Yield‐to‐Maturity  •  OQen used to compare bonds with diﬀerent  maturi=es, issuers, etc.  •  Can be misleading:   –  The yield on a three year bond depends on the  borrowing rate at three years, as well as at all  previous coupon dates.  –  The curve does not reveal year‐by‐year  informa=on about borrowing costs.  Spot Rates  •  Let P(t,t+k) denote the price (at =me t) of a  zero coupon bond with face value 1, with k  periods un=l maturity.  •  Let t=0. The spot rate for k periods to maturity  is the yield‐to‐maturity of a zero coupon bond  with k periods to maturity. It is the solu=on sk  of the equa=on:  P (0, k ) = (1 + sk ) −k Spot rate curve for diﬀerent maturi=es of Canadian Treasuries.  Source: www.bandofcanada.ca  Spot Rates  •  We can also price a bullet bond based on  observed spot rates:  n P = cF ∑ (1 + sk )− k + F (1 + sn )− n k =1 •  More generally, if an instrument pays the  (determinis=c) cash‐ﬂow Ck at the =me k  € periods in the future, k=1,…,n, its price is:  n P = ∑ Ck (1 + sk )− k k =1 Forward Rates  •  Rates for contracts made today for borrowing  in future periods.  •  fj,k is the nota=on for the forward rate covering  period j to k  •  For example, f3,5 is the interest rate for  agreeing today to borrow money 3 years from  now and repay it 5 years from now.  Forward Rates and Arbitrage  •  Arbitrage is the opportunity to earn a riskless  proﬁt by taking advantage of mispricing in one  or more markets.  •  In ﬁnancial theory, we generally assume that  the market does not permit arbitrage  opportuni=es.   –  Otherwise, investors would immediately invest in  arbitrage opportuni=es in huge amounts.  •  Supply and demand would alter prices un=l the  arbitrage opportuni=es no longer existed.  Forward Rates and Spot Rates  •  To prevent arbitrage, we must have the  following rela=onship between forward and  spot rates:  (1 + f j ,k ) k − j (1 + sk ) k = (1 + s j ) j •  In terms of bond prices:  € (1 + f j ,k ) k− j P (0, j ) = P (0, k ) Forward Rates and Spot Rates  •  One period forward rates fk,k+1 are simply  denoted by fk.  •  It is easy to see that:  •  Spot rates are geometric averages of forward  rates.  €•  There is a one‐to‐one rela=onship between  the spot curve and the forward curve.  –  Spot rates uniquely determine forward rates and  vice‐versa  1 + sk = ((1 + f 0 )(1 + f1 ) (1 + f k −1 ))1 / k Short Rates  •  One‐period interest rates that apply for  borrowing at future =mes.  •  rk is the rate for borrowing between =me k and  =me k+1 that prevails in the market at =me k.  •  fk rate for borrowing between =me k and k+1  agreed upon at -me zero. •  rk rate for borrowing between =me k and k+1  agreed upon at -me k (it is the spot rate that  prevails in the future).  •  Viewed from today, rk is random.   Bootstrapping  •  The most useful representa=on of the term  structure of interest rates is a spot‐rate curve.  •  We don’t observe spot rates directly in the  market.  –  Bond prices (oQen for coupon bearing bonds) are  observed.  •  The process of inferring spot rates from  observed prices of bonds is called  bootstrapping.  Shape of the Term Structure  Normal  sT   sT   Inverted  T  Flat  sT   T  T  Theories of the Term Structure  •  Expecta=on Theory (Forward rates represent  market future expecta=ons of interest rates).  –  Pure/Unbiased Expecta=ons Theory  –  Liquidity Preference Theory  –  Preferred Habitat Theory  •  Market Segmenta=on Theory  Pure/Unbiased Expecta=ons Theory  •  Forward rates represent expected future spot  rates:   f k = E [ rk ] •  The slope of the curve represents expecta=ons  of future rates:   –  Upward sloping means ‘the market expects’ rates  € to go up.  –  Downward sloping means ‘the market expects’  rates to go down.  Liquidity Preference Theory  •  Investors favour liquidity.  •  Forward rates are expected future spot rates plus  a liquidity premium.   •  The liquidity premium Lk increases with k.  f k = E [ rk ] + Lk , Lk > 0 € –  Reﬂects the fact that lenders prefer to lend for short  horizons (borrowers prefer to borrow for long  horizons).  •  An upward sloping curve may reﬂect only  increasing liquidity premiums (not necessarily an  expected increase in rates).  Preferred Habitat Theory  •  Same as Liquidity Preference Theory, except that  it allows for Lk to be posi=ve, nega=ve, or zero.  f k = E [ rk ] + Lk •  Borrowers and investors have preferred maturity  ranges.  •  If supply/demand for a given maturity sector  € does not match, borrowers/investors may be  driven outside their preferred habitat if they are  compensated by an appropriate risk premium.  Market Segmenta=on Theory  •  Neither investors nor borrowers are willing to  shiQ from one maturity sector to the other to  take advantage of opportuni=es arising  between expecta=ons and forward rates.  •  The shape of the yield curve is determined by  supply and demand for securi=es within each  maturity sector, and independently from other  maturi=es.  –  Each sector (or segment) is a separate market  ...
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## This note was uploaded on 01/13/2011 for the course ACTSC 445 taught by Professor Christianelemieux during the Spring '09 term at Waterloo.

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