EMBAF

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Factors Affecting Bond Yields and the Term
Structure of Interest Rates

• Zvi Wiener
• Based on Chapter 5 in Fabozzi
• Bond Markets, Analysis and Strategies

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Base Interest Rate

• Treasury
• Libor
• Prime

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Term Structure of Interest Rates
Yield curve
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• http//bond.yahoo.com/rates.html
• http//www.ratecurve.com/yc2.html

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Yield Curve Term Structure of IR
Flat
r
maturity
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Types of Issuers

• Governments
• Agencies
• Corporate
• Municipals
• Others

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Factors affecting Bond yields and TS

• Base interest rate - benchmark interest rate
• Risk Premium - spread
• Expected liquidity
• Market forces - Demand and supply

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Taxability of interest

• qualified municipal bonds are exempts from
  federal taxes.
• After tax yield pretax yield (1- marginal tax
  rate)

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Do not use yield curve to price bonds

• Period A B
• 1-9 6 1
• 10 106 101
• They can not be priced by discounting cashflow
  with the same yield because of different
  structure of CF.
• Use spot rates (yield on zero-coupon Treasuries)
  instead!

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• On-the-run Treasury issues
• Off-the-run Treasury issues
• Special securities
• Lending
• Repos and reverse repos

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30 yr US Treasuries
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10 yr US Treasuries
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Duration and Term Structure of IR
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Partial Duration
Key rate duration
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Determinants of the Yield Curve

• Federal Reserve sets a target level for the fed
  funds rate - the rate at which depository
  institutions make uncollaterized overnight loans
  to one another.
• Long-term rates reflect expectations of future
  rates and can be influenced by the outlook for
  monetary policy.

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Liquidity

• Bid-offer spread 1-2 cents per 100 face
• Corporate bonds for example 13 cents
• High yield bonds 19 cents
• on-the-run - recently issued in a particular
  maturity class. With time became off-the-run.
• Flight to Quality (fall 98) bid-ask 16-25 cents.

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Term Structure of IR

• If we knew the future IR
• 0(Today) 8
• 1 10
• 2 11
• 3 11

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Term Structure of IR

• If we knew the future IR
• 0(Today) 8
• 1 10
• 2 11
• 3 11

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r1 8 r1 10 r3 11 r4 11

• Spot rate is the yield to maturity on zero-coupon
  bonds.

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Future versus Spot Rates
r1 8 r1 10 r3 11 r4 11
y1 8
y2 8.995
y3 9.66
y4 9.993
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Forward Rates

• Suppose you will need a loan in two years from
  now for one year.
• How one can create such a loan today?
• Go short a three-year zero coupon bond.
• Go long a two-year zero coupon bond.

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• Suppose you will need a loan in two years from
  now for one year.
• How one can create such a loan today?
• Go short a three-year zero coupon bond.
• Go long a two-year zero coupon bond.
• 1 0 0 -1.3187
• -1 0 1.188 0

0 1 2 3
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Forward Rates

• (1 yn)n (1 yn-1)n-1(1 fn)
• (1 yn)n
• (1 yn-1)n-1
• 1 -1.3187
• -1 1.188

0 1 2 3
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Forward Rates

• (1 yn)n (1 yn-1)n-1(1 fn)
• (1 yn)n
• (1 yn-1)n-1
• 1 -1.3187
• -1 1.188

0 1 2 3
fn
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Forward Rates

• In other words we can lock now interest rate for
  a loan which will be taken in future.
• To specify a forward interest rate one should
  provide information about
• todays date
• beginning date of the loan
• end date of the loan

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Forward Rates

• Buy a two years bond
• Buy a one year bond and then use the money to buy
  another bond (the price can be fixed today).

(1r2)(1r1)(1f12)
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Forward Rates

• (1r3)(1r1)(1f13) (1r1)(1f12)(1f13)
• Term structure of instantaneous forward rates.

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Forward Rates - Advanced

• Let P(t,s) be the price at time t of a pure
  discount bond maturing at time s gt t. Then the
  yield to maturity R(t,T) is the internal rate of
  return at time t on a bond maturing at tT.
• P(t, tT) Exp-R(t,T)T
• Then
• R(t,T) - LogP(t, tT)/T

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Forward Rates - Advanced

• The integral of the forward rates gives the yield
  to maturity

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Forward Rates - Advanced

• The integral of the forward rates gives the yield
  to maturity
• or alternatively

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FRA Forward Rate Agreement

• A contract entered at t0, where the parties (a
  lender and a borrower) agree to let a certain
  interest rate R, act on a prespecified
  principal, K, over some future time period S,T.
• Assuming continuous compounding we have
• at time S -K
• at time T KeR(T-S)
• Calculate the FRA rate R which makes PV0
• hint it is equal to forward rate

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The Expectations Hypothesis

• Suggested by Lutz.
• Forward interest rates is the expected future
  spot rate.
• Cox-Ingersoll-Ross have investigated this
  hypothesis and find that it is not consistent
  with an economic equilibrium.
• However it gives often a right direction for
  expectations.

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Liquidity Preference

• Hicks (1939) suggested that lenders demand a
  premium for locking up their money for long
  period of time.
• This implies that the term structure will be
  always upward sloping.
• The theory ignores the borrowing side of the
  market.

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Market Segmentation and Preferred Habitat Theories

• Modigliani and Sutch
• The market is segmented, investors absolutely
  prefer one maturity over another.
• This means that there is no connection between
  interest rates for different maturities.

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Modern Theories

• Equilibrium Theories CIR, BP
• Non-equilibrium Theories Dothan, Vasicek,
• Ho-Lee, Hull-White, HJM
• Most of them are based on a Brownian Motion as a
  source of market uncertainty.

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Brownian Motion
B
Time
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Brownian Motion

• Starts at the origin
• Is continuous
• Is normally distributed at each time
• Increments are independent
• Markovian property
• Technical conditions

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Home AssignmentChapter 5

• Ch. 5 Questions 2, 3, 10, 13.

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Measuring the Term Structure

• There are too many data plus some noise.
• The easiest way to measure the TS is with liquid
  zero coupon bonds.
• We obtain a series of points.

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Measuring the Term Structure
rzero
Time to maturity
0 3m 6m 1yr 3yr 5yr 10yr 30yr
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First Order Spline
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Second Order Spline
rzero
Time to maturity
0 3m 6m 1yr 3yr 5yr 10yr 30yr
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Measuring the Term Structure

• There are too many data plus some noise.
• The easiest way to measure the TS is with liquid
  zero coupon bonds.
• We obtain a series of points.
• One can connect them with a spline.
• First order is good for pricing simple bonds.
• For swaps one need a very high precision.