EWMBA 232 Money Markets and Financial Institutions

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EWMBA 232Money Markets and Financial Institutions

• Prof. Greg Duffee
• Haas School of Business
• Fall 2004

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The mathematics of interest rates

• What is the value today of 1.20 to be received
  in one year?
• i is current one-year discount rate, yield,
  or interest rate
• Logic Hypothetical investment of PV at rate i
  for one year results in 1.20

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• Calculate yield for 1.20 to be paid in 2 years
  with PV1
• Calculate yield for 5 to be paid every year with
  PV80 (a perpetuity, or consol bond)

Hypothetical investment of 80 at 6.25 interest,
withdrawing Interest annually, produces 5/year.
More generally, iC/P for an annuity.
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• What is the monthly payment on a 30 year mortgage
  for 1 million at an annual interest rate of 7?
• Answer depends on compounding frequency used in
  interest rate calculation
• For a monthly compounding frequency
• To solve, use calculator, mortgage table, or next
  slide

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• Solution pay 6653.03/month
• Compare to payment for infinitely-lived mortgage

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• Given future payments and PV, interest rate is
  not unique
• No compounding simple interest rate
• Market conventions use different compounding
  frequencies for different instruments
• More frequent compounding corresponds to lower
  yield

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• Questions
• PV1, F1.1, maturity 1. What are simple,
  semiannual-compounding, and continuous
  compounding yields?

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• Are yield differences created by different
  compounding conventions larger or smaller when
  interest rates are low?

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A focus on bonds

• Bonds pay a face amount at maturity and (perhaps)
  coupons at fixed intervals
• Zero-coupon bond discount bond
• In US, most bonds pay coupons every six months
• Interest rate often called yield to maturity
• Key issue is relation between a bonds price and
  yield

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Example

• Price versus yield for a 10-year ZCB (F1000)

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Notation

• Price of ZCB paying 1 in T years is PT
• Yield on a T-maturity bond is YT (use annual
  compounding)
• Price of coupon bond paying C at the end of each
  year and 1 in T years is Pc,T. Yield is Yc,T.

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• Coupon bond yield is the discount rate that, when
  applied to each cash flow, produces the bond
  price
• Coupon bond yields are defined implicitly
• Definition Par bond
• A bond with price equal to face amount
• For a par bond, yield coupon/face

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• How should we interpret a bonds yield?
• ZCB Return to holding bond over life of bond
• Not true for coupon bondreinvestment risk
• Technically, internal rate of return
• For horizons shorter than life of bond, yield of
  bond and return to holding bond can differ widely
  
• Returns are uncertain

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Measuring interest rate risk

• Interest rate risk The risk of price changes
  owing to changes in interest rates
• Bonds are subject to other risks (default,
  taxation, liquidity) here we abstract from these
• We focus on the return on a bond (?P/P) given a
  change in the bonds yield (?Y)

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• Differential calculus approach
• Intuition Time to cash flow measures sensitivity
  of bonds return to yield

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Examples

• 10 to 11, 1 year ZCB
• 10 to 11, 10 year ZCB

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Duration

• Main measure of interest rate risk
• For ZCB equals maturity
• For coupon bond equals weighted average of
  time-to-payments
• Weights are PVs of payments calculated using
  bonds yield

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• Modified duration includes fractional adjustment
  (neg of derivative)
• Formula linking MD to price changes
• ? P - MD x P x (percentage point ? in yield) x
  (1/100)

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Uses of duration

• Measure risk of a bond portfolio
• Construct portfolio to achieve desired bet (if
  any) on changes in yields
• Methodology Duration is a weighted average of
  durations of individual instruments or cash flows
  
• Implicit assumption parallel shifts in the
  yield curve

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• Duration is a linear way to measure a nonlinear
  risk

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• Two reasons why linear formula is only an
  approximation
• Nonlinear price, yield relation
• ? P - MD x P x (percentage point ? in
  yield) x (1/100)
• MD varies with yield

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• Both reasons imply formula underestimates bond
  values
• Nonlinearity more pronounced for higher durations
• A more precise formula uses gains from a
  convexity adjustment

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Effective duration

• What is the interest rate risk of instruments
  with cash flows that depend on interest rates?
• Callable/putable bonds, mortgages, inverse
  floaters
• Measure is effective duration
• Straightforward to implement with cash flows that
  are deterministic functions of interest rates
  (e.g., inverse floaters)
• Implementation with more complicated instruments
  (e.g., mortgages) requires mathematical models of
  term structure, cash flows