Agenda

1
Agenda

• Some duration formulas (CT1, Unit 13, Sec. 5.3)
• An aside on annuity bonds (Lando Poulsen Sec.
  3.3 attached to hand-out)
• Convexity (CT1, Unit 13, Sec. 5.4)
• Immunisation (CT1, Unit 13, Sec. 5.5)

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Duration

• Measures the sensitivity of present values/prices
  to changes in the interest rate.
• It has dual meaning
• A derivative wrt. the interest rate
• A value-weighted discounted average of payment
  times (so its unit is years)

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• Set-up
• Cash-flows at tk
• Yield curve flat at i (or continuously
  compounded/on force form )
• Present value of cash-flows

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Macauley Duration

• The Macauley duration (or discounted mean term)
  is defined by
• Clearly a weighted average of payment dates.
• But also Sensitivity to changes in the force of
  interest.
• Or put differently To parallel shifts in the
  (continuously compounded) yield curve.

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Duration of an Annuitiy

• The duration of an n-year annuity making payments
  D is (independent of D and) equal to
• (Note On the Oct. 21 slides there was either a D
  too much or a D too little)
• where as usual with

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• and (IA) is the value of an increasing annuity

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Annuity Bonds

• The remaining principal (outstanding notional) of
  an annuity bond with (nominal) coupon rate r and
  (annual) payment D satisfies
• Repeated substitution gives

8

• Suppose (wlog) p0100.
• For the loan to be paid off after n periods we
  must have pn0, i.e.

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• This we can rewrite to solve for the yearly
  payment
• In finance people will often refer to this as the
  annuity formula.
• The yearly payments (or instalments) consist of
  interest payments and repayment of principal.

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Duration of a Bullet Bond

• Using similar reasoning, the duration of a bullet
  bond w/ coupon payments D and notional R is
• (Note On the Oct. 21 slides D was missing in the
  denominator.)

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Convexity

• The convexity of is defined as
• This is the effective (or volatility
  version) could also do Macauley or Fisher-Weil
  style.

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• Convexity and (effective) duration give a 2nd
  order accurate (Taylor expansion) approximation
  to changes in present value for (small) interest
  rate changes

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A classical picture of duration and convexity
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Convexity is a measure of dispersion around the
duration

• We can write Macauley duration as
• and Macauley-style convexity as
• A measure of dispersion around the duration is

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Immunisation

• Consider a pension fund that has assets
  and liabilities .
• We say that the fund is immunised against
  
• movements in the interest rate around
  if
• and

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• By Taylor expanding the difference between assets
  and liabilities (known as the surplus) we get
  that the inequality condition is fulfilled if
• - the assets and the liabilities have the same
  duration,
• and
• - convexity of the assets is higher than the
  convexity of the liabilities

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• These are known as Redingtons conditions.
• It doesnt matter whether we use effective or
  Macaluey duration.
• Typical exercise approach The equality
  conditions give two linear equations in two
  unknows the convexity condition follows from a
  dispersion consideration.

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Immunisation isnt the be-all-and-end-all of
interest rate risk management

• Note that an immunised portfolio is looks very
  much like an arbitrage.
• That tells us that considering only parallel
  shifts to flat yield curves isnt the perfect way
  to model interest rate uncertainty.
• And models with genuinely random behaviour is the
  next topic.