Chapter 13: Advanced Bond topics Duration

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Chapter 13 Advanced Bond topics Duration
Convexity
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Chapter Summary

• Objective To examine various strategies
  available to fixed-income portfolio managers.
• Review of bond pricing relationships
• Duration
• Convexity
• Passive Fixed Income Management
• Active Fixed Income Management

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Managing Fixed Income Securities Basic Strategies

• Active strategy
• Trade on interest rate predictions
• Trade on market inefficiencies
• (ie. buy underpriced bonds)
• Passive strategy
• Control risk
• Balance risk and return
• Dont attempt to outsmart the market

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Bond pricing

• Bondholders have interest rate risk even if
  coupons are guaranteed
• Why?
• Unless they hold the bond to maturity, the price
  of the bond will change as interest rates in the
  economy change

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Bond Pricing Relationships
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Bond Pricing Relationships

• Inverse relationship between price and yield
• An increase in a bonds yield to maturity results
  in a smaller price decline than the gain
  associated with a decrease in yield (convexity)
• Long-term bonds tend to be more price sensitive
  than short-term bonds

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Bond Pricing Relationships (contd)

• As maturity increases, price sensitivity
  increases at a decreasing rate
• Price sensitivity is inversely related to a
  bonds coupon rate
• Price sensitivity is inversely related to the
  yield to maturity at which the bond is selling

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Interest Rate Sensitivity
0
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Chapter Summary

• Objective To examine various strategies
  available to fixed-income portfolio managers.
• Review of bond pricing relationships
• Duration
• Convexity
• Passive Fixed Income Management
• Active Fixed Income Management

10
Duration

• Summary statistic of a bond
• A measure of the effective maturity of a bond
• When bonds pay coupons, its better to receive
  coupons sooner rather than later so maturity
  isnt a good statistic
• Coupon bonds make pmts before bond maturity it
  can be seen that each coupon has its own maturity
  and the effective maturity is a weighted average
  of all pmts.

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Duration

• The weighted average of the times until each
  payment is received, with the weights
  proportional to the present value of the payment
• Duration is shorter than maturity for all bonds
  except zero coupon bonds
• Duration is equal to maturity for zero coupon
  bonds

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Example 8-year, 9 annual coupon bond
Actual cash flows
Area where PV of CF before and after balance out
PV of cash flows
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Duration Calculation
PV of cash flows as a of bond price
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Duration Calculation example
Eg. Coupon 8, yield 10, years to maturity
2
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Time
Payment
PV of CF
Weight
C1 X
Bond
years
(10)
C4
.5
40
38.095
.0395
.0198
1
40
36.281
.0376
.0376
1.5
40
34.553
.0358
.0537
2.0
1040
855.611
.
8871
1.7742
sum
964.540
1.000
1.8853
DURATION
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Why is duration a key concept?

• Its a simple summary statistic of the effective
  average maturity of the portfolio
• It is an essential tool in immunizing portfolios
  from interest rate risk
• It is a measure of interest rate risk of a
  portfolio
• Equal duration assets are equally sensitive to
  changes in interest rates

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Duration/Price Relationship

• Price change is proportional to duration and not
  to maturity

• Or, if we denote D modified duration

D is the 1st derivative of bonds price with
respect to yield ie. D (-1/P)(dP/dY)
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Duration/Price Relationship
P P
-D i
eg. What would be the percentage change in the
price of a bond with a modified duration of 9,
given that interest rates fall 50 basis points
(ie. 0.5)? (-9)(-.05) 4.5 eg. What would
be the change in price of a bond with a
Macaulay Duration of 10 if interest rates rise by
50 basis points (ie. 0.5) The current YTM is
4. D 10/1.04 9.615 Therefore , change
in price (-9.615)(.5) -4.81
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Notes on duration

• Duration provides a measure to quantify the
  bonds sensitivity to interest rates
• The duration formula is derived from the bond
  price-yield relationship
• Calculate dP/dy and divide with P
• Approximating the bond price change using
  duration is equivalent to moving along the slope
  of the bond price-yield curve
• Bonds of equal duration are equally sensitive to
  interest rate movements

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Rules for Duration

• Rule 1 The duration of a zero-coupon bond equals
  its time to maturity
• Rule 2 Holding maturity constant, a bonds
  duration is higher when the coupon rate is lower
• Rule 3 Holding the coupon rate constant, a
  bonds duration generally increases with its time
  to maturity

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Rules for Duration (contd)

• Rule 4 Holding other factors constant, the
  duration of a coupon bond is higher when the
  bonds yield to maturity is lower
• Rule 5 The duration of a level perpetuity is
  equal to

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Rules for Duration (contd)

• Rule 6 The duration of a level annuity is equal
  to
• Rule 7 The duration for a coupon bond is equal
  to

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Chapter Summary

• Objective To examine various strategies
  available to fixed-income portfolio managers.
• Review of bond pricing relationships
• Duration
• Convexity
• Passive Fixed Income Management
• Active Fixed Income Management

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Convexity

• Duration approximates price change but isnt
  exact
• For small changes in yields, duration is close
  but for larger changes in yields, there can be a
  large error
• Duration always underestimates the value of bond
  price increases when yields fall and
  overestimates declines in price when yields rise

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Duration and Convexity
Price
Pricing error from convexity
Yield
Duration (approximates a line vs a curve)
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Convexity of Two Bonds
A is more convex than B If rates inc ? As price
falls less than Bs If rates dec ? As price
rises more than Bs Convexity is desirable for
investors so they will pay for it (ie. As yield
is probably less than Bs)
Bond B
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Convexity

• Definition of convexity
• The rate of change of the slope of the
  price/yield curve expressed as a fraction of the
  bonds price.

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Correction for Convexity
Correction for Convexity
Convexity is the rate of change of the slope of
the price/yield curve 2nd derivative

• Convexity is computed like duration, as a
  weighted average of the terms (t²t) (rather than
  t) divided by (1y)²
• For non-callable bonds, convexity is always
  positive

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Convexity calculationeg. Coupon 8, YTM10,
n2
Convexity 18.4759/1.05² 16.7582
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Properties of Convexity

• Inverse relationship between convexity and coupon
  rate
• Direct relationship between maturity and
  convexity
• Inverse relationship between yield and convexity

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Examples

• ST, high coupon bond ? low convexity
• LT, high coupon bond ? high convexity
• Eg. If a bond has a modified duration of 10 and
  convexity of 120, if interest rates decrease by
  75 basis points, what will be the percentage
  change in price?

(-10)(-.0075) (.5)(120)(.0075)2 7.83
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Duration and Convexity of Callable Bonds

• The max price of the bond is the call price bec
  when the price reaches that value, it will be
  called by the company
• We say the value is compressed to the call
  price

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Duration and Convexity of Callable Bonds
Unattractive so investor is compensated with
call premium
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Chapter Summary

• Objective To examine various strategies
  available to fixed-income portfolio managers.
• Review of bond pricing relationships
• Duration
• Convexity
• Passive Fixed Income Management
• Active Fixed Income Management

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Passive Management

• Definition
• Does not attempt to search for undervalued bonds
• Attempts only to control the risk of fixed income
  investments
• Passive does not mean do nothing!

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Passive Management

• Index strategy
• Matches interest rate risk to bond indicies
• Matches the risk associated with interest rate
  fluctuations to that of index
• Immunization techniques
• Reduces interest rate risk to zero
• Shields portfolio from interest rate fluctuations

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Passive ManagementStrategy 1 Indexing strategy

• Objective is to create a bond portfolio that
  mirrors the composition and performance of an
  established bond index
• Scotia Capital Universe Index (Canada)
• Salomon Brothers Broad Investment Grade (US)
• Lehman Brothers Aggegate Index (US)
• Merrill Lynch Domestic Master Index (US)
• Note Bond indicies hold bonds with maturities gt
  1 yr so as time passes, bonds get dropped from
  the index

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Passive ManagementIndexing strategy - issues

• Portfolio must be constantly rebalanced
• Bond index contains a large number of securities
  (eg. US indicies have more than 5000 bonds so
  hard to buy all bonds)
• Bonds are very thinly traded in Canada
• As bond maturies fall to lt 1yr, bonds are dropped
  from the portfolio and as new bonds are issued,
  they are added
• The securities in the bond index change
  constantly
• Impossible to replicate index exactly
• Stratified sampling or cellular approach is used

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Passive ManagementIndexing strategy

• General Idea
• Bond mkt is divided into several classes which
  are more or less homogeneous
• Weights are calculated for each class
• The bond portfolio has items from each class in
  the given weight

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Passive ManagementIndexing strategy cellular
approach
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Passive ManagementStrategy 2 Immunization

• Immunization of interest rate risk
• Net worth immunization
• Duration of assets Duration of liabilities
• (eg. banks matched maturity funding)

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Passive ManagementStrategy 2 Immunization

• Target date immunization
• ie. future value of portfolio is protected
  against changes in interest rates
• (eg. pension funds)
• Price risk
• bond price changes inversely with interest
  changes
• Reinvestment risk
• Coupons are reinvested at different rates
• If Holding Period matches Duration
• the two risks will exactly offset each other

c
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Passive ManagementStrategy 2 Immunization

• In practice, we cant rebalance the portfolio
  constantly becase of transaction costs

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Notes on duration of bond portfolios

• The duration of a bond portfolio is equal to the
  weighted average of the durations of the bonds in
  the portfolio
• The portfolio duration, however, does not change
  linearly with time. The portfolio needs,
  therefore, to be rebalanced periodically to
  maintain target date immunization

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Chapter Summary

• Objective To examine various strategies
  available to fixed-income portfolio managers.
• Review of bond pricing relationships
• Duration
• Convexity
• Passive Fixed Income Management
• Active Fixed Income Management

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Active Bond Management Swapping Strategies
Taking advantage of interest rate forecasts or
bond underpricing to make profits ie.
outsmarting the market

• Substitution swap
• Inter-market swap
• Rate anticipation swap
• Pure yield pickup
• Tax swap

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Contingent Immunization

• Combination of active and passive management
• Strategy involves active management with a floor
  rate of return
• As long as the rate earned exceeds the floor, the
  portfolio is actively managed
• Once the floor rate or trigger rate is reached,
  the portfolio is immunized