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Chapter 16 Revision of the Fixed-Income Portfolio. Outline. Introduction Passive versus active management strategies Duration re-visited Bond convexity. Introduction. Fixed-income security management is largely a matter of altering the level of risk the portfolio faces: Interest rate risk

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  • Introduction
  • Passive versus active management strategies
  • Duration re-visited
  • Bond convexity
  • Fixed-income security management is largely a matter of altering the level of risk the portfolio faces:
    • Interest rate risk
    • Default risk
    • Reinvestment rate risk
  • Interest rate risk is measured by duration
passive versus active management strategies
Passive Versus Active Management Strategies
  • Passive strategies
  • Active strategies
  • Risk of barbells and ladders
  • Bullets versus barbells
  • Swaps
  • Forecasting interest rates
  • Volunteering callable municipal bonds
passive strategies
Passive Strategies
  • Buy and hold
  • Indexing
buy and hold
Buy and Hold
  • Bonds have a maturity date at which their investment merit ceases
  • A passive bond strategy still requires the periodic replacement of bonds as they mature
  • Indexing involves an attempt to replicate the investment characteristics of a popular measure of the bond market
  • Examples are:
    • Salomon Brothers Corporate Bond Index
    • Lehman Brothers Long Treasury Bond Index
indexing cont d
Indexing (cont’d)
  • The rationale for indexing is market efficiency
    • Managers are unable to predict market movements and that attempts to time the market are fruitless
  • A portfolio should be compared to an index of similar default and interest rate risk
active strategies
Active Strategies
  • Laddered portfolio
  • Barbell portfolio
  • Other active strategies
laddered portfolio
Laddered Portfolio
  • In a laddered strategy, the fixed-income dollars are distributed throughout the yield curve
  • For example, a $1 million portfolio invested in bond maturities from 1 to 25 years (see next slide)
laddered portfolio cont d
Laddered Portfolio (cont’d)

Par Value Held ($ in Thousands)

Years Until Maturity

barbell portfolio
Barbell Portfolio
  • The barbell strategy differs from the laddered strategy in that less amount is invested in the middle maturities
  • For example, a $1 million portfolio invests $70,000 par value in bonds with maturities of 1 to 5 and 21 to 25 years, and $20,000 par value in bonds with maturities of 6 to 20 years (see next slide)
barbell portfolio cont d
Barbell Portfolio (cont’d)

Par Value Held ($ in Thousands)

Years Until Maturity

barbell portfolio cont d14
Barbell Portfolio (cont’d)
  • Managing a barbell portfolio is more complicated than managing a laddered portfolio
  • Each year, the manager must replace two sets of bonds:
    • The one-year bonds mature and the proceeds are used to buy 25-year bonds
    • The 21-year bonds become 20-years bonds, and $50,000 par value are sold and applied to the purchase of $50,000 par value of 5-year bonds
other active strategies
Other Active Strategies
  • Identify bonds that are likely to experience a rating change in the near future
    • An increase in bond rating pushes the price up
    • A downgrade pushes the price down
risk of barbells and ladders
Risk of Barbells and Ladders
  • Interest rate risk
  • Reinvestment rate risk
  • Reconciling interest rate and reinvestment rate risks
interest rate risk
Interest Rate Risk
  • Duration increases as maturity increases
  • The increase in duration is not linear
    • Malkiel’s theorem about the decreasing importance of lengthening maturity
    • E.g., the difference in duration between 2- and 1-year bonds is greater than the difference in duration between 25- and 24-year bonds
interest rate risk cont d
Interest Rate Risk (cont’d)
  • Declining interest rates favor a laddered strategy
  • Increasing interest rates favor a barbell strategy
reinvestment rate risk
Reinvestment Rate Risk
  • The barbell portfolio requires a reinvestment each year of $70,000 par value
  • The laddered portfolio requires the reinvestment each year of $40,000 par value
  • Declining interest rates favor the laddered strategy
  • Rising interest rates favor the barbell strategy
reconciling interest rate reinvestment rate risks21
Reconciling Interest Rate & Reinvestment Rate Risks
  • The relationships between risk and strategy are not always applicable:
    • It is possible to construct a barbell portfolio with a longer duration than a laddered portfolio
      • E.g., include all zero-coupon bonds in the barbell portfolio
    • When the yield curve is inverting, its shifts are not parallel
      • A barbell strategy is safer than a laddered strategy
bullets versus barbells
Bullets Versus Barbells
  • A bullet strategy is one in which the bond maturities cluster around one particular maturity on the yield curve
  • It is possible to construct bullet and barbell portfolios with the same durations but with different interest rate risks
    • Duration only works when yield curve shifts are parallel
bullets versus barbells cont d
Bullets Versus Barbells (cont’d)
  • A heuristic on the performance of bullets and barbells:
    • A barbell strategy will outperform a bullet strategy when the yield curve flattens
    • A bullet strategy will outperform a barbell strategy when the yield curve steepens
  • Purpose
  • Substitution swap
  • Intermarket or yield spread swap
  • Bond-rating swap
  • Rate anticipation swap
  • In a bond swap, a portfolio manager exchanges an existing bond or set of bonds for a different issue
purpose cont d
Purpose (cont’d)
  • Bond swaps are intended to:
    • Increase current income
    • Increase yield to maturity
    • Improve the potential for price appreciation with a decline in interest rates
    • Establish losses to offset capital gains or taxable income
substitution swap
Substitution Swap
  • In a substitution swap, the investor exchanges one bond for another of similar risk and maturity to increase the current yield
    • E.g., selling an 8% coupon for par and buying an 8% coupon for $980 increases the current yield by 16 basis points
substitution swap cont d
Substitution Swap (cont’d)
  • Profitable substitution swaps are inconsistent with market efficiency
  • Obvious opportunities for substitution swaps are rare
intermarket or yield spread swap
Intermarket or Yield Spread Swap
  • The intermarket or yield spread swap involves bonds that trade in different markets
    • E.g., government versus corporate bonds
  • Small differences in different markets can cause similar bonds to behave differently in response to changing market conditions
intermarket or yield spread swap cont d
Intermarket or Yield Spread Swap (cont’d)
  • In a flight to quality, investors become less willing to hold risky bonds
    • As investors buy safe bonds and sell more risky bonds, the spread between their yields widens
  • Flight to quality can be measured using the confidence index
    • The ratio of the yield on AAA bonds to the yield on BBB bonds
bond rating swap
Bond-Rating Swap
  • A bond-rating swap is really a form of intermarket swap
  • If an investor anticipates a change in the yield spread, he can swap bonds with different ratings to produce a capital gain with a minimal increase in risk
rate anticipation swap
Rate Anticipation Swap
  • In a rate anticipation swap, the investor swaps bonds with different interest rate risks in anticipation of interest rate changes
    • Interest rate decline: swap long-term premium bonds for discount bonds
    • Interest rate increase: swap discount bonds for premium bonds or long-term bonds for short-term bonds
forecasting interest rates
Forecasting Interest Rates
  • Few professional managers are consistently successful in predicting interest rate changes
  • Managers who forecast interest rate changes correctly can benefit
    • E.g., increase the duration of a bond portfolio is a decrease in interest rates is expected
volunteering callable municipal bonds
Volunteering Callable Municipal Bonds
  • Callable bonds are often retied at par as part of the sinking fund provision
  • If the bond issue sells in the marketplace below par, it is possible:
    • To generate capital gains for the client
    • If the bonds are offered to the municipality below par but above the market price
properties of duration
Properties of Duration
  • We already saw that the concept of duration can be seen as a time-weighted average of the bonds discounted payments as a proportion of the bond price, or as a weighted average of the cash flows “times”.
  • Duration can also be interpreted as a risk measure for bonds, however.

Example: Bond A has a 10-year maturity, and bears a 7% coupon rate. Bond B has 10 years left to maturity, and a coupon rate of 13%. The current market interest rate is 7%.

The price of bonds A and B are $1,000 and $1,421.41 respectively. What happens to these prices if the market rate changes from 7% to 7.7% ?

duration of a portfolio
Duration of a Portfolio
  • The duration of a portfolio is the weighted average of the durations of the individual assets making up the portfolio.
  • Proof: suppose you hold N1 units of security 1 and N2 units of security 2. Let P1 and P2 be the prices of the two securities, and let D1 and D2 be their respective durations.
bond convexity
Bond Convexity
  • The importance of convexity
  • Calculating convexity
  • General rules of convexity
  • Using convexity
the importance of convexity
The Importance of Convexity
  • Convexity is the difference between the actual price change in a bond and that predicted by the duration statistic
  • In practice, the effects of convexity are relevant if the change in interest rate level is large.
the importance of convexity cont d
The Importance of Convexity (cont’d)
  • The first derivative of price with respect to yield is negative
    • Downward sloping curves
  • The second derivative of price with respect to yield is positive
    • The decline in bond price as yield increases is decelerating
    • The sharper the curve, the greater the convexity
the importance of convexity cont d46
The Importance of Convexity (cont’d)

Greater Convexity

Bond Price

Yield to Maturity

the importance of convexity cont d47
The Importance of Convexity (cont’d)
  • As a bond’s yield moves up or down, there is a divergence from the actual price change (curved line) and the duration-predicted price change (tangent line)
    • The more pronounced the curve, the greater the price difference
    • The greater the yield change, the more important convexity becomes
the importance of convexity cont d48
The Importance of Convexity (cont’d)

Error from using

duration only

Bond Price

Current bond


Yield to Maturity

calculating convexity
Calculating Convexity
  • The percentage change in a bond’s price associated with a change in the bond’s yield to maturity:
calculating convexity cont d
Calculating Convexity (cont’d)
  • The second term contains the bond convexity:
general rules of convexity
General Rules of Convexity
  • There are two general rules of convexity:
    • The higher the yield to maturity, the lower the convexity, everything else being equal
    • The lower the coupon, the greater the convexity, everything else being equal
  • Recall the previous immunization example.
  • Bond 2 (asset) has the same duration as the liability.
  • However, there are other ways to select a portfolio of assets with a duration matching the liability’s duration.
building portfolio of given duration
Building Portfolio of given Duration
  • Instead of using Bond 2 (with duration of 10) to match the obligation’s liability, let us build a portfolio made up of bond 1 and 3.
  • We want duration =10, therefore we need:
  • wD1+(1-w)D2=10, (D1=7.665 and D2=14.636)
  • This implies w=0.66509
if interest rates change to say 7
If interest rates change to, say, 7%:

The Portfolio’s payoff remains more or less intact, just like Bond 2, and would thus allow us to meet the $ 1,790.85 obligation.

bond 2 vs portfolio
Bond 2 vs. Portfolio
  • Last slide’s graph of Terminal values shows that both Bond 2 and the carefully chosen Portfolio (of Bonds 1 and 3) have a slope of zero around 6%.
  • This indicates that both have been immunized, i.e. they both have a duration of 10 in this case.
  • However, their curvature is different: the Portfolio is more convex than Bond 2.
Since the graph represents terminal values, convexity here is a good thing. We get more over funding (extra $ after having paid the obligation) from the portfolio if the interest rate departs from 6% than we get from Bond 2.
  • Therefore, when comparing two immunized portfolios, the portfolio whose terminal value is more convex with respect to a change in interest rates is more desirable.
making a portfolio completely insensitive to changes in yields
Making a Portfolio Completely Insensitive to Changes in Yields
  • There are situations when it may be desirable to render a portfolio as insensitive to interest rate changes as possible.
  • The way to achieve this is to not only match the assets and liabilities durations, but to also match their convexities.
Recall our earlier immunization problem where interest rates change from r to r+Dr. The new values of the future obligation and of the bond are:
Equating the two and recalling that we have already matched the first and second terms in the expansion yields the following requirement:
  • This is the constraint that must be met in order for the assets (bonds) and the liabilities (obligations) to have matching convexities (in addition to already having matching durations)
convexity matching example
Convexity Matching Example
  • You need to immunize an obligation whose present value V0 is $1,000. The payment is to be made 10 years from now, and the current interest rate is 6%. The payment is thus the future value of 1,000 at 6%, therefore it is:

1,000(1.06)10 = $1,790.85

  • The Excel spreadsheet on the next slide shows four bonds that you have at your disposition to immunize the liability.
what happens if rates go up to 7
What happens if rates go up to 7%?

We notice that only Bond 2 preserves its terminal value close to $1,791: it is the only bond with matching duration.

It worked because the change in interest rate was small. What happens if rates go up to 10% ? (a large shift)

None of the bonds maintained their terminal values now. The change in interest rate was too large.

how to build a portfolio of bonds with matching convexity
How to build a portfolio of bonds with matching convexity?
  • Set up the following system (Example with three bonds):
  • where
  • and
The numbers 1, 10 and 110 on the right-hand side come from the fact that the weights must sum to one, that the weighted average duration must match the liability (obligation) duration of 10, and finally that the weighted average convexity constraint must match the liability convexity value of N(N+1), i.e. 10(10+1)=110.
  • Using the “secondDur” Visual Basic function in Excel (for convenience, but not required) for the convexity constraints and solving for the weights by inverting the matrix yields the weights of a portfolio that is fully immunized.