1. Chapter 16 Revision of the Fixed-Income Portfolio

2. Outline Introduction Passive versus active management strategies
Duration re-visited
Bond convexity

3. Introduction
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

4. 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

5. Passive Strategies
Buy and hold
Indexing

6. 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

7. Indexing
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

8. 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

9. Active Strategies Laddered portfolio Barbell portfolio
Other active strategies

10. 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)

11. Laddered Portfolio (cont’d)
Par Value Held ($ in Thousands)
Years Until Maturity

12. 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)

13. Barbell Portfolio (cont’d)
Par Value Held ($ in Thousands)
Years Until Maturity

14. 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

15. 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

16. Risk of Barbells and Ladders
Interest rate risk
Reinvestment rate risk
Reconciling interest rate and reinvestment rate risks

17. 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

18. Interest Rate Risk (cont’d)
Declining interest rates favor a laddered strategy
Increasing interest rates favor a barbell strategy

19. 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

20. Reconciling Interest Rate & Reinvestment Rate Risks
The general risk comparison:
Rising Interest Rates
Falling Interest Rates
Interest Rate Risk
Barbell favored
Laddered favored
Reinvestment Rate Risk

21. 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

22. 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

23. 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

24. Swaps Purpose Substitution swap Intermarket or yield spread swap
Bond-rating swap
Rate anticipation swap

25. Purpose
In a bond swap, a portfolio manager exchanges an existing bond or set of bonds for a different issue

26. 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

27. 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

28. Substitution Swap (cont’d)
Profitable substitution swaps are inconsistent with market efficiency
Obvious opportunities for substitution swaps are rare

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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.

39. Example: Bond A has a 10-year maturity, and bears a 7% coupon rate
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, respectively. What happens to these prices if the market rate changes from 7% to 7.7% ?

40. Answer:

41. 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.

43. Bond Convexity The importance of convexity Calculating convexity
General rules of convexity
Using convexity

44. 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.

45. 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

46. The Importance of Convexity (cont’d)
Greater Convexity
Bond Price
Yield to Maturity

47. 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

48. The Importance of Convexity (cont’d)
Error from using
duration only
Bond Price
Current bond
price
Yield to Maturity

49. Calculating Convexity
The percentage change in a bond’s price associated with a change in the bond’s yield to maturity:

50. Calculating Convexity (cont’d)
The second term contains the bond convexity:

51. 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

52. Example 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.

54. 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=

55. 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, obligation.

57. 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.

58. 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.

59. 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.

60. 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:

61. 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)

62. 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.

64. 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.

65. It worked because the change in interest rate was small
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.

66. How to build a portfolio of bonds with matching convexity?
Set up the following system (Example with three bonds):
where
and

67. 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.

68. Solution for the Weights

69. Verifying that it Works by Computing Portfolio Terminal Values for Various Rates

70. Plotting it Graphically: