1. Managing Bond Portfolio
CHAPTER 16

2. Managing Bond Portfolios
Objectives:
Analyze the features of a bond that affect the sensitivity of its price to interest rates.
Compute the duration of bonds.
Formulate fixed-income immunization strategies for various investment horizons.
Analyze the choices to be made in an actively managed fixed-income portfolio.

3. 16.1 INTEREST RATE RISK
Interest rate risk is uncertainty related to bond prices due to uncertainty in interest rate

4. Time to maturity Coupon rate Interest rate sensitivity:
Interest Rate Risk
Interest rate sensitivity:
Time to maturity
Coupon rate

5. Interest Rate Sensitivity
Inverse relationship between price and yield
Long-term bonds tend to be more price sensitive than short-term bonds
Interest rate risk is inversely related to bond’s coupon rate

6. Table 16.1 Prices of 8% Coupon Bond (Coupons Paid Semiannually)

7. Table 16.2 Prices of 8% Coupon and Zero-Coupon Bond (Semiannually Compounding)

8. Duration
Change in price of 8% coupon bond < change in price of 0% coupon bond or 8% bond has higher interest rate risk than 0% bond
We also know that long-term bond has higher interest rate risk than short-term bond.
This implies that 0% bond is a longer term investment than 8% bond.
This implies that regular maturity is not good enough to tell about the short term/long term nature (or interest rate sensitivity) of the bonds.
We need a maturity that can tell 0% bond is really longer term investment than 8% bond
Hence we have concept of effective maturity or duration that takes into account both regular maturity and coupon payments.

9. Duration 8% coupon bond Time 0 1 2 3 ..... 20
cash flow
0% coupon bond
cash flow
If we view a bond is a portfolio of all coupon payments and principal payment, and each payment has its own maturity, and the effective maturity of the bond is the weighted average of all maturity of all payments, then obviously, the maturity of 0% coupon bond is different from maturity of 8% coupon bond.

10. Duration A measure of the effective maturity of a bond
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

11. Duration
Measures the effective maturity by weighting the payments by their proportion of the bond value.
where t =1, 2, 3, ... T are the times to maturity of payments
y is the bond's yield to maturity (current market rate)

12. Spreadsheet 16.1 Calculating the Duration of Two Bonds

13. Duration: example
A pension plan is obligated to make disbursements of $1 million, $2 million, and $1 million at the end of each of the next three years, respectively. Find the duration of the plan’s obligations if the interest rate is 10% annually.

14. Duration Duration measure does three things:
It measures the effective average maturity of a bond.
It measures interest rate sensitivity correctly.
It provides the necessary information for immunization.

15. Duration and Interest Rate Sensitivity
Sensitivity of prices to interest rate changes:
where y is the yield to maturity

16. Duration
Example: The duration for a bond (6% coupon rate, semi-annual payment, 2 years to maturity), currently priced at $929.08, with a yield-to-maturity (YTM) of 10% is years. If interest rates rise by 0.5 percentage points (50 basis points), what will be the percentage change in the price of the bond?
If interest rates rise by 0.1 percentage points (10 basis points), what will be the percentage change in the price of the bond?

17. Convexity
The relationship between bond prices and yields is not linear
Duration rule is a good approximation for only small changes in bond yields

18. Figure 16.3 Bond Price Convexity: 30-Year Maturity, 8% Coupon; Initial Yield to Maturity = 8%

19. Correction for Convexity

20. Bond Price Sensitivity
Determinants of a bond’s price sensitivity to interest rate changes:
the time to maturity
(Duration not always increasing in time to Maturity)
the coupon rate
(Duration always decrease with high Coupon)
the yield to maturity
(Duration always decrease if YTM increase)

21. Rules for Duration
Rule 1 The duration of a zero-coupon bond equals its time to maturity
Rule 2 Holding maturity constant, a bond’s duration is higher when the coupon rate is lower
Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity
Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower
Rules 5 The duration of a level perpetuity is equal to: (1+y) / y

22. Figure 16.2 Bond Duration versus Bond Maturity

23. Table 16.3 Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons)

24. Use of duration Key concept in bond management
Simple summary measure of effective maturity of bond. It is the effective maturity that tells the short-term/long- term nature of bond, not regular maturity
Measure of interest sensitivity of a bond portfolio
Bond A: 8% semi-annual coupon, 2 years to maturity, D =
Bond B: 0% coupon, 2 years to maturity, D = 2
Bond B has more interest rate risk than bond A
Bond B: 0% coupon, years to maturity, D =
Bond B and Bond A has same interest rate risk
It provides the necessary information for immunization.

25. Trading Strategies Using Duration
Longest-duration security provides the maximum price variation
If you expect a decline in interest rates, increase the average duration of your bond portfolio to experience maximum price increase
If you expect an increase in interest rates, reduce the average duration to minimize your price decline
FIN Lecture 4 9/6/2007

26. 10.2 PASSIVE BOND MANAGEMENT

27. Passive Bond Management
Takes prices as given and tries to control the risk
of the fixed-income portfolio.
Measures:
Net Worth Immunization (Present)
(e.g. Banks: Asset/Liability Management)
2. Target Date Immunization (Future)
(e.g. Pension Funds: meet future obligations)

28. Main Idea Behind Immunization
Net Worth Immunization:
Match duration of asset and liabilities by adjusting their maturity structure (Gap Management)
Target Date Immunization:
Set the duration of a portfolio equal to the target date. This guarantees that at this date reinvestment risk and price risk exactly cancel out.

29. Target Date Immunization
Example. An insurance company issue a 5-years Guaranteed investment contract (GIC) at 8%, nominal value $10,000. The insurance company decides to meet this obligation by investing $10,000 in 8% annual coupon bonds with maturity in 6yrs. Can the firm meets its obligation at time 5? What if interest rate drops to 7% ? What if interest rate increases to 9% ?

30. Table 16.4 Terminal value of a Bond Portfolio After 5 Years (All Proceeds Reinvested)

31. Figure 16.9 Growth of Invested Funds

32. Figure Immunization

33. Problem 7, Chapter 16(p.544)
An insurance company must make payments to a customer of $10 mil in 1 year and $4 mil in 5 years. The market interest rate is 10%
If it wants to fully fund and immunize its obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase?
What must be the face value and market value of that zero- coupon bond?

34. Net Worth Immunization
Given a liability currently worth L and with duration DL
Match it with an asset currently worth L and with duration DL.
This guarantees that, for small changes in the interest rate the net worth will always be approximately zero.

35. Net Worth Immunization
Current Value of
Asset and
Liabilities
Current of Coupon Bond (Asset)
(YTM=8%)
Present Value
of CIG (Liability)
(YTM=8%)
8%=YTM
Interest rate

36. Immunization: Rebalancing
Even if a position is immunized, there is still need for rebalancing for duration because of
Change in interest rate in market
Passage of time
Cannot rebalance continuously because of transaction cost involved
In practice, must establish a compromise between the desire for perfect immunization which requires continuous rebalancing and the need to control for trading cost which dictates less frequent rebalancing

37. 10.3 ACTIVE BOND MANAGEMENT

38. Active Bond Management
Sources of potential profits:
Interest rate forecasts
Identification of mispriced bonds

39. Swapping Strategies Substitution swap Intermarket swap
Rate anticipation swap
Pure yield pickup
Tax swap

40. Swapping Strategies Substitution swap Intermarket swap
an exchange of bond for nearly identical substitute (coupon, maturity, quality, call features, etc)
Toyota bond, 20 years to maturity, 8% coupon, YTM = 8.05%
Honda bond, 20 years to maturity, 8% coupon, YTM = 8.15%
Honda looks more attractive
Intermarket swap
the yield spread between 2 sectors of bond market is temporarily out of line.
Example: currently, the yield spread between 10-year T- bond and 10 year BBB corporate bond is 3%, the historical spread is 2%, should consider selling T-bond and buy BBB corporate bond

41. Swapping Strategies Rate anticipation swap Pure yield pickup Tax swap:
expected rate to fall, swap into bonds of longer duration. Expected rate to rise, swap into shorter duration
Pure yield pickup
the strategy depends on the shape of the yield curve in the market
If yield curve is upward sloping, should move into long-term bonds to get higher yield. However, at the same time, your portfolio is exposed to higher interest rate risk.
Tax swap:
Swap a capital gain bond to a capital loss bond to avoid tax

42. Contingent Immunization
A combination of active and passive management
The 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

43. Figure 16.11 Contingent Immunization