1. Managing Bond Portfolios
Topic 8 (Ch. 16)
Managing Bond Portfolios
Interest rate risk
 Interest rate sensitivity
 Duration
Convexity
Immunization

2. Interest rate sensitivity
 Interest Rate Risk
An inverse relationship exists between bond prices and yields, and interest rates can fluctuate substantially.
As interest rates rise and fall, bondholders experience capital losses and gains. These gains or losses make fixed-income investments risky.
Determinants of interest rate risk:
E.g. The percentage change in price corresponding to changes in YTM for 4 bonds that differ according to coupon rate, initial YTM, and time to maturity.
Interest rate sensitivity

4. Observations:
Bond prices and yields are inversely related: as yields increase, bond prices fall; as yields fall, bond prices rise.
An increase in a bond’s YTM results in a smaller price decline than the price gain associated with a decrease of equal magnitude in yield.
The price curve is convex.

5. Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of short-term bonds.
Compare the interest rate sensitivity of bonds A and B, which are identical except for maturity.
Bond B, which has a longer maturity than bond A, exhibits greater sensitivity to interest rate changes.

6. The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases
(i.e. interest rate risk is less than proportional to bond maturity).
Although bond B has six times the maturity of bond A, it has less than six times the interest rate sensitivity.
Although interest rate sensitivity seems to increase with maturity, it does so less than proportionally as bond maturity increases.

7. Interest rate risk is inversely related to the bond’s coupon rate.
Prices of high-coupon bonds are less sensitive to changes in interest rates than prices of low-coupon bonds.
Consider bonds B and C, which are alike in all respects except for coupon rate.
The lower-coupon bond C exhibits greater sensitivity to changes in interest rates.

8. The sensitivity of a bond’s price to a change in its yield is inversely related to the YTM at which the bond currently is selling.
Consider bonds C and D are identical except for the YTM at which the bonds currently sell.
Bond C, with a higher YTM, is less sensitive to changes in yields.
Conclusions:
Maturity, coupon rate, and initial YTM are major determinants of interest rate risk.

9. Duration
Why do we need duration?
Consider a 20-year 8% coupon bond and a 20-year zero-coupon bond.
The 20-year 8% bond makes many coupon payments, most of which come years before the bond’s maturity date.
Each of these payments may be considered to have its own “maturity date,” and the effective maturity of the bond is thus some sort of average of the maturities of all the cash flows paid out by the bond.

10. By contrast, the zero-coupon bond makes only one payment at maturity
By contrast, the zero-coupon bond makes only one payment at maturity. Its time to maturity is thus a well-defined concept.
The time to maturity of a bond is not a perfect measure of the long- or short-term nature of the bond.
We need a measure of the average maturity of the bond’s promised cash flows to serve as a useful summary statistic of the effective maturity of the bond.
We would like also to use the measure of as a guide to the sensitivity of a bond to interest rate changes, because we know that price sensitivity depends on time to maturity.

11. Macaulay’s duration:
The weighted average of the times to each coupon or principal payment made by the bond.
The weight associated with each payment time clearly should be related to the “importance” of that payment to the value of the bond.
Thus, the weight applied to each payment time should be the proportion of the total value of the bond accounted for by that payment.
This proportion is just the present value of the payment divided by the bond price.

12. Macaulay’s duration:
where
t: the time to each cash flow (coupon or principal)
: the weight associated with the
cash flow made at time t (CFt)
y: the bond’s YTM
T: time to maturity.

13. Example 1:
An 8% coupon, 2-year maturity bond with par value of $1,000 paying 4 semiannual coupon payments of $40 each.
The YTM on this bond is 10% (i.e. 5% per half-year).

14. Note: D < T (time to maturity) = 2 years

15. Example 2:
An zero-coupon, 2-year maturity bond with par value of $1,000.
The YTM on this bond is 10% (i.e. 5% per half-year).
Note: D = T (time to maturity) = 2 years

16. When interest rates change, the proportional change in a bond’s price (P) can be related to the change in its YTM (y) according to the following rule:
Notes:
1. Modified duration: 2.

17. The percentage change in bond price is just the product of modified duration and the change in the bond’s YTM.
Because the percentage change in the bond price is proportional to modified duration, modified duration is a natural measure of the bond’s exposure to changes in interest rates.

18. Example:
An 8% coupon, 2-year maturity bond with par value of $1,000 paying 4 semiannual coupon payments of $40 each.
The initial YTM on this bond is 10% (i.e. 5% per half-year).
Now, suppose that the bond’s semiannual yield increases by 1 basis point (i.e. 0.01%) to 5.01%.
Calculate the new value of the bond and the percentage change in the bond’s price.

20. Alternatively, using the equation:
Note:
Because we use a half-year interest rate of 5%, we also need to define duration in terms of a number of half-year periods to maintain consistency of units.
half-yeas.

21. What determines duration? Rules:
The duration of a zero-coupon bond equals its time to maturity.
We have seen that a zero-coupon, 2-year maturity bond has D = 2 years.

22. Bond duration versus bond maturity:

23. Holding maturity constant, a bond’s duration is higher when the coupon rate is lower.
We have seen that a coupon bond has a lower duration than a zero with equal maturity because coupons early in the bond’s life lower the bond's weighted average time until payments.
This property is attributable to the impact of early coupon payments on the average maturity of a bond’s payments. The higher these coupons, the higher the weights on the early payments and the lower is the weighted average maturity of the payments.
e.g. Compare the durations of the 3% coupon and 15% coupon bonds, each with identical yields of 15%.

24. Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity.
Duration always increases with maturity for bonds selling at par or at a premium to par.
For some deep-discount bonds, duration may fall with increases in maturity.
e.g. Compare the relations between bond duration and bond maturity for the following 3 bonds:
(i) 15% coupon YTM = 6% (premium bond)
(ii) 15% coupon YTM = 15% (selling at par)
(iii) 3% coupon YTM = 15% (deep-discount bond)

25. Note:
For the zero-coupon bond, maturity and duration are equal.
However, for coupon bonds duration increases by less than a year with a year’s increase in maturity. The slope of the duration graph is less than 1.0.
Although long-maturity bonds generally will be high-duration bonds, duration is a better measure of the long-term nature of the bond because it also accounts for coupon payments.
Time to maturity is an adequate statistic only when the bond pays no coupons; then, maturity and duration are equal.

26. Holding other factors constant, the duration of a coupon bond is higher when the bond’s YTM is lower.
At lower yields the more distant payments made by the bond have relatively greater present values and account for a greater share of the bond's total value.
Thus, in the weighted-average calculation of duration the distant payments receive greater weights, which results in a higher duration measure.
e.g. Compare (i) the duration of the 15% coupon, 6% YTM bond and (ii) the duration of the 15% coupon, 15% YTM bond.

27. The duration of a level perpetuity is:
D = (1 + y)/y.
e.g. At a 10% yield, the duration of a perpetuity that pays $100 once a year forever is:
(1 + 10%)/(10%) = 11 years
At an 8% yield, the duration is:
(1 + 8%)/8% = 13.5 years.

28. The duration of a level annuity is:
where T: the number of payments
y: the annuity’s yield per payment period.
e.g. A 10-year annual annuity with a yield of 8% will
have duration:

29. The duration of a coupon bond equals:
where c: the coupon rate per payment period
T: the number of payment periods
y: the bond’s yield per payment period.

30. e.g. A 10% coupon bond (par = $1,000) with 20 years until maturity, paying coupons semiannually, would have a 5% semiannual coupon and 40 payment periods.
If the YTM were 4% per half-year period, the bond’s duration would be:
= half-years = 9.87 years

31. For coupon bonds selling at par value (i. e
For coupon bonds selling at par value (i.e. c = y), the duration becomes:
where T: the number of payment periods
y: the bond’s yield per payment period.

32.   Convexity
The duration rule for the impact of interest rates on bond prices is only an approximation.
Recall:
The percentage change in the value of a bond approximately equals the product of modified duration times the change in the bond’s yield:

33. If this were exactly so, a graph of the percentage change in bond price as a function of the change in its yield would plot as a straight line, with slope equal to -D*.
Yet, the relationship between bond prices and yields is not linear.
The duration rule is a good approximation for small changes in bond yield, but it is less accurate for larger changes.

34. Bond price convexity: 30-year maturity, 8% coupon bond; initial yield to maturity = 8%.

35. The two lines are tangent at the initial yield.
Thus, for small changes in the bond’s YTM, the duration rule is quite accurate.
However, for larger changes in yield, the duration rule becomes progressively less accurate.

36. The duration approximation (the straight line) always understates the value of the bond.
It underestimates the increase in bond price when the yield falls, and it overestimates the decline in price when the yield rises.
This is due to the curvature of the true price-yield relationship.
Curves with shapes such as that of the price-yield relationship are said to be convex, and the curvature of the price-yield curve is called the convexity of the bond.

37. Mathematically:
Thus, D* is the slope of the price-yield curve expressed as a fraction of the bond price.

38. Similarly, the convexity of a bond equals the second derivative of the price-yield curve divided by bond price:
The convexity of noncallable bonds is positive since the slope increases (i.e. becomes less negative) at higher yields; that is,

39. The formula for the convexity of a bond:
where
P: bond price
y: the bond’s yield per payment period
CFt: cash flow (coupon or par value) paid to the
bondholder at time t
T: the number of payment periods

40. Adjustments for frequency of coupon payments per year:
If the coupons occur every 1/2 year, the convexity measure is in number of (i.e. 1/4) years.
To adjust the convexity to number of years, the convexity must be divided by 4.
In general, if the cash flows occur m times per year, the convexity is adjusted by dividing by m2.

41. Example 1:
An 8% coupon, 30-year maturity bond with par value of $1,000 paying 30 annual coupon payments of $80 each.
The bond sells at an initial YTM of 8% annually.
 bond price = $1,000.

43. Example 2:
A 6% coupon, 5-year maturity bond with par value of $1,000 paying 10 semiannual coupon payments of $30 each.
The bond sells at an initial YTM of 9% (i.e. 4.5% per half-year).

45. Convexity allows us to improve the duration approximation for bond price changes:
The first term on the right-hand side is the same as the duration rule and the second term is the modification for convexity.
For a bond with positive convexity, the second term is positive, regardless of whether the yield rises or falls.
This insight corresponds to the fact that the duration rule always underestimates the new value of a bond following a change in its yield.

46. Recall the example:
An 8% coupon, 30-year maturity bond with par value of $1,000 paying 30 annual coupon payments of $80 each.
The bond sells at an initial YTM of 8% annually.
 bond price = $1,000.

47. If the bond’s yield increases from 8% to 10%, the bond price will fall to:
= $

48. The duration rule:
which is considerably more than the bond price actually falls.
The duration-with-convexity rule is more accurate:

49. Note:
If the change in yield is small, the convexity term, which is multiplied by , will be extremely small and will add little to the approximation.
In this case, the linear approximation given by the duration rule will be sufficiently accurate.
Thus, convexity is more important as a practical matter when potential interest rate changes are large.

50. Why do investors like convexity?
Bonds A and B have the same duration at the initial yield, but bond A is more convex than bond B.
Bond A enjoys greater price increases and smaller price decreases when interest rates fluctuate by larger amounts.

52. Duration and convexity of callable bonds
When interest rates are high, the price-yield curve for a callable bond is convex, as it would be for a straight bond.
But, as rates fall, there is a ceiling on the possible price: The callable bond cannot be worth more than its call price.
In this region, the price-yield curve lies below its tangency line, and the curve is said to have negative convexity.

53. Price-yield curve for a callable bond:

54. In the region of negative convexity, interest rate increases result in a larger price decline than the price gain corresponding to an interest rate decrease of equal magnitude.
If rates rise, the bondholder loses, as would be the case for a straight bond.
But, if rates fall, rather man reaping a large capital gain, the investor may have the bond called back from her.

55. Effective duration for callable bonds:
where P: bond price
r: interest rate
Note:
We do not compute effective duration relative to a change in the bond’s own YTM. (The denominator is ; not )
This is because for callable bonds, which may be called early, the YTM is often not a relevant statistic.

56. Example:
A callable bond with a call price of $1,050 is selling today for $980.
If the interest rate moves up by 0.5%, the bond price will fall to $930.
If it moves down by 0.5%, the bond price will rise to $1,010.
 = assumed increase in rates
- assumed decrease in rates
= 0.5% - (-0.5%)
= 1% = 0.01

57.  = price at 0.5% increase in rates
- price at 0.5% decrease in rates
= $930 - $l,010
= -$80 
i.e. the bond price changes by 8.16% for a 1% swing
in rates around current values.

58.   Immunization
A strategy used by a firm to meet its future obligations which fluctuate with interest rates.
Example:
An insurance company issues a guaranteed investment contract (GIC) for $10,000.
Essentially, GICs are zero-coupon bonds issued by the insurance company to its customers.
If the GIC has a 5-year maturity and a guaranteed interest rate of 8%, the insurance company is obligated to pay $10,000  (1.08)5 = $14, in 5 years.

59. Suppose that the insurance company chooses to fund its obligation with $10,000 of 8% annual coupon bonds, selling at par value ($1,000), with 6 years to maturity.
As long as the market interest rate stays at 8%, the company has fully funded the obligation.

60. Years Remaining until Obligation Accumulated Value of Invested Payment
Payment Number
Years Remaining until Obligation
Accumulated Value of Invested Payment
A. Rates remain at 8% 1 4
800 × (1.08)4 ＝
1,088.39 2 3
800 × (1.08)3
1,007.77
800 × (1.08)2
933.12
800 × (1.08)1
864.00 5
800 × (1.08)0
800.00
Sale of bond
10,800/1.08
10,000.00
14,693.28

61. However, if interest rates change, two offsetting influences will affect the ability of the fund to grow to the targeted value of $14,
If interest rates rise, the fund will suffer a capital loss, impairing its ability to satisfy the obligation.
However, at a higher interest rate, reinvested coupons will grow at a faster rate, offsetting the capital loss.
In other words, fixed-income investors face two offsetting types of interest rate risk: price risk and reinvestment rate risk.

62. If the portfolio duration is chosen appropriately, these two effects will cancel out exactly.
When the portfolio duration is set equal to the investor’s horizon date, the accumulated value of the investment fund at the horizon date will be unaffected by interest rate fluctuations.
For a horizon equal to the portfolio’s duration, price risk and reinvestment risk exactly cancel out.

63. Recall:
For coupon bonds selling at par value, the duration becomes:
 The duration of the 6-year maturity bonds used to fund the GIC is:

65. We can also analyze immunization in terms of present values as opposed to future values:

66. If the obligation was immunized, why is there any surplus in the fund?
The answer is convexity.
The coupon bond has greater convexity than the obligation it funds.
Thus, when rates move substantially, the bond value exceeds the present value of the obligation by a noticeable amount.

68. Rebalancing immunized portfolios
As interest rates and asset durations change, a manager must rebalance the portfolio of fixed-income assets continually to realign its duration with the duration of the obligation.
Even if interest rates do not change, asset durations will change solely because of the passage of time.
Thus, even if an obligation is immunized at the outset, as time passes the durations of the asset and liability will generally fall at different rates.
Without portfolio rebalancing, durations will become unmatched and the goals of immunization will not be realized.

69. Example:
A portfolio manager faces an obligation of $19,487 in 7 years, which at a current market interest rate of 10%, has a present value of $10,000 [= $19,487/(1 + 10%)7].
Right now, suppose that the manager wishes to immunize the obligation by holding only 3-year zero-coupon bonds and perpetuities paying annual coupons.
At current interest rates, the perpetuities have a duration of (1 + 10%)/10% = 11 years.
The duration of the zero is simply 3 years.

70. Let w: the zero’s weight
1 – w: the perpetuity’s weight
The duration of a portfolio is the weighted average of the durations of the assets comprising the portfolio.
To achieve the desired portfolio duration of 7 years, the manager would have to choose appropriate values for the weights of the zero and the perpetuity in the overall portfolio:

71.  The manager invests:
(i) $5,000 (= $10,000  0.5) in the zero-coupon bond
[the face value of the zero will be $5,000  (1.10)3
= $6,655]
(ii) $5,000 (= $10,000  0.5) in the perpetuity,
providing annual coupon payments of $500 per year
indefinitely.
The portfolio duration is then 7 years, and the position is immunized.

72. Next year, even if interest rates do not change, rebalancing will be necessary.
The present value of the obligation has grown to $11,000 [= $19,487/(1 + 10%)6], because it is 1 year closer to maturity.
The manager’s funds also have grown to $11,000:
(i) The zero-coupon bonds have increased in value
from $5,000 to $5,500 [= $6,655/(1.10)2] with the passage of time.
(ii) The perpetuity has paid its annual $500 coupon
and still is worth $5,000 [= $500/10%].

73. However, the portfolio weights must be changed.
The zero-coupon bond now will have duration of 2 years, while the perpetuity remains at 11 years.
The obligation is now due in 6 years.
The weights must now satisfy the equation:

74. The manager must invest a total of $11,000  5/9 = $6,111
The manager must invest a total of $11,000  5/9 = $6, in the zero.
This requires that the entire $500 coupon payment be invested in the zero and that an additional $ of the perpetuity be sold and invested in the zero in order to maintain an immunized position.

75. Note 1:
Rebalancing of the portfolio entails transaction costs as assets are bought or sold, so one cannot rebalance continuously.
In practice, an appropriate compromise must be established between the desire for perfect immunization, which requires continual re­balancing, and the need to control trading costs, which dictates less frequent rebalancing.

76. Note 2:
Why not simply buy a zero-coupon bond that provides a payment in an amount exactly sufficient to cover the projected cash outlay?
If we follow the principle of cash flow matching, we automatically immunize the portfolio from interest rate movement because the cash flow from the bond and the obligation exactly offset each other.
Once the cash flows are matched, there is no need for rebalancing.

77. However, cash flow matching is not more widely pursued probably because:
 Immunization strategies are appealing to firms that do not wish to bet on general movements in interest rates, but these firms may want to immunize using bonds that they perceive are undervalued.
However, cash flow matching places so many more constraints on the bond selection process that it can be impossible to pursue a strategy using only “underpriced” bonds.

78.  Sometimes, cash flow matching is simply not possible.
To cash flow match for a pension fund that is obligated to pay out a perpetual flow of income to current and future retirees, the pension fund would need to purchase fixed-income securities with maturities ranging up to hundreds of years.
Such securities do not exist, making exact cash flow matching infeasible.