A great example of a passive fixed-income strategy is immunization. The objective is to try to lock in a target rate of return over a known investment horizon. This is accomplished by balancing the two opposing sources of interest rate risk over the holding period – cash flow reinvestment risk (on coupons and principal received prior to the horizon) and market price risk (on bonds that need to be liquidated at the horizon). If rates rise and remain risen, the future value of reinvested cash flow goes up but the sale price on the remaining bonds goes down. If, however, rates fall and remain fallen, the future value of reinvested cash flow goes down but the sale price on the remaining bonds goes up. As you no doubt are thinking, immunization requires assumptions about the timing and extent of interest rate volatility.

A numerical example here really helps. Assume that an investor has a firm 12-year investment horizon and buys on February 15, 2014, the Chapter 9 Treasury portfolio comprising some short-term notes, intermediate-term notes, and long-term bonds and zero-coupon STRIPS. For instance, this 12-year horizon could be the time until planned retirement or when payment is due for a child to attend a high-tuition (but very good) private university somewhere in Massachusetts. I'll lop off three digits of principal for each security to make the example more realistic. Note that all the summary statistics remain the same – bond math is invariant to scale.

So, we have a fixed-income bond portfolio valued at $345,701.56. The cash flow yield is 3.364%; the annual Macaulay and modified durations are 12.030 and 11.831; and the annual cash flow dispersion and convexity are 120.1 and 262.0. These are the “true” portfolio statistics based on the big- bundle-of-cash-flow approach and the cash flow yield, that is, the overall internal rate of return, and are not market-value-weighted averages.

The investor achieves a 12-year horizon yield of 3.364% if all coupon interest and principal cash flows received before February 15, 2026, are reinvested at 3.364% and if all bonds remaining in the portfolio at that time are sold at prices to yield 3.364%. This is the standard assumption about an internal rate of return that we first saw in Chapter 3. Essentially, if you stay on the constant-yield price trajectory, you can get off at any point and realize the original portfolio yield. Another way of saying this is that future transactions are made along the original implied forward curve. This assumes no default, of course. Also, this rate of return is in nominal terms. I'll get to immunization for real yields using linkers later.

Table 10.1 is an abridged version of the spreadsheet I use for this example. The third column is the cash flow for each semiannual period. The internal rate of return – that is, the solution for YieldPORT using equation 9.1 from Chapter 9 – is 1.682% per semiannual period; annualized it is 3.364%.

TABLE 10.1 Total Returns for a 12-Year Horizon on February 15, 2026

Date

Date

Cash Flow

Total Return at 3.364%

Total Return at 2.364%

Total Return

at 4.364%

0

02/15/14

-345,701.56

1

08/15/14

2,787.50

4,090.99

3,652.50

4,579.58

2

02/15/15

2,787.50

4,023.32

3,609.83

4,481.79

3

08/15/15

2,787.50

3,956.77

3,567.66

4,386.09

4

02/15/16

122,787.50

171,409.84

155,317.13

189,078.49

5

08/15/16

2,562.50

3,518.05

3,203.51

3,861.69

16

02/15/22

2,562.50

2,928.31

2,815.07

3,045.50

17

08/15/22

2,562.50

2,879.87

2,782.19

2,980.47

18

02/15/23

102,562.50

113,358.24

110,054.59

116,744.11

19

08/15/23

1,562.50

1,698.40

1,657.05

1,740.57

20

02/15/24

1,562.50

1,670.31

1,637.70

1,703.40

21

08/15/24

1,562.50

1,642.68

1,618.56

1,667.03

22

02/15/25

1,562.50

1,615.50

1,599.66

1,631.43

23

08/15/25

1,562.50

1,588.78

1,580.97

1,596.59

24

02/15/26

1,562.50

1,562.50

1,562.50

1,562.50

25

08/15/26

1,562.50

1,536.65

1,544.25

1,529.13

26

02/15/27

1,562.50

1,511.23

1,526.21

1,496.48

55

08/15/41

1,562.50

931.65

1,085.47

800.23

56

02/15/42

1,562.50

916.24

1,072.79

783.14

57

08/15/42

1,562.50

901.08

1,060.26

766.42

58

02/15/43

226,562.50

128,495.91

151,941.78

108,757.31

515,896.08

519,138.65

518,660.65

The fourth column is the value of each cash flow as of period 24, the horizon date on February 15, 2026. For example, the period-1 cash flow of $2,787.50 on August 15, 2014, grows to $4,090.99 over the 23 semiannual periods at the rate of 1.682%.

The period-58 cash flow of $226,562.50, which includes the redemption of principal on the two long-term bonds on February 15,2043, has a discounted value of $128,495.91 over the 34 remaining semiannual periods between February 2026 and February 2043.

The sum of the fourth column is $515,896.08. That's the total return on the horizon date under the strong assumption of reinvesting cash and selling bonds at the unchanged rate of 3.364%. It's no surprise that the realized holding-period rate (HPK) over the 12 years, as calculated in Chapter 3, is that same rate when annualized.

To see immunization at work, suppose that after purchasing the portfolio on February 15,2014, the entire Treasury curve shifts upward or downward. In the first case, as shown in the fifth column, the new portfolio cash flow yield is 2.364%, a 100-basis-point decrease. Notice that I do not need to assume necessarily a parallel shift to the yield curve. That is, all I need for the exercise is a new, lower rate that applies when cash flows are reinvested and when bonds are sold at the horizon date. Alternatively, I can assume that future transactions are made at the new implied forward rates. To get the entries for each date, I subtract 50 basis points from the original yield per semiannual period and redo the time-value-of-money calculations.

The total return on the horizon date is $519,138.65 and the horizon yield is 3.417%.

The sixth column shows the results from repeating the exercise for the increase in the rate up to 4.364% (s.a.). Again, a parallel shift is not required. We only need the assumption that cash flow reinvestment and bond sales can be made along the new, post-event implied forward curve that produces a portfolio yield of 4.364%. I add 50 basis points to the original yield per semiannual period and let the spreadsheet do the work. The total return as of February 15, 2026, is $518,660.65 and the horizon yield is 3.409% (s.a.).

If this is the first time you are seeing this demonstration of immunization, I hope that you are suitably impressed. Our investor essentially has locked in a total rate of return (in nominal terms) over the 12-year horizon. The holding- period rate of return remains close to the original cash flow yield of 3.364% (s.a.), in fact it is a little above, for a 100-basis-change in the interest rate. It's a bit higher because of convexity – this portfolio is more convex, that is, has a greater dispersion of cash flows, than a 12-year zero-coupon bond that would have provided “perfect” immunization. You can change the yield however you like – it works, as long as you keep that new rate constant going forward. Moreover, the yield curve event does not have to be instantaneous. The investor is immunized for the time until the first cash flow reinvestment date in six months.

How does immunization work? Why did I pick a 12-year investment horizon? It's because this portfolio happens to have an annual Macaulay duration of 12.030. The idea is that when the Macaulay duration of the portfolio matches (or, at least, is close to) the investment horizon, the cash flow reinvestment and market price risks offset. This is apparent in the last three columns of Table 10.1. When the yield goes up, the investor benefits from reinvesting the early (pre-horizon) cash flows but suffers from selling the late (post-horizon) coupon and principal payments. The opposite happens if yields go down.

Macaulay duration is quite remarkable. Not only does it provide a good first-order approximation for the change in market value following a sudden change in the yield, it also provides a measure of total return risk looking out into the future. When the investor's horizon is less than the duration, the risk is that yields go up because the market price effect dominates cash flow reinvestment. When the horizon is more than the duration, the risk is that yields fall because reinvestment dominates the market price risk. Remember this next time you win a big lottery.

Immunization is not without risk, however. Recall the underlying assumption that yields rise and remain risen or that yields fall and remain fallen. Surely, that's not likely. For example, suppose that the rate for reinvesting cash flows drops to 2.364% but then jumps up to 4.364% on the horizon date when the remaining bonds need to be sold. Summing the results for the first 24 periods from column 5 in Table 10.1 and then for the next 36 periods from column 6 give a total return of $469,947.62 and a realized rate of return of only 2.575%.

The amount of the interest rate risk in a duration-matching immunization strategy depends on the timing of the cash flows. This particular portfolio has significant cash flow dispersion. That short-term T-note maturing in 2016 is the major source of reinvestment risk. Where will 10-year notes be trading in two years? The long-term T-bond and STRIPS are a major source of market price risk. Where will 17-year Treasury prices be when those securities are scheduled to be sold in 12 years? The interest rate risk is mitigated by choosing bonds that match the Macaulay duration for the portfolio to the horizon and also minimize the dispersion statistic.

You probably are thinking that there is an obvious solution to this risk – just buy a zero-coupon bond that matures at the investment horizon. There is no cash flow subject to reinvestment risk, no market price risk (because the bond is redeemed at par value and not sold on the open market), and the dispersion is zero. In fact, it is productive to think of an immunization strategy using a portfolio of coupon bonds as “zero replication.” This is illustrated in Figure 10.4. If the yield remains constant, the price of the

FIGURE 10.4 Immunization as Zero Replication

zero moves smoothly up the trajectory reaching its destination at maturity, which by design matches the horizon date. However, the actual price path getting there might be quite volatile as yields bounce around. That doesn't matter – the hold-to-maturity zero-coupon bond achieves its objective independent of the price path.

The idea of immunization is to structure and manage the portfolio of coupon bonds to reach the same destination as the maturity-matching zero. By “structure,” I mean build the portfolio so that its Macaulay duration is close to the investment horizon. By “manage,” I mean rebalance the portfolio regularly to stay on duration target. In sum, keep the relevant summary statistic close to that of the zero-coupon bond that provides for perfect immunization.

Suppose in Figure 10.4 that the first event is a lower yield on the zero; its price moves up. The bond portfolio should track that price movement closely assuming the change in the portfolio yield is the same as the change on the zero. If the next market movement is a higher yield, both should fall in market value by the same amount as long as their durations continue to match. But we need to rebalance regularly to deal with “duration drift.” Remember from Chapter 6 that Macaulay duration on a coupon bond is inversely related to its yield to maturity. Also, as time passes duration declines in the saw-tooth pattern during the coupon period and then spikes upward on payment dates. The problem is that, as time passes and yields change, the durations of the portfolio of coupon bonds and the zero that provides perfect immunization do not change by the same amount.

The portfolio of coupon bonds needs to be managed actively to stay on duration target. How frequently the portfolio is rebalanced depends on transactions costs. Note that this can be done using derivative overlays such as interest rate swaps. Rebalancing means that the risk management problem starts over again each period. While the numerical examples assumed that rates rise and remain risen or that rates fall and remain fallen, immunization works in principle as long as the portfolio stays duration- matched – and as long as the yield curve is reasonably well behaved. That means generally parallel shifts to a generally flat curve. The real danger is a steepening twist, especially causing higher yields as the horizon nears and remaining bonds need to be sold.

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