Holdingperiod Immunization

The Online Learning Center (www.mhhe.com/bkm) contains a spreadsheet that is useful in understanding the concept of holding-period immunization. The spreadsheet calculates duration and holding-period returns on bonds of any maturity. The spreadsheet shows how price risk and reinvestment risk offset if a bond is sold at its duration.

A 1 B 1 C

D

E

F

G

1

Holding Period Immunization

2

Example:

3

YTM

0.1158

Mar Price

1111.929

4

Coupon R

0.14

5

Maturity

7

6

Par Value

1000

7

Holding P

5

8

Duration

5.000251

9

10

11

If Rates Increase by 200 basis points

If Rates Increase by 100 basis points

12

Rate

0.1358

Rate

0.1258

13

FV of CPS

917.739

FV of CPS

899.7046

14

SalesP

1006.954

SalesP

1023.817

15

Total

1924.693

Total

1923.522

16

IRR

0.115981

IRR

0.115845

17

18

19

20

If Rates Decrease by 200 basis points

If Rates Decrease by 100 basis points

21

Rate

0.0958

Rate

0.1058

22

FV of CPS

847.5959

FV of CPS

864.6376

23

SalesP

1077.145

SalesP

1058.897

24

Total

1924.741

Total

1923.534

25

IRR

0.115987

IRR

0.115847

This example highlights the importance of 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. Moreover, even if interest rates do not change, asset durations will change solely because of the passage of time. Recall from Figure 16.4 that duration generally decreases less rapidly than does maturity. Thus, even if an obligation is immunized at the outset, as time passes the durations of the asset and liability will fall at different rates. Without portfolio rebalancing, durations will become unmatched and the goals of immunization will not be realized. Obviously, immunization is a passive strategy only in the sense that it does not involve attempts to identify undervalued securities. Immunization managers still actively update and monitor their positions.

As another example of the need for rebalancing, consider a portfolio manager facing an obligation of $19,487 in seven years, which, at a current market interest rate of 10%, has a present value of $10,000. Right now, suppose that the manager wishes to immunize the obligation by holding only three-year zero-coupon bonds and perpetuities paying annual

PART IV Fixed-Income Securities

Figure 16.10 Immunization. The coupon bond fully funds the obligation at an interest rate of 8%.

Moreover, the present value curves are tangent at 8%, so the obligation will remain fully funded even if rates change by a small amount.

coupons. (Our focus on zeros and perpetuities helps keep the algebra simple.) At current interest rates, the perpetuities have a duration of 1.10/.10 = 11 years. The duration of the zero is simply three years.

For assets with equal yields, 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 seven years, the manager would have to choose appropriate values for the weights of the zero and the perpetuity in the overall portfolio. Call w the zero's weight and (1 - w) the perpetuity's weight. Then w must be chosen to satisfy the equation w x 3 years + (1 - w) x 11 years = 7 years which implies that w = i/2. The manager invests $5,000 in the zero-coupon bond [the face value of the zero will be $5,000 X (1.10)3 = $6,655] and $5,000 in the perpetuity, providing annual coupon payments of $500 per year indefinitely. The portfolio duration is then seven years, and the position is immunized.

Next year, even if interest rates do not change, rebalancing will be necessary. The present value of the obligation has grown to $11,000, because it is one year closer to maturity. The manager's funds also have grown to $11,000: The zero-coupon bonds have increased in value from $5,000 to $5,500 with the passage of time, while the perpetuity has paid its annual $500 coupon and still is worth $5,000. However, the portfolio weights must be changed. The zero-coupon bond now will have duration of 2 years, while the perpetuity

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