Bond Immunization

Finance Notes: Dr. Kashefi Bond Portfolio Immunization Techniques A bond portfolio is immunized if its investment performance is not sensitive to changes in interest rates. A bond portfolio manager can use the concept of duration and to immunize the portfolio. The immunization techniques fall into two categories: (1) the bank immunization case and the planning period case. The primary focus of this paper is on the planning period case. The Planning Period Case With this technique the bond portfolio manager is concerned with managing a portfolio toward an horizon period. Many bond portfolios have a definite planning period, with the goal being to achieve a target value for the portfolio at the end of the planning period. A typical problem occurs in pension fund management, with the bond manager managing the bonds in the pension fund toward a horizon date set when pensions become payable. The problem confronting the bond manager in this case concerns the effect of changing interest rates on the immediate value of the bond portfolio and on the reinvestment rate, the rate at which cash thrown off by the bond portfolio can be reinvested. Overall, a manager investing toward a future horizon period tries to maximize the value of the portfolio on that future date, subject to risk constraints. Equivalent to maximizing the future value of a portfolio, a manger might attempt to maximize the Realized Compound Yield to Maturity (RCYTM). RCYTM = T

TV −1 IV

TV= Terminal Value IV= Initial Value T= Horizon date According to this definition, the RCYTM is the compound yield realized on an investment over T periods. For example, assume that a $1,000 bond (bond A) portfolio consists of one 10% annual coupon rate with a face value of $1,000, which matures in five years. If interest rates are currently 10% and remain steady for the five-year period, all of the coupon payments may be invested at 10%. In this case, the RCYTM will be 10% over that five-year period. Bond A

R=10%

Period Investment 0

Coupon Future Value Payment (1+R)t

$1,000

$1,000

1

$100

$146

2

$100

$133

3

$100

$121

4

$100

$110

5

$100 TV

$100 $1,611

1

RCYTM = 5

$1,611 −1 = 10% $1,000

However, a change in interest rates during planning period can dramatically affect the RCYTM. If interest rates had suddenly dropped at the beginning of the investment period from 10% to 6%, the RCYTM would have fallen to 9.35%. Table 1

Bond A

R=6% Coupon Future Value Payment (1+R)t

Period Investment 0

$1,000

$1,000

1

$100

$126

2

$100

$119

3

$100

$112

4

$100

$106

5

$100 TV

$100 $1,564

RCYTM = 5

$1,546 −1 = 9.35% $1,000

Planning Period Immunization allows the bond manager to avoid this result. If the duration of the portfolio equals the number of years in the planning period, the portfolio will be immunized. This means a shift in interest rates will not affect the RCYTM or the terminal wealth achieved over a given planning period. In this example, the problem arose from the fact that the duration of the portfolio (4.17) was less than the planning period (5years). Table below shows the calculation of the duration Calculation Table 2 of Duration Bond A Initial Coupon Period (t) Investment Payment C 0 $1,000 1 $100 2 $100 3 $100 4 $100 5 $100 5 $1,000

txC

1/(1+10%)

$100 $200 $300 $400 $500 $5,000

0.9091 0.8264 0.7513 0.6830 0.6209 0.6209

t

Present Value at 10%

$90.91 $165.29 $225.39 $273.21 $310.46 $3,104.61 $4,169.87 Duration=$4,169.87/$1,000 = 4.17

Assume now that a longer duration bond (bond B) was also available, one having 8 years, a 10% coupon rate, and yield-to-maturity of 10%. With initial yields at 10%, this bond would have a price of $1,000 and duration of 5.87. For the planning period of five years, and with availability of theses two bonds, it is possible to create a portfolio with duration of 5.0, to match the 5 years horizon period. To create the new portfolio, funds must be contributed to both bonds so that the average duration of the portfolio is 5.0. If 51.18 2

percent of the portfolio’s value is committed to the five-year bond, and 48.82 percent is committed to the eight-year bond, the resulting portfolio will have an average duration of 5.0. Portfolio’s Duration = 0.5118(4.17) + 0.4882(5.87)=5.0 The percentages for each bond, 51.18% and 48.82%, is calculated by solving for the XA and XB from the two equations. XA (4.17) + XB (5.87)=5.0 XA + XB =1.0 To see how this works, assume that the bonds are divisible so that 51.18% of $1,000, ($511.80) is invested in the five-year bond (bond A) and $488.20 is committed to eightyear bond (bond B). To see how the immunization works in this case, assume once again that interest rates change from 10% to 6% as soon as the portfolio is established. In this case the future value of the investment in bond B is $1671 and for bond A is estimated to be $1564 in Table 1. Table 3

Bond B

Period Investment

Coupon Future Value Payment (1+R)t

1

$100

$126

2

$100

$119

3

$100

$112

4

$100

$106

5 5

$100

$100

TV

$1,107 $1,671

Sell Bond B

The weighted future value of two bonds is equal to $1615 (this higher number is due to rounding) as shown below. Weight Future Value Weighted Value 51.18% 1,563.71 $800.31 48.82% $1,671 $815.60 100.00% Total value $1,615.91

The rate of return on this two-bonds portfolio is equal to 10%, which is the target rate of the portfolio. RCYTM = 5

$1,615 −1 = 10% . $1,000

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