FIXED INCOME SECURITIES
LECTURE OUTLINE: IMMUNIZATION
PROFESSOR DAVID T. BROWN
WARRINGTON COLLEGE OF BUSINESS
UNIVERSITY OF FLORIDA
1. Objective.
In the last class we showed how duration is both the average maturity of a bond and a measure of
bond’s sensitivity to interest rate changes. Thinking about duration as the average maturity of a
bond develops the foundations for using duration to manage interest rate risk, i.e. in creating
immunized portfolios.
2. Price Risk and Reinvestment Risk.
A cash flow to be received in the future is subject to price risk if it is received after the end of the
investors holding period. Cash flows received after the end of the investors holding period must
be sold at an uncertain price – hence the term price risk. Unanticipated increases in interest rates
lower the end of holding value of cash flows subject to price risk.
A cash flow to be received in the future is subject to reinvestment risk if it is received before the
end of the investors holding period. Cash flows received before the end of the investors holding
period must be reninvested at uncertain interest rates – hence the term reinvestment risk.
Unanticipated decreases in interest rates lower the end of holding period value of cash flows
subject to reinvestment risk.
3. Immunization.
Definition.
A bond portfolio is said to be immunized if the duration of the portfolio equals the investor’s
holding period.
What does “immunized” mean?
Immunization eliminates interest rate risk: if the portfolio is immunized, then an unanticipated
change in interest rates will have no effect on the investors holding period return.
Intuition.
Immunization occurs because price risk offsets reinvestment risk. If the duration (average
maturity) of your portfolio equals your holding period, then in some sense “half” payments are
subject to reinvestment risk and “half” are subject to price risk.
Example. Suppose you buy a four year $100 annuity in a 10% rate environment. The annuity
sells for $316.99. Now suppose that some time during the coming year interest rates rise to 11%.
The following shows the holding period returns for various holding periods.
One year holding period. With a one year holding period, your wealth in one year is $100 plus
the sale price of the bond. Thus, your holding period return is
[(100 + 244.37)/316.99] – 1 = 8.64%.
Not surprisingly, the increase in interest rates hurts you. Your return is 8.64% while it would
have been 10% had rates not changed.
Two year holding period. With a two year holding period, the first cash flow is reinvested for
one year, the second cash flow is received at the end of the holding period and the last two cash
flows are sold. Thus, the two year holding period return is
[(100(1.11) + 100 + 171.25)/316.99]1/2 – 1 = 9.81%.
Notice you are still subject to price risk, the return is less than 10%, however the effect of the
interest rate increase is much less than with a one year holding period return.
Three year holding period. With a three year holding period, the interest rate increase will
actually increase your holding period return. You have much more cash flows subject to
reinvestment risk than price risk. The three year holding period return is
[(100(1.11)2 + 100(1.11) + 100 + 90.09)/316.99]1/3 – 1 = 10.21%.
Four year holding period. A four year holding period obtains the largest gain from the rate
increase. The four year holding period return is
[(100(1.11)3 + 100(1.11)2 + 100(1.11) + 100)/316.99]1/4 – 1 = 10.40%.
The duration of this bond is
(.2868)(1) + (.2607)(2) + (.2370)(3) + (.2155)(4) = 2.38 years.
If you held this bond for 2.38 years, your wealth in 2.38 years would be the following function of
interest rates, r,
(100)(1+r)1.38 + (100)(1+r).38 + 100/(1+r).62 + 100/(1+r)1.62 = Wealth.
The beauty of immunization is that for any value of r that you pick, W will equal
316.99*(1.10)2.38.
4. Rebalancing.
A portfolio is immunized when the duration of the assets equals the holding period. In order to
remain immunized over time, the portfolio must be rebalanced. The following two examples
show that the need to rebalance occurs when ever the portfolio provides cash flows before the end
of the holding period.
Example One. Consider a two-year investment horizon. A portfolio with equal amounts
invested in a one and three year zero is immunized. After one year the duration of the portfolio is
(.5)(2) + (.5)(0) = 1 year.
This is great since your portfolio has a duration of one which equals your holding period. The
problem is the cash (the zero duration asset) needs to be reinvested into an instrument with a
duration greater than 0. Thus, some of the position in the two year must be sold as well.
Example Two. Suppose I have $1000 today to invest and rates are 10%. I want to guarantee
myself (100)(1.10)2 or $1210 in two years. Suppose I invest $500 in a one year zero and $500 in
a three year zero. Given the market prices of the $100 zeros (price of one year = 90.91 and price
of three year = 75.13), I buy 6.655 three year bonds and 5.500 one year bonds.
Suppose that rates fall to 9% during the year, then at the end of the year, my wealth is
(100)(5.500) + (6.655)[100/(1.09)2] = $1110.13.

Notice that market rates are now 9%, so that if invest the value of my portfolio at 9% for the
coming year, my wealth will be (1.09)(1110.13) = $1210.04. Thus, I have achieved my goal.

Notice I need to rebalance my portfolio since at the end of one year I have $550 in cash.