Duration and Portfolio
Immunization
Portfolio Immunization
• Portfolio immunization
– An investment strategy that tries to protect the
expected yield from a security or portfolio of
securities by acquiring those securities whose
duration equals the length of the investor’s
planned holding period.
Portfolio Immunization
• If the average duration of a portfolio equals
the investor’s desired holding period, the
effect is to hold the investor’s total return
constant regardless of whether interest rates
rise or fall.
– In the absence of borrower default, the
investor’s realized return can be no less than
the return he has been promised by the
borrower.
Example
• Assume we are interested in a $1,000 par
value bond that will mature in two years.
• The bond has a coupon rate of 8 percent and
pays $80 in interest at the end of each year.
• Interest rates on comparable bonds are also
at 8 percent but may fall to as low as 6
percent or rise as high as 10 percent.
Example
• The buyer knows he will receive $1000 at
maturity, but in the meantime he faces the
uncertainty of having to reinvest the annual
$80 in interest earnings at 6%, 8%, or 10%.
Example: Case 1
• Let interest rates fall to 6%.
– The bond will earn $80 in interest payments for
year one, $80 for year two, and $4.80 ($80 x
0.06) when the $80 interest income received the
first year is reinvested at 6% during year 2.
Example: Case 1
• How much will the investor earn over the
two years?
– First year’s interest earnings + Second year’s
interest earnings + Interest earned reinvesting
the first year’s interest earnings at 6% + Par
value of the bond at maturity.
– $80 + $80 + $4.80 + $1,000 = $1,164.80
Example: Case 2
• Let interest rates rise to 10%.
– The bond will earn $80 in interest payments for
year one, $80 for year two, and $8.00 ($80 x
0.10) when the $80 interest income received the
first year is reinvested at 10% during year 2.
Example: Case 2
• How much will the investor earn over the two
years?
– First year’s interest earnings + Second year’s
interest earnings + Interest earned reinvesting the
first year’s interest earnings at 10% + Par value
of the bond at maturity.
– $80 + $80 + $8 + $1,000 = $1,168.00
Immunization and Duration
• The investor’s earnings could drop as low
as $1,164.80 or rise as high as $1,168.
• But, if the investor can find a bond whose
duration matches his or her planned holding
period, he or she can avoid this fluctuation
in earnings.
– The bond will have a maturity that exceeds the
investor’s holding period, but its duration will
match it.
Example: Case 1
• Let interest rates fall to 6%.
– The bond will earn $80 in interest payments for
year one, $80 for year two, and $4.80 ($80 x
0.06) when the $80 interest income received the
first year is reinvested at 6% during year 2.
– But, the bond’s market price will rise to
$1,001.60 due to the drop in interest rates.
Example: Case 1
• How much will the investor earn over the
two years?
– First year’s interest earnings + Second year’s
interest earnings + Interest earned reinvesting
the first year’s interest earnings at 6% + Market
price of the bond at the end of the investor’s
planned holding period.
– $80 + $80 + $4.80 + $1,001.60 = $1,166.40
Example: Case 2
• Let interest rates rise to 10%.
– The bond will earn $80 in interest payments for
year one, $80 for year two, and $8.00 ($80 x
0.10) when the $80 interest income received the
first year is reinvested at 10% during year 2.
– But, the bond’s market price will fall to
$998.40 due to the rise in interest rates.
Example: Case 2
• How much will the investor earn over the two
years?
– First year’s interest earnings + Second year’s
interest earnings + Interest earned reinvesting the
first year’s interest earnings at 10% + Par value
of the bond at maturity.
– $80 + $80 + $8 + $998.40 = $1,166.40
Conclusion
• The investor earns identical total earnings
whether interest rates go up or down.
– With duration set equal to the buyer’s planned
holding period, a fall (rise) in the reinvestment
rate is completely offset by an increase (a
decrease) in the bond’s market price.
Conclusion
• Immunization using duration seems to work
reasonably well because the largest single
element found in most interest rate movements
is a parallel change in all interest rates
(explains about 80% of all interest rate
movements over time).
• So, investors can achieve reasonably effective
immunization by approximately matching the
duration of their portfolios with their planned
holding periods.
Opportunity Cost
• Duration is not free. There is an
opportunity cost.
– If the investor had simply bought a bond with a
calendar maturity of two years and interest rates
rose, he or she would have earned $1,168.
• The opportunity cost of immunization is a lower,
but more stable, expected return.
Limits of Duration
• In reality it can be difficult to find a
portfolio of securities whose average
portfolio duration exactly matches the
investor’s planned holding period.
– As the investor grows older, his planned
holding period grows shorter, as does the
average duration of his portfolio, but they may
not decline at the same rate.
• Portfolio requires constant adjustments.
Limits of Duration
• Many bonds are callable so bondholders
may find themselves with a sudden and
unexpected change in their portfolio’s
average duration.
• The future path of interest rates cannot be
perfectly forecast; therefore, immunization
with duration cannot be perfect without the
use of complicated models.