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.