# Duration and Immunization of Portfolio

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```					Duration and Immunization of
Portfolio
MATH 384, Review lecture 10
Andrei Ordine

Duration and Immunization of Portfolio – p.1/23
In this lecture
Modiﬁed duration
Macaulay duration
Duration of a portfolio
Asset-Liability Matching
Duration and Convexity
Hedging interest rate risk

Duration and Immunization of Portfolio – p.2/23
Bond pricing equation

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(1)

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Assume that the term structure is ﬂat at any moment (for
simplicity).
We are interested in the sensitivity of the price to the
yield-to-maturity .
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Duration and Immunization of Portfolio – p.3/23
Modiﬁed duration

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(2)

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Duration and Immunization of Portfolio – p.4/23
Modiﬁed duration
Check:
Does         depend on the face value?
When       grows, does      grow?

Duration and Immunization of Portfolio – p.5/23
Macaulay duration
Longer-term bonds are more sensitive to changes in
the rate of interest.
Another parameter used is Macaulay duration:

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(3)

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It measures “average” maturity of the bond.

Duration and Immunization of Portfolio – p.6/23
Example
Problem 1 Calculate the Macaulay duration of a
zero-coupon bond and of a bond with coupon rate .

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Show that the duration is a weighted average of times of
bond payments.

Duration and Immunization of Portfolio – p.7/23
Duration of a portfolio
Problem 2 Show that the duration of a portfolio of
bonds is equal to a weighed sum of durations of
individual bonds.

Duration and Immunization of Portfolio – p.8/23
Application
Asset-Liability Matching and immunization:
An organization would like to make sure that the
present value of its assets less the present value of its
liabilities (net position) is hedged against
movements in the rates of interest.

Duration and Immunization of Portfolio – p.9/23
ALM
What is the ﬁrst order condition?
What is the second order condition?

Duration and Immunization of Portfolio – p.10/23
ALM
The ﬁrst order condition is:

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(4)

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The second order condition is:
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(5)

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For the second-order condition to be formulated in a
short way, we need to deﬁne convexity.

Duration and Immunization of Portfolio – p.11/23
ALM
The convexity of a bond is

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(6)

Duration and Immunization of Portfolio – p.12/23
ALM
Suppose that     are the assets and                                                                                the liabilities.

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The ﬁrst order condition is:
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(7)
The second order condition is:
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(8)

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Duration and Immunization of Portfolio – p.13/23
Example
million investment in 30-year zero-coupon Brazil bonds.
To hedge interest rate risk, some bonds need to be sold
short. The following duration data is available for
different bonds:
Napping Beauty

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30-year coupon bonds

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30-year zero coupon
¢

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Duration and Immunization of Portfolio – p.14/23
Example
Problem 4 Suppose that the manager hedges its
investment in the Brazil bonds by selling short the
30-year coupons. How much will it need to sell short?

Duration and Immunization of Portfolio – p.15/23
\$23,734m

Duration and Immunization of Portfolio – p.16/23
Example
Problem 5 Selling bonds short requires ﬁnding a
current owner to loan the bonds. The manager has found
an owner willing to lend \$20 million worth of the 30-year
coupon bonds to the fund, which the ﬁrm can then sell
short. By itself, however, this short position will not
hedge the total interest rate risk. How many additional
Napping Beauties (in dollar terms) will the fund have to
sell short to hedge its total exposure in the Brazil bonds?

Duration and Immunization of Portfolio – p.17/23
\$6.404

Duration and Immunization of Portfolio – p.18/23
Example
Problem 6 The bonds have convexities as follows:
Napping Beauty

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30-year coupon bonds

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30-year zero coupon

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Does the second-order condition hold for the portfolio in
the last problem?

Duration and Immunization of Portfolio – p.19/23
Second approximation
Using Taylor series, the change of the value of a

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bond due to interest rate change      is

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(9)

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Duration and Immunization of Portfolio – p.20/23
Example
Problem 7 Compute the second-order approximation of
changes in the value of portfolio in the previous problem
assuming +1% and -1% changes in the rate of interest.
The current rate of interest is 7.55%

Duration and Immunization of Portfolio – p.21/23
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Duration and Immunization of Portfolio – p.22/23
Summary
Thank you!

Duration and Immunization of Portfolio – p.23/23

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