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Chapter 11
Interest Rate Sensitivity
(Duration we will cover in Finc420)
The concept:
• Any security that gives an investor more money back
sooner (as a % of your investment) will have lower price
volatility when interest rates change.
Maturity is a major determinant of bond price sensitivity
to interest rate changes, but
It is not the only factor; in particular the coupon rate and
the current ytm are also major determinants.
More on Duration
1. Duration increases with maturity
2. A higher coupon results in a lower duration
3. Duration is shorter than maturity for all bonds except
zero coupon bonds
4. Duration is equal to maturity for zero coupon bonds
5. All else equal, duration is shorter at higher interest rates
Duration/Price Relationship
• Price change is proportional to duration
and not to maturity
D = Duration
DP/P = -D x [Dy / (1+y)]
D* = modified duration
D* = D / (1+y)
DP/P = - D* x Dy
Interest Rate Risk
Interest rate risk is the possibility that an investor does not earn
the promised ytm because of interest rate changes.
A bond investor faces two types of interest rate risk:
1.Price risk: The risk that an investor cannot sell the bond for as
much as anticipated. An increase in interest rates reduces the
sale price.
2.Reinvestment risk: The risk that the investor will not be able to
reinvest the coupons at the promised yield rate. A decrease in
interest rates reduces the future value of the reinvested
The two types of risk are potentially offsetting.
• Immunization: An investment strategy
designed to ensure the investor earns the
promised ytm.
• A form of passive management, two
1. Target date immunization
• Attempt to earn the promised yield on the bond
over the investment horizon.
• Accomplished by matching duration of the bond to
the investment horizon
2. Net worth immunization
• The equity of an institution can be
immunized by matching the duration of
the assets to the duration of the liabilities.
Cash Flow Matching and
• Cash flow from the bond and the obligation exactly offset
each other
– Automatically immunizes a portfolio from interest rate
• Not widely pursued, too limiting in terms of choice of
• May not be feasible due to lack of availability of
investments needed
Problems with Immunization
1. May be a suboptimal strategy
2. Does not work as well for complex portfolios with option
components, nor for large interest rate changes
3. Requires rebalancing of the portfolio periodically, which
then incurs transaction costs
– Rebalancing is required when interest rates move
– Rebalancing is required over time
The Need for Convexity
(calculation in Finc 420)
• Duration is only an approximation
• Duration asserts that the percentage price
change is linearly related to the change in
the bond’s yield
– Underestimates the increase in bond prices
when yield falls
– Overestimates the decline in price when the
yield rises
Convexity: Definition and Usage
 CFt
Convexity 
(t  t )
2 
P  (1 y) t 1  (1 y)
Where: CFt is the cash flow (interest and/or principal) at
time t and y = ytm
The prediction model including convexity is:
 D 
 1/ 2  Convexity  Dy 2
(1  y )
Swapping Strategies - active
1. Substitution swap
– Exchanging one bond for another with very similar
characteristics but more attractively priced
2. Intermarket spread swap
– Exploiting deviations in spreads between two market segments
3. Rate anticipation swap
– Choosing a duration different than your investment horizon to
exploit a rate change.
• Rate increase: Choose D > Investment horizon
• Rate decrease: Choose D < Investment horizon
Swapping Strategies
4. Pure yield pickup
– Switching to a higher yielding bond, may be longer
maturity if the term structure is upward sloping or
may be lower default rating.
5. Tax swap
– Swapping bonds for tax purposes, for example
selling a bond that has dropped in price to realize a
capital loss that may be used to offset a capital gain
in another security