Download Duration and Convexity: The Price / Yield

Duration and Convexity: The Price / Yield Relationship
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Investors who own fixed income securities should be aware of the relationship between interest rates
and a bond’s price. As a general rule, the price of a bond moves inversely to changes in interest rates: a
bond’s price will increase as rates decline and will decrease as rates move up. Macaulay duration is the
weighted-average maturity of a bond’s cash flows, which is measured in years. Modified duration
attempts to estimate how the price of a bond will change in response to a change in interest rates and is
stated in terms of a percentage change in price. Typically when duration is quoted it is referring to a
bond’s modified duration rather than Macaulay duration. Taking this concept one step further, a bond’s
convexity is a measurement of how duration changes as yields change. These two measurements can
provide insight into how a bond is expected to perform should interest rates change and can help
investors understand the price risk of fixed income securities in different interest rate environments.
What is Duration?
In simple terms, modified duration gives an idea of how the price of a bond will be affected should
interest rates change. A higher duration implies greater price sensitivity upwards (downwards) should
rates move down (up). Duration is quoted as the percentage change in price for each given percent
change in interest rates. For example, the price of a bond with a duration of 2 would be expected to
increase (decline) by about 2.00% for each 1.00% move down (up) in rates.
The duration of a bond is primarily affected by its coupon rate, yield, and remaining time to maturity.
The duration of a bond will be higher the lower its coupon, lower its yield, and longer the time left to
maturity. The following scenarios of comparing two bonds should help clarify how these three traits
affect a bond’s duration:
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Example:
If the coupon and yield are the same, duration increases with time left to maturity
If the maturity and yield are the same, duration increases with a lower coupon
If the coupon and maturity are the same, duration increases with a lower yield
5% Coupon Bond at Par: Price Change for a Given Rise in Interest Rates
Example:
Price/Yield Relationship for Bonds with Differing Maturities (5% Coupon)
These are hypothetical examples for illustrative purposes only. They are not intended to reflect the actual performance of any security.
Convexity
As the yield on a bond changes so too does its duration, a bond’s convexity measures the sensitivity of a
bond’s duration to changes in yield. Duration is an imperfect way of measuring a bond’s price change, as
it indicates that this change is linear in nature when in fact it exhibits a sloped or “convex” shape. A
bond is said to have positive convexity if duration rises as the yield declines. A bond with positive
convexity will have larger price increases due to a decline in yields than price declines due to an increase
in yields. Positive convexity can be thought of as working in the investor’s favor, since the price becomes
less sensitive when yields rise (prices down) than when yields decline (prices up). Bonds can also have
negative convexity, which would indicate that duration rises as yields increase and can work against an
investor’s interest. The table below highlights the types of bonds that exhibit each type of convexity.
Examples of Bonds with Positive and Negative Convexity
A useful way to visualize a bond’s convexity is to plot the potential price change against various yields. If
two bonds have the same duration and yield but differing convexities, a change in interest rates will
affect each bond differently. For example, the chart below shows three bonds: a bond with higher
positive convexity (Bond A) will be less affected by interest rates than a bond with lower positive
convexity (Bond B). On the other hand, a bond with negative convexity (Bond C) will exhibit larger price
fluctuations should rates rise than if they were to fall.
Bonds Can Have Very Different Convexities: Positive vs. Negative
Conclusion
Duration and convexity are two metrics used to help investors understand how the price of a bond will
be affected by changes in interest rates. How a bond’s price responds to changes in interest rates is
measured by its duration, and can help investors understand the implications for a bond’s price should
interest rates change. The change in a bond’s duration for a given change in yields can be measured by
its convexity.
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If rates are expected to increase, consider bonds with shorter durations. These bonds will be less
sensitive to a rise in yields and will fall in price less than bonds with higher durations.
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If rates are expected to decline, consider bonds with higher durations. As yields decline and
bond prices move up, higher duration bonds stand to gain more than their lower duration
counterparts.
This piece was prepared by Fixed Income Services of Raymond James
The author of this material is a Trader in the Fixed Income Department of Raymond James & Associates (RJA), and is not an Analyst.
Any opinions expressed may differ from opinions expressed by other departments of RJA, including our Equity Research Department, and are subject to change
without notice. The data and information contained herein was obtained from sources considered to be reliable, but RJA does not guarantee its accuracy and/or
completeness. Neither the information nor any opinions expressed constitute a solicitation for the purchase or sale of any security referred to herein. This material
may include analysis of sectors, securities and/or derivatives that RJA may have positions, long or short, held proprietarily. RJA or its affiliates may execute
transactions which may not be consistent with the report’s conclusions. RJA may also have performed investment banking services for the issuers of such securities.
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liquidity, credit quality, volatility, and duration. Past performance is no assurance of future results.
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levels of credit and liquidity risk than investments in investment grade securities. Investors who own fixed income securities should be aware of the relationship
between interest rates and the price of those securities. As a general rule, the price of a bond moves inversely to changes in interest rates. Diversification does not
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