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LOS h: Describe the convexity measure of a bond and estimate a bond's percentage price change, given the bond's duration and convexity and a specified change in interest rates.

A 7% coupon bond with semiannual coupons has a convexity in years of 80. The bond is currently priced at a yield to maturity (YTM) of 8.5%. If the YTM decreases to 8%, the predicted effect due to convexity on the percentage change in price would be:

Convexity adjustment: +(Convexity)(change in i)²

Convexity adjustment = +(80)(-0.005)(-0.005) = +0.0020 or 0.20% or +20 basis points.

Why is convexity a good thing for a bond holder? Because when compared to a low convexity bonds a high convexity bond:

A)	is usually underpriced.

B)	has better price changes regardless of the direction of the yield change.

C)	is more sensitive to interest rate changes, increasing the potential payoff.

Relative to a bonds with low convexity, the price of a bond with high convexity will increase more when rates decline and decrease less when rates rise.

How does the convexity of a bond influence the yield on the bond? All else the same, for a bond with high convexity investors will require:

A)	a lower yield.

B)	a higher or lower yield depending on the bond's duration.

C)	a higher yield.

Convexity is to the advantage of the bond holder because a high-convexity bond's price will decrease less when rates increase and will increase more when rates decrease than a low-convexity bond's price.

For a given change in yields, the difference between the actual change in a bond’s price and that predicted using the duration measure will be greater for:

A)	a bond with less convexity.

B)	a short-term bond.

C)	a bond with greater convexity.

Duration is a linear measure of the relationship between a bond’s price and yield. The true relationship is not linear as measured by the convexity. When convexity is higher, duration will be less accurate in predicting a bond’s price for a given change in interest rates. Short-term bonds generally have low convexity.

With respect to an option-free bond, when interest-rate changes are large, the duration measure will overestimate the:

A)	fall in a bond's price from a given increase in interest rates.

B)	increase in a bond's price from a given increase in interest rates.

C)	final bond price from a given increase in interest rates.

When interest rates increase by 50-100 basis points or more, the duration measure overestimates the decrease in the bond’s price

Since interest rates and the price of bonds are inversely related, unstable interest rates will lead to larger price fluctuations in bonds. The larger the change in the price of a bond the more error will be introduced in determining the new price of the bond if only duration is used because duration assumes the price yield relationship is linear when in fact it is a curved convex line. If duration alone is used to price the bond, the curvature of the line magnifies the error introduced by yield changes, and makes the convexity adjustment even more important.

If a bond has a convexity of 120 and a modified duration of 10, what is the convexity adjustment associated with a 25 basis point interest rate decline?

Convexity adjustment: +(C) (Δi)²

Con adj = +(120)(-0.0025)(-0.0025) = +0.000750 or 0.075%

A bond’s duration is 4.5 and its convexity is 43.6. If interest rates rise 100 basis points, the bond’s percentage price change is closest to:

Recall that the percentage change in prices = Duration effect + Convexity effect = [-duration × (change in yields)] + [convexity × (change in yields)²] = (-4.5)(0.01) + (43.6)(0.01)² = -4.06%. Remember that you must use the decimal representation of the change in interest rates when computing the duration and convexity adjustments.

A bond has a modified duration of 6 and a convexity of 62.5. What happens to the bond's price if interest rates rise 25 basis points? It goes:

ΔP/P = (-)(MD)(Δi) +(C) (Δi)²

=(-)(6)(0.0025) + (62.5) (0.0025)² = -0.015 + 0.00039 = - 0.01461

A bond has a modified duration of 7 and convexity of 50. If interest rates decrease by 1%, the price of the bond will most likely:

Percentage Price Change = –(duration) (?i) + convexity (?i)²

therefore

ercentage Price Change = –(7) (–0.01) + (50) (–0.01)²=7.5%.

A bond has a modified duration of 6 and a convexity of 62.5. What happens to the bond's price if interest rates rise 25 basis points? It goes:

?P = [(-MD × ?y) + (convexity) × (?y)²] × 100

?P = [(-6 × 0.0025) + (62.5) × (0.0025)²] × 100 = -1.461%