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Bond Price Volatility. Chapter 4. Price Volatility Characteristics. Price Volatility of Option-Free Bond. Although the prices of all option-free bonds move in opposite direction from the change in yield required, the % price change is not the same for all bonds.

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Price Volatility of Option-Free Bond

  • Although the prices of all option-free bonds move in opposite direction from the change in yield required, the % price change is not the same for all bonds.

  • For very small changes in the yield required, the % price change for a given bond is roughly the same, whether the yield required increases or decreases.

  • For large changes in the required yield, the % price change is not the same for an increase in the required yield as it is for a decrease in the required yield.

  • For a given large change in basis points, the % price increase is greater than the % price decrease.


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Price Volatility

  • For a given term to maturity and initial yield, the price volatility of a bond is greater, the lower the coupon rate.

  • For a given coupon rate and initial yield, the longer the term to maturity, the greater the price volatility.

  • The higher the YTM at which a bond trades, the lower the price volatility.


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Measures of Price Volatility

  • price value of a basis point – gives dollar price volatility not %

  • yield value of a price change

  • duration






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Duration

  • duration is less than (coupon bond) or equal to (zero coupon bond) the term to maturity

  • all else equal,

    • the lower the coupon, the larger the duration

    • the longer the maturity, the larger the duration

    • the lower the yield, the larger the duration

  • the longer the duration, the greater the price volatility


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Duration

  • dollar duration = (-MD) * P

  • spread duration – measure of how a non-Treasury bond’s price will change if the spread sought by the market changes

    • spread duration = 0 for Treasury

    • for fixed rate security it is the approximate change in the price of a fixed-rate bond for a 100 bp change in the spread

    • for a floater, a spread duration of 1.4 means that if the spread the market requires changes by 100 bp, the floater’s price will change by about 1.4%

  • portfolio duration – weighted average of bonds’ durations







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Convexity

  • second derivative of price-yield is dollar convexity measure of bond

  • convexity measure

  • convexity measure in terms of periods squared so to convert to annual figure, divide by 4




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Convexity (P=88.1309)

consider the 25-year 6% bond selling at 70.357 to yield 9%


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% Price Change (P=88.1309)

  • consider a 25 year 6% bond selling to yield 9%

    • MD = 10.62, convexity = 182.92

    • required yield increases 200 bp from 9% to 11%

    • estimated price change due to duration and convexity is -21.24% + 3.66% = -17.58%


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implication of convexity for bonds when yields change (P=88.1309)

market takes convexity into account when pricing bonds

but to what extent should there be difference?

Convexity


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As the required yield increases (decreases), the convexity of a bond decreases (increases). This property is referred to as positive convexity.

For a given yield and maturity, the lower the coupon, the greater the convexity of a bond.

For a given yield and modified duration, the lower the coupon, the smaller the convexity.

Convexity


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Approximating Duration of a bond decreases (increases). This property is referred to as positive convexity.

  • Use the 25 year, 6% bond trading at 9%. Increase the yield by 10bp from 9% to 9.1%. So ∆y = 0.001. The new price is P+ = 69.6164.

  • Decrease the yield on the bond by 10 bp from 9% to 8.9%. The new price is P- = 71.1105.

  • Because the initial price, P0, is 70.3570, the duration can be approximated as follows


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Approximating Duration of a bond decreases (increases). This property is referred to as positive convexity.

  • Increase the yield on the bond by a small number of bp and determine the new price at this higher yield level. New price is P+.

  • Decrease the yield on the bond by the same number of bp and calculate the new price. P-

  • Letting P0 be the initial price, duration can be approximated using the following where ∆y is the change in yield used to calculate the new prices. This gives the average % price change relative to the initial price per 1-bp change in yield.


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Approximating Convexity of a bond decreases (increases). This property is referred to as positive convexity.


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