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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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### Bond Price Volatility

Chapter 4

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.

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.

Measures of Price Volatility

- price value of a basis point – gives dollar price volatility not %
- yield value of a price change
- duration

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

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

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

Convexity (P=88.1309)

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

% 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%

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?

ConvexityAs 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.

ConvexityApproximating 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

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.

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

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