**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. • 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**

**Duration**

**Modified 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

**Portfolio Duration**

**Portfolio Duration**

**Measures of Bond Price Volatility**

**Price-Yield Relationship**

**Price Approximation using Duration**

**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

**Calculation of Convexity for 5 Year, 9%, Selling to Yield 9%** (Price = 100)

**Calculation of Convexity for 5 Yr, 6%, Selling to Yield 9%** (P=88.1309)

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

**% Price Change** • 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** market takes convexity into account when pricing bonds but to what extent should there be difference? Convexity

**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

**Approximating Duration** • 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** • 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**