1. Bond Price Volatility

2. Price Yield Relationship Recall the earlier discussion… –Inverse relationship between Price and Yield Price Yield

3. Price Yield Relationship What do you observe in the given graph? Price Yield

4. Price Yield Relationship % Change in Price is not equal for increase in the yield as it is for decrease in the required yield Price Yield

5. Price Volatility Lower the coupon rate – greater the volatility Longer the term to maturity- greater the volatility So what should you be doing if you expect a decline in interest rate?

6. Price Volatility Higher the YTM bond trades at- lower its price volatility An implication of this is: For a given change in yield, price volatility is greater when yield levels are low…

7. Measures of Bond Price Volatility Price Value of a Basis Point (PVBP): Change in the price of a bond if the required yield changes by 1 basis point. Duration –Macaulay Duration (Weighted average) –Modified Duration which is (dp/dy)/p, also equal to [-modified duration/(1+y)]

8. Approximate Duration (P 1 -P 2 )/(2* P 0 * Change in Yield)

9. Properties of Duration Duration is less than term to maturity Duration of a zero coupon bond is equal to its maturity Lower the coupon greater the duration Greater the yield lower the modified duration Doesn’t this sound consistent?

10. Portfolio Duration How do we calculate the portfolio duration? How effective is it to its purpose? –Parallel shift in the yield curve –Non- parallel shift in yield curve Concept of key rate duration

11. Concerns on Duration We have assumed flat yield curve- how appropriate is that ? We have assumed that shift in yield curve is parallel- how appropriate is that? Misapplication of duration to bonds with embedded options

12. Don’t think of Duration as measure of Time Don’t get carried by the weighted average TTM as implied by Macaulay Duration A bond can have duration in excess of its maturity (leverage effect) A bond can have negative duration!! Call can have a duration of 30 even if its TTM is 1 year!

13. Convexity Duration fails to capture the entire price change Price Yield Actual Price Tangent line (estimated price)

14. Convexity There is an error in estimating price based only on duration Price Yield Actual Price Tangent line (estimated price)

15. Measuring Convexity Dollar Convexity Measure = (d 2 P)/(dy 2 ) Convexity Measure = Dollar Convexity Measure / Price Percentage Price change due to convexity = 0.5 * Convexity Measure * (dy) 2

16. Approximate Value of Convexity (P + + P - -2P 0 ) P 0 *(Change Yld) 2

17. Value of Convexity Can two bonds have equal duration but different convexities? When would convexity be attributed a high value?