What is this Table intended to depict? Step by step derive the formula for duration and convexity. What is the difference between effective duration, modified duration, and duration?

518 PART IV Fixed-Income Securities Table 16.1 1 Year T = 10 Years -20 Years 1,000.00 907.99 Yield to Maturity (APR) Prices of 8% coupon bond (coupons paid semiannually) 1,000.00 990.64 1,000.00 934.96 18% 9% Fall in price (%)* 0.94% 6.50% 9.20% I Equals value of bond at a 9% yield to maturity divided by value of bond at (the original) 8% yield, minus 1. rate sensitivity, let's start with a simple numerical example. Table 16.1 gives IThe interest rates are expressed as annual percentage rates (APRs), meaning value by less than 1% when the interest rate increases from 8% to 9%. The 10-ye To see why bond characteristics such as coupon rate or vield to maturity affect interst bond prices emiannual coupon bonds at different yields to maturity and times to matungj meaning that the tve 6-month yield is doubled to obtain the stated annual yield.] The shortest-term boed (-term bond falls in ar falls by 6.5%, and the 20-year bond by over 9%. Now look at a similar computation using a zero-coupon bond rather than the rice of n the 8% co ty, the ve to interest rate bond. zero-coupon bond falls by a greater proportional amount than the price ol tne ne price of the bond. Because we know that long-term bonds are more sensitive to interest iab than are short-term bonds, this observation suggests that in some sense a zero- must represent a longer-term bond than an equal-time-to-maturity coupon In fact, this insight about the effective maturity of a bond is a useful Onen bos make mathematically precise. To start, note that the times to maturity this example are not perfert mon ul one that we car aturity of the two bonds honds. The