10.

For an option-free bond, prices will fall as yields rise, and more important, perhaps, prices will rise unabated as yields fall - in other words, they'll move in line with yields. For the callable bond, the decline in yield will reach the point where the rate of increase in the price of the bond will start slowing down and eventually level off; this is known as negative convexity. Such behavior is due to the fact that the issuer has the right to retire the bond prior to maturity at some specified call price. That call price, in effect, acts to hold down the price of the bond (as rates fall) and causes the price/yield curve to flatten! The point where the curve starts to flatten is at (or near) a yield level of y'. And note that so long as yields remain above that level (y'), a callable bond will behave like any option-free (non-callable) issue and exhibit positive convexity! That's because at high yield levels, there is little chance of the bond being called.

f: Compute the duration of a bond, given information about how the bond's price will increase and decrease for a given change in interest rates.

Suppose that there is a 15-year option free bond with an annual coupon of 7% trading at par. If interest rates rise by 50 basis points (0.50%), the estimated price of the bond is 95.586%. (N = 15; PMT = 7.00; FV = 100; I/Y = 7.50%; CPT PV = -95.586). If interest rates fall by 50 basis points, the estimated price of the bond is 104.701%. Therefore, we can calculate the duration as:

(104.701 - 95.586) / (2)(100)(.005) = 9.115

What this tells us is that for a 100 basis point, or a 1.00% change in required yield, the expected price change is 9.115%.

g: Compute the approximate percentage price change for a bond, given the bond's duration and a specified change in yield.

For an option-free bond, prices will fall as yields rise, and more important, perhaps, prices will rise unabated as yields fall - in other words, they'll move in line with yields. For the callable bond, the decline in yield will reach the point where the rate of increase in the price of the bond will start slowing down and eventually level off; this is known as negative convexity. Such behavior is due to the fact that the issuer has the right to retire the bond prior to maturity at some specified call price. That call price, in effect, acts to hold down the price of the bond (as rates fall) and causes the price/yield curve to flatten! The point where the curve starts to flatten is at (or near) a yield level of y'. And note that so long as yields remain above that level (y'), a callable bond will behave like any option-free (non-callable) issue and exhibit positive convexity! That's because at high yield levels, there is little chance of the bond being called.

f: Compute the duration of a bond, given information about how the bond's price will increase and decrease for a given change in interest rates.

Suppose that there is a 15-year option free bond with an annual coupon of 7% trading at par. If interest rates rise by 50 basis points (0.50%), the estimated price of the bond is 95.586%. (N = 15; PMT = 7.00; FV = 100; I/Y = 7.50%; CPT PV = -95.586). If interest rates fall by 50 basis points, the estimated price of the bond is 104.701%. Therefore, we can calculate the duration as:

(104.701 - 95.586) / (2)(100)(.005) = 9.115

What this tells us is that for a 100 basis point, or a 1.00% change in required yield, the expected price change is 9.115%.

g: Compute the approximate percentage price change for a bond, given the bond's duration and a specified change in yield.